Manufacturers Mergers and Product Variety in
Vertically Related Markets
Chrysovalantou Milliou and Joel Sandonisy
March 2014
Abstract
In this paper, we study manufacturers mergers and their impact on product variety.
We consider a market with two manufacturers that sell their products to consumers
through two competing multi-product retailers and can invest in product innovation in
order to expand their product range. We show that a merger between manufacturers
leads to higher wholesale prices but that it can also lead to an increase in product
variety. However, even when it increases product variety, a manufacturers merger is
harmful to consumers and welfare.
Keywords: horizontal mergers; product variety; vertical relations
JEL classi…cation: L11; L13; L41; L42
The …rst author gratefully acknowledges …nancial support from the Research Support Program - Action
1 of the Athens University of Economics and Business. The second author gratefully acknowledges …nancial
support from Ministerio de Ciencia e Innovación and FEDER funds under SEJ 2007-62656, Generalitat
Valenciana under grant PROMETEO/2013/037 and from Ministerio de Economía y Competitividad under
project ECO2012-34928.
y
Milliou: Department of International and European Economic Studies, Athens University of Economics
and Business, Athens 10434, Greece, e-mail: [email protected];
1
Introduction
A common business strategy among manufacturers of branded consumer goods is product
proliferation, that is, the o¤ering of di¤erent variations of the same product by a single
…rm. Kellogg’s, for instance, o¤ers more than 20 di¤erent varieties of cereals, Ben & Jerry’s
sells 39 varieties of ice-cream, and Colgate and Crest each delivers more than 35 types of
toothpaste. Product proliferation is not restricted to supermarket goods. It takes place in
several product categories, such as sunglasses, apparel, watches, and consumer electronics.1
Besides investing in the extension of their product lines, …nal product manufacturers
often decide to merge among them. This occurs in various industry sectors. For instance,
in the automobile industry, German carmaker Volkswagen is on track to …nalize the full
takeover of sports car manufacturer Porsche. They said in a recent statement that Porsche
will become another fully integrated brand of the Volkswagen group as of August 1, 2014.
Moreover, mobile phone manufacturers merge with each other (Sony/Ericsson), and so
do producers of personal care goods (Colgate-Palmolive/Sanex), software programs (Oracle/PeopleSoft), electronics (Panasonic-Sanyo) and so on.2 One of the recent concerns of
the U.S. antitrust authorities regarding mergers between manufacturers is their potential
impact on product variety. That is, whether and how a manufacturers merger will alter
the merged …rms’ decisions regarding their product lines, and in turn, will a¤ect product
variety in the market. In fact, this concern has been formally expressed in the most recent
U.S. Merger Guidelines (2010) which mention explicitly that in the evaluation of horizontal
mergers, the authorities should focus not only on the impact of mergers on cost-related ef…ciencies, but also on their potential impact on product variety. Moreover, in practise, the
U.S. antitrust authorities have taken into account the merger’s impact on product variety
in the evaluation of a number of recent merger cases. For example, in 2003, the U.S. Federal Trade Commission (FTC) challenged the merger of Nestlé Holdings, Inc. and Dreyer’s
Grand Ice Cream, Inc., in the super-premium ice cream market, citing that "the market
for superpremium ice cream is already highly concentrated, and this deal will reduce the
number of signi…cant competitors from three to two" and it would "lead to anticompetitive
1
In most of these product categories, …rms’product lines di¤er, not in quality, but in other features such
as scent, color, or design, i.e., their products are horizontally di¤erentiated.
2
For additional examples see Froeb et al. (2007) or visit http://www.manufacturing.net/topics/mergersand-acquisitions.
1
e¤ects . . . including less product variety and higher prices".3
In this paper, we study the relationship between manufacturers mergers and product
variety. A key novelty of our approach is that we consider a setting where manufacturers
operate in a vertically related industry, i.e., in an industry where production and distribution
of the goods take place at di¤erent industry levels. In such a setting, a manufacturers
merger can a¤ect not only the number of goods produced but also the terms of vertical
trade which, in turn, determine the e¢ ciency of downstream …rms and the …nal prices
paid by consumers. Our purpose is to address a number of fundamental questions of both
theoretical and practical importance, such as: How product innovation a¤ects and it is
a¤ected by the intensity of market competition? What is the relationship between product
variety and wholesale prices? Whether and how a merger among manufacturers alters their
product innovation decisions? Is a manufacturers merger pro…table and welfare improving?
In our model, two upstream manufacturers produce initially two horizontally di¤erentiated goods that they distribute to consumers through two competing multiproduct retailers.
Manufacturers decide, …rst, whether or not they will merge, and second, whether they will
invest in product innovation. More speci…cally, they decide whether they will introduce additional in the market after incurring the respective …xed costs. The resulting total number
of products in the market can be two, three or four. Next, manufacturer(s) set the wholesale
prices of their goods and, in turn, the downstream retailers choose their quantities. As a
benchmark, we consider also, what happens when …rms operate in a one-tier market, i.e.,
when product manufactures sell directly their products to consumers. The consideration of
this benchmark case allows us to highlight the role of vertical relations and the potential
importance of accounting for them and for vertical trading in the analysis of the merger
implications on product variety.
The introduction of a new product in the market gives rise to two opposite e¤ects
regardless of whether manufacturers have merged or not. First, it results into an increase
in the market size. We refer to this e¤ect, which encourages product innovation, as the
market expansion e¤ect. Second, product innovation causes the appearance of intrabrand
competition: a new product that a manufacturer introduces steals away - "cannibalizes"
- demand from its already existing products. Clearly, this intrabrand competition e¤ect
3
Similar arguments were used also in decisions regarding the merger among Whole Foods Market and
Wild Oats Market in the supermarket industry, the merger among Oracle and PeopleSoft in the software
industry, and the merger of Kimberly-Clark Corp and H.J. Heinz Co. in the food industry.
2
discourages product innovation. Furthermore, product innovation has an additional e¤ect
in the non-merger case only, an interbrand competition e¤ect: the new product competes
against the product(s) of the rival manufacturer. The latter e¤ect intensi…es when product
innovation takes place.
In light of the above mentioned e¤ects, we show that in the absence of a merger upstream, as the total number of products in the market increases, the wholesale prices decrease. However, when the merger materializes, the number of products has no e¤ect on the
wholesale prices. Intuitively, in the non-merger case, the introduction of additional products in the market implies that interbrand competition among manufacturers gets stronger,
and thus, the incentives of each manufacturer to behave more aggressively by lowering the
wholesale prices of its product(s) get reinforced. In the merger case instead, the interbrand
competition e¤ect is fully internalized by the merged …rms. Importantly, we …nd that the
equilibrium wholesale prices when two manufacturers merge are always higher than when
they remain independent. This holds even in the cases where we …nd that there is more
product variety in the merger case than in the non-merger case. Stated in other words,
double marginalization is always more severe when the merger materializes.
We show also that the equilibrium number of products depends crucially on the intensity
of product market competition, captured by the degree of product di¤erentiation.[[THIS
IS A GENERAL STATEMENT - BUT THE ANALYSIS THAT FOLLOWS IS
ONLY for THE CASE OF NO MERGER and not for the merger case.]] More
speci…cally, when manufactures remain separated and products are very close substitutes,
innovation incentives are completely absent. When, instead, products are not so close
substitutes, at least one of the manufacturers has incentives to innovate as long as the cost
of doing so is not too high. This occurs because when products are very close substitutes,
both the intrabrand competition e¤ect and the interbrand competition e¤ect are strong and
o¤set the positive market expansion e¤ect. Otherwise and as long as product innovation is
not too costly, the market expansion e¤ect can dominate and result into product innovation.
Interestingly, when goods are su¢ ciently di¤erentiated and the cost of product innovation
takes intermediate values, manufacturers are trapped into a prisoners’dilemma situation:
although they would be better o¤ if they could both commit not to innovate, since product
innovation is a unilaterally dominant strategy, in equilibrium they both end up introducing
a new variety to the market. [[PERHAPS WE SHOULD USE ALSO SOMEHOW
AS AN ARGUMENT ALSO THE WHOLESALE PRICE EFFECT]]
3
Importantly, we …nd that a manufacturers merger can a¤ect product variety. In fact, we
can have more product variety when manufacturers merge than when they remain separated.
This holds only when products are su¢ ciently close substitutes and product innovation is
not too costly. Why is that? As mentioned previously, the market expansion e¤ect and
the intrabrand competition e¤ect are present both with and without a merger. Instead, the
interbrand competition e¤ect is present only when manufacturers remain separated. When
products are close substitutes, the interbrand competition e¤ect is strong and reduces the
product innovation incentives of the non-merged manufacturers. The opposite holds when,
instead, products are not too close substitutes, for two reasons. The …rst reason is the
prisoners’ dilemma situation in which the manufacturers are trapped in the non-merger
case. The second reason is the fact that the intrabrand competition e¤ect is stronger in the
merger case. This is so because by investing in product innovation, an upstream merged …rm
increases its products from two to three or to four products, while an upstream separated
…rm increases its products only from one to two.
Concerning the mergers welfare implications, we …nd that although a merger always
bene…ts the upstream manufacturers, it hurts both the downstream retailers, the consumers
and overall welfare. This is because of the higher wholesale prices in the merger case that
worsen the double marginalization problem and result into higher …nal prices. Thus, even
in the cases where a merger upstream leads to more variety in the market, its overall e¤ect
on social welfare is negative. This suggests that a merger’s potentially positive impact on
product variety, cannot be used as an argument in favor of the merger in its evaluation
by the antitrust authorities in cases in which the merger enhances signi…cantly the market
power.
Our paper is related to several di¤erent strands of the literature. First of all, it is
related to the strand of the literature that studies the relationship between market concentration and product variety. A large proportion of this literature is empirical. In particular,
Alexander (1997) and Watson (2009) study respectively the music distribution industry and
the eyewear retailing market and …nd a non-monotonic relationship between concentration
and product variety. George (2007), instead, studies the U.S. daily newspapers market
and concludes that more concentrated markets tend to have more variety both in terms of
the number and of the variety of topics covered. Focusing on the issue of product variety
in response to a merger, Berry and Waldfogel (2001) …nd that mergers in the U.S. radio
broadcasting market prompted an increase in both variety per station and overall variety
4
in the market. In contrast, according to Gotz and Gugler (2006) and Fan (2003) mergers in
the Austrian gasoline market and in the U.S. daily newspapers market caused a decrease in
their product variety. Summing up, the empirical literature documents the existence of a
relationship between product variety and market structure and suggests that the assumption that a merger leaves product variety una¤ected is questionable. This, in turn, suggests
that the analysis of mergers in markets with di¤erentiated products and multi-product …rms
that does not take into account the mergers’impact on product variety can be incomplete.
Although the empirical evidence indicates that mergers can a¤ect product variety, the
existing theoretical work on the e¤ects of horizontal mergers focuses either on the mergers’
price e¤ects (e.g., Reynolds et al., 1983, Davidson and Deneckere, 1985) and/or on their
cost related e¢ ciencies (e.g., Farrell and Shapiro, 1988, Perry and Porter, 1985) and ignores,
to a large extent, their impact on product variety.4 A notable exception is the paper of
Lommerud and Sorgard (1997) which studies how a merger a¤ects …rms’decisions regarding
the expansion of their product lines. More speci…cally, Lommerud and Sorgard consider a
market with three …rms and di¤erentiated products. They assume that initially each …rm
o¤ers one product and that there is a …xed non-sunk cost of marketing a brand. They
…nd that whenever a merger between two out of the three …rms is pro…table, it either has
a negative impact or no impact at all on product variety and that it is often detrimental
to welfare. When instead the merger triggers the introduction of a new product by the
outsider, and thus, causes an increase in product range, the merger is unpro…table. Another
exception is a recent paper by Chen and Schwartz (2013) which also studies a merger’s
implications on product innovation.5 In contrast to Lommerud and Sorgard (1997), Chen
and Schwartz (2013) consider mergers to monopoly and assume that initially only a single
product is o¤ered by all the market participants. They …nd that when product innovation
is su¢ ciently non-drastic, the incentives to innovate are stronger when the merger takes
place, and subsequently, that consumer welfare and overall welfare can then be higher under
monopoly than under other more rivalrous regimes.6 Importantly, all the above mentioned
4
We should note that there exists a vast theoretical literature on the product line decisions of competing
…rms (see, e.g., Brander and Eaton, 1984, Raubitschek, 1987, Martínez-Giralt and Neven, 1988, Gilbert
and Matutes, 1993, De Fraja, 1993, Dobson and Waterson, 1996, Avenel and Caprice, 2006, Gandhi et al.
