ISSN (Print) 0473-453X

Discussion Paper No. 1162

ISSN (Online) 2435-0982

HORIZONTAL MERGER ANALYSISWITH ENDOGENOUS
PRODUCT RANGE CHOICE
Nisvan Erkal
Lijun Pan

February 2022

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Horizontal Merger Analysis with Endogenous
Product Range Choice
Nisvan Erkaly

Lijun Panz

February 2022

Abstract
We consider mergers between multi-product …rms in a market with monopolistically competitive fringe of single-product …rms. Aggregate productvariety is determined by product variety choices of multi-product …rms andentry/exit decisions of single-product …rms. Mergers can generate marginal
cost synergies (a¤ecting marginal cost of quantity) or …xed cost synergies (affecting marginal cost of variety). We show that with marginal cost synergies,
consumer welfare decreases whenever aggregate variety increases following
a merger. However, with …xed cost synergies, an increase in aggregate variety can indicate that the merger is bene…cial. Our results also show highsynergies do not necessarily improve consumer welfare.
Keywords: Antitrust policy; merger analysis; multiproduct …rms; product range choice; entry; cost synergies
JEL classi…cation: L11; L41; D43

We would like to thank Makoto Hanazono, Akifumi Ishihara, Patrick Legros, Leslie Marx,
Noriaki Matsushima, Debby Minehart, Takeshi Murooka, Volker Nocke, Yoshiyasu Ono, Mathieu
Parenti, Daniel Piccinin, Nicolas Schutz, Jacques-Francois Thisse and the participants at EARIE
2018, EEA-ESEM 2019, APIOC 2019, IIOC 2021, and RIEB workshop at Kobe University for
their constructive comments. Dang Song Nguyen Lam has provided excellent research assistance.
yDepartment of Economics, University of Melbourne, VIC 3010, Australia. Email: nisvan@unimelb.edu.au.
zSchool of Economics, Nanjing University, China; Institute of Social and Economic Research,
Osaka University, Japan. Email: pan.lijun@hotmail.com

1

1

Introduction

Antitrust authorities regard post-merger product diversity as an important consideration in determining the social desirability of mergers.1 Indeed, many mergers
involve multi-product …rms and may result in a readjustment of the merged entity’s
product range. For example, after the merger in 2007 between Whole Foods Market and Wild Oats, the two leading retailers of organic and natural food in the US,
Whole Foods Market sold 35, closed 12, and converted one-third of the remaining
Wild Oats stores into Whole Foods outlets.2 Another example is the consolidationof the American radio broadcasting industry prompted by the Telecommunications Act of 1996, which raised both the per-station and overall variety in terms ofprogramming formats (Berry and Waldfogel, 2001). As a result, it is important tounderstand under what circumstances mergers resulting in changes in the merged
entity’s product range can harm consumer welfare.
We study this issue in a model which allows for merger-generated e¢ ciencies
and post-merger entry. Merger-generated e¢ ciencies and post-merger entry are
two features evaluated positively by antitrust authorities while assessing the welfare impact of a merger. The general thinking is that a merger is likely to be
procompetitive if merger e¢ ciencies are substantial or the prospect of entry in therelevant market is high.3
Combining merger-generated e¢ ciencies, product range change and entry in a
1

For example, the 2018 OECD document Considering non-price e¤ ects in merger control –Background note by the Secretariat (paragraph 86) states that “a merger that results in changes
to the available variety of di¤erentiated products can have implications for consumer welfare.”
The 2010 U.S. Horizontal Merger Guidelines (Section 6.4) states that “reductions in variety
following a merger may or may not be anticompetitive.”2
See Harvard Business Review Case Study called “Whole Foods and Wild Oats Merger,”available at https://store.hbr.org/product/whole-foods-market-and-wild-oats-merger/UV1019.
3
For example, it is stated in the US Horizontal Merger Guidelines that “a primary bene…t ofmergers to the economy is their potential to generate signi…cant e¢ ciencies and thus enhance
the merger …rm’s ability and incentive to compete, which may result in lower prices, improved
quality, enhanced service, or new products.” Similarly, the European Commission’s Horizontal
Merger Guidelines state the following: “The relevant benchmark in assessing e¢ ciency claims
is that consumers will not be worse o¤ as a result of the merger. For that purpose, e¢ ciencies
should be substantial ....” As far as potential entrants are concerned, the US Horizontal Merger
Guidelines consider entry into the relevant market as part of the full assessment of competitivee¤ect: “The prospect of entry into the relevant market will alleviate concerns about adverse
competitive e¤ects only if such entry will deter or counteract any competitive e¤ects of concern
so the merger will not substantially harm customers.”2coherent theoretical framework is necessary to determine how they interact witheach other. The interaction raises a number of questions. How do multi-product…rms adjust their product range following a merger in the face of e¢ ciency gains?
What are the reactions of non-merging …rms, including both incumbents and potential entrants? What are the impacts of such mergers on aggregate productvariety in the marketplace and welfare?
To address these questions, we consider a horizontally di¤erentiated goods
market where a few multi-product …rms and a host of single-product …rms compete in quantity. We assume that the multi-product …rms act strategically while
the single-product …rms are monopolistically competitive.4 Such a mixed market
structure consisting of multi-product and single-product …rms has been shown tobe prevalent in many industries.5
The aggregate product range in the market is determined endogenously in ourmodel through the multi-product …rms’ product range adjustments and singleproduct …rms’ entry/exit decisions. We assume that each multi-product …rm
strategically supplies a continuum of varieties, and the variety range it o¤ers isa choice variable. A single-product …rm is in…nitesimal and can freely enter or exit
the market.
In our analysis, we allow for two kinds of merger-generated synergies. We
assume that a merger may result in marginal cost savings or …xed cost savings. The…xed cost in our model is a per-variety …xed cost. Since we allow the multi-product…rms to determine their product range endogenously, the …xed cost represents themarginal cost of producing one more variety. Hence, it is di¤erent from costs
related to back-o¢ ce functions. For example, in his analysis of …xed cost e¢ ciencies
from mergers in the US radio industry, Jeziorski (2014) identi…es “large withinformat cost synergies.”6 Speci…cally, he …nds that “operating an extra station in
4

The theoretical framework with large oligopolists and small monopolistic competitors is …rst
considered by Shimomura and Thisse (2012), who assume that all the …rms are single-product…rms and examine the impact of a large …rm’s entry. Pan and Hanazono (2018) and Parenti (2018)consider the coexistence of multi-product large …rms and single-product small …rms, examining
the impact of large …rms’entry and trade liberalization, respectively. We draw on this frameworkwith particular attention to merger analysis.
5
For instance, in the US manufacturing industry, 91% of total output is supplied by 41% of…rms, 89% of which adjust product range every …ve years (Bernard et al., 2010).
6
Each radio station is characterized by a format which includes information about type ofmusic, the number of news and talk shows, as well as information about being inactive.

3

the already-owned format costs more than 60% less.”
For mergers with marginal cost synergies, we show that if such synergies aresu¢ ciently high, the merging multi-product …rms expand their variety range andthe mass of single-product …rms shrinks. The net e¤ect of these two changes onaggregate product variety in the market depends on whether the multi-product…rms have a …xed cost advantage over the single-product …rms. If the multiproduct …rms have a …xed cost advantage, then high synergies result in an increasein aggregate product variety. Interestingly, the merger, despite generating highsynergies and increasing the aggregate product variety, harms consumer welfare.
This is because of a reallocation e¤ect: the varieties produced by the single-product…rms are replaced by the varieties produced by the multi-product …rms. Since thelatter charges a higher price for its products (both before and after the merger),the reallocation hurts consumers.
Hence, our analysis reveals that if the multi-product …rms have a …xed costadvantage over the single-product …rms, then high marginal cost synergies harm
consumers, but a moderate level of marginal cost synergies increases consumer
welfare despite causing aggregate product variety to decrease. If the multi-product…rms do not have a …xed cost advantage, the results are reversed. In this case, high(low) marginal cost synergies cause aggregate product variety to decrease (increase)and consumer welfare to increase (decrease). These results imply that in mergers
with marginal cost synergies, any consumer welfare gain does not originate froman increase in aggregate variety. Rather, it is the lower prices that consumers face
as a result of the reallocation of production between the multi-product and singleproduct …rms that bene…t the consumers. Moreover, contrary to the view taken
in merger guidelines, entry is not always a necessary condition for a merger toincrease consumer welfare. When single-product …rms have …xed cost advantage,
consumer welfare increases despite exit by single-product …rms which is caused bysu¢ ciently high merger-generated synergies.
We next consider the impact of mergers which generate …xed cost synergies.
Fixed cost synergies reduce the cost of introducing new varieties. An importantdi¤erence between mergers with marginal-cost synergies and mergers with …xed
cost synergies is that mergers with …xed cost synergies also have a price e¤ect. Thisis because as the …xed cost level decreases and the multi-product …rm introduces4more varieties, it charges higher prices in order to mitigate the cannibalization
between their own varieties.
Due to this price e¤ect which increases with the level of synergies, mergers withsu¢ ciently high …xed cost synergies are harmful to consumers. We further showthat if multi-product …rms are more e¢ cient to start with, both in terms of their…xed and marginal cost of production, then all mergers are harmful for consumers.
Otherwise, some mergers with moderate levels of synergies may increase consumer
welfare. Importantly, in contrast with the case of marginal cost synergies, anincrease in aggregate variety can be an indication that the merger is bene…cial for
consumers in this case if the multi-product …rms have a …xed cost advantage overthe single-product …rms before the merger.
These results imply that in terms of antitrust analysis, an increase in aggregate
product variety or high synergies should not be seen as reasons for favorable treatment of mergers. Even mergers which result in both increased aggregate productvariety and high synergies can still reduce consumer welfare. Mergers which resultin an increase in aggregate product variety can still harm consumers if producersof lower-priced varieties are forced out of the market and the varieties they o¤er
are replaced by the varieties o¤ered by the merged …rms. Mergers with high synergies can still harm consumers if they result in higher prices for the merged …rms’
varieties.
Although there exists a large literature on mergers, most of the theoretical literature rely on models with single-product …rms. Two exceptions are Nocke andSchutz (2018) and Johnson and Rhodes (2021).7 Nocke and Schutz (2018) analyze
horizontal mergers in a model of multi-product …rms and price competition with
nested CES or nested logit demands. They show that the Her…ndahl index provides
an adequate measure of the oligopoly distortions to consumer surplus and aggregate surplus. However, they do not explicitly consider endogenous product range
choice by multi-product …rms. Johnson and Rhodes (2021) investigate mergers
where …rms may reposition their product lines by adding or removing products ofdi¤erent qualities following a merger. They identify a product-mix e¤ect by which
7

See also Lommerud and Sogard (1997) who examine merger pro…tability in a model withthree …rms where brand withdrawal by the merged entity and new brand introduction by thenon-merging …rm are allowed. The …rms are assumed to be single-product …rms before themerger.

