Multiproduct Firm Competition under Horizontal and Vertical Di … · 2005. 3. 16. · Rafael Moner-Colonques and Jos e J. Sempere-Monerrisy March 2005 Abstract This paper examines

Multiproduct Firm Competition under Horizontal

and Vertical Di�erentiation�

Rafael Moner-Colonques and Jos�e J. Sempere-Monerrisy

March 2005

Abstract

This paper examines the incentives of duopolistic �rms to become mul-

tiproduct in a setting where products are both horizontally and vertically

di�erentiated. A two-stage game is developed in which �rst �rms decide on

the number of variants to o�er and then compete in prices. Quality is exoge-

nous and consumers are heterogenous in income and location. It is shown that

both �rms do not �nd it optimal to o�er two variants when they are to com-

pete directly with those of the rival - the interleaved con�guration. However,

if �xed costs per variant are negligible, this is the only equilibrium when any

variant competes directly with just one of the rival's - the neighbour con�gura-

tion. Most interestingly, asymmetric patterns where only the low-quality �rm

supplies two variants may arise. Broadly speaking, and other things equal,

this is more likely to occur for large enough transport costs. In our model, it

is the size of transport costs relative to the degree of vertical di�erentiation

which makes the competitive e�ect dominate or not the associated costs per

variant.

JEL Classi�cation: D43, L13

Keywords: Multiproduction, Vertical Di�erentiation, Circular City.

It is not competing to the Young Economist Award

�Financial support from Spanish Ministerio de Educaci�on y Ciencia under the project SEJ2004-

07554/ECON and support fron the Generalitat Valenciana under the project GRUPOS04/13.yDepartment of Economic Analysis, University of Valencia. Campus dels Tarongers. Avda.

dels Tarongers s/n. E-46022-Valencia. [email protected]; [email protected]

1

1 Introduction

This paper examines the incentives of duopolistic �rms to become multiproduct in a

setting where products are both horizontally and vertically di�erentiated. Products

have very many attributes, some of which can be classi�ed as horizontal while other

as vertical. Consumers perceive that a certain brand car is of a better quality than

another, and at the same time, some consumers may prefer cars with hatchbacks

and others prefer cars with boots. Thus, suppose there are two chain stores each

o�ering a product of a particular quality and that these �rms are given the chance

to open one or two outlets. Alternatively, let us think of two airline companies that

have to choose whether to o�er one or two ights and where each airline's trip is of

a di�erent quality, say for example because of its duration. In addition, �rms may

have their particular location choices constrained by, for example, physical or ad-

ministrative reasons. Clearly, infrastructure imposes a maximum number of ights

per hour and two planes cannot take o� at the same time from the same runway.

These features draw our attention to the analysis of �rms' incentives to become

multiproduct rather than the modelling of the location decision. However, in an

e�ort to provide some robustness to the analysis, two alternative con�gurations will

be considered when both �rms are multiproduct: one in which any variant competes

directly with those of the rival and one in which any variant competes directly with

just one of the rival's.

Multiproduction can be explained by a number of reasons. Firms may want to

reach new markets, to generate demand, to exploit economies of scope or to deter

entry. This paper does not incorporate any of these motives to better isolate the

strategic rationale behind �rms' decision to become multiproduct. A two-stage game

is developed in which �rst �rms decide on the number of variants (outlets or ights,

if you wish) to o�er and then compete in prices. The horizontal characteristics space

is assumed to be a circumference, variants are located equidistantly from each other

and there is a �xed cost per variant. Quality is exogenous and each �rm's variants

are of the same quality but di�erent from the rival's. It is our purpose to establish

conditions under which multiproduct competition arises endogenously as an equilib-

rium outcome of �rms' maximizing behaviour and where products are di�erentiated

in two dimensions.