2008). However, this literature does not examine the impact of market structure changes on these decisions.
Moreover, there is an extensive theoretical literature on product variety driven not by product proliferation
but by …rms’market entry decisions (see e.g., Salop, 1979, Schmalensee, 1978, Lancaster, 1979, Eaton and
Lipsey, 1979 and Innes, 2008).
5
Greenstain and Ramey (1998) analyze a similar topic in a vertical product di¤erentiation framework.
6
Note that there is a number of papers (see e.g., Gandhi et al., 2008 and Mazzeo et al., 2012) which exam-
5
papers consider only one-tier markets. Therefore, in contrast to us, they do not take into
account the fact that product manufacturers often operate in two-tier markets and, by not
doing so, clearly, they do not explore the role of vertical relations and trading for the merger
implications on product variety.
Our paper is also related to the recently growing literature on horizontal mergers in
vertically related industries. Horn and Wolinsky (1988), Ziss (1995), O’Brien and Sha¤er
(2003), Inderst and Wey (2003), Froeb et al. (2007), Milliou and Petrakis (2007) and Milliou and Pavlou (2013), similarly to us, study mergers in the upstream market.7 Within this
literature, only Milliou and Pavlou (2013), in line with us, analyze their potential e¢ ciency
gains of upstream mergers. However, Milliou and Pavlou (2013) do so considering a market
with exclusive upstream-downstream relations pre-merger and cost-related e¢ ciency gains.
We complement Milliou and Pavlou (2013) work by using a less restrictive market structure, and importantly, by analyzing the product variety related e¢ ciency gains of upstream
mergers. The only papers that, to the best of our knowledge, do consider product variety
issues within this literature are the papers by Inderst and Sha¤er (2007) and Faulí-Oller
(2008), which focus on downstream mergers and not on upstream mergers. Both of these
papers demonstrate that a merger among retailers allows them to commit not to sell one of
the goods supplied by manufacturers, and thus, that such a merger can result into a welfaredetrimental decrease in product variety. These papers di¤er from ours not only because of
their di¤erent focus, but also because in their setting there is no product innovation; the
number of products manufactured by the upstream manufacturers is exogenous and the
downstream …rm(s) choose how many of them they will distribute.8
Finally, our work is also related to the branch of the literature that endogenizes the
distribution systems that arise when there is market power at both the manufacturing and
the retailing levels (see, e.g., O’Brien and Sha¤er, 1997, Bernheim and Whinston, 1998,
Lin, 1990, O’Brien and Sha¤er, 1993, Chang, 1992, Moner-Colonques et al., 2004 and 2011
and Mauleon et al., 2011). In our paper, in contrast to this literature, we do not focus on
ine how a horizontal merger can a¤ect product variety not though through altering the number of products
but through altering the product characteristics. In other words, they examine whether post-merger, the
merged …rm has incentives to reposition its products in terms of their degree of product di¤erentiation.
7
Berry and Waldfogel (2001), von Ungern-Sternberg (1996), Dobson and Waterson, (1997), Inderst and
Wey (2003), Lommerud et al. (2005), Fauli-Oller and Bru (2008), Symeonidis (2008) and (2010), Faulí-Oller
(2008), Faulí-Oller and Sandonís (2012) and Inderst and Sha¤er (2007), study, instead, downstream mergers.
8
In a related paper, Milliou, Petrakis and Sloev (2010), similarly to us, endogenize upstream product
innovation. However, they focus on the role played by the economies of scope and upstream entry and not
by the upstream mergers.
6
the choice of distribution system. In fact, we take as given the distribution system, in the
sense that we assume that retailers are multiproduct …rms that distribute all the varieties
produced upstream, and focus instead on manufacturers mergers and on manufacturers
incentives to invest in product innovation.9
The rest of the paper has the following structure. In Section 2, we describe our model.
In Section 3, we perform the equilibrium analysis. In Section 4, we examine the implications
of merger on the wholesale prices and product variety. In Section 5, we explore the merger
incentives and the merger’s impact on downstream …rms, consumers and welfare. In Section
6, we discuss the role of vertical relations. In Section 7, we enrich our analysis by considering
what happens when there are three upstream …rms in the market and the merger does not
lead to monopoly. Finally, in Section 8, we conclude.
2
The Model
We consider a market consisting of two upstream product manufacturers, M1 and M2 , and
two downstream retailers, R1 and R2 . Each Mi , with i = 1; 2, produces, initially, a singleproduct at zero marginal cost.10 Each Rk , with k = 1; 2, sells to the …nal consumers the
products of both manufacturers after paying the per unit of product wholesale prices, w1
and w2 , respectively to M1 and M2 .
Manufacturers can invest in product innovation. In particular, each Mi can introduce
an additional product variety after incurring a …xed cost, F > 0. Moreover, manufacturers
can merge among them. The order of their decisions is as follows. In stage one, M1 and
M2 decide whether or not they will merge.11 In stage two, if the merger has not taken
place, each Mi decides whether it will introduce an additional product. Depending on their
decisions, we will have two, three or four products in the market. In case of merger, the
merged entity, M , decides respectively whether it will continue producing two products or it
will invest in product innovation and increase the number of products to three or four.12 In
9
This is not a restrictive assumption however. Moner-Colonqués et al. (2011) show, in a similar setting, that retailers would choose to overlap their product lines whenever their retail margins are not too
asymmetric.
10
Note that we would obtain qualitatively similar results if we had assumed instead a positive and constant
marginal production cost.
11
In Section 7, we extend our analysis to the case in which there are 3 mannufacturers in the upstream
market and the merger takes place among just two of them, i.e., it does not create an upstream monopoly.
12
We implicitly assume that the …xed cost of producing the initial variety is already sunk for the …rms. In
other words, the withdrawal of one of the two initial brands by the merged …rm would not result into any
7
stage three, the manufacturer(s) set the wholesale prices for each of their products. Finally,
in stage four, the retailers choose the quantities of all the products.
The retailers face a continuum of identical consumers. Following Singh and Vives (1984),
the representative consumer has a quadratic utility function separable in income given by:
u(Q1 ; :::; QN ; I) = a
N
X
N
X
1 X 2
(
Qn + 2
Qn Qj ) + I;
2
Qn
n=1
n=1
n6=j
where Qn denotes the total quantity of good n sold in the market, with n 2 f1; 2; 3; 4g, N
denotes the total number of products in the market, i.e., N 2 f2; 3; 4g depending on the
manufacturers’product innovation decisions, and I stands for consumer’s income. From the
utility maximization program we directly obtain the following (inverse) demand function
faced by Rk for each product n:
pnk (qnk ; qnl ; Q
n)
=a
qnk
qnl
(Q
n );
where pnk and qnk are respectively the price and the quantity of product n sold by Rk , qnl
is the quantity of the same product sold by the rival retailer Rl , with k; l = 1; 2 and k 6= l,
and Q
n
is the quantity of the rest of the product(s).13 The parameter , with 0 <
< 1,
denotes the degree of product substitutability; namely, the higher is , the closer substitutes
are the products of the manufacturers.14;15
Our notational convention for the rest of the paper will be as follows. The …rst superscript, M or S, will denote respectively whether the manufacturers have merged or
remained separated and the second superscript will denote the total number or products in
the market.
…xed cost saving. Given this, it can be shown that the merged …rm has no incentives to withdraw one of its
products from the market and become single-product.[[WE SHOULD THINK ABOUT INCLUDING
THIS IN THE MAIN TEXT] In Section 7, we discuss, however, how this result can change if we consider
a setting with three upstream manufacturers.
13
Observe that qnk + qnl = Qn with n 2 f1; 2; 3; 4g:
14
Note that with this formulation of utility, the two retailers are not di¤erentiated, i.e., consumers do not
perceive di¤erently a product sold at one retailer than at the other retailer.
15
Note that for the sake of simpli…city denotes the degree of product substitutability both among the
products of di¤erent manufacturers and among the products of the same manufacturer.
8
3
Equilibrium Product Innovation
In the last stage of the game, independently of whether the manufacturers have merged or
not, each Rk chooses the quantity of each of the products in order to maximize its pro…ts
which are given by:
Rk
=
N
X
[pnk (qnk ; qnl ; Q
n)
wn ]qnk :
n=1
Solving the system of …rst order conditions, we obtain the equilibrium quantities of each
product n sold by R1 and R2 , qn1 (wn ; w
n)
and qn2 (wn ; w
n)
respectively.16 Clearly, these
quantities depend not only on the wholesale prices but also on the total number of products
in the market. The respective total market demand for product n is Qn = qn1 (wn ; w
qn2 (wn ; w
3.1
n)
+
n ).
Upstream Separated Firms and Product Innovation
When M1 and M2 remain separated, in stage three, they choose the wholesale prices of
their products. There are three possible third-stage subgames depending on the number of
products that they manufacture: (i) the "no product innovation" subgame where both M1
and M2 produce one product each, (ii) the "partial product innovation" subgame where M1
produces two products and M2 one, and (iii) the "full product innovation" subgame where
both M1 and M2 produce two products each.17 Next, we analyze each subgame separately.
When both M1 and M2 produce one product each, and thus, N = 2, they face respectively the following maximization problems:
M ax
w1
M1
= w1 Q1 (w1 ; w2 ) and M ax
M2
w2
= w2 Q2 (w1 ; w2 ):
(1)
Solving (1), we obtain the equilibrium wholesale prices:
w1S2 = w2S2 =
a(1
2
)
:
(2)
When, instead, M1 produces two products and M2 produces one, their maximization
16
The exact expressions are included in the Appendix.
There is a fourth subgame in which M1 produces one product and M2 produces two products. However,
the analysis for this subgame is the same as the one for the subgame in which M1 produces two products
and M2 one, with the roles of M1 and M2 reversed.
17
9
problems are:18
M ax
w1 ;w3
M1
= w1 Q1 (w1 ; w2 ; w3 ) + w3 Q3 (w1 ; w2 ; w3 ) and M ax
w2
M2
= w2 Q2 (w1 ; w2 ; w3 ):
(3)
The resulting equilibrium wholesale prices are:
w1S3 = w3S3 =
a(2 +
3 2)
a(1
and w2S3 =
4 + 2 (2
)
2 + (2
2)
)
:
(4)
It is easy to see that w1S3 > w2S3 . That is, the multi-product manufacturer charges higher
wholesale prices than its single-product rival.
Finally, when both manufacturers produce two products, they solve the following problems:
M ax
M1
= w1 Q1 (w1 ; w2 ; w3 ; w4 ) + w3 Q3 (w1 ; w2 ; w3 ; w4 ) and
(5)
M ax
M2
= w2 Q2 (w1 ; w2 ; w3 ; w4 ) + w4 Q4 (w1 ; w2 ; w3 ; w4 ):
(6)
w1 ;w3
w2 ;w4
>From the solution of (5) and (6), we have:
w1S4 = w3S4 = w2S4 = w4S4 =
a(1
)
2
.
(7)
Comparing the di¤erent wholesale prices, we reach the following conclusion:
Proposition 1 When the upstream manufactures are separated, the wholesale prices decrease with the total number of products in the market.
According to Proposition 1, when the upstream manufacturers are separated, the higher
the number of products in the market, the lower the wholesale prices. This …nding is mainly
driven by the fact that both upstream manufacturers have incentives to behave aggressively
and reduce their wholesale prices in order to favour the sales of their own products relative
to the respective sales of their rival’s products. Their incentives to do so are stronger when
competition is more intense and competition is indeed more intense when the number of
products in the market increases. In particular, while when manufacturers produce only one
product each, there is only interbrand competition, when either one of the manufacturers or
18
Note that we implicitly assume in this case that products 1 and 3 are manufactured by M1 and product
2 by M2 :
10
both of them produce more products, then not only interbrand is …ercer, but also intrabrand
competition is present. In what follows, we will refer to the negative impact of product
innovation on the wholesale prices as the wholesale pricing e¤ect.