5

mergers without synergies may raise consumer surplus whereas mergers with synergies may lower consumer surplus. We complement their work by considering amarket with horizontally di¤erentiated products, where multi-product and singleproduct …rms coexist. Our paper also di¤ers from both of these two papers byallowing for entry by the single-product …rms.
Post-merger entry constitutes an important factor in merger evaluation andhas been investigated in several studies, such as Werden and Froeb (1998), Cabral(2003), Spector (2003), Davidson and Mukherjee (2007), Erkal and Piccinin (2010),
Anderson et al. (2020) and Caradonna et al. (2021).8 Nevertheless, all of these
papers assume an exogenous product range for each …rm. Endogenous productrange choice by multi-product …rms is central to our paper.
Our paper is also related to the literature which use merger simulation or
structural methods to investigate the impact of product choice and product repositioning on merger outcomes. This literature has considered mergers in di¤erent
varieties of markets, such as radio stations (Berry and Waldfogel, 2001; Sweeting,2010), airlines (Li et al., 2019), smartphones (Fan and Yang, 2020a), and craft
beer (Fan and Yang, 2020b). One key message from this literature is that price effects and welfare consequences of mergers emerging from models with a …xed set of
products may be quite di¤erent from those emerging from models with endogenous
product choice or product repositioning.9
The rest of the paper is organized as follows. Section 2 introduces the model.
Section 3 conducts the equilibrium analysis without a merger. Section 4 analyzes
the equilibrium with a merger and examines the impact of marginal cost synergies
on market performance and welfare. Section 5 considers the case of …xed costsynergies. Section 6 concludes.
8

Gowrisankaran (1999) develops a dynamic Cournot game with endogenous investment,
merger, entry and exit decisions. His computational analysis suggests that mergers’anticompetitive e¤ects are unlikely to be reversed by entry.
9
For example, while Gandhi et al. (2008) and Sweeting (2010) show that allowing for productrepositioning in merger analysis makes mergers appear less anticompetitive, Mazzeo et al. (2019)…nd the opposite.

6

2

Model

We draw on Parenti (2018) and Pan and Hanazono (2018) to describe the market
structure with multi-product (MP) and single-product (SP) …rms.10 Consider aclosed economy with two goods, a homogeneous good and a horizontally di¤erentiated good. In the di¤erentiated good market, there are a discrete number N 2of MP …rms and a continuum [0; S] of symmetric SP …rms. MP …rm n = 1; :::; N
supplies a continuum of varieties, with its product range represented by ! n .

2.1

Preferences and DemandSpeci…cally, the utility of the representative consumer is described by
U =

"Z

S

qsp (i)di +

0

2

"Z

0

N Z !nX
n=1

S

qsp (i)di +

#

n
qmp
(x)dx

0

N Z !n
X
n=1

n
qmp
(x)dx

0

2
#2

"Z

0

S

(qsp (i))2 di +N Z !n
X
n=1

n
(qmp
(x))2 dx0

+ q0 ;n(x) denotes
where qsp (i) is the quantity produced by SP …rm i 2 [0; S], and qmp
the quantity of variety x 2 [0; ! n ] for MP …rm n = 1; :::; N . We treat ! n asa continuous variable and let q0 stand for the consumption of the homogeneous
good which is the numeraire.
We assume that the demand parameters , and are all positive. The parameters and capture the degree of substitutability between the varieties ofthe di¤erentiated product and the numeraire: an increase in or a decrease inshifts out the demand for the di¤erentiated good relative to the numeraire. The
parameter represents the degree of di¤erentiation between the di¤erentiated varieties. The degree of di¤erentiation increases with since a higher corresponds
to a stronger preference for diversi…ed consumption.
10

The seminal work by Shimomura and Thisse (2012) is the …rst paper that characterizes thecoexistence of large oligopolists and and small monopolistic competitors, but they assume that
all the …rms are single-product. Also assuming single-product …rms, Anderson et al. (2020)
validate the results in Shimomura and Thisse (2012) in a general framework via the aggregative
games approach.

7

#

(1)

The representative consumer’s budget constraint is given by
Z S

psp (i)qsp (i)di +0

N Z !nX
n=1npnmp (x)qmp(x)dx + q0 = I,

(2)0

where psp (i) and pnmp (x) are the prices of SP …rm i’s product and MP …rm n’svariety x, respectively. The price of the numeraire is normalized to 1. The representative consumer’s income is I, which is exogenously given.
The inverse demand function facing each SP and MP …rm is determined by the
maximization of the consumer’s utility function subject to the budget constraint.
The inverse demand functions are given bypnmp (x) =

n
qmp
(x)

Q

n = 1;; N; x 2 [0; ! n ]

(3)

and
psp (i) =
where
Q=

Z S

qsp (i)qsp (i)di +0i 2 [0; S]

Q
N Z !n
X
n=1

n
qmp
(x)dx

(4)

(5)

0

is the aggregate output of the di¤erentiated good.

2.2

Firms

In the absence of a merger, we assume that each MP …rm has a constant marginal
cost C and each SP …rm has a constant marginal cost c. In addition, both MP
and SP …rms incur …xed costs. SP …rms incur a …xed cost f , which consists of a…xed cost of production and an entry cost. As incumbent …rms, MP …rms only
incur a …xed cost of production per variety that they produce, denoted by F . In
the following analysis, our results will critically depend on whether the per-variety…xed cost of the MP …rms is smaller or larger than that of SP …rms. For example,
if the entry cost of SP …rms is su¢ ciently high or MP …rms enjoy economies ofscope, then we may have F < f . On the other hand, if producing multiple
varieties means increased coordination costs, then we may have F > f .
The …rms play a non-cooperative game in which they choose their actions si8

multaneously. As incumbent …rms, each MP …rm produces a range of varieties andstrategically chooses both its product range and the quantity of each variety. SP…rms freely enter or exit the market, and once entering the market, each SP …rm
decides on its quantity. Hence, the aggregate product range in the model is determined by both the product range choices of the MP …rms and the entry decisionsof the SP …rms. Note that there is an essential di¤erence in the strategic behavior
between MP and SP …rms. Supplying a continuum of varieties and occupying asubstantial market share, each MP …rm generates a non-negligible market impact
and competes in an oligopolistic manner. In contrast, each SP …rm is negligible
in the market and hence behaves non-strategically as a monopolistic competitor.
We next explain the pro…t functions of the two type of …rms.
2.2.1

MP Firms

Each MP …rm n optimally chooses the range ! n of varieties it will provide and then(x) for variety x 2 [0; ! n ]. The pro…t of MP …rm n is expressed byquantity qmpnnmp (! n ; qmp ( )) =Z !n(pnmp (x)

n
C)qmp
(x)dx

(6)

! n F:0

Substituting pnmp (x) from equation (3) into (6) gives usnnmp (! n ; qmp ( ))

= (

Z !n

n
qmp
(x)dx

C)
0
Z !n
n
qmp
(x)dx
0

2

Z !n

n
(qmp
(x))2 dx
0
Z !n
n
n
Q
qmp
(x)dx

(7)
! n F:

0RS
Each MP …rm takes as given the total output of other …rms, Q n = 0 qsp (i)di+
P R !k kn(x). Hence,qmp (x)dx, and maximizes its pro…t with respect to ! n and qmp
0

k6=n

it takes its impact on the aggregate output into account while solving its maximization problem.

92.2.2SP Small Firms

The pro…t of a SP …rm is expressed by
sp (i) = (psp (i)c)qsp (i)f.

(8)

Plugging psp (i) from equation (4) into (8) yields
sp (i) = (c)qsp (i)[qsp (i)]2

Qqsp (i)f:(9)

As di¤erent from a MP …rm, a SP …rm is negligible in the market and understands that its decision does not a¤ect the aggregate output Q. Therefore, each SP…rm maximizes its pro…t with respect to its quantity qsp (i), treating the aggregate
output Q parametrically.
2.2.3

Equilibrium

In the following analysis, we compare equilibria with and without a merger, by
taking into account the impact of the merger on the MP …rms’product choice andthe mass of SP …rms. We focus on the equilibrium where both ! > 0 and S > 0.
The equilibrium size S of the monopolistically competitive fringe is endogenously
determined by the zero-pro…t condition:sp (i) = (c)qsp (i)[qsp (i)]2

Qqsp (i)f = 0:(10)

Hence, in equilibrium, (i) each …rm chooses its own action to maximize its own
pro…ts, (ii) the MP …rms earn positive pro…ts with a positive product range, and(iii) the mass of SP …rms is positive and such that the zero-pro…t condition holds.
A market structure with MP and SP …rms exists if and only if Assumption 1
holds.
Assumption 1
8
>
>
>
<
(i) > max
>
>
>
:

9
p i
p
p h
(N + 1) f N F ; >c(N + 1) CN + 2
>
>
=
h
i
p
p
p
cN C(N 2) + 2
N f (N 1) F ;
.
>
p i >
p h
p
>
cN C(N 1) + 2(N 1) f (N 3) F ;10(ii) c + 2

pf >C +2p

F.

We show the derivation of the conditions stated in Assumption 1 in Appendix
A. As will become clear later on, Assumption 1(i) is a lower-bound condition on
the demand intercept such that the mass of SP …rms S > 0 in equilibrium.
The assumption ensures that S > 0 in equilibrium in all the market structures
we consider in this paper: when there is no merger, when there is a merger with
marginal cost synergies, and when there is a merger with …xed cost synergies.
Assumption 1(ii) is a condition on the product range of MP …rms and ensuresthat MP …rms earn a positive pro…t with a positive product range in equilibrium.
This condition is su¢ cient both in the case without a merger and in the case witha merger. A closer look at Assumption 1(ii) shows that if the SP and MP …rms
have the same costs (i.e., c = C and f = F ), there will be no MP …rms in themarket. If f F , then Assumption 1(ii) requires that c > C. On the other hand,
if c C, then Assumption 1(ii) requires that f > F . Hence, the MP …rms musthave either a marginal or a …xed cost advantage in comparison to the SP …rms.
This is because a MP …rm maximizing the pro…ts from all its varieties ends upinternalizing externalities. This implies that at any given total output level, theyearn less than they would if they were acting as a single-product …rm. Hence, itwould not be pro…table for MP …rms to exist if they had the same cost structure
as or higher costs than the SP …rms.