1

When both dimensions are present it is important to determine which one is

relatively most important. Ireland (1987) and Neven and Thisse (1990) have in-

troduced the concepts of horizontal and vertical dominance. Their analyses are

con�ned to single-product �rms. We will initially consider the case of horizontal

dominance for all possible market structures. This implies a high enough per unit of

distance transportation cost as compared with an expression that is a function of the

degree of vertical di�erentiation, and such an expression is greater when products

are highly vertically di�erentiated. The model that we examine also accounts for

heterogeneous consumers in income. Our results include the following. Both �rms

do not �nd it optimal to o�er two variants when they are to compete directly with

those of the rival - the interleaved con�guration. However, if �xed costs per variant

are negligible, this is the only equilibrium when any variant competes directly with

just one of the rival's - the neighbour con�guration. Typically, when �xed costs

per variant are rather large both �rms remain single product. Furthermore, such

an outcome will show up not only for large enough values of �xed costs but also

for high levels of vertical di�erentiation and transport costs su�ciently large. Most

interestingly, asymmetric equilibria may arise. Broadly speaking, and other things

equal, an equilibrium where only the low-quality �rm supplies two variants is more

likely to occur for large enough transport costs. However, for very low values of

transport costs and low levels of vertical di�erentiation, the only asymmetric equi-

librium is the one where the high-quality �rm o�ers two variants. In our model,

which is the equilibrium pattern is not explained by the growth in aggregate de-

mand e�ect underlying the representative consumer approach when a new variant

is introduced, since we assume that all consumers purchase for any possible market

structure. Rather it is the size of transport costs relative to the degree of vertical

di�erentiation which makes the competitive e�ect dominate or not the associated

costs per variant. Before going through the literature in the �eld, we would like

to point out, in the single-product case, that whenever vertical di�erentiation is

removed (same quality for both �rms) both our approach and Neven and Thisse's

(1990) are similar. Yet they are not whenever horizontal di�erentiation is taken out

(both �rms same location); in Neven and Thisse (1990), the consumer indi�erent in

income does not depend on horizontal characteristics of consumers whereas in ours

the farther away the consumer the richer the indi�erent consumer must be.

2

Review of the literature.

There exist a number of contributions modelling competition between multiprod-

uct �rms. Some references, using a representative consumer approach, are those by

Brander and Eaton (1984), Shaked and Sutton (1990) and Dobson and Waterson

(1996). Other papers consider address models to look at product line rivalry. Thus,

Mart��nez-Giralt and Neven (1988) consider the circle and the line models of hori-

zontal product di�erentiation. These authors assume that there are two �rms each

running two outlets and show that both �rms will choose the same location for their

outlets and therefore do not become multiproduct. Klemperer (1992), in a circu-

lar model, studies whether duopolists prefer interlaced product lines rather than

head-to-head competition. Important features of his model are that every consumer

desires to purchase the entire range of existing products, that there are shopping

costs of using additional suppliers and that the number of products per �rm is ex-

ogenous. In contrast with the standard intuition, it is shown that there are cases

for which competition is less intense under head-to-head product lines. Another

interesting contribution is that of De Fraja (1993) where �rms choose locations on

a segment line, how many products to o�er and delivered prices. Interestingly,

he demonstrates that �rms choose close substitute goods, which makes competition

strong, and force themselves to locate far apart (or viceversa). Finally, the modelling

of location-then-price in a multi-characteristics space �a la Hotelling is undertaken

by Irmen and Thisse (1998). They show that �rms only maximize di�erentiation in

the dominant characteristic while minimizing di�erentiation in the others.

As for vertical di�erentiation, the papers by Champsaur and Rochet (1989) and

De Fraja (1996) merit to be cited. The former paper studies a two-stage game

where duopolists �rst choose quality and then compete in prices. At equilibrium,

both �rms become multiproduct for they choose a range of (di�erent) qualities. The

latter reference considers quantity competition instead and concludes that �rms be-

come multiproduct yet they will select to exactly match each other's product line.

The combination of both horizontal and vertical di�erentiation in a single model

is not an easy task. Two relevant contributions in a setting with single-product �rms

are those of Neven and Thisse (1990) and Dos Santos Ferreira and Thisse (1996).