Next, we analyze each manufacturer’s product innovation decision in stage two. There
are three candidate second-stage equilibria, each of them corresponding to one of the subgames analyzed above. First, we substitute the equilibrium wholesale prices into the manufacturers’respective (gross from the cost of product innovation) pro…ts in each subgame. We
note that the latter can be negative under "full product innovation" unless product innovaS
S
tion is not too costly, and in particular, unless F < F ( ), with F ( )
a2 (1 )(2+3 )2
.
3(1+2 )[2+2 (2 )]2
Next, we check for possible deviations from each of the candidate equilibrium and we …nd
the following:
S
Proposition 2 When the upstream manufacturers are separated, F < F ( ), and
(i)
> 0:9164, the unique equilibrium is no product innovation (i.e., N = 2),
(ii) 0:8346 <
< 0:9164, the unique equilibrium is no product innovation if F > F1S ( )
and partial product innovation (i.e. N = 3) if F < F1S ( ),
(iii)
< 0:8346, the unique equilibrium is no product innovation if F > F1S ( ), partial
product innovation if F2S ( ) < F < F1S ( ) and full product innovation (i.e., N = 4) when
F < F2S ( ),
S
with F ( ) > F1S ( ) > F2S ( ) for all .
Proposition 2 informs us that the equilibrium number of products depends not only on
how costly is to innovate, but importantly, on the intensity of product market competition.
In particular, when products are very close substitutes, the upstream manufacturers have
no incentives to innovate. When, instead, products are not too close substitutes, at least
one of the manufacturers will innovate as long as the cost of doing so is not too high. In
fact, both of them will innovate when the cost of product innovation is su¢ ciently low and
products are su¢ ciently di¤erentiated.
Intuitively, product innovation gives rise to four main e¤ects. First, there is the "wholesale pricing" e¤ect that we identi…ed above (Proposition 1) that corresponds to the decrease
in double marginalization when the number of products increases. This e¤ect clearly discourages product innovation. Second, there is a market expansion e¤ect which refers to the
increase in the market size/demand caused by the introduction of an additional product
variety. Clearly, this e¤ect encourages product innovation. Third, there is an intrabrand
11
competition e¤ect. This refers to the fact that when an upstream manufacturer extends its
product range, its already existing product faces additional competition. The new product
partly cannibalizes the demand for the existing product. Clearly, this e¤ect discourages
product innovation. Finally, the fourth e¤ect is an interbrand competition e¤ect. This effect corresponds to the intensi…cation of competition with the rival upstream manufacturer
driven by product innovation and also discourages product innovation. When products are
su¢ ciently close substitutes, competition is tough and the wholesale pricing e¤ect, the intrabrand competition e¤ect and the interbrand competition e¤ect are very strong and dominate
the positive market expansion e¤ect. Otherwise and as long as product innovation is not
too costly, the market expansion e¤ect can dominate and result into product innovation.
As mentioned above, when products are not very close substitutes and product innovation is not too costly both upstream …rms innovate. Interestingly though, when both
…rms innovate, they can be trapped into a prisoner’s dilemma situation, i.e., they would
be better o¤ if none of them innovated unless products are not su¢ ciently di¤erentiated
and the cost of product innovation is too low. This is so because under partial innovation,
the non-innovating upstream …rm has incentives to deviate and innovate as well, so that
it expands its own market and steals away market share not only from its own product
(cannibalization) but mainly from its rival’s products.
3.2
Upstream Merger and Product Innovation
Next, we analyze the merger case. After the merger, the newly formed upstream monopolist
chooses, in stage three, the wholesale prices for the products that it manufactures. In
particular, M solves the following:
M ax
w1 ;::;wN
M
=
N
X
wn Qn (wn ; w
n ),
(8)
n=1
where N = 2 when it does not undertake any product innovation, and thus, manufactures
only two products, and N = 3 and N = 4 when respectively it manufactures three and four
products. From the …rst order conditions, we …nd:
w1M 2 = w2M 2 = w1M 3 = w2M 3 = w3M 3 = w1M 4 = w2M 4 = w3M 4 = w4M 4 =
12
a
.
2
(9)
Two important observations are in place for the equilibrium wholesale prices (9) under
upstream monopoly. First, we note that the wholesale prices are independent of the degree
of product di¤erentiation. And second, we identify an additional "indi¤erence result", stated
formally in Proposition 3 below: the upstream multi-product monopolist’s pricing decisions
do not depend on the number of its products. This result is similar to the indi¤erence result
identi…ed in the literature (see e.g., Petrakis and Dhillon, 2002) regarding the independence
of the wholesale prices charged by a single-product upstream monopolist from the number
of downstream …rms.
Proposition 3 When the upstream manufactures are merged, the wholesale prices are independent of the number of products in the market.
In order to analyze the merged …rm’s product innovation decision in stage two, …rst,
we substitute (9) into its respective (gross from the cost of product innovation) pro…ts (8)
for N = 2, N = 3 and N = 4 and, second, we subtract the associated …xed cost(s) of
product innovation from them. Doing so, we note that we need to assume that F < F
with F
M
a2
2+4
M
,
, in order to make sure that M ’s pro…ts are positive in all cases under
consideration. Finally, comparing M ’s equilibrium pro…ts,
M 2,
M
M3
M
M 4,
M
and
we reach
the following conclusion:
Proposition 4 When the upstream manufacturers are merged and F < F
F1M ( ) and F2M ( ) which decrease in
M
and satisfy F1M ( ) > F2M ( ) > F
( ), there exist
M
( ), such that
the unique equilibrium is
(i) no product innovation (i.e., N = 2) if F > F1M ( ),
(ii) partial product innovation (i.e., N = 3) if F2M ( ) < F < F1M ( ), and
(iii) full product innovation (i.e., N = 4) if F < F2M ( ), with F1M ( ) > F2M ( ) >
F
M
( ).
As Proposition 4 states, the upstream monopolist has incentives to invest in product
innovation unless the cost of doing so is too high (F > F1M ( )). In fact, when the cost
of product innovation is su¢ ciently low (F < F2M ( )), it will introduce two, and not
just one, products. Intuitively, under upstream monopoly, the market expansion e¤ect
that encourages product innovation and the intrabrand competition e¤ect that discourages
product innovation are both present. The market expansion e¤ect dominates the intrabrand
competition e¤ect as long as product innovation is not too costly. Otherwise, the positive
13
impact of the market expansion e¤ect cannot compensate the upstream monopolist for the
cost of product innovation.
Proposition 4 also states that the critical values for product innovation decrease - they
become stricter, when product substitutability increases; hence, product innovation is less
likely to take place then. This is so because, when products are more similar among them,
the intrabrand competition e¤ect is reinforced, i.e., a product tends to cannibalize more
the other(s) product(s), and as a result, it has a stronger negative impact on the upstream
monopolist’s product innovation incentives.
4
Merger Implications on Wholesale Prices and Product Variety
In this Section, we examine the implications of an upstream merger on the wholesale prices
and product variety. Starting with the wholesale prices, our main conclusion is:
Proposition 5 The wholesale prices are higher when upstream …rms merge than when they
remain separated.
Note that the above result holds independently of the number of products manufactured
in the presence and in the absence of the merger. Hence, it holds even when there is less
product innovation in the non-merger case than in the merger case. This is not so surprising:
once the upstream manufacturers merge, they stop competing among them. This, in turn,
means that they internalize the externality that they would otherwise impose on each other
by o¤ering lower wholesale price(s) for their own product(s) so that their downstream sales
are enhanced.
Combining Propositions 2 and 4, we are able to evaluate the impact of an upstream
merger on product innovation, and hence, on product variety:
Proposition 6 Product variety is:
(i) higher when upstream …rms merge than when they remain separated if
and F is not too high as well as if 0:6931 <
> 0:8346
< 0:8346 and F takes intermediate values,
(ii) lower when upstream …rms merge than when they remain separated if
< 0:7261
and F takes intermediates values,
(iii) the same when upstream …rms merge and when they remain separated in all other
cases.
14
A merger among upstream manufacturers can enhance product variety. This holds
mainly when products are su¢ ciently close substitutes and product innovation is not too
costly. Why is that? As mentioned previously, the market expansion e¤ect and the intrabrand competition e¤ect are present both in the non-merger case and in the merger case.
Instead, the interbrand e¤ect, which weakens the product innovation incentives, is present
only in the non-merger case. When products are close substitutes, the interbrand competition e¤ect is strong and leads to less product innovation then. Still, one might wonder why
its presence does not always lead to less product innovation in the non-merger case than in
the merger case. This could be due to the prisoner’s dilemma situation in the former case,
which occurs when products are not too di¤erentiated and product innovation is neither
too costly nor too cheap, and results into overinvestment in product innovation. Further,
it could be due to the fact that for a given level of product di¤erentiation, the intrabrand
competition e¤ect is stronger when the merger materializes. This is so because the merged
…rm produces from the outset two products and product innovation can increase its products from two to three or to four products. Instead, product innovation by a separate …rm
can increase its products only from one to two. Since the intrabrand competition e¤ect
is stronger in the merger case, its negative impact on product innovation incentives is, in
turn, also more severe then. Finally, notice that the wholesale pricing e¤ect, that is, anticipating higher wholesale prices after a merger, encourages manufacturers to merge and
to introduce new varieties after the merger. Moreover, whereas in absence of a merger the
wholesale prices decrease with variety, after a merger they are independent of the number of
products, which again, provides extra incentives to increase variety in case of a merger compared with the no merger case. VALANTA, I HAVE ADDED TWO SENTENCES
WITH THE WHOLESALE PRICNG EFFECT HERE. SEE IF YOU LIKE
IT OR YOU CAN IMPROVE IT.[[WE SOULD SOMEHOW INCLUDE THE
WHOLESALE PRICING EFFECT HERE .. IN ORDER TO BE ABLE TO
DIFFERENTIATE IT LATER ON FROM THE ONE-TIER CASE]]
We should mention that there are cases in which, while we have no product innovation
in the non-merger case, we can have full innovation, and not only partial innovation in the
merger case and the reverse. That is, there are cases in which the number of products
between the two cases di¤ers by one product and cases in which it di¤ers by two products.
For instance, when
> 0:9164, the total number of products without the merger is always
2 while with the merger it is 3 if F takes intermediate values and 4 if F is su¢ ciently
15
low.19 Moreover, we should clarify that when we say in the statement of Proposition 6 that
product innovation is the same with and without the merger "in all other cases", we do not
mean that it leads to no product innovation but that it leads to the same level of product
innovation as pre-merger.
5
Merger Incentives and Welfare Implications
In what follows, we analyze, …rst, the upstream …rms’ merger incentives, and then the
merger’s impact on downstream pro…ts, consumer surplus and total welfare.
Proposition 7 An upstream merger always arises in equilibrium.
A merger upstream allows the manufacturers to eliminate interbrand competition, and
thus, to eliminate the negative externality that they impose on each other. This allows
them to charge higher wholesale prices. Further, it allows them to coordinate their product
innovation decisions, and thus, to avoid the prisoner’s dilemma situation which results into
over-investment. In light of these, it is not surprisingly that, as Proposition 7 formally
states, merger incentives are always present or, in other words, an upstream merger always
has a positive impact on upstream pro…ts.
What about though the merger’s impact on downstream pro…ts? It is important to note
that in the merger case as well as in the non-merger case, the pro…ts of downstream …rms
increase with product variety. This is quite intuitive. When there are more products in
the market, the market size increases, and thus, the sales and pro…ts of the retailers also
increase. Actually, in the case of upstream separated …rms, there is an additional reason
for which retailers are better o¤ when there is more product variety: an increase in product
variety leads to lower wholesale prices and so to increased downstream e¢ ciency. However,
we already know that a merger upstream increases the wholesale prices even when it leads
to more variety in the market. So, how is this trade o¤ resolved? Notice that an upstream
merger leads to higher variety in the market only when the goods are su¢ ciently close
substitutes. But for close substitute goods, competition downstream is stronger and this
explains that the positive market expansion e¤ect of the introduction of additional goods to
the market is smaller. So a merger upstream could potentially bene…t retailers only when
19
The detailed description for all the cases that arise in equilibrium and the speci…c conditions under
which they hold are all available in the Appendix, in the proof of Proposition 6.