3

Market Outcome without a Merger

3.1

SP Firms’Entry and Pro…t Maximization

Since each SP …rm’s impact on the market is negligible, it does not internalize its
impact on the market output. Maximizing the pro…t given in (9) with respect to
qsp (i) yields
c
Q
:
(11)
qsp (i) = qsp =2

11

Due to symmetry, all SP …rms choose the same level of output. We can solve forthe price of the SP …rms by using (4). This yields:
+c
2

psp =

Q

(12)

:

After substituting (10) in (11) and using that Q = Qmp + Sqsp , where Qmp isthe aggregate quantity of MP …rms, we can …nd the equilibrium mass of SP …rms
as a function of Qmp :S(Qmp ) =

1

"s

f

(

c

Qmp )

2

#

:

(13)

The equilibrium mass of SP …rms decreases with Qmp .
Expressing Q = Qmp + Sqsp , we can now substitute (13) into (11). Simplifying
yields the optimal quantity of each SP …rm as
qsp =

s

f

.

(14)

Finally, substituting for (11) and (13) in (12) yields the equilibrium price of SP…rms aspf.
(15)
psp = c +Note that psp , qsp and the revenue psp qsp are positively correlated with f . At ahigher …xed cost, the equilibrium mass of SP …rms will be lower. Each SP …rm,
expecting less competition, charges a higher price.
Furthermore, substituting (14) back into (10), the aggregate quantity in equilibrium is pinned down by the zero-pro…t condition of the SP …rms:
Q =

c

p
2 f

:

(16)

Owing to the free entry and exit of SP …rms, the aggregate quantity of the differentiated good depends on the technology of SP …rms, but is independent of
the technology of MP …rms. Furthermore, the aggregate output is negatively correlated with the marginal and …xed costs of SP …rms. In other words, higher
12e¢ ciencies of SP …rms would expand the market at the aggregate level.

3.2

MP Firms’Pro…t Maximization

Unlike SP …rms, MP …rms impose a non-negligible impact on the market. MP…rm n maximizes its pro…t given in (7) with respect to its per-variety quantityand product range. The …rst-order condition of MP …rm n with respect to then(x) of its xth variety isquantity qmp
C

n
2 qmp
(x)

2

Z !n

n
qmp
(v)dv

Q n = 0;(17)0which implies symmetric production behavior across the varieties within …rm n.
nnThat is, qmp(x) = qmp
for x 2 [0; ! n ]: Hence, we have
Z !n

n
n
qmp
(v)dv = ! n qmp:0

Substituting the above symmetry property into equation (17), we can solve for thenof each variety as a function of MP …rm n’s product range
optimal quantity qmp! n and the expected aggregate output Q n of other …rms:
n
qmp
=

C
Q n
:
2( + ! n )(18)

Everything else being equal, an increase in …rm n’s product range (larger ! n )nresults in a decrease in the quantity of each variety (smaller qmp) produced due toa cannibalization e¤ect.
Large …rm n also maximizes its pro…t with respect to its product range, ! n .
The …rst-order condition isCn( + 2 ! n ) qmp

13n
Q n qmp= F:(19)

Since Q n = Sqsp +P! k qmp , (19) can be expressed as

k6=n

2 ( + !n) =

r

F

"

Sqsp +CX! k qmp

k6=n

!#

:

(20)

From equations (18) and (19), we obtain the optimal per-variety output for theMP …rm:s
Fn= qmp =
qmp
:
(21)
p
p
Substituting for qmp = F= and qsp = f = in (20) yields
!n =

r

F

"

C

S

s

f

+

N
X

!k

k=1

s !#
F

2 ;

(22)

which implies that ! n = ! for each n = 1; ::; N . Simplifying gives an expression
for ! which indicates that as the mass of SP …rms increases, MP …rms react by
decreasing their variety range:!(S) =

3.3

p

=F

C

p
S f=

(N + 1)2:(23)

Mixed Market Equilibrium

A mixed market equilibrium exists if and only if S > 0 and ! > 0, which are
ensured by the conditions given in Assumption 1. In such an equilibrium, thevariety range is endogenously determined by the total mass of SP …rms and the
product range of each MP …rm. In this section, we characterize the mixed market
equilibrium.
p
From symmetry, we have Qmp = N !qmp = N ! F= . Therefore, (13) can bere-expressed as
p
p
=f (
c
N ! F= ) 2
S(!) =:(24)
Equations (23) and (24) jointly determine the equilibrium mass of SP …rms S
and the equilibrium product range of a MP …rm ! . Solving these two equations
14

simultaneously for S and ! yields

S =

s

! =

r

[c(N + 1)p hp2(N + 1) f

CN ]

f
(c

C) + 2

p

p

f

p

(25)

F
.

F

p i
N F

(26)

It is straightforward to check that both the mass of SP …rms and the per-…rm
variety of MP …rms are decreasing in own …xed cost and increasing in rival’s …xed
cost.11
We can now solve for the equilibrium price and pro…ts of the MP …rms using(3) and (6). We obtain
pmp = c + 2p= psp +
and
mp =

h p
2

p

f

p

p
f

F
p

p
F + (c

p
f

(27)
F
i2
C)

.

(28)

The MP …rms’ equilibrium price is increasing in the SP …rms’ …xed cost anddecreasing in MP …rms’ per-variety …xed cost. As f decreases, the MP …rms
face more competition from the SP …rms. This increase in competitive pressurecauses them to decrease their prices. As F decreases, each MP …rm produces morevarieties because it pays less for each variety it produces. This causes the …rm tocharge a higher price per unit of each variety in order to soften the competition
between its varieties.
Lemma 1 compares the equilibrium per-variety quantity and price of SP …rms
and MP …rms. It plays a key role in the welfare analysis of mergers with marginal
cost synergies that we conduct in the next section.
Lemma 1 qsp > qmp and psp < pmp if and only if f > F .
MP …rms produce less (more) per variety and charge a higher (lower) price if
11n
In Appendix A, we show that qL
and ! n derived from (21) and (26) are locally optimal.

15

their …xed cost of production is lower (higher) than the per-variety …xed cost of SP…rms. If f > F , then the cost for an MP …rm to add a new variety is lower thanthe cost for a SP …rm to enter the market and supply a variety. Since consumers
prefer a diversi…ed consumption, each MP …rm takes advantage of a lower …xed
cost by expanding its variety range. With an increased product range, each MP…rm …nds it pro…table to reduce the per-variety quantity and increase the mark-up
on each variety to ease the competition between its products.

4

Market Outcome with a Merger: Marginal Cost
Synergies

If two MP …rms merge, then there are (N 1) MP …rms in the market, including
the merged …rm (insider) and (N 2) MP outsiders. We start by examining therole of marginal cost synergies. We use the superscript M C to denote the variables
after a merger with marginal cost synergies.
We assume the merged entity enjoys a marginal cost synergy of (1)C,
so its marginal cost is reduced to C, with 0 < < 1. Following the merger,
the SP …rms simultaneously decide whether or not to enter the market and whatquantity to produce. The MP incumbent …rms make their product range and pervariety quantity choices. This determines their total …xed cost of production.12
The merging …rms maximize their joint pro…ts while the rest maximize individual
pro…ts.

4.1

Outsiders

We …rst examine the behavior of non-merging …rms. In a market structure witha merger, each SP outsider still maximizes (9) and its optimal choice still satis…es(11). Hence, the equilibrium total output level with and without the merger ispinned down by the same zero-pro…t condition given in (10) and is equal to (16).13
12

Hence, we assume that after merging, the merged entity experiences a change in its marginal
cost of production due to synergies and a change in its …xed cost of production due to the change
in its product range.
13
More generally, this result follows from the fact that under the twin assumptions of quadratic
utility and Cournot competition, the …rms play an aggregative game, where their payo¤ functions

16

It follows from (11) and (12) that the SP outsiders’equilibrium quantity and priceare the same with and without the merger. Moreover, it is straightforward to seefrom (3), (17) and (19) that since QM C = Q , the merger has no impact on the
behavior of the MP outsiders either. We summarize these observations in the next
lemma:
Lemma 2 A bilateral merger between two MP …rms has no impact on (i) theequilibrium total output level (i.e., QM C = Q ); (ii) the SP …rms’ behavior (i.e.,
CMCpM= psp and qsp= qsp ); and (iii) the MP outsiders’behavior and pro…ts (i.e.,
sp
C
MC
MC
C
= ! , qmp
= qmp and M
pM
mp = pmp , !
mp =
mp ).

4.2

MP Insider

The merging MP …rms choose their product range and quantity for each variety to
MC
C
C
= M
jointly maximize their pro…ts. Let M
I1 + I2 stand for the joint pro…ts
ICC
of the merging parties, where I stands for insider, and M
and M
represent
I1
I2
the pro…ts of the two parties of the merged …rm. We have
MC
=
I

Z !M
C
I1

MC
qI1
(v)

C

0

+

Z !M
C
I2

Z !M
C
I1

MC
qI1
(x)dx

0

MC
qI2
(w)

C

0

Z !M
C
I2

Z !M
C
I2

MC
qI2
(y)dy

Q I

0

MC
qI2
(y)dy

0

Z !M
C
I1

MC
qI1
(x)dx

0

C
MC
!M
I1 F + ! I2 F ;CMCwhere ! M
I1 and ! I2 represent the product range of the two parties of the mergedMCCMC(v) represents the quantity of variety v 2 [0; ! M…rm, qI1
I1 ], and qI2 (w) repreCsents the quantity of variety w 2 [0; ! M
I2 ]:
MC
C
The …rst-order conditions with respect to qI1(v) and ! M
I1 are

C

2

MC
(v)
qI1

2

Z !M
C
I1

MC
(x)dx
qI1

0

2

Z !M
C
I2

MC
(y)dy
qI2

Q I = 0 (29)0can be written as a function of their own action and an aggregate of all players’actions (Q in thiscase). See Anderson et al. (2020) on the long-run equilibrium properties of aggregative games.

17

!

Q I

!