16
competition downstream is soft (that is, when the degree of product substitutability is low).
But in that case, the merger upstream does not lead to more variety in the market. So as
a conclusion, an upstream merger hurts retailers’pro…ts regardless of its e¤ect on product
innovation: the negative e¤ect of the higher wholesale prices always dominates the positive
market expansion e¤ect of the (potential) increased variety, leading to the clear-cut result
that the upstream merger hurts the downstream pro…ts.
What about the merger’s impact on consumer surplus and social welfare? Concerning
consumer surplus, when the goods are su¢ ciently di¤erentiated, a merger hurts consumers
…rst, because it increases the wholesale prices, which translates into higher …nal prices for
consumers and second, because either it reduces or does not change the number of goods in
the market. However, when the goods are su¢ ciently good substitutes and for intermediate
values of the cost of product innovation, a merger upstream leads to more variety through
product innovation by the merged entity, which is bene…cial to consumers. However, even
when the merger increases variety in the market, the negative anticompetitive e¤ect of the
merger through higher wholesale prices always dominates, implying that consumer surplus
is always reduced by the merger.
Finally, concerning the e¤ect of the merger on social welfare, a trade o¤ also arises
because the merger is always pro…table for the upstream …rms but it hurts downstream
…rms and consumers. As the following proposition shows, the anticompetitive e¤ect of
a "merger to monopoly" upstream is so pronounced that it cannot be compensated by
the potential merger generated e¢ ciency gains, which in our setting arise via product innovation. Let us summarize the previous discussion in the following proposition:I HAVE
MERGED A LITTLE BIT THESE PARAGRAPHS TO AVOID REPETITION.
SEE IF YOU LIKE IT MORE NOW.[[I THINK THAT THE ABOVE 2 PARAGRAPHS REPEAT WHAT WAS SAID IN THE PARAGRAPH ABOVE... SO
WE SHOULD MERGE THEM AND SHORTEN THE ANALYSIS]]
Proposition 8 An upstream merger reduces downstream pro…ts as well as consumer surplus and social welfare.
Our result regarding the merger’s impact on social welfare, allows us to provide a theoretical justi…cation for the view expressed in the recent U.S. Merger Guidelines (2010) according to which "e¢ ciencies almost never justify a merger to monopoly or near-monopoly.
Just as adverse competitive e¤ects can arise along multiple dimensions of conduct, such
17
as pricing and new product development, so too can e¢ ciencies operate along multiple
dimensions".
6
The Role of Vertical Relations
In this Section, we explore the role of vertical relations for the merger’s implications. In
order to do so, …rst, we analyze brie‡y what happens in the absence of vertical relations,
i.e., in an one-tier market where product manufacturers sell directly their products to the
consumers, and then, we compare our results with the ones that we have already found in
the presence of vertical relations.
In a one-tier market, stage three does not exist, i.e., w1 and w2 are zero by assumption,
and the quantities are chosen directly by the product manufacturer(s). Performing the
equilibrium analysis, we conclude that in such a market, similarly to what happens in
a vertically related market, the merged …rm never withdraws one of its already existing
products from the market. Importantly, we conclude that a merger in a one-tier market
never enhances product variety. This is formally stated in the following Proposition:
Proposition 9 In a one-tier market, a merger decreases product innovation when F takes
intermediate values. Otherwise, it does not have any e¤ ect on product innovation.
Independently of whether the merger has no e¤ect or a negative e¤ect on product variety,
since the merger results into monopoly, it is always pro…table. Moreover, a straightforward
implication of the decreased product variety and increased market power under merger, is
that the merger in a one-tier market is harmful to consumers and to welfare.
It follows from the above that the consideration of vertical relations is of signi…cant
importance. If one does not take them into account, it reaches the conlcusion that a
merger always restricts product variety. In contrast, if one allows for such relations, it can
conclude that, under certain conditions, a merger can enhance, instead of decrease, product
variety.[[WE NEED TO EXPLAIN THIS MORE....BUT I DID NOT THINK
OF AN EXPLANATION YET]] Still, independently of the merger’s impact on product
variety, we conclude that a manufacturers merger to monopoly, either in a one-tier market
or in a vertically related market, is always welfare-detrimental.
[[IN THE FUTURE, WE SHOULD THINK ABOUT COMPARING THE
CRITICAL VALUES OF Fs BETWEEN THE TWO MARKETS IN ORDER
18
PERHAS TO GET ADDITIONAL RESULTS - WE SHOULD THINK ABOUT
THIS MORE]]
7
Extension: 3 Upstream Firms
In this Section, we explore whether the manufacturers merger’s negative impact on social
welfare obtained so far is driven by our simplifying assumption of a duopolistic upstream
market that allows us to consider only mergers to monopoly. In order to do so, we extend
our main model to the case in which pre-merger there are three, instead of two, upstream
manufacturers. As before, manufacturers can invest in product innovation, so each Mi ,
i = 1; 2; 3 can introduce an additional product variety after incurring a …xed cost, F > 0.
This implies that in absence of a merger we can have now 3, 4, 5 or 6 products in the market.
In the case of a merger, we consider only a merger among two out of the three manufacturers.
In this case, the merged entity, M , decides whether it will continue producing the two initial
products, it will withdraw one of its two products or it will invest in product innovation
and increase the number of its varieties to three or four. That is, after a 2-…rms merger
we can have 2, 3, 4, 5 or 6 goods in the market. SOLVED [[WHY INCLUDE THIS
FOOTNOTE HERE SINCE YOU MENTION IT LATER ON IN THE MAIN
TEXT?]]
Given that the complete resolution of this extended model is cumbersome and for the
sake of simplicity in the exposition, we will describe the main results and the di¤erences of
this extended model with respect to the main model.20
Adding a third manufacturer is interesting because it allows to analyze the e¤ect of a
manufacturers merger not only on the merged …rms incentives to alter the product range
but also on the outsider’s incentives to do the same. In contrast with the previous model,
in this extended model, a 2-…rms merger does not monopolize the upstream sector and
the existence of interbrand competition post-merger reduces the merged …rms’ incentives
to add new varieties to the market. Indeed, we get that whereas in the 2 manufacturers
setting and for close enough substitute goods, a merger could lead the merged entity to
increase the number of varieties, in a 3 manufacturers setting, a 2-…rms merger never leads
the merging …rms to increase the number of varieties. Indeed, it can be shown that for
close enough substitutes ( > 0:91), it could lead them to withdraw one of the two varieties
20
The detailed analysis is available from the authors upon request.
19
that were already produced before the merger (this never occurs in the main model). The
explanation for the latter result is just the existence of interbrand competition post-merger,
which reduces upstream …rms’incentives to invest in product innovation, especially when
the goods are close substitutes: in this latter case, the reduction of interbrand competition
produced by the withdrawal of one of the already existing brands may even compensate the
merging …rms for the reduction in sales (notice that this occurs even if there is no …xed cost
saving, given that the …xed cost of the initially produced brand is assumed to be already
sunk).
On the other hand and not surprisingly, by reducing interbrand competition, the merger
leads to an increase in the equilibrium wholesale prices, which hurts downstream …rms
(and consumers). This implies that the only way for a merger to be welfare enhancing in
this setting would be through a positive e¤ect on variety. We do …nd a region (for very
di¤erentiated products (
< 0:39) and intermediate values of the …xed cost F ) where a
2-…rms merger increases total variety (and social welfare). But, interestingly, this increase
in variety is not produced by the merged …rm; it is the non-merging …rm who responds
aggressively to the merger by adding one more variety. And, as it is intuitive to understand,
this merger is not pro…table for the merging …rms so it will not take place (this is another
di¤erence with the main model, where the manufacturers merger is always pro…table).
Lommerud and Sorgard (1997) …nd a similar result in their setting with a one tier industry
and three …rms.
Apart from this particular region, as long as the …xed cost F of introducing a new
variety is not too large, the merger always leads to less variety compared with the no
merger case. Of course, for su¢ ciently large values of F and regardless of the degree of
product di¤erentiation, we …nd a region where there would be no product innovation in
both the merger and no merger cases. So we can conclude that in the extended setting with
three producers, any pro…table merger must reduce consumer surplus and social welfare
(because it increases the wholesale prices (which translate into higher …nal prices) and it
does not increase variety in the market).
Summarizing, the comparison between the two models seems to suggest that adding
more manufacturers goes in the direction of reducing the merged …rms’ incentives to add
new varieties after the merger due to increased competition, which makes di¢ cult to …nd
welfare improving mergers. On the other hand, talking about merger pro…tability, whereas
in the setting with two manufacturers an upstream merger is always pro…table, in the
20
extended setting we can …nd unpro…table mergers, the reason being always an aggressive
response by the outsider …rm that reacts to the merger by adding a new brand.
8
Concluding Remarks
We have investigated the product innovation incentives of competing manufacturers and
how they are a¤ected by a merger among them. We have taken into account the fact
that most product manufacturers instead of selling their products to consumers directly,
they sell them through downstream multi-product retailers. In such a setting, a merger
among product manufacturers can change not only the upstream market structure but also
the trading with the downstream retailers. Both of these changes can, in turn, a¤ect the
incentives of manufacturers to introduce additional products, and thus, they can a¤ect
product variety.
We have found that the incentives of manufacturers to introduce new products depends
on the degree of product substitutability. In particular, when manufacturers are separated,
product innovation takes place only when goods are su¢ ciently di¤erentiated and product innovation is not too costly. Interestingly, as the number of products in the market
increases, and thus, market competition intensi…es, the wholesale prices charged by the
manufacturers decrease. An upstream merger allows the manufacturers to internalize the
negative externality that they impose on each other in the absence of a merger, and as
a result, it allows them to charge higher wholesale prices. Moreover, by eliminating the
interbrand competition e¤ect, the merger can causes an increase, instead of a decrease, in
total product variety relative to the non-merger case. This holds when the cost of product
innovation is not too high and the goods are close enough substitutes since then the negative
impact of the interbrand competition under no merger is very pronounced. In light of these,
it is not surprising that an upstream merger is always welcomed by the upstream …rms and
it is never welcomed by the downstream retailers. [[THIS NEED TO BE IMPROVED
IN TERMS OF EXPOSITION]]
Most of the literature on horizontal mergers and their potential e¢ ciency gains has
focused so far on cost related e¢ ciencies. However, in practice …rms often invest in order not
only to reduce their costs but also to expand their product lines and increase their product
variety. Our paper adds value to the existing literature by examining horizontal mergers’
potential product variety related e¢ ciencies. We point out that although an upstream
21
merger could give rise to such e¢ ciency gains, i.e., it could result into more product variety,
these e¢ ciency gains are not strong enough to overturn its negative impact on consumers and
total welfare through the increased wholesale prices. Stated in di¤erent words, we provide a
theoretical justi…cation for the view expressed in the recent U.S. Merger Guidelines (2010)
according to which "e¢ ciencies almost never justify a merger to monopoly or near-monopoly.
Just as adverse competitive e¤ects can arise along multiple dimensions of conduct, such
as pricing and new product development, so too can e¢ ciencies operate along multiple
dimensions." We have shown that this anticompetitive aspect of manufacturers mergers is
shown to remain true when we increase the number of producers in the upstream level from
two to three, the reason being an increasing level of interbrand competition that reduces the
merged …rm incentives to introduce new varieties into the market. Intuitively, this result
should hold also for a higher number of producers. I HAVE TRIED....[[WE HAVE NOT
CHECKED THAT THIS IS THE CASE, WE HAVE CHECKED IT ONLY
WITH 3 FIRMS, WITH 4 IT COULD BE DIFFERENT ETC.. SO PERHAPS
WE SHOULD WRITE THIS DIFFERENTLY IN THE FUTURE]]
We should stress that our paper constitutes just a …rst step in the direction of understanding the relation between upstream horizontal mergers and product innovation. In a
following step, one could examine the role of alternative contract types, such as two-part
tari¤s.21 Moreover, one could also explore how the downstream market structure in‡uences
the manufacturers merger’s impact on product variety or it could incorporate economies of
scale in the analysis. These extensions are left for future research.