MC
qI1
(v)dv

MC
qI2
(w)dw

and
C

MC
(v)
qI1

2

Z !M
C
I1

MC
qI1
(x)dx

2

Z !M
C
I2

MC
qI2
(y)dy

0

0

Q I

!

MC
qI1
(v) F = 0

(30)
MC
MC
where Q I = 0
qsp(i)di + k=1 0
qmpk(v)dv denotes the total output ofMCC
all outsider …rms. The …rst-order conditions with respect to qI2(v) and ! M
I2 canbe written similarly.
MC
MC
The …rst-order conditions with respect to qI1(v) and qI2(w) imply symmetryMCMC
across the quantities produced of the di¤erent varieties: qI1= qI2= qIM C .
CCMCDe…ne ! M
= !M
as the product range of the merged …rm. The …rstI
I1 + ! I2order conditions could be rewritten as
R SM CPN 2 R !M C

C

2 qIM C

C MC
2 !M
qI
I

Q I =0

(31)

and
C

qIM C

C MC
2 !M
qI
I

Q I qIM C

F = 0:(32)

It is immediate from these two equations thatqIM C =

s

F

:

(33)

Comparing with (21), we note that the merger has no impact on the per-variety
quantity choice of the merged entity.
C
Since qIM C = qmp and QM C = Q , we obtain from (3) that pM= pmp .
I
That is, the merger has no impact on the price of the merged entity. However,as we show below, the merger causes the merged entity to change the productrange it o¤ers. This …nding stands in contrast to what would happen in a set-up
where product variety was not endogenously determined. In such a set-up, the full
impact of the merger would be felt through a price change.14
To …nd the insider’s equilibrium product range from (32), we …rst note that
14

The result we obtain here is speci…c to marginal cost synergies. As we show in the next
section, with …xed cost synergies, mergers a¤ect consumer welfare through the twin channels of
price and product variety changes.

18

Q I = QM C
obtainC MC!M
qI . Then, after substituting for QM C and qIM C in (33), we
I

C
!M
I

=

r

p
2

p

p

f

F + (c

C)

F

:

(34)

:

(35)

Consequently, the insider’s pro…t is

MC
I

=

A merger is pro…table if
p
( 2
<1

h p
2
MC
I

p
F + (c

p
f

>2

h p
1) 2

i2
C)

mp . Simplifying givespfp
F + (c

C

C)

i

= .

Hence, while the merger has no impact on the pro…ts of the outsider …rms, it has
a positive impact on the insider …rms if and only if < .
C
The merged …rm expands its product range after the merger (i.e., ! M> 2! )
I
if
p
p p
2
f
F + (c C)= .
<1C
We make the following assumption which ensures that 0 < :pAssumption 2 2

p

f

p

F + (c2C) < 0:Since 0 < < , this assumption allows us to consider pro…table mergers withand without a product range expansion. Any merger which results in productexpansion must be pro…table. However, pro…table mergers may be accompanied
by a decrease in the range of products o¤ered by the merged …rms if the level ofsynergies is such that < < .
In what follows, we refer to the case of < as low synergies, < < asmoderate synergies, and < as high synergies. The following lemma summarizes
the impact of di¤erent levels of marginal cost synergies on the merged entity.
Lemma 3 A merger with marginal cost synergies is pro…table if < : Although
the merger has no impact on the per-variety quantity or price of the merged …rms,
19

it has a positive impact on the product range of the merged …rms if < <(high synergies), and has a negative impact on the product range of the merged…rms if < < (moderate synergies).
Lemma 2 and 3 show that although a merger between two MP …rms has no
impact on the SP …rms’equilibrium quantity, the MP …rms’equilibrium quantity
and the MP outsiders’product range, it changes the product range of the merged…rms if the level of synergies is su¢ ciently high. Intuitively, a marginal increase inproduct variety has two kinds of e¤ects on the merged entity’s pro…ts. The …rst
e¤ect is negative due to cannibalization of its own products and the second one ispositive due to business stealing from outsiders. In the absence of marginal costsynergies, the …rst e¤ect dominates and the merged …rms end up reducing theirproduct range. When the cost synergies are su¢ ciently high, the second e¤ect
dominates and the net result is an increase in product variety.

4.3

Market Performance

We are now in a position to evaluate the welfare implications of the merger. We
start by analyzing the impact of the merger on the mass of SP …rms. The premerger aggregate output Q and post-merger aggregate output QM C can be expressed as
Q= S qsp + N ! qmp ;

QM C

MC
= S M C qsp
+ (N

MC
C MC
2)! M C qmp
+ !M
qI :
I

As shown in Lemma 2, the aggregate output QM C , the quantity of each SP …rm
MCMC, and the quantity of each MP outsider ! M C qmp
do not change with theqspMC.
merger. In addition, Lemma 3 states that for the MP insider, qIM C = qmp
Hence, rearranging these two expressions gives us
S M C qsp +MC
Vmp
qmp = 0;(36)MCwhere S M C = S M C
S is the change in the mass of SP …rms, and Vmp=
C!M
2! is the change in the variety range of MP …rms. Equation (36) implies
I

20

MC
that S M C > 0 if and only if Vmp< 0, i.e., any product variety expansion
by the merged entity displaces SP …rms. Alternatively, when the insider chooses
to shrink its product range (with moderate merger e¢ ciencies from Lemma 3), it
produces less than the sum of the two merging …rms’outputs without the merger.
This leaves room in the market for more SP …rms to enter.
We can solve for the equilibrium number of SP …rms with a merger as a functionof the merger-generated marginal cost synergies:

S

MC

( )=

s

[cN

C(N

2 + )]

f

p hp
2
fN

p

F (N

i
1)

.

(37)

It is straightforward to note that this expression is increasing in . Hence, a larger
level of synergies (i.e., lower ) implies a smaller mass of SP …rms.
Lemma 4 summarizes the impact of merger-generated marginal cost synergies
on the mass of SP …rms.
<< ), which induce a
Lemma 4 Moderate merger e¢ ciencies (that is,shrinkage of the insider’s product range, increase the mass of SP …rms. By contrast, high merger e¢ ciencies, (that is, 0 <), which induce an expansion of
the insider’s product range, reduce the mass of SP …rms.
The change in consumer welfare from a merger can be expressed as:
U MC =S M C qsp (qsp|2{zqmp ):}

Reallocation E¤ect

The reallocation e¤ect represents the reallocation of production between the MP
and SP …rms. A change in the MP insider’s product range after the merger results in the reallocation of production across MP and SP …rms. The sign of thereallocation e¤ect depends on the change in the mass of SP …rms, S F C , and thecomparison of per-variety quantity between SP and MP …rms, qsp and qmp .
Consider …rst the case when a SP …rm’s …xed cost is larger than that of a MP
…rm, i.e., f > F: As shown in Lemma 1, if f > F , then a SP …rm produces more
at a lower price than a MP …rm for each variety, i.e., qsp > qmp and psp < pmp .
21

Furthermore, by Lemmas 3 and 4, with a moderate marginal cost synergy, the MP
insider shrinks its production and the mass of SP …rms increases ( S M C > 0).
Thus, a portion of MP …rms’production is reallocated to the SP …rms. Since SP…rms supply a larger per-variety quantity at a lower price than MP …rms, the entryof SP …rms generates a positive reallocation e¤ect on consumer welfare. Hence,taking the endogenous post-merger product choice of the MP insider into account,
our welfare results imply that a moderate level of merger-generated marginal costsynergies may improve consumer welfare (and high synergies may not) in a market
where the SP …rms are more aggressive.
In contrast, if a SP …rm’s …xed cost is smaller than a MP …rm’s, i.e., f < F ,15
then a SP …rm behaves less aggressively than a MP …rm, i.e., qsp < qmp andpsp > pmp . In this case, moderate marginal cost synergies, which increase the massof SP …rms, generate a negative reallocation e¤ect on consumer welfare. Therefore,
when f < F , moderate marginal cost synergies that invite more SP …rms into themarket are detrimental to consumer welfare, but high marginal cost synergies thatcause SP …rms to exit are bene…cial to consumer welfare.
Proposition 1 summarizes the impact of mergers with marginal cost synergies
on consumer welfare.
Proposition 1 When the …xed cost of a SP …rm is higher than the per-variety…xed cost of a MP …rm, i.e., f > F , all pro…table mergers with moderate marginal
cost synergies ( < < ) are bene…cial to consumer welfare, but all pro…tablemergers with high marginal cost synergies (0 <) are detrimental to consumer
welfare. In contrast, when the …xed cost of a SP …rm is lower than the per-variety…xed cost of a MP …rm, i.e., f < F , all pro…table mergers with moderate marginal
cost synergies ( < < ) are detrimental to consumer welfare, but all pro…tablemergers with high marginal cost synergies (0 <) are bene…cial to consumer
welfare.
Proposition 1 underlines that while analyzing the potential impact of a merger
with marginal cost synergies, it is important to consider …rms’ per-variety …xed
15

By the second condition in Assumption 1, this is satis…ed only if c > C, i.e. a SP small…rm’s marginal cost is strictly higher than a MP large …rm’s.

22costs of production. The existence of high marginal cost synergies is not su¢ cient to reach a conclusion on the welfare impact of a merger. Mergers with highmarginal cost synergies are detrimental to consumer welfare if f > F .16
As a corollary to Proposition 1, we emphasize the link between a merger’s
impact on the entry/exit of SP …rms and its impact on consumer welfare.
Corollary 1 When f > F , all pro…table mergers which induce entry (exit) of SP…rms are bene…cial (harmful) to consumer welfare. In contrast, when f < F , all
pro…table mergers which induce entry (exit) of SP …rms are harmful (bene…cial)
to consumer welfare.
On the policy front, Corollary 1 has signi…cant implications. We know from
Lemma 4 that mergers with su¢ ciently high marginal cost synergies cause SP…rms to exit. Corollary 1 states that such exit may sometimes (i.e., when f < F )be an indication that the merger is bene…cial to consumer welfare. Hence, using
Corollary 1, policy makers can adopt a “fringe test” to evaluate the impact ofa merger on consumer welfare. The impact of a merger on the fringe …rms (SP…rms) may help them understand whether the merger is likely to harm or hurt
consumers. Importantly, Corollary 1 shows that, contrary to the view taken inmerger guidelines, entry is not always a necessary condition for a merger to increase
consumer welfare.
We next investigate how a change in aggregate variety a¤ects consumer welfare. As a …rst step, the following lemma establishes the link between the level ofsynergies and aggregate variety. Rearranging (36) gives
qsp

qmp
qmp

SMC =

V MC,

where V M C = S M C + VLM C is the change in aggregate variety. By Lemma
1, if f > F , then qsp > qmp . That is, the per-variety quantity of a SP …rm islarger than that of a MP …rm. Since aggregate quantity does not change withthe merger, qsp > qmp implies that Sign( V M C ) = Sign( S M C ). As a result,an increase in the mass of SP …rms is accompanied with a decrease in aggregate
16

Covarrubias et al. (2020) present evidence in line with this condition.