9
Appendix
Third stage equilibrium outputs
The equilibrium quantities obtained after solving the last stage of the game for given wholesale prices are:
21
We should note that serious complications arise in situations in which rival upstream …rms trade through
non-linear contracts with the same competing downstream …rms. As mentioned in a review article by
Miklos-Thal et al. (2010, p.345) "The formal modeling of such "interlocking" vertical relations has proved
di¢ cult... and we still know relatively little about many basic questions... Interlocking relationships cause
modeling issues such as either the inexistence or a large multiplicity of equilibria even in simple competition
games." Also, Inderst (2010, p. 343) states that "... the benchmark model where competing upstream
…rms simultaneously make take-it-or-leave-it o¤ers to competing downstream …rms, may fail to have an
equilibrium in pure strategies."
22
(i) when N = 2:
q11 (w1 ; w2 ) = q12 (w1 ; w2 ) =
q21 (w1 ; w2 ) = q22 (w1 ; w2 ) =
a(1
) w1 + w2
2)
3(1
) w2 + w1
;
2)
3(1
a(1
(ii) when N = 3 :
q11 (w1 ; w2 ; w3 ) = q12 (w1 ; w2 ; w3 ) =
q21 (w1 ; w2 ; w3 ) = q22 (w1 ; w2 ; w3 ) =
q31 (w1 ; w2 ; w3 ) = q32 (w1 ; w2 ; w3 ) =
a(1
)
(1 +
3(1
) (1 +
3(1
) (1 +
3(1
a(1
a(1
)w1 + (w2 + w3 )
) 6 2
)w2 + (w1 + w3 )
) 6 2
)w3 + (w1 + w2 )
;
) 6 2
(iii) when N = 4:
q11 (w1 ; w2 ; w3 ; w4 ) = q12 (w1 ; w2 ; w3 ; w4 ) =
q21 (w1 ; w2 ; w3 ; w4 ) = q22 (w1 ; w2 ; w3 ; w4 ) =
q31 (w1 ; w2 ; w3 ; w4 ) = q32 (w1 ; w2 ; w3 ; w4 ) =
q41 (w1 ; w2 ; w3 ; w4 ) = q42 (w1 ; w2 ; w3 ; w4 ) =
a(1
)
a(1
)
a(1
)
a(1
)
(1 + 2
3(1
(1 + 2
3(1
(1 + 2
3(1
(1 + 2
3(1
)w1 +
2 )
)w2 +
2 )
)w3 +
2 )
)w4 +
2 )
(w2 + w3 + w4 )
9 2
(w1 + w3 + w4 )
9 2
(w1 + w2 + w4 )
9 2
(w1 + w2 + w3 )
:
9 2
Proof of Proposition 2
Just by plugging the third stage equilibrium wholesale prices into the manufacturers pro…t
functions in the three di¤erent sub-games we get:
S2
M1
=
S3
M1
=
S4
M1
=
2a2 (1 )
S2 =
:
M2
3(1+ )(2 )2
a2 (1 )(2+3 )2
; S3
M2
3(1+2 )(2+(2 ) )2
2
2)
S4 = a (1
:
M2
3+9
=
2a2 (1 )(1+ )3
:
3(1+2 )(2+(2 ) )2
A necessary condition for "partial product innovation" (three goods) to be an equilibrium
S
is that F
F ( )=
S3
M1
=
a2 (1 )(2+3 )2
.
3(1+2 )(2+(2 ) )2
A necessary condition for "full product innovation" (four goods) to be an equilibrium is
that F
S
F ( )=
S4
Mi
=
a2 (1
3+9
2)
; i=1,2.
When is "no innovation" an equilibrium? The only possible deviation by, let’s say, M1
S
is to "partial product innovation". Now, if F > F ( ), then such incentives do not exists.
If F
S
F ( ); M1 deviates only if
S3
M1
S2 :
M1
F
23
This is satis…ed as long as F
F1S ( ),
where F1S ( ) =
a2 (1
3
)
( (1+2
(2+3 )2
)(2+(2 ) )2
S
2
(1+ )(2
)2
S
) and F1S ( ) < F ( ) < F ( ): But
it is direct to see that F1S ( ) < 0 when g > 0:9164: So if g > 0:9164; no deviation is
pro…table and "no innovation" is an equilibrium. If g
0:9164; however, "no innovation"
is an equilibrium only if F > F1S ( ):
When is "partial product innovation an equilibrium? First of all, we know that it is not
S
S
an equilibrium when F > F ( ): Also when F
and F > F1S ( ):When g
0:9164 and F
F ( ) and g > 0:9164; or when g
F1S ( );the possible deviation is for M2 to produce
two goods instead of 1 good. This deviation is pro…table for M2 as long as
which holds if F
0:9164
F2S ( ), where F2S ( ) =
a2 (1 )(2+3 )2
3(1+ (5+6 ))(2+(2 ) )2
S4
M2
S3 ,
M2
F
and F1S ( ) > F2S ( ): But
it is direct to see that F2S ( ) < 0 when g > 0:8346: So if 0:8346 < g
0:9164; this deviation
is not pro…table and "partial product innovation" is an equilibrium. Finally, if g
0:8346
and F > F2S ( ); "partial product innovation" is also an equilibrium.
S
When is "full innovation" an equilibrium? It is not an equilibrium when F > F ( ):
S
When F
F ( ); the only possible deviation is for M2 to produce 1 good instead of 2
goods. We know that this deviation is pro…table for M2 as long as g > 0:8346, or when
g
0:8346 and F > F2S ( ):So we conclude that "full innovation" is an equilibrium only
when g
F2S ( ):
0:8346 and F
Proof of Proposition 3
We have to compare social welfare under the three possible scenarios, namely, 4, 3, or 2
goods in the market.
We need …rst to compute the equilibrium quantities for each scenario by plugging the
equilibrium wholesale prices (equations 2 and 4) into the third stage equilibrium outputs
(above in the Appendix). Doing so we get:
S2 = q S2 = q S2 = q S2 =
q11
12
21
22
S3 = q S3 = q S3 = q S3 =
q11
12
31
32
S4 = q S4 = q S4 = q S4 =
q11
12
21
21
a
;
3 2
a(1+ )2
a(2+3 )
S3
S3
6(2+ (6+(3 2 ) )) ; q21 = q22 = 6+3 (6+(3 2 ) ) ;
S4 = q S4 = q S4 = q S4 = a(1+ )
q31
32
41
42
6+18
6+3
By using these equilibrium outputs we can compare social welfare associated to each scenario:
(i) Welfare comparison between "full product innovation and "no product innovation":
S4
S4
S4
u(QS4
1 ; Q2 ; Q3 ; Q4 )
2(9F (2
S ( )=
FW
42
2F
S2
u(QS2
1 ; Q2 ; 0; 0) =
)2 (1+ )(1+3 )+a2 ( 1+ )(10+ (2+
9(2 )2 (1+ )(1+3 )
a2 (1 )(10+ (2+ 6 2 +g 3 ))
:
9(2 )2 (1+ )(1+3 )
6
2 +g 3 )))
24
: This expression equals zero if F =
(ii) Welfare comparison between "partial product innovation and "no product innovaS3
S3
tion": u(QS3
1 ; Q2 ; Q3 ; 0)
S2
u(QS2
1 ; Q2 ; 0; 0) =
F
(9F (1+ )(1+2 )(4+ (2+ ( 4+ ))2 +a2 ( 1+ )(40+ (72+ ( 14+ ( 40+ ( 1+ )( 5+6 ))))))
. This
9(1+ )(1+2 )(4+ (2+ ( 4+ ))2 )
2
S ( ) = a (1 )(40+ (72+ ( 14+ ( 40+ ( 1+ )( 5+6 )))))) :
pression equals zero if F = FW
32
9(1+ )(1+2 )(4+ (2+ ( 4+ ))2
ex-
(iii) Welfare comparison between "full product innovation and "partial product innovaS4
S4
S4
tion": u(QS4
1 ; Q2 ; Q3 ; Q4 )
=
S3
S3
u(QS3
1 ; Q2 ; Q3 ; 0) =
F
(9F ( 2+ ( 2+ ))2 (1+ (5+6 ))+a2 ( 1+ )(10+ (36+ (55+ (37+2 ( 8+ ( 13+2 )))))))
:
9( 2+ ( 2+ ))2 (1+ (5+6 ))
This ex-
pression equals zero if
a2 ( 1+ )(10+ (36+ (55+ (37+2 ( 8+ ( 13+2 )))))))
:
9( 2+ ( 2+ ))2 (1+ (5+6 ))
S
S
S
FW 43 ( ) < FW 42 ( ) < FW 32 ( ) which implies that
S ( )=
F = FW
43
We check that
S ( ); the
if F > FW
32
S ( )<F
…rst best would be to have 2 goods in the market (no product innovation); if FW
43
S ( );the …rst best would be to have 3 goods in the market (partial product innovation);
FW
32
S ( ); the …rst best would be to have 4 goods in the market (full product
FW
43
…nally, if F
innovation). Now, by comparing the …rst best with the market equilibrium number of
varieties we obtained in Proposition 2, we can see that the following holds: F2S ( ) <
S ( ) < F S ( ) < F S ( ):Finally, by taking into account the fact that
F1S ( ) < FW
43
W 42
W 32
F1S ( ) < 0 when g > 0:9164 and that F2S ( ) < 0 when g > 0:8346;Proposition 3 results.
Proof of Proposition 5
Just by plugging the third stage equilibrium wholesale prices into the manufacturers pro…t
functions in the three di¤erent sub-games we get:
M2
M
a2
3+3
=
;
M3
M
=
a2
2+4
;
M4
M
2a2
3+9
=
:
A necessary condition for "partial product innovation" (three goods) to be an equilibrium
is that F
F
M
( )=
M3
M
=
a2
2+4
.
A necessary condition for "full product innovation" (four goods) to be an equilibrium is
that F
F
M
( )=
M4
M
=
2a2
3+9
with F
M
( )<F
M
( ):
When is "no innovation" an equilibrium? It is always equilibrium when F > F
When F
M
( ) < F
F
M
M
( ):
( ) the only possible deviation by the merged …rm is to full
F
innovation (four goods). If instead F
M
( ); there could be deviations to both "partial
product innovation (three goods) and "full product innovation" (four goods). If F
M deviates to three goods as long as
a2 (1 )
and F1M (
6(1+3 +2 2 )
M 4 2F
M 2 : This is
M
M
M3
M
M
F
M 2:
M
F1M ( ) =
)<F
only if
satis…ed as long as F
This is satis…ed if F
F
M
( );
F1M ( ), where
( ): On the other hand, M deviates to four goods
25
F3M ( ), where F3M ( ) =
a2 (1 )
6(1+4 +3 2 )
and F1M ( ) > F3M ( ): In short, "no innovation" is an equilibrium if and only if F > F1M ( ).
When is "partial product innovation" an equilibrium? First of all, we know that it is
not an equilibrium when F > F
M
( ): When F
F
M
( ), the possible deviation is for M
to produce two goods or four goods instead of three goods. We know that a deviation
to two goods is pro…table for M as long as F > F1M ( ). On the other hand, M deviates
to four goods only if
a2 (1 )
6(1+ (5+6 )
F2M ( ) =
M4
M
F
M 3:
M
This is satis…ed as long as F
F2M ( ), where
with F1M ( ) > F3M ( ) > F2M ( ): So summarizing, "partial product
innovation is an equilibrium if and only if F2M ( ) < F
F1M ( ):
When is "full innovation" an equilibrium? It is not an equilibrium when F > F
When F
M
( ) < F
F
M
M
( ):
( ); the only possible deviation is for M to produce 2 goods
instead of 4 goods, whereas if F
F
M
( ); the possible deviations are for M to produce
two or three goods respectively. So, summarizing, "full innovation" is an equilibrium if and
only if F
F2M ( ) (recall that the ranking between the di¤erent relevant threshold values
of F is F2M ( ) < F3M ( ) < F1M ( ) < F
F
M
( ) < F
M
( ) and so F
F2M ( ) also implies
F3M ( )):
Finally, it is direct to see that F1M ( ), F2M ( ) and F
M
( ) are all decreasing in :
Proof of Proposition 7
Combining the results of Propositions 2 and 5 we can easily rank the di¤erent threshold
values of F in order to be able to compare the equilibrium number of varieties pre-merger
and post-merger for di¤erent values of the product di¤erentiation parameter :
It is direct to see that when
> 0:9164 we have F2S ( ) < F1S ( ) < 0 < F2M ( ) <
F1M ( ), which implies that (i) for F < F2M ( ); the merger upstream leads to more product
innovation (4 varieties) compared with the pre-merger case (only two varieties); (ii) for
F2M ( )
F < F1M ( ) the merger upstream leads to more product innovation (3 varieties)
compared with the pre-merger case (only 2 varieties); …nally, (iii) for F
F1M ( ) the merger
does not a¤ect the number of varieties (2 goods in each case).