23

variety, and vice versa. On the contrary, if f < F , which implies that qsp < qmp ,then Sign( V M C ) = Sign( S M C ). Lemma 5 summarizes the impact of a merger
on aggregate variety.
Lemma 5 When the …xed cost of a SP …rm is higher than the per-variety …xed
cost of a MP …rm, i.e., f > F , a merger with moderate marginal cost synergies( < < ) decreases aggregate variety while a merger with high marginal costsynergies (0 <) increases aggregate variety. In contrast, when the …xed costof a SP …rm is lower than the per-variety …xed cost of a MP …rm, i.e., f < F ,a merger with moderate marginal cost synergies ( < < ) increases aggregate) decreases
variety while a merger with high marginal cost synergies (0 <aggregate variety.
Finally, combining Proposition 1 with Lemma 5 yields that in mergers with
marginal cost synergies, there exists an inverse relationship between aggregate
variety and consumer welfare. Whenever aggregate variety increases, consumer
welfare decreases.
Proposition 2 A merger with marginal cost synergies increases consumer welfare
if and only if it decreases aggregate variety.
This result implies that in mergers with marginal cost synergies, any consumer
welfare gain does not originate from an increase in aggregate variety. Rather, it is
the lower prices that consumers face as a result of the reallocation of production
between MP and SP …rms that bene…t the consumers.

5

Market Outcome with a Merger: Fixed Cost
Synergies

In this section, we analyze the impact of a merger between two MP …rms that
results in …xed cost synergies. While marginal cost synergies decrease the perunit production cost, …xed cost synergies reduce the per-variety production cost.
Speci…cally, suppose that with the merger, the merged …rm’s per-variety …xed costis reduced to 2 F where 2 (0; 1).
24

As in the case of marginal cost synergies, a merger with …xed cost synergies
has no impact on the aggregate quantity or the individual behavior of non-merging…rms. To see this, note …rst that each SP outsider still maximizes (9) and the freeentry condition is still given by (10). This implies that the equilibrium behaviorof SP outsiders is given by the same expression with and without a merger, andFC
pFspC = psp and qsp= qsp where the superscript F C denotes …xed cost synergies.
As the behavior of SP outsiders does not change with a merger, by the free entry
condition (10), the equilibrium total output does not change either: QF C = Q .
Finally, since QF C = Q and the …rst-order conditions of MP outsiders are still
given by (17) and (19), the merger has no impact on the behavior of the MPFC= qmp and FmpC = mp . Therefore, Lemma
outsiders. That is, pFmpC = pmp , qmp
2 continues to hold.
We next consider the impact of …xed cost synergies on the merged entity. The
merged …rm’s pro…t is
FC
I

=
=

FC
I1 +
Z !FI1C

FC
I2
FC
qI1
(v)

C

Z !FI2C

FC
qI2
(w)

C

Z !FI2C

FC
qI2
(y)dy

0

! FI1C F + ! FI2C F .

FC
qI2
(y)dy

QF C
I

0

Z !FI2C

0

2

FC
qI1
(x)dx

0

0

+

Z !FI1C

Z !FI1C

FC
qI1
(x)dx

!

(38)
FC
qI1
(v)dv

QF C
I

0

!

FC
The …rst-order conditions with respect to qI1(v) and ! FI1C are

C

2

FC
qI1
(v)

2

Z !FI1C

FC
qI1
(x)dx

2

0

Z !FI2C

FC
qI2
(y)dy

QF C
I = 0 (39)

0

and
C

FC
qI1
(v)

2

Z !FI1C

FC
qI1
(x)dx

0

R SF C2

Z !FI2C
0C
PN 2 R !Fmpk

FC
qI2
(y)dy

QF C
I

!

FC
(v)
qI1

(40)

FC
FC
where QF C
qsp(i)di + k=1 0
qmpk(v)dv denotes the total output of
I = 0FC
all outsider …rms. The …rst-order conditions with respect to qI2(v) and ! FI2C can

25

2

F =0

FC
qI2
(w)dw

be written similarly. Imposing symmetry across the quantities produced of theFCFCdi¤erent varieties and setting qI2= qI1= qIF C gives us
C

2 qIF C

2 ! FI1C qIF C

qIF C

2 ! FI1C qIF C

2 ! FI2C qIF C

QF C
I = 0

(41)

2

(42)

and
(

C

2 ! FI2C qIF C

FC
QF C
I )qI

F = 0:

It is immediate from these two equations thatqIF C =

s

F

(43)

:

Substituting qIF C into the inverse demand, the insider’s price is
pFI C = c + 2

p

f

p

F > pmp = c + 2

p

f

p

F:

(44)

In contrast with the case of marginal cost synergies, we immediately observethat …xed cost synergies a¤ect both the per-variety quantity and price of themerged entity. Speci…cally, per-variety quantity decreases and price increases as…xed cost synergies increase. An increase in …xed cost synergies gives the merged
entity an advantage in the cost of introducing new varieties. As the merged entity
introduces more varieties, it …nds it optimal to increase the price of each variety
in order to reduce the competitive pressure between them.
pMoreover, since psp = c +f , we can also write (44) as pFI C = psp +p
p pf
F . Hence, the price di¤erence between the merged entity andthe SP …rms depends on the di¤erence between their (post-merger) …xed costs.
This implies that the ranking between the prices of the MP and SP …rms may bereversed if it is the case that f < F before the merger and the magnitude of …xed
cost synergies are such that f > 2 F after the merger.
FC
Note that QF C
! FI1C qIF C! FI2C qIF C and that ! FI1C = ! FI2C underI = Q
symmetry. De…ne ! FI C = ! FI1C + ! FI2C as the variety range of the merged …rm.
Substituting (43) into (42) and using Lemma 2, the equilibrium variety range of26

the merged entity can be expressed as! FI C=1

Its pro…t is
FC
I

=

r

p
2

p

p
F + (c

f

C)

h p
2

(45)

:

F
p

p

f

F + (c

i2
C)

(46)

:

We next consider how the merged entity’s pro…ts, product range and total
production depend on the level of synergies. From (45) and (46), we can readily
verify that both ! FI C and FI C decrease with . That is, the merged entity’s
product range and pro…t increase as it enjoys larger …xed cost synergies. A merger
is pro…table if and only if FI C > 2 mp . Simplifying yields:<1p

2 1 h p
p
2
2 F

p
f

p

F + (c

i
C) =

1:

(47)

A merger increases its total variety range (i.e., ! FI C > 2! ) if and only if

<1
2

p
2
hp

p

f
p
2 f

p

F + (c C)i=p
F + (c C)

2:

(48)

Finally, a merged entity increases its total quantity (i.e., ! FI C qIF C > 2! qmp ) if
and only if
p
p p
2
f
F + (c C)
p
<1
= 3:
(49)
2 F
It is readily veri…able that 3 < 2 < 1 < 1, where the last inequality follows
from Assumption 1(ii). We summarize the impact of di¤erent levels of …xed costsynergies on the behavior of the merged entity in the next lemma.
Lemma 6 A merger with …xed cost synergies causes the per-variety quantity of themerged …rms to decrease and the per-variety price of the merged …rms to increase.
Moreover, there exist threshold values 0 < 3 < 2 < 1 < 1 such that (i) all
mergers with 0 < < 1 are pro…table, (ii) all mergers with 0 < < 2 result27

in an expansion of variety range of the merged entity, and (iii) all mergers with
0 < < 3 result in an increase in the total production of the merged entity.
Similar to Lemma 4, the next lemma establishes the impact of a merger with…xed cost synergies on the mass of SP …rms. As discussed above, the merger has no
impact on the aggregate quantity, the SP …rms’behavior, and the MP outsiders’behavior. Using
Q = QF C
and expanding both sides, we get
S qsp + 2! qmp + (N2)! qmp = S F C qsp + ! FI C qIF C + (N2)! qmp ;which implies
S F C qsp = 2! qmp! FI C qIF C ;where S F C = S F C
S denotes the change in the mass of SP …rms. As statedin the next lemma, this expression implies that the mass of SP …rms decreases
whenever the MP insider expands its total production.
Lemma 7 A pro…table merger with merger-generated …xed cost synergies 0 < <3 causes the mass of SP …rms to decrease. A pro…table merger with mergergenerated …xed cost synergies 3 < < 1 causes the mass of SP …rms to increase.
This result is consistent with the one we have for marginal cost synergies(Lemma 4). Since a merger with marginal cost synergies has no impact on themerged entity’s per-variety quantity, an increase in total variety is equivalent to anincrease in total production with marginal cost synergies. In the case with …xed
cost synergies, an increase in total variety is not equivalent to an increase in total
production because a merger reduces the per-variety quantity of the MP insider.