When 0:8415 <
< 0:9164 we have F2S ( ) < 0 < F1S ( ) < F2M ( ) < F1M ( ), which
implies that (i) for F < F1S ( ), the merger upstream leads to more product innovation (4
varieties) compared with the pre-merger case (only 3 varieties); F1S ( )
F < F2M ( ), the
merger upstream leads to more product innovation (4 varieties) compared with the premerger case (only 2 varieties); (iii) for F2M ( )
F < F1M ( ) the merger upstream leads to
more product innovation (3 varieties) compared with the pre-merger case (only 2 varieties);
26
…nally, (iv) for F
F1M ( ) the merger does not a¤ect the number of varieties (2 goods in
each case).
When 0:8346 <
< 0:8415 we have F2S ( ) < 0 < F2M ( ) < F1S ( ) < F1M ( ), which
implies that (i) for F < F2M ( ), the merger upstream leads to more product innovation (4
varieties) compared with the pre-merger case (only 3 varieties); (ii) for F2M ( )
F < F1S ( )
the merger does not a¤ect the number of varieties (3 goods in each case); (iii) for F1S ( )
F < F1M ( ) the merger upstream leads to more product innovation (3 varieties) compared
with the pre-merger case (only 2 varieties); …nally, (iv) for F
F1M ( ) the merger does not
a¤ect the number of varieties (2 goods in each case).
When 0:7261 <
< 0:8346 we have 0 < F2S ( ) < F2M ( ) < F1S ( ) < F1M ( ), which
implies that (i) for F < F2S ( ) the merger does not a¤ect the number of varieties (4
goods in each case); (ii) for F2S ( )
F < F2M ( ), the merger upstream leads to more
product innovation (4 varieties) compared with the pre-merger case (only 3 varieties); (iii)
for F2M ( )
F < F1S ( ) the merger does not a¤ect the number of varieties (3 goods in
each case); (iv) for F1S ( )
F < F1M ( ) the merger upstream leads to more product
innovation (3 varieties) compared with the pre-merger case (only 2 varieties); …nally, (v)
for F
F1M ( ) the merger does not a¤ect the number of varieties (2 goods in each case).
When 0:6931 < F < 0:7261 we have 0 < F2S ( ) < F2M ( ) < F1M ( ) < F1S ( ), which
implies that (i) for F < F2S ( ) the merger does not a¤ect the number of varieties (4
goods in each case); (ii) for F2S ( )
F < F2M ( ), the merger upstream leads to more
product innovation (4 varieties) compared with the pre-merger case (only 3 varieties); (iii)
for F2M ( )
F < F1M ( ) the merger does not a¤ect the number of varieties (3 goods in each
case); (iv) for F1M ( )
F < F1S ( ) the merger upstream leads to less product innovation
(2 varieties) compared with the pre-merger case (3 varieties); …nally, (v) for F
F1S ( ) the
merger does not a¤ect the number of varieties (2 goods in each case).
When 0:4046 < F < 0:6931 we have 0 < F2M ( ) < F2S ( ) < F1M ( ) < F1S ( ), which
implies that (i) for F < F2M ( ) the merger does not a¤ect the number of varieties (4
goods in each case); (ii) for F2M ( )
F < F2S ( ), the merger upstream leads to less
product innovation (3 varieties) compared with the pre-merger case (4 varieties); (iii) for
F2S ( )
F < F1M ( ) the merger does not a¤ect the number of varieties (3 goods in each
case); (iv) for F1M ( )
F < F1S ( ) the merger upstream leads to less product innovation
(2 varieties) compared with the pre-merger case (3 varieties); …nally, (v) for F
merger does not a¤ect the number of varieties (2 goods in each case).
27
F1S ( ) the
When F < 0:4046 we have 0 < F2M ( ) < F1M ( ) < F2S ( ) < F1S ( ), which implies that
(i) for F < F2M ( ) the merger does not a¤ect the number of varieties (4 goods in each
case); (ii) for F2M ( )
F < F1M ( ), the merger upstream leads to less product innovation
(3 varieties) compared with the pre-merger case (4 varieties); (iii) for F1M ( )
F < F2S ( ),
the merger upstream leads to less product innovation (2 varieties) compared with the premerger case (4 varieties); (iv) for F2S ( )
F < F1S ( ), the merger upstream leads to less
product innovation (2 varieties) compared with the pre-merger case (3 varieties); …nally, (v)
F1S ( ) the merger does not a¤ect the number of varieties (2 goods in each case).
for F
Proof of Proposition 8
In order to prove Proposition 8 we proceed by comparing the post-merger upstream pro…ts
with the sum of the pre-merger upstream …rms’ pro…ts for each possible combination of
(equilibrium) number of varieties pre and post-merger that arise for di¤erent values of the
product di¤erentiation parameter :
Let us write here the equilibrium upstream pro…ts expressions after a merger upstream
takes place for the three possible subgames:
M2
M
=
a2
3+3
M3
M
;
=
a2
2+4
;
M4
M
=
2a2
3+9
:
On the other hand, the equilibrium separated upstream …rms’pro…ts are given by:
S2
M1
=
S3
M1
=
S4
M1
=
2a2 (1 )
S2 =
:
M2
3(1+ )(2 )2
2
2
a (1 )(2+3 )
; S3
M2
3(1+2 )(2+(2 ) )2
2
2)
S4 = a (1
:
M2
3+9
So let start with the case
=
2a2 (1 )(1+ )3
:
3(1+2 )(2+(2 ) )2
< 0:4046: In this case, we can have equilibria (2, 2), (2, 3),
(2, 4), (3, 4) and (4, 4).22 Let’ s compare the upstream pro…ts for all these cases. Let’s
F1S ( ): We have to compute:
start by equilibrium (2,2), which occurs if F
a2 2
> 0, so the merger is pro…table in this region.
3(1+ )(2 )2
Equilibrium (2,3) occurs if F2S ( ) F < F1S ( ): We have to compute in this case:
2
M 2 ( S3 + S3 F ) = a ( 2+ ( 2+ (7+4 (3+ (2+ ))))) +F
0 as long as F F 23 ( )
M
M1
M2
3(1+ )(1+2 )( 2+ ( 2+ )2
a2 ( 2+ ( 2+ (7+4 (3+ (2+ )))))
: It is direct to check that F 23 ( ) < F2S ( ) regardless of
3(1+ )(1+2 )( 2+ ( 2+ )2
M2
M
(
S2
M1
+
S2 )
M2
=
=
,
which implies that the merger is pro…table in this region.
Equilibrium (2,4) occurs if F1M ( )
M2
M
(
S4
M1
+
S4
M2
2F ) =
F < F2S ( ): We have to compute in this case:
a2 (1+ (1+2 (1+ )))
3(1+ )(1+3 )
+ 2F
0 as long as F
F 24 ( ) =
22
The …rst number represents the number of varieties post-merger and the second one represents the
number of varieties pre-merger.
28
a2 ( 1+ (1+2 (1+ )))
:
6(1+ )(1+3 )
It is direct to check that F 24 ( ) < F1M ( ) regardless of
, which
implies that the merger is pro…table in this region.
Equilibrium (3,4) occurs if F2M ( )
F < F1M ( ): We have to compute in this case:
2
1+ +4 2 +8 3 )
M3
S4
( S4
F ) = a (6(1+
M
M1 + M2
(5+6 ))
a2 ( 1+ +4 2 +8 3 )
: It is direct to check that F 34 ( )
6(1+ (5+6 ))
+F
0 as long as F
F 34 ( ) =
< F2M ( ) regardless of , which implies
that the merger is pro…table in this region.
Equilibrium (4,4) occurs if F < F2M ( ): We have to compute in this case:
M4
M
(
S4
M1
+
S4 )
M2
=
2a2 2
3+9
> 0; which implies that the merger is pro…table in this
region.
Let’s go on with the case 0:4046 <
< 0:6931: In this case, we can have equilibria (2,
2), (2, 3), (3, 3), (3, 4) and (4, 4). Let’s compare the upstream pro…ts for all the cases not
analyzed previously (for equilibria (2,2), (3,4) and (4,4) the proof is already done above).
Equilibrium (2,3): it su¢ ces to check that F 23 ( ) < F2M ( ), which holds regardless of
, so the merger is pro…table in this region.
Equilibrium (3,3) occurs if F2S ( )
M3
M
(
S3
M1
+
S3 )
M2
=
F < F1M ( ): We have to compute in this case:
a2 2 (6+7 )
6(1+2 )( 2+ ( 2+ ))2
> 0, which implies that the merger is
pro…table in this region.
Let’s go on with the case 0:6931 <
< 0:7261: In this case, we can have equilibria (2,
2), (2, 3), (3, 3), (4, 3) and (4, 4). Let’ s compare the upstream pro…ts for all the cases
not analyzed previously (for equilibria (2,2), (2,3), (3,3) and (4,4) the proof is already done
above).
Equilibrium (4,3) occurs if F2S ( )
F < F2M ( ): We have to compute in this case:
2
M 4 F ( S3 + S3 ) = a (2+ (1+ (2+ ( 3+ (17+10 )))) F
M
M1
M2
3( 2+ ( 2+ )2 (1+ (5+6 ))
a2 (2+ (1+ (2+ ( 3+ (17+10 ))))
: It is direct to check that F 43 (
3( 2+ ( 2+ )2 (1+ (5+6 ))
0 as long as F
) >
F2M (
F 43 ( ) =
) regardless of ,
which implies that the merger is pro…table in this region.
Let’s go on with the case 0:7261 <
< 0:8346: In this case, we can have equilibria (2,
2), (3, 2), (3, 3), (4, 3) and (4, 4). Let’ s compare the upstream pro…ts for all the cases
not analyzed previously (for equilibria (2,2), (3,3), (4,3) and (4,4) the proof is already done
above).
Equilibrium (3,2) occurs if F1S ( )
M3
F
( S2
M
M1
a2 (4+ ( 8+ (7+3 )))
: It
6(1+ )(1+2 )(2 )2
+
S2 )
M2
=
F < F1M ( ): We have to compute in this case:
a2 (4+ ( 8+ (7+3 )))
6(1+ )(1+2 )(2 )2
F
0 as long as F
F 32 ( ) =
is direct to check that F 32 ( ) > F1S ( ) regardless of , which implies
that the merger is pro…table in this region.
29
Let’s go on with the case 0:8346 <
< 0:8425: In this case, we can have equilibria (2,
2), (3, 2), (3, 3) and (4, 3). All the cases have been previously analyzed above.
Let’s go on with the case 0:8425 <
< 0:9164: In this case, we can have equilibria (2, 2),
(3, 2), (4, 2) and (4, 3). Let’s compare the upstream pro…ts for all the cases not analyzed
previously (for equilibria (2,2), (3,2) and (4,3) the proof is already done above).
Equilibrium (4,2) occurs if F1S ( )
F < F2M ( ): We have to compute in this case:
2a2 (2+ ( 1+ )(4+ ))
M4
S3
2F ( S3
M
M1 + M2 ) = 3( 2+ )2 (1+ )(1+3 )
a2 (2+ ( 1+ )(4+ ))
:It is direct to check that F 42 ( ) >
3( 2+ )2 (1+ )(1+3 )
2F
0 as long as F
F 42 ( ) =
F2M ( ) regardless of , which implies
that the merger is pro…table in this region.
Equilibrium (4,3): it su¢ ces to check that F 43 ( ) > F1S ( ), which holds if
(and this is satis…ed as we are analyzing the range 0:8425 <
> 0:3965
< 0:9164) so the merger is
pro…table in this region.
Let’s go on …nally with the case
> 0:9164:In this case, we can have equilibria (2, 2),
(3, 2) and (4, 2). All of them have been already analyzed above, so the merger is pro…table
in this last region also.
Proof of Proposition 9
The equilibrium downstream pro…ts under separated upstream …rms for each possible subgame (2, 3 or 4 goods) are given by:
S2
R1
=
S2
R2
=
S3
R1
=
S3
R2
=
S4
R1
=
S4
R2
=
2a2
;
9(2 )2 (1+ )
2
a (1+ )(6+7 (2+ ))
;
18(1+2 )( 2+ ( 2+ ))2
2
2
a (1+ )
9+27 :
The equilibrium downstream pro…ts under merged upstream …rms for each possible subgame
(2, 3 or 4 goods) are given by:
M2
R1
=
M2
R2
=
M3
R1
=
M3
R2
=
M4
R1
=
M4
R2
=
a2
18+18 ;
a2
12+24 ;
a2
9+27 :
Now we have to compare the downstream pro…ts in the merger and no merger cases
for all the possible combinations of varieties that arise in equilibrium, that is: (2,2), (3,3),
(4,4), (3, 2), (4,2), (4,3), (2,3), (2, 4), (3,4). The comparisons are straightforward for all
the cases where after the merger we have equal or less variety, that is, for the cases (2,2),
(3,3), (4,4), (2,3), (2, 4), (3,4). For the rest of the cases we have:
Equilibrium (3,2):
M3
R1
S2
R1
=
a2 (3(2 )2 (1+ ) 8a2 (1+2 ))
36(2 )2 (1+ )(1+2 )
30
< 0 only if
> 0:2239: But we
know that equilibrium (3,2) can arise only if
> 0:7261; so the merger reduces downstream
…rms’pro…ts.
Equilibrium (4,2):
(4,2) can arise only if
Equilibrium (4,3):
(4,3) can arise only if
M4
R1
S2
R1
< 0 only if
> 0:2942: But we know that equilibrium
> 0:8415; so the merger reduces downstream …rms’pro…ts.
M4
R1
S3
R1
< 0 only if
> 0:1388: But we know that equilibrium
> 0:6931; so the merger reduces downstream …rms’pro…ts.
Proof of Proposition 10
Social welfare: Given that we assume that there are no positive marginal production costs,
it is direct to see that we can de…ne the (gross of any product innovation costs) social welfare
function simply as:
W (Q1 ; :::; QN ) = u(Q1 ; :::; QN ):
We have to compare social welfare under a merged upstream …rm and under separated
upstream …rms for all possible combinations of varieties in equilibrium. We need …rst to
compute the equilibrium quantities for each possible scenario by plugging the equilibrium
wholesale prices (equations 2, 4 and 7) into the third stage equilibrium outputs. Doing so
we get:
S2 = q S2 = q S2 = q S2 =
q11
12
21
22
S3 = q S3 = q S3 = q S3 =
q11
12
31
32
S4 = q S4 = q S4 = q S4 =
q11
12
21
21
a
;
3 2
a(2+3 )
a(1+ )2
S3
S3
6(2+ (6+(3 2 ) )) ; q21 = q22 = 6+3 (6+(3 2 ) ) ;
S4 = q S4 = q S4 = q S4 = a(1+ ) ;
q31
32
41
42
6+18
6+3
M 2 = qM 2 = qM 2 = qM 2 =
q11
12
21
22
M 3 = qM 3 = qM 3 = qM 3 =
q11
12
21
22
a
6+6
M3
q31
;
M3 =
= q32
M 4 = qM 4 = qM 4 = qM 4 = qM 4 = qM 4 =
q11
12
21
22
31
32
a
6+12 ;
M 4 = qM 4
q41
42
=
a
6+18
:
By using these equilibrium outputs we can compare the net social welfare associated to each
scenario:
We start in the region
< 0:4046: In this region we can have equilibria (2,2), (2,3),
(2,4), (3,4), (4,4).
Equilibrium (2,2): we have to check the sign of:
2a
2a
u( 6+6
; 6+6
; 0; 0)
u( 6+3 2a 3 2 ; 6+3 2a 3 2 ; 0; 0) =
a2 ( 8+5 )
9(2 2 )(1+ )
< 0, which implies that
in this case the merger reduces social welfare.
Equilibrium (2,3): we have to sign:
2a
2a
u( 6+6
; 6+6
; 0; 0)
u( 3(2+
a(2+3 )
2a(1+ )2
a(2+3 )
(6+(3 2 ) )) ; 6+3 (6+(3 2 ) ) ; 3(2+ (6+(3 2 ) )) ; 0)
31
+F =
9F (1+ )(1+2 )( 2+ ( 2+ ))2 +a2 ( 10+ ( 26+ ( 29+4 ( 7+ ( 1+4 )))))
< 0 if
9(1+ )(1+2 )( 2+ ( 2+ ))2
a2 (10+ (26+ (29+4 (7+ 4 2 ))))
F < FW 23 ( ) = 9(1+ )(1+2 )( 2+ ( 2+ ))2 : But we are in equilibrium (2,3) if F2S (
< F1S ( ): It is direct to check that FW 23 ( ) > F1S ( ); which implies that in this case
=
F
)
the
merger reduces social welfare.
Equilibrium (2,4): we have to sign:
2a
2a
u( 6+6
; 6+6
; 0; 0)
=
F
) 2a(1+ ) 2a(1+ ) 2a(1+ )
u( 2a(1+
6+18 ; 6+18 ; 6+18 ; 6+18 ) + 2F =
18F (1+ )(1+3 )+a2 ( 5+ ( 3+2 ( 3+ )))
<
9(1+ )(1+3 )
2
(3 2 ( 3+ )))
< FW 24 ( ) = a (5+
: But
18(1+ )(1+3 )
0 if
we are in equilibrium (2,4) if F1M ( )
F <
F2S ( ): It is direct to check that FW 24 ( ) > F2S ( ); which implies that in this case the
merger reduces social welfare.
Equilibrium (3,4): we have to sign:
2a
2a
2a
; 6+12
; 6+12
; 0)
u( 6+12
=
F
) 2a(1+ ) 2a(1+ ) 2a(1+ )
u( 2a(1+
6+18 ; 6+18 ; 6+18 ; 6+18 ) + F =
18F (1+2 )(1+3 )+a2 ( 5+ ( 11+4 ( 7+2 )))
18(1+ )(5+6 )
2
(11+4 (7 2 )))
: But
< FW 34 ( ) = a (5+
18(1+ )(5+6 )
< 0 if
we are in equilibrium (3,4) if F2M ( )
F <
F1M ( ): It is direct to check that FW 34 ( ) > F1M ( ); which implies that in this case the
merger reduces social welfare.
Equilibrium (4,4): we have to sign:
2a
2a
2a
2a
u( 6+18
; 6+18
; 6+18
; 6+18
)
=
2a2 ( 4+ )
9+27
) 2a(1+ ) 2a(1+ ) 2a(1+ )
u( 2a(1+
6+18 ; 6+18 ; 6+18 ; 6+18 ) =
< 0, which implies that in this case the merger reduces social welfare.
Let’s move to the region 0:4046 <
< 0:6931:In this region we can have equilibria (2,2),
(2,3), (3,3), (3,4), (4,4). We have already done (2,2), (2,3) and (4,4).
Equilibrium (3,3): we have to sign:
2a
2a
2a
u( 6+12
; 6+12
; 6+12
; 0)
=
u( 3(2+
a2 ( 32+ ( 66+ ( 10+27 )))
18(1+2 )( 2+ ( 2+ ))2
a(2+3 )
2a(1+ )2
a(2+3 )
(6+(3 2 ) )) ; 6+3 (6+(3 2 ) ) ; 3(2+ (6+(3 2 ) )) ; 0)
=
< 0, which implies that in this case the merger reduces
social welfare.
Equilibrium (3,4): we are in equilibrium (3,4) if F2M ( )
F < F2S ( ):It is direct to
check that FW 34 ( ) > F2S ( ), which implies that in this case the merger reduces social
welfare.
Let’s move to the region 0:6931 <
< 0:7261:In this region we can have equilibria (2,2),
(2,3), (3,3), (4,3), (4,4). We have already done (2,2), (2,3) and (3,3) and (4,4).
Equilibrium (4,3): we have to sign:
2a
2a
2a
2a
; 6+18
; 6+18
; 6+18
) F u( 3(2+
u( 6+18
a(2+3 )
2a(1+ )2
a(2+3 )
(6+(3 2 ) )) ; 6+3 (6+(3 2 ) ) ; 3(2+ (6+(3 2 ) )) ; 0)
32
=
9F (1+2 )(1+3 )( 2+ ( 2+ ))2 +a2 (10+ ( 6+ ( 101+ ( 114+ (11+38 )))))
< 0 if F > FW 43 ( )
9(1+ )(5+6 )( 2+ ( 2+ ))2
a2 (10+ ( 6+ ( 101+ ( 114+ (11+38 )))))
: But we are in equilibrium (4,3) if F2S ( )
F <
9(1+ (5+6 ))( 2+ ( 2+ ))2
F2M ( ): It is direct to check that FW 43 ( ) < F2S ( ) if > 0:1464; which holds in this region
=
=
so this implies that in the region under consideration the merger reduces social welfare.
Let’s move to the region 0:7261 <
< 0:8346:In this region we can have equilibria (2,2),
(3,2), (3,3), (4,3) and (4,4). We have already done (2,2), (3,3), (4,3) and (4,4).
Equilibrium (3,2): we have to sign:
2a
2a
2a
u( 6+12
; 6+12
; 6+12
; 0)
=
18F (2
But we are in
F1S ( ) if
F
u( 6+3 2a 3 2 ; 6+3 2a 3 2 ; 0; 0) =
)2 (1+2 )(1+ )+a2 (20+ ( 56+3 (1+5 )))
< 0 if F
18(1+2 )(1+ )(2 )2
equilibrium (3,2) if F1S ( ) F < F1M ( ): It
> FW 32 ( ) =
a2 (20+ ( 56+3 (1+5 )))
:
18(1+2 )(1+ )(2 )2
is direct to check that FW 32 ( ) <
> 0:1954; which holds in this region so this implies that in this case the merger
reduces social welfare.
Let’s move to the region 0:8346 <
< 0:8415:In this region we can have equilibria (2,2),
(3,2), (3,3) and (4,3). We have already done (2,2), (3,2), (3,3).
Equilibrium (4,3): the merger reduces welfare if F > FW 43 ( ):But we check that
FW 43 ( ) < 0 if g > 0:32 so in the region under consideration F > FW 43 ( ) and so the
merger is anticompetitive.
Let’s move to the region 0:8415 <
< 0:9164:In this region we can have equilibria (2,2),
(3,2), (4,2) and (4,3). We have already done (2,2) and (4,3).
Equilibrium (3,2): it is anticompetitive if F > FW 32 ( ) =
are in equilibrium (3,2) if F2M ( )
if
F < F1M ( ): It is direct to
a2 (20+ ( 56+3 (1+5 )))
:But we
18(1+2 )(1+ )(2 )2
check that FW 32 ( ) < F2M ( )
> 0:2922; which holds in this region, so in the region under consideration F > FW 32 ( )
and so the merger is anticompetitive.
Equilibrium (4,2): we have to sign:
2a
2a
2a
2a
; 6+18
; 6+18
; 6+18
)
u( 6+18
=
2F
u( 6+3 2a 3 2 ; 6+3 2a 3 2 ; 0; 0) =
)2 (1+3 )(1+ )+2a2 (10+ ( 24+ (3+5 )))
<0
9(1+3 )(1+ )(2 )2
equilibrium (4,2) if F1S ( )
F < F2M ( ):
18F (2
we are in
F1S ( ) if
if F > FW 42 ( ) =
a2 (10+ ( 24+ (3+5 )))
:But
9(1+3 )(1+ )(2 )2
It is direct to check that FW 42 ( ) <
> 0:2061; which holds in this region so in the region under consideration
F > FW 42 ( ) and so the merger is anticompetitive.