5.1

Market performance

Since the price of the merged entity changes with the level of …xed cost synergies,
as shown in (44), the impact of a merger on consumer welfare can be decomposed28

into the following two e¤ects:17
UFC =S F C qsp qsp|2{z

qmp + ! I qIF C} |Reallocation E¤ect

pmp{z

Price E¤ect

pFI C :}

(50)

As before, the reallocation e¤ect represents the reallocation of production between
the MP and SP …rms. As in the case of marginal cost synergies, the sign of thereallocation e¤ect depends on the change in the mass of SP …rms, S F C , and thecomparison of per-variety quantity between SP and MP …rms, qsp and qmp . The
following lemma establishes when the reallocation e¤ect is positive. The result
follows from Lemma 7 and the expressions for qsp and qmp given in (11) and (21).
Lemma 8 For f > F; the reallocation e¤ect is positive if and only if the massof SP …rms increases, i.e., S F C > 0; which occurs for 3 < . For f < F; thereallocation e¤ect is positive if and only if the mass of SP …rms decreases, i.e.,
S F C < 0; which occurs for 0 < < 3 .
Lemma 8 implies that when f < F (f > F ), a necessary condition for mergers
to bene…t consumer welfare is to have exit (entry) by SP …rms. When f < F , eventhough society pays a lower …xed cost per variety with each SP …rm, the SP …rms
charge a higher price per unit (Lemma 1). Due to the higher price charged by the
SP …rms, it is bene…cial to have less of them.
The second e¤ect in (50), the price e¤ect, is about the impact of the merger
on the price of the merging …rms. From (44), we observe that the price e¤ect isnegative whenever the merged entity incurs …xed cost synergies since …xed costsynergies increase the price of the MP insider, i.e., pmp < pFI C . Furthermore, theprice e¤ect strengthens as the merged entity enjoys higher …xed cost synergies,
as both ! I qIF C and (pmp pFI C ) increase with . Therefore, the negative pricee¤ect may dominate whenever the merged entity enjoys su¢ ciently high …xed costsynergies. The existence of the negative price e¤ect suggests that mergers with…xed cost synergies may be more likely to decrease consumer welfare in comparison
to marginal cost synergies.
17

C
Speci…cally, using that QF C = Q , U = 2 [ S F C (qsp )2 + ! F(qIF C )2 2! (qmp )2 ] =
I
FC
C
C FC
(qsp )2 + ! F(qIF C )2
qmp ( S F C qsp + ! F
qI )] = 2 [ S F C qsp (qsp
qmp ) +
I
I
2[ S
C FC
C FC
C
qI (pmp pF)]!F
qI (qIF Cqmp )] = 2 [ S F C qsp (qsp qmp ) + ! F
I
I
I

29

We next examine more closely whether a pro…table merger with …xed costsynergies improves or harms consumer welfare. To do this, we …rst establish howdi¤erent levels of …xed cost synergies impact consumer welfare without taking into
account merger pro…tability.
The change in consumer welfare with …xed cost synergies can be expressed as
UFC =

p (
2

hp

C) f
p
f 2 f F (2

(c
p
+2

p

F (2

)

i

) + F (22which has a negative sign with respect to the quadratic termto U F C = 0 are given by

4 =

5 =

p
4 f + (cC)

qp
32 ( fpp
4 f + (cqpC) + 32 ( f

p

p p
( f

p

p p
F )2 + 16 ( f
p
4 F

p

F )2 + 16
p
4 F

2

)

)

;

(51)

. The two solutions

F )(cC) + (c

C)2
(52)

F )(c

C) + (cC)2(53)
where 4 < 5 . Therefore, U F C > 0 if and only if 4 < < 5 . It is readily veri…ed
p
p
p p
p
that 5 > 0.18 4 > 0 if and only if
f 2 F 2 ( fF ) + (c C) <p
2 fF:
We summarize the results in Lemma 9.
p
p
p p
p
p
Lemma 9 (i) If
f 2 F 2 ( fF ) + (c C)
2 f F , then 4
0: In this case, consumer welfare increases if 2 (0; 5 ), and decreases if 2 ( 5 ; 1).
p
p
p p
p
p
(ii) If
f 2 F 2 ( fF ) + (c C) < 2 f F , then 4 > 0: In
this case, consumer welfare increases if 2 ( 4 ; 5 ), and decreases if 2 (0; 4 ) [( 5 ; 1):
Assuming 4 > 0, we de…ne 2 (0; 4 ) as high …xed cost synergies, 2 ( 4 ; 5 )as moderate …xed cost synergies, and 2 ( 5 ; 1) as low …xed cost synergies. Then,pp p
p
By Assumption 1(ii), 2
f
F + (c C) > 0, which is equivalent to 2 f +pp(c C) > 2 F > 0.
This is su¢ cient for 4 f + (c C) > 0.
Furthermore,
wer assume that the roots 4 and 5 are real, which is guaranteed by (55). Therefore,
p 2
p
p
p p
2
32
f
F + 16f
F (c C) + (c C) > 0; and 5 > 0 holds.
18

30

;

;

Lemma 9 implies that mergers with moderate …xed cost synergies increase consumer welfare, whereas mergers with high or low …xed cost synergies decrease
consumer welfare. Because the price e¤ect is increasing in the level of synergies,
its e¤ect makes mergers with high …xed cost synergies consumer welfare decreasing.
We are interested in exploring how pro…table mergers a¤ect consumer welfare.
Since Lemma 6 demonstrates that a merger is pro…table if 2 (0; 1 ), we makethe following assumption to ensure that 1 > 0 and there exist some pro…tablemergers.
Assumption 3 (cppC) < 2( 2 + 2) Fp
2 f:

In order to examine the impact of pro…table mergers on consumer welfare, weneed to compare 1 with 4 and 5 . Suppose that the conditions in Lemma 9-(ii)are satis…ed such that 4 and 5 exist and are both positive. The next proposition
establishes the impact of mergers with …xed cost synergies on consumer welfare.
Proposition 3 Consider a merger with …xed cost synergies and suppose that Assumptions 1 and 3 hold. Then:(i) If f > F and c C, then all pro…table mergers decrease consumer welfare.
(ii) If f > F and c < C, then there exist some pro…table mergers with moderate…xed cost synergies which increase consumer welfare.
(iii) If f < F and c > C, then there are two subcases depending on how e¢ cient
the MP …rms are:pp p pF ): Then all pro…tablea. Suppose (c C) > ( 8 + 4 2) ( fmergers decrease consumer welfare.
pp p pF ): Then there exist someb. Suppose (c C) < ( 8 + 4 2) ( fpro…table mergers with moderate …xed cost synergies which increase consumer welfare.
Proof. See Appendix C.
In Proposition 3(i), MP …rms are more e¢ cient than SP …rms in terms of boththeir …xed cost and their marginal cost. As shown in Appendix C, we have 1
4.
Hence, mergers are pro…table under high synergies only. We know from Lemma
9 that high synergies are not consumer welfare increasing because they result in31

a strong price e¤ect which is always negative. Proposition 3(i) implies that anecessary condition for mergers with …xed cost synergies to increase consumer
welfare is that SP …rms are more e¢ cient in terms of either their …xed cost or
their marginal cost. Recall that if SP …rms are more e¢ cient in terms of bothtypes of costs, we get a contradiction with Assumption 1 and a mixed market
structure would not exist.
In Proposition 3(ii), MP …rms are more e¢ cient than SP …rms in terms of
their …xed cost but not in terms of their marginal cost. Since 4 < 1 < 5 , somemergers with moderate level of synergies can be pro…table and consumer welfare
increasing at the same time. In Proposition 3(iii), MP …rms are less e¢ cient interms of their …xed cost but more e¢ cient in terms of their marginal cost than SP…rms. Subcase (a) in Proposition 3(iii) shows that even if the MP …rms do nothave a …xed cost advantage, if their marginal cost advantage is su¢ ciently large,
then all mergers harm consumer welfare. Otherwise, there exist pro…table mergers
with moderate synergies which increase consumer welfare.
Figure 1 provides a graphical representation of Proposition 3. On the horizontalpaxis is f and on the vertical axis is the marginal cost di¤erence between SP and
MP …rms, c C: We divide the parameter space which satisfy Assumptions 1
and 3 (represented by the orange and blue lines) into four subregions. The green
line represents the cut-o¤ condition for case (iii) in Proposition 3. In subregions(i) and (iiia), all pro…table mergers decrease consumer welfare. In subregions (ii)and (iiib), mergers may increase consumer welfare only when the synergies aremoderate.
[Figure 1 around here]
In summary, Proposition 3 and Figure 1 imply that antitrust agencies should
be cautious of mergers between MP …rms which generate …xed cost synergies if
the MP …rms are more e¢ cient than the SP …rms. Moreover, they should becautious of the e¢ ciency defense when merger related e¢ ciencies are substantial.
This is because the bene…t from …xed cost e¢ ciencies is not necessarily passed on
to consumers. Mergers with …xed cost synergies (unlike mergers with marginal
cost synergies) result in higher prices. That is, with a low per-variety …xed cost,the merging parties …nd it optimal to supply a wide range of varieties and charge
32a higher price per variety. This negative price e¤ect of a merger is increasingin the level of synergies and may dominate whenever the merged entity enjoys
su¢ ciently high …xed cost synergies. At the same time, the potential expansionof production by the merging parties may lead to the exit of some SP …rms who
supply di¤erentiated goods at lower prices. Therefore, mergers with high …xed costsynergies may generate anticompetitive e¤ects, raising the average market priceand decreasing consumer welfare.
We next investigate the link between a merger’s impact on aggregate variety
and its impact on consumer welfare. We know from Proposition 2 that with
marginal cost synergies, consumer welfare decreases whenever aggregate variety
increases. The next proposition states that this is not necessarily the case with…xed cost synergies.
Proposition 4 Consider a merger with …xed cost synergies and suppose that Assumptions 1 and 3 hold.
(i) Suppose f < F . If there is an increase in aggregate variety, then consumer
welfare always decreases. If there is a decrease in aggregate variety, then consumer
welfare may increase or decrease.
(ii) Suppose f > F . If there is a decrease in aggregate variety, then consumer
welfare always increases. If there is an increase in aggregate variety, then consumer
welfare may increase or decrease.
Proof. See Appendix D.
Proposition 4 implies that if the SP …rms have a …xed cost advantage over theMP …rms before the merger, then an increase in aggregate variety always implies
that the merger is detrimental to consumer welfare. In this case, any consumer
welfare increasing merger is accompanied with a decrease in aggregate variety.
This result is consistent with the result for mergers with marginal cost synergies.
However, if the MP …rms have a …xed cost advantage over the SP …rms before themerger, then an increase in aggregate variety can be an indication that the merger
is bene…cial for consumers. In this case, it is possible that a merger with …xed
cost synergies increases both the aggregate variety and consumer welfare. This
result di¤ers from the result for mergers with marginal cost synergies, in whichcase consumer welfare decreases whenever aggregate variety increases.
33

6Conclusion

We consider bilateral mergers between multi-product …rms in a market structure with a monopolistically competitive fringe of single-product …rms. Aggregate
product variety in the market is determined by the endogenous product variety
choices of the multi-product …rms and entry/exit decisions of single-product …rms.
We consider and compare the impact of mergers when there are merger-generated
marginal cost synergies or …xed cost synergies. Fixed cost synergies in our model
correspond to savings on the cost of producing one more variety .
The consumer welfare impact of mergers with marginal cost synergies originatesfrom a reallocation e¤ect. Due to this reallocation e¤ect, when the multi-product…rms have a …xed cost advantage over the single-product …rms, mergers with su¢ ciently high marginal cost synergies harm consumers, but mergers with a moderatelevel of cost synergies increase consumer welfare. However, if single-product …rms
have a …xed cost advantage, then mergers with su¢ ciently high marginal cost synergies increase consumer welfare while mergers with moderate synergies decrease
consumer welfare.
The consumer welfare impact of mergers with …xed cost synergies originatesfrom a reallocation e¤ect as well as a price e¤ect. Because the price e¤ect isnegative and increasing in the level of synergies, mergers with su¢ ciently highsynergies always hurt consumer welfare. Similarly, mergers which take place inindustries where the multi-product …rms are more e¢ cient than the single-product…rms also always hurt consumer welfare.
These results indicate that in general, we should be cautious of the e¢ ciency
defense when merger-generated e¢ ciencies are substantial. Moreover, it is of policy
interest whether a merger’s impact on aggregate variety can be used as an indication of its impact on consumer surplus. For example, are those mergers which
result in an increase in aggregate variety more likely to increase consumer welfare?
Our …ndings reveal that in mergers with marginal cost synergies, consumer welfare
decreases whenever aggregate variety increases. Hence, any consumer welfare gain
does not originate from an increase in aggregate variety. In contrast, an increase inaggregate variety can be an indication that the merger is bene…cial for consumers
in the case of mergers with …xed cost synergies if the multi-product …rms have a34

…xed cost advantage over the single-product …rms before the merger.