Let’s move to the region
> 0:9164: In this region we can have equilibria (2,2), (3,2),
(4,2). We have already done (2,2) and (3,2).
Equilibrium (4,2): it is anticompetitive if F > FW 42 ( ) =
check that FW 42 ( ) < 0 if
a2 (10+ ( 24+ (3+5 )))
:
9(1+3 )(1+ )(2 )2
We
> 0:4645, which implies that in the region under consideration
33
F > FW 42 ( ) and so the merger is anticompetitive.
Consumer surplus:
We will compare consumer surplus under both merged and separated upstream …rms
only for the cases where a merger increases variety (from 3 to 4, from 2 to 4 or from 2 to
3) because they are the only possible cases where consumer surplus could increase after a
merger upstream.
Equilibrium (4,3): this equilibrium arises if 0:6931 <
2a
2a
2a
2a
u( 6+18
; 6+18
; 6+18
; 6+18
)
< 0:9164: We have to sign:
M4
M
M4
M4
R1
R2
2a(1+
a(2+3 )
a(2+3 )
S3
S3
S3
(u( 3(2+ (6+(3 2 ) )) ; 6+3 (6+(3 2 ) ) ; 3(2+ (6+(3 2 ) )) ; 0)
M1
M2
R1
2
a (2+ ( 6+ ( 49+ ( 78+ ( 35+4 )))))
< 0 if > 0:1388, which implies that in
=
9(1+ (5+6 ))( 2+ ( 2+ ))2
)2
S3 ))
R2
=
the region
under consideration the merger upstream reduces consumer surplus.
Equilibrium (4,2): this equilibrium arises if
> 0:8415:We have to sign:
2a
2a
2a
2a
; 6+18
; 6+18
; 6+18
)
u( 6+18
M4
M
M4
R1
M4
R2
(u( 6+3 2a 3 2 ; 6+3 2a 3 2 ; 0; 0)
S2
M1
S2
M2
S2
R1
=
2a2 (2+
( 6+ ( 3+ )))
9(1+3 )(1+ )(2 )2
< 0 if
S2 ))
R2
=
> 0:2942, which implies that in the region under consid-
eration the merger upstream reduces consumer surplus.
Equilibrium (3,2): this equilibrium arises if
2a
2a
2a
u( 6+12
; 6+12
; 6+12
; 0)
M3
M
(u( 6+3 2a 3 2 ; 6+3 2a 3 2 ; 0; 0)
=
a2 (4+ ( 16+3 ( 3+ )))
18(1+2 )(1+ )(2 )2
< 0 if
M3
R1
S2
M1
> 0:7261: We have to sign:
M3
R2
S2
M2
S2
R1
S2 ))
R2
=
> 0:2239, which implies that in the region under
consideration the merger upstream reduces consumer surplus.
10
References
Alexander, P.M., 1997. Product Variety and Market Structure: A New Measure and a
Simple Test. Journal of Economic Behavior and Organization 32, 207-214.
Avenel, E. and Caprice, S., 2006. Upstream market power and product line di¤erentiation in retailing. International Journal of Industrial Organization 24 (2), 319-334.
Bernheim, B.D. and Whinston, M.D., 1998. Exclusive Dealing. Journal of Political
Economy 106, 64-103.
Berry, S. and Waldfogel, J., 2001. Do Mergers Increase Product Variety? Evidence from
Radio Broadcasting. Quarterly Journal of Economics 116, 1009-1025.
34
Brander, J.A. and Eaton, J., 1984. Product line rivalry. American Economic Review
74, 323-334.
Chang, M.H., 1992. Exclusive Dealing Contracts in a Successive Duopoly with Side
Payments. Southern Economic Journal 59, 180-193.
Chen, Z. , 2004. Countervailing Power and Product Diversity, unpublished manuscript.
Chen, Y. and Schwartz, M., 2013. Product Innovation Incentives: Monopoly versus
Competition. Journal of Economics and Management Science 22, 513-528.
Davidson, C. and Deneckere, R., 1985. Incentives to Form Coalitions with Bertrand
Competition. Rand Journal of Economics 16, 473-486.
De Fraja, G., 1993. Relaxing Spatial Competition through Product Line Choice (or vice
versa). Regional Science and Urban Economics, 23, 461-486.
Dhillon, A. and Petrakis, E., 2002. A Generalized Wage Rigidity Result. International
Journal of Industrial Organization 20, 285-311.
Dobson, P.W. and Waterson, M., 1996. Exclusive Trading Contracts in Successive
Di¤erentiated Duopoly. Southern Economic Journal 63, 361-377.
Dobson, P. and Waterson, M., 1997. Countervailing Power and Consumer Prices. Economic Journal 107, 418-430.
Eaton, B. and Lipsey, R., 1979. The Theory of Market Pre-emption: The Persistence
of Excess Capacity and Monopoly in Growing Spatial Markets’Economica, 46, 149-58.
Fan, Y., 2013. Ownership Consolidation and Product Characteristics: A Study of the
US Daily Newspaper Market. American Economic Review 103, 1598-1628.
Farrell, J. and Shapiro, C., 1988. Horizontal Mergers: An Equilibrium Analysis. American Economic Review 80, 107-126.
Fauli-Oller, R., 2008. Capacity Restrictions by Retailers. Mimeo. Universidad de
Alicante.
Fauli-Oller, R. and Bru, L., 2008. Horizontal Mergers for Buyer Power. Economics
Bulletin 12, 1-7.
Faulí-Oller, R., Sandonís, J. and J. Santamaría, 2011. Downstream Mergers and Upstream Investment. The Manchester School 79, 884-898.
Faulí-Oller, R. and Sandonís, J., 2014. Welfare E¤ects of Downstream Mergers and
Upstream Market Concentration. The Singapore Economic Review, forthcoming.
Froeb, L., Tschantz, S. and Werden, G., 2007. Vertical Restraints and the E¤ects of
Upstream Horizontal Mergers, in The Political Economy of Antitrust, V. Ghosal, V., and
35
Stennek, S. (Eds.), Amsterdam: North-Holland Publishing.
Gabrielsen, T.S. and Sørgard, L., 1999. Discount Chains and Brand Policy. Scandinavian Journal of Economics 101, 127-142.
Gandhi, A., Froeb, L., Tschantz, S. and Werden, G., 2008. Post-merger Product Repositioning. Journal of Industrial Economics 56, 49-67.
George, L., 2007. What’s Fit to Print: The E¤ect of Ownership Concentration on
Product Variety in Daily Newspaper Markets. Information Economics and Policy 19, 285303.
Gilbert, R. and Matutes, C., 1993. Product Line Rivalry and Brand Di¤erentiation.
Journal of Industrial Economics 41, 223-240.
Gotz, G. and Gugler, K., 2006. Market Concentration and Product Variety under
Spatial Competition: Evidence from Retail Gasoline. Journal of Industry, Competition and
Trade 6, 225-234.
Greenstain S. and Ramey G., 1998. Market Structure, Innovation and Vertical Product
Di¤erentiation. International Journal of Industrial Organization 16, 285-311.
Horn, H. and Wolinsky, A., 1988. Bilateral Monopolies and Incentives for Mergers.
Rand Journal of Economics 19, 408-419.
Inderst, R. and Sha¤er, G., 2004. Retail Merger, Buyer Power and Product Variety.
Economic Journal 117, 45-67.
Inderst, R. and Wey, C., 2003. Bargaining, Mergers and Technology Choice in Bilaterally
Oligopolistic Industries, Rand Journal of Economics 34, 1-19.
Inderst, R. and Wey, C., 2011. Countervailing Power and Dynamic E¢ ciency. Journal
of the European Economic Association 9, 702-720.
Innes, R., 2008. Entry for Merger with Flexible Manufacturing: Implications for Competition Policy. International Journal of Industrial Organization 26, 266-287.
Lancaster, K., 1990. The Economics of Product Variety: a Survey. Marketing Science
9, 189-206.
Lin, Y.J., 1990, The Dampening-of-Competition E¤ect of Exclusive Dealing. Journal of
Industrial Economics 39, 209-223.
Lommerud, K. and Sørgard, L., 1997. Merger and Product Range Rivalry. International
Journal of Industrial Organization 16, 21-42.
Lommerud, K., O. Straume and Sørgard, L., 2005. Downstream Merger with Upstream
Market Power. European Economic Review 49, 717-743.
36
Martínez-Giralt, X. and Neven, D., 1988. Can Price Competition Dominate Market
Segmentation? Journal of Industrial Economics 36, 431-443.
Mauleon, A., J.J. Sempere-Monerris, and Vannetelbosch, V., 2011, Networks of Manufacturers and Retailers. Journal of Economic Behavior and Organization 77, 351-367.
Mazzeo, M., Seim, K. and Varela, M., 2012. The Welfare Consequences of Mergers with
Product Repositioning. Mimeo, Kellogg School of Management.
Miklós-Thal, J., Rey, P. and Vergé, T., 2010. Vertical Relations. International Journal
of Industrial Organization 28, 345-349.
Milliou, C. and Petrakis, E., 2007, Upstream Horizontal Mergers, Vertical Contracts,
and Bargaining. International Journal of Industrial Organization 25, 963-987.
Milliou, C., Petrakis, E. and Sloev, I., 2013. Equilibrium Product Variety and Market
Structure in Successive Oligopolies. Mimeo, Athens University of Economics and Business.
Milliou, C. and Pavlou, A., 2013. Upstream Mergers, Downstream Competition and
R&D Investments. Journal of Economics and Management Strategy 22, 787-809.
Moner-Colonques, R., J.J. Sempere-Monerris, and Urbano, A., 2004, The Manufacturers’ Choice of Distribution Policy under Successive Duopoly. Southern Economic Journal
70, 532-548.
Moner-Colonques, R., Sempere-Monerris, J.J. and Urbano, A., 2011. Product Line
Choice in Retail Duopoly. Journal of Economics and Management Strategy 20, 777-802.
O’Brien, D. and Sha¤er, G., 1993. On the Dampening-of-Competition E¤ects of Exclusive Dealing. Journal of Industrial Economics 41, 215-221.
O’Brien, D. and Sha¤er, G., 1997. Nonlinear Supply Contracts, Exclusive Dealing and
Equilibrium Market Foreclosure. Journal of Economics and Management Strategy 6, 755785.
O’Brien, D. and Sha¤er, G., 2003. Bargaining, Bundling and Clout: The Portfolio
E¤ects of Horizontal Mergers. Rand Journal of Economics 36, 573-595.
Perry, M. and Porter, R., 1985. Oligopoly and the Incentives for Horizontal Merger.
American Economic Review 75, 219-227.
Petrakis, E. and Dhillon, A., (2002). A Generalized Wage Rigidity Result withCentralized Negotiations, International Journal of Industrial Organization 20, 285-311.
Raubitschek, R.S., 1987. A Model of Product Proliferation with Multiproduct Firms.
Journal of Industrial Economics 3, 269-279.
Reynolds, R., Salant, S. and Switzer, S,. 1983. Losses from Horizontal Merger: The
37
E¤ects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium. Quartely Journal of Economics 98, 185-199.
Salop, S., 1979. Monopolistic Competition with Outside Goods. Bell Journal of Economics 10, 141-156.
Schmalensee, R., 1978. Entry Deterrence in the Ready-to-Eat Breakfast Cereal Industry.
Bell Journal of Economics IX, 305-327.
Shaked, A. and Sutton, J., 1990. Multiproduct Firms and Market Structure, Rand
Journal of Economics 21, 45-62.
Singh, V. and Vives, X., 1985. On the E¢ ciency of Bertrand and Cournot Equilibria
with Product Di¤erentiation. Journal of Economic Theory 34, 166-175.
Symeonidis, G., 2008. Downstream Competition, Bargaining and Welfare. Journal of
Economics and Management Strategy 17, 247-270.
Symeonidis, G., 2010. Downstream Merger and Welfare in a Bilateral Oligopoly. International Journal of Industrial Organization 28, 230-243.
Von Ungern-Sternberg, T.,1996. Countervailing Power Revisited. International Journal
of Industrial Organization 14, 507-520.
Ziss, S., 1995. Vertical Separation and Horizontal Mergers. Journal of Industrial Economics 43, 63-75.
38