References[1] Anderson, S., N. Erkal and D. Piccinin, 2020. Aggregative games and oligopoly
theory: short-run and long-run analysis, RAND Journal of Economics, 51,
470-495.
[2] Bernard A. B., S. J. Redding and Schott P. K., 2010. Multiple-product …rms
and product switching, American Economic Review, 100, 70-97.
[3] Berry, S. and Waldfogel, J. 2001. Do mergers increase product variety? Evidence from radio broadcasting. Quarterly Journal of Economics, 116, 10091025.
[4] Cabral, L.M.B., 2003. Horizontal mergers with free-entry: why cost e¢ ciencies
may be a weak defense and asset sales a poor remedy, International Journalof Industrial organization, 21, 607–623.
[5] Caradonna, P., Miller, N. H. and Sheu, G. (2021). Mergers, Entry, and Consumer Welfare. Georgetown University, mimeo.
[6] Covarrubias, M., Gutiérrez, G., Philippon, T. (2020). From Good to Bad
Concentration? US Industries over the past 30 years. NBER Macroeconomics
Annual, 34(1), 1-46.
[7] Davidson, C. and Mukherjee, A., 2007. Horizontal mergers with free entry,
International Journal of Industrial Organization, 25, 157–72.
[8] Erkal, N. and Piccinin, D., 2010. Welfare-reducing mergers in di¤erentiatedoligopolies with free entry. Economic Record, 86(273), 178-184.
[9] Fan, Y., and Yang, C., 2020a. Competition, product proliferation, and welfare: A study of the US smartphone market. American Economic Journal:
Microeconomics, 12(2), 99-134.

35

[10] Fan, Y., and Yang, C., 2020b. Merger, product variety and …rm entry: theretail craft beer market in California. Working paper.
[11] Gandhi, A., L. Froeb, S. Tschantz and G. J. Werden, 2008. Post-merger
product repositioning. Journal of Industrial Economics, 56(1), 49-67.
[12] Gowrisankaran, G., 1999. A dynamic model of endogenous horizontal mergers.
RAND Journal of Economics, 30, 56–83.
[13] Jeziorski, P. 2014. Estimation of Cost E¢ ciencies from Mergers: Application
to US Radio. RAND Journal of Economics, 45(4), 816-846.
[14] Johnson, J. and Rhodes, A., 2021. Multiproduct ,mergers and quality competition. RAND Journal of Economics, forthcoming.
[15] Li, S., J. Mazur, Y. Park, J. Roberts, A. Sweeting, and J. Zhang, 2019.
Repositioning and market power after airline mergers. Working paper.
[16] Lommerud, K. and Sogard, L., 1997, Merger and product range rivalry. International Journal of Industrial Organization, 16, 21-42.
[17] Mazzeo, M. J., K. Seim, and M. Varela, 2018. The welfare consequences ofmergers with endogenous product choice. Journal of Industrial Economics,
66(4), 980–1016.
[18] Nocke, V. and Schutz, N., 2018. An aggregative games approach to merger
analysis in multiproduct-…rm oligopoly. Working paper.
[19] Pan, L., and Hanazono, M., 2018. Is a big entrant a threat to incumbents?
The role of demand substitutability in competition among the big and thesmall. Journal of Industrial Economics, 66(1), 30-65.
[20] Parenti, M., 2018. Large and small …rms in a global market: David vs. Goliath.
Journal of International Economics, 110, 103-118.
[21] Shimomura, K. I., and Thisse, J. F., 2012. Competition among the large andthe small. RAND Journal of Economics. 43(2), 329-347.

36

[22] Spector, D., 2003. Horizontal mergers, entry, and e¢ ciency defences. International Journal of Industrial Organization, 21, 1591-1600.
[23] Sweeting, A., 2010. The e¤ects of mergers on product positioning: evidence
from the music radio industry. RAND Journal of Economics, 41(2), 372-397.
[24] U.S. Department of Justice and FTC, 2010, Horizontal Merger Guidelines,
Washington, DC.
[25] Werden, G.J. and Froeb, L.M., 1998. The entry-inducing e¤ects of horizontal
mergers: an exploratory analysis. Journal of Industrial Economics, 46, 525–
43.

A

Derivation of Assumption 1

In this appendix, we show the derivation of the expressions in Assumption 1.
Assumption 1(i) ensures that the mass of SP …rms is positive in equilibrium.
S ; S M C and S F C denote the equilibrium mass of SP …rms when no merger takes
place, when there is a merger with marginal cost synergies, and when there is amerger with …xed cost synergies, respectively. Hence, we need to …nd the condition
such that min S ; S M C ; S F C > 0.
p i
p
p h
(N + 1) f N F :From (25), the condition for S > 0 is > c(N +1) CN +2
From (37), we see that S M C is increasing in . Therefore, a su¢ cient condition hfor S M C ( ) > 0 is iS M C (0) > 0, which requires> cN C (N 2)ppp
N f (N 1) F . Finally, when there is merger with …xed cost syner2
gies, the equilibrium mass of small …rms is given by
SF C =

s

[cN
f

C(N

1)]

p hp
2
f (N

1)

p
F (N

i
3+ )

;

which is increasing inh . Therefore, a su¢ cienti condition for S F C > 0 is >pp p
cN C(N 1)+2
f (N 1)
F (N 3) : Combining all three conditions
gives us the expression in Assumption 1(i).
Assumption 1(ii) ensures that N > 0, or equivalently, ! > 0 in equilibrium.
The equilibrium ! values (with and without a merger) are given by (26), (34) and37

(45). Comparing these three expressions reveals that (34) is decreasing in and(45) is decreasing in , so it is su¢ cient to show that (26) > 0, which corresponds
to the condition assumed in Assumption 1(ii).

B

Second-order conditions for a MP …rm

The …rst-order derivatives of MP …rm n’s pro…t with respect to its per-varietynand product range ! n are given byquantity qmp
@ nmp
= (
n
@qmp
@ nmp
= (
@! n

C

n
2 qmp

n
2 ! n qmp

Cn
( + 2 ! n )qmp

Q n )! n ;nQ n )qmp

F:

Therefore, the Hessian matrix is given byH =C

2( + ! n )n2( + 2 ! n )qmp

C

2(

Q n

n
+ 2 ! n )qmp
n 2
2 (qmp
)

Q

n

!

;

with determinant
jH j = 4

n 2( + ! n )(qmp)

Cn2( + 2 ! n )qmp

Q n

2

:

n 2
We can readily show that 2( + ! n ) < 0 and 2 (qmp
) < 0.
Substituting the …rst-order conditions in (17) and (22), we haveCn
2 ! n qmpn
Q n = 2 qmp:

Therefore,
jH j = 4n 2! n (qmp) > 0:n
Therefore, qmp
and ! n are locally optimal.19
19

The second order conditions for the MP large outsiders are also satis…ed, following the same
proof. As for the MP large insider, we can modify the proof by replacing C with C.

38

C

Proof of Proposition 3

p
p p
( f
F ) and = c C: and represent the relative e¢ ciency of
Let =a SP and a MP …rm. Recall that 2 + > 0 by Assumption 1(ii).
First, suppose 32 2 +16 + 2 < 0. Joint with Assumption 1(ii), this conditionis equivalent to< 0 and2pp
4 2; 8 + 4 2 :

8

(54)

In this case, the roots 4 and 5 , given in (52) and (53), are imaginary, and any
merger decreases consumer welfare.
Now suppose 32 2 +16 + 2 > 0. Joint with Assumption 1(ii), this conditionis equivalent to

(b)

(a)
< 0 and 2

0 and > 2 ; andpp1; 8 4 2 [ 8 + 4 2; 2 :(55)

To examine the impact of a pro…table merger with …xed cost synergies on consumer
welfare, we compare 1 with 4 and 5 .
Comparing 1 and 4 , we have

1

4 =

p
32 2 + 16

+

p
2 2
p
4 F

2

1

p
+4 2

p
p
which is negative if and only if (2 2 1) + 4 2 > 0 andconditions hold if (a)0 and > 0, or (b) < 0 and = <Combined with (55-b), we …nd that 1 < 4 if and only if
(a)

0 and

> 0; or (b)It is straightforward to check that

< 0 and =8;> 0.20 Thesep
16 + 4 2 =7.

p
4 2:

(56)

4 > 1 under condition (56-b).

p
p
p
2
2
1)
+
4
2
>
32
+ 16 +p2 > 0.
Speci…cally, 1
<
0
if
and
only
if
(2
4
p
2
2
2
Given thatp 2 + > 0 by Assumption 1(ii), (32 + 16 + ) [(2 2 1) + 4 2 ] =4(22) (2 + ) is negative if and only if > 0.
20

39

Comparing

5

1 and

1 =

5 , we havep
32 2 + 16

+

p
(2 2
p
4 F

2+

p
1) + 4 2;pwhich is negative if and only if < 0 and = 2(16 + 4 2)=7; 2 . Note thatppby condition (55-b), 5 does not exist if < 0 and = 2(16 + 4 2)=7; 8 + 4 2 .
Therefore, 5 < 1 if and only if< 0 and

p
8+4 2

= <

2:

(57)

In summary, we obtain the following three possible rankings of 1 , 4 and 5 :p(i) If0 and > 0, or < 0 and =8 4 2, then 1 < 4 < 5 .
(ii) If > 0 and = 2 ( 2; 0], then 4 < 1 < 5 .
p(iii) If < 0 and = 2
8 + 4 2; 2 , then 4 < 5 < 1 < 1:
Combined with Lemma 9, we are able to establish the impact of mergers withdi¤erent …xed cost synergies on consumer welfare in di¤erent parts of the parameter
space:(i) If0 and > 0; then 1 < 4 < 5 . In this case, all pro…table mergers
decrease consumer welfare. This gives Proposition 3(i).
(ii) If > 0 and = 2 ( 2; 0], then 4 < 1 < 5 . In this case, pro…table mergers with synergies satisfying 2 ( 4 ; 1 ) increase consumer welfare, and pro…tablemergers with synergies satisfying 2 (0; 4 ) decrease consumer welfare. This gives
Proposition 3(ii).
p(iii) If < 0 and =8 4 2; then 1 < 1 < 4 < 5 . If < 0 andpp= 2
8 4 2; 8 + 4 2 , then the roots of 4 and 5 are imaginary. In bothcases, any pro…table merger decreases consumer welfare. This gives Proposition
3(iii)a.
p(iv) If < 0 and = 2
8 + 4 2; 2 , then 4 < 5 < 1 : In this case, profitable mergers with synergies satisfying 2 ( 4 ; 5 ) increase consumer welfare, andpro…table mergers with synergies satisfying 2 (0; 4 ) [ ( 5 ; 1 ) decrease consumer
welfare. This gives Proposition 3(iii)b.

40

D

Proof of Proposition 4

p
p p
( f
F ) and = c C: and represent the relative e¢ ciency of
Let =a SP and a MP …rm.
We …rst note that, using Q = QF C , we have
S F C qsp + ! FI C qIF C, ! FI C=

2! qsp = 0;
S F C qsp

2! qmpqIF C:

Therefore, we can express the change in aggregate variety as
V

=
=
=

S F C + ! FI C
2!
2! qmpS F C qsp
FC
2!
S +
qIF C
1
[
S F C (qsp qIF C ) + 2! (qmpFC

qI

qIF C )];

(58)

and the change in consumer welfare as
U =
=
=

2
2

[ S F C (qsp )2 + ! FI C (qIF C )2

2! (qmp )2 ][ S F C (qsp )2

2! (qmp )2 + qIF C (2! qmp

qsp[ S F C (qsp2

qIF C )

2!

qmp
(q
qsp mpS F C qsp )]qIF C )]:

(59)

We next note the following two properties that will be useful later. Property 1
follows from (43) and Property 2 follows from Lemma 1.
Property 1: A merger with …xed cost synergies results in a reduction in thepper-variety quantity of the insider, i.e., qmp qIF C = (1
) F= > 0.
p
Property 2: If f < F , then qmp =qsp = F=f > 1. If f > F , then qmp =qsp < 1.
Proof of Proposition 4(i):The …rst part of Proposition 4(i) states that when f < F , if there is an increasein aggregate variety, then consumer welfare always decreases. To see this, note thatby (58), if V > 0; then S F C (qsp qIF C ) 2! (qmp qIF C ) < 0. Property 1

41

and 2 imply
2! (qmp =qsp )(qmp

qIF C ) <2! (qmp

qIF C ).

qIF C )

2! (qmp

qIF C )] < 0:Then, by (59) we haveU < ( qsp =2)[ S F C (qsp

Therefore, when f < F , if V > 0; then U < 0.
The second part of Proposition 4(i) states that when f < F; if there is adecrease in aggregate variety, then consumer welfare may increase or decrease. We
proceed in two steps.
p
Step 1: We …rst show that when f < F and ( 8 + 4 2) < , a decrease inaggregate variety ( V < 0) is always accompanied with a decrease in consumer
welfare ( U < 0). This follows from Proposition 3(iii)a, which states that allppro…table mergers decrease consumer welfare if f < F and ( 8 + 4 2) < :p
Step 2: Next we prove that when f < F and 2 < < ( 8 + 4 2) , if
V < 0, then U may be positive or negative. Note that these are the conditions
given in Proposition 3(iii)b.
First, we can readily show that V is U-shaped in .21 Let 6 and 7 , with
V . Hence, V < 0 if and only if 2 ( 6 ; 7 ).
6 < 7 , stand for the two roots of
In addition, we have shown in (51) that U is an inverted U-shaped function of, with roots 4 and 5 . U > 0 if and only if 2 ( 4 ; 5 ). Therefore, in order toestablish the link between V and U , we need to rank 1 , 4 , 5 , 6 , and 7 . We
establish the ranking in the following two sub-steps.
Step 2.1: In this step, we show that 6 < 4 < 5 < 7 .
At 4 and 5 , U = 0; and by (59), S F C qsp qIF C = 2! (qmp =qsp )(qmp
qIF C ). Substituting this expression into (58), we get V = (1=qIF C )2! (qmp
qIF C )(1 qmp =qsp ): By Property 1, qmp qIF C > 0. By Property 2, if f < F , then
1 < qmp =qsp . Therefore, V < 0:
Since V is U-shaped in , and U is inverted U-shaped in , V ( 4 ) < 0,
V ( 5 ) < 0, and U ( 4 ) = U ( 5 ) = 0 imply that 6 < 4 < 5 < 7 .
21Speci…cally,

V =

[

p

f

4 (f

p

f F +F )]+
p
fF

42

p

f +2 F2+2 f.

Step 2.2: We next show that 7 < 1 . Note that for > 7 , V > 0: Therefore,
to show that 7 < 1 , we show that V ( 1 ) > 0:p
When 2 < < ( 8 + 4 2) , the denominator of V ( 1 ) is always negative and the numerator of V ( 1 ) is quadratic in .22 Denote the numerator of V ( 1 ) as N U V 1 : If f < F and= 2 , then N U V 1 = 0 andppp2
F < 0. If f < F and=
8+4 2 ,@(N U V 1 )=@ = 2 2ppthen N U V 1
8 2 3=2 (0:01 f + 0:05 F ) < 0. Therefore, when f < F andp
2 < <8 + 4 2 , the numerator of V ( 1 ) is always negative. Since thedenominator of V ( 1 ) is also negative, ( 1 ) > 0. Thus, we prove that 1 > 7 :By Steps 2.1 and 2.2, we conclude that when f < F and 2 < < ( 8 +p
4 2) , 6 < 4 < 5 < 7 < 1 : Hence, if aggregate variety decreases, which
implies that …xed cost synergies satisfy 2 ( 6 ; 7 ), then consumer welfare may
increase, which happens when 2 ( 4 ; 5 ), or decrease, which happens when 2(0; 6 ) [ ( 5 ; 7 ).
p
By Steps 1 and 2, we conclude that when f < F and ( 8 + 4 2) < , ifaggregate variety decreases, then consumer welfare decreases. When f < F andp
2 < < ( 8 + 4 2) , if aggregate variety decreases, then consumer welfare
may increase or decrease.
This completes the proof of Proposition 4(i).
Proof of Proposition 4(ii):
The proof technique is analogous to that of Proposition 4(i). The …rst partof Proposition 4(ii) states that when f > F , if there is a decrease in aggregate
variety, then consumer welfare always increases. To see this, note that if V < 0;then S F C (qsp qIF C ) 2! (qmp qIF C ) > 0 by (58). Property 1 and 2 imply
that
2! (qmp =qsp )(qmp qIF C ) > 2! (qmp qIF C ).
From (59) we have
U > ( qsp =2)[ S F C (qsp22

Speci…cally,V ( 1) =

8
>
>
<
>
>
:

qIF C )

2! (qmp

qIF C )] > 0:

p p i
p h p
p
2( 2 1) f + (4 3 2) F
h p
p p i
p
p p
4 2 1 f+ 6 5 2 F
8
2
2

2

p

p

p

[ f F ( 2 1)( +2

43p pf) 2 2
F]

1

9
>
>
=

3 >
>
;

Therefore, when f > F , if V < 0; then U > 0:
The second part of Proposition 4(ii) states that when f > F; if there is anincrease in aggregate variety, then consumer welfare may increase or decrease. As
stated above, V is U-shaped in with roots 6 and 7 . Therefore, in order toestablish the relationship between V and U , we need to rank 1 , 4 , 5 , 6 , and
7 . We proceed in two steps.
Step 1: In this step, we show that when f > F; 4 < 6 < 7 < 5 .
When = 4 or = 5 , U = 0 and by (59), S F C qsp qIF C = 2! (qmp =qsp )(qmp
qIF C ): Substituting this expression in (58), we get V = (1=qIF C )2! (qmp
qIF C )(1 qmp =qsp ): By Property 1, qmp qIF C > 0: By Property 2, if f > F ,then qmp =qsp < 1. Therefore, V > 0:
Since V is U-shaped in , and U is inverted U-shaped in , V ( 4 ) > 0,
V ( 5 ) > 0; and U ( 4 ) = U ( 5 ) = 0 implying that 4 < 6 < 7 < 5 .
Step 2: In this step, we show that 6 < 1 < 7 :When = 1, (43) implies that qIF C = qmp , and Lemma 7 implies that S F C >0: In addition, Property 2 implies that when f > F , qsp qmp > 0: Thus, by (58),
we have
V (1) = (1=qIF C )[=S F C (qsp

S F C (1=qIF C )(qsp

qIF C )]
qmp )

< 0:
Since V < 0 if and only if 6 < < 7 , V (1) < 0 implies that 6 < 1 < 7 :By Steps 1 and 2, we conclude that when f > F , 4 < 6 < 1 < 7 < 5 :Furthermore, from the proof of Proposition 3, we know that when f > F; 4 < 1 .
As a result, if aggregate variety increases, which implies that …xed cost synergies
satisfy < 6 , then consumer welfare may increase, which happens when 2( 4 ; minf 1 ; 6 g); or decrease, which happens when 2 (0; 4 ). This completes theproof for Proposition 4(ii).

44

(iiia)(iiib)(Assumption 3)

(i)

(ii)
(Assumption 1)

All profitable mergers decrease consumer welfare.
Mergers with moderate fixed cost synergies increase consumer welfare.

Figure 1 The Impact of Fixed Cost Synergies on Consumer Welfare (Proposition 3)