3

Neven and Thisse (1990) examine a two-stage game where single-product �rms �rst

select location and quality of their products and then compete in prices. They dis-

tinguish whether the quality di�erence dominates the di�erence in variety. If this

is the case then the situation is one of vertical dominance; if the opposite occurs

then it is one of horizontal dominance. Their basic �nding is that duopolists choose

maximal di�erentiation in one dimension and minimal di�erentiation in the other

dimension and which one is the equilibrium is determined by the type of dominance.

This statement about a max-min combination of characteristics is quali�ed by Dos

Santos Ferreira and Thisse (1996) in a model where �rms can choose transportation

technologies prior to competition in prices. The interested reader may also refer to

Anderson et al. (1992).

There are some papers dealing with multiproduct competition in which products

are horizontally and vertically di�erentiated. It is worth mentioning the work by

Gilbert and Matutes (1993) and Canoy and Peitz (1997).1 The treatment of product

line along with quality decisions in a single model can turn out complicated as far as

existence of equilibrium is concerned. Thus, these two papers take qualities as given

and duopolists select whether to o�er a low-quality variant, a high-quality variant

or both and then compete in prices. Both these papers propose a sequential game

and show that, under certain conditions, �rms �nd it optimal to specialize on only

one variant, a niche strategy, and do not become multiproduct. However, product

proliferation can be an equilibrium strategy before the threat of entry.2 Finally,

Ireland (1991) might fall in this category of papers by considering that the choice

of the frequency bus services is indeed a choice of product line.

The analysis that follows can be seen as a further step in modelling horizontal

and vertical di�erentiation by letting �rms select how many products to o�er. In

contrast with Gilbert and Matutes (1993) and Canoy and Peitz (1997) the number

of variants (or outlets) is chosen simultaneously by �rms, there is no entry threat

1The reader might like to see Ma~nez and Waterson (2001) for a good survey on multiproduction

and product di�erentiation.2The discrete choice theory is a benchmark to treat in a uni�ed way both the representative

consumer and the address approaches to product di�erentiation. Please refer to Anderson et al.

(1992) and to Ma~nez and Waterson (2001) for some references, some of them empirical ones, that

combine vertical and horizontal dimensions.

4

and all the variants of the same �rm have the same quality. Our setting is more

in the spirit of Neven and Thisse (1990) and will model consumer preferences as in

Ireland (1991). The next section sets out the model. A distinction is made when

both �rms choose two variants, the interleaved competition game and the neighbour

competition game. Section 3 studies �rms' incentives to become multiproduct and

provides the complete characterization of the �rst-stage equilibrium choice in the

number of variants. Some concluding remarks close the paper.

2 The Model

Assume two �rms, the low quality and the high quality �rm. Each �rm can o�er

one or two variants (open one or two outlets). Each �rm's products have the same

quality but di�erent from the rival's. A product is de�ned as a pair (xi; h); denoting

its position in the variety and quality spaces. Varietal characteristic xi lies on a

circumference of length 1 denoted by C; the quality level is normalized such that for

the high quality �rm good h is set equal to one; while for the low quality �rm good

h belongs to the interval (0; 1) and is exogenous. Then, the parameter h captures

a quality di�erential between both goods. The space of product characteristics is

thus a cylinder C � [0; 1]: Consumers do not rank the product varieties in the sameway so that the �rst characteristic corresponds to some horizontal di�erentiation.

The second characteristic portrays vertical di�erentiation since all consumers prefer

a high quality to a low quality product.

Each consumer k is de�ned by two parameters, zk and yk: zk is interpreted as his

most preferred variety and lies on C; the circumference of length 1. yk denotes the

income of consumer k, and it lies on the interval [y1; y1+1]: The space of consumers'

characteristics (z; y) is the cylinder C� [y1; y1+1]: Then for each particular zk thereis a set of consumers with income uniformly distributed on [y1; y1 + 1]; and this

happens for all zk 2 z: Assume that y1 is large enough for all consumers to �nda product for which their utility is positive in equilibrium, i.e. all consumers buy.

Consumers are uniformly distributed over C� [y1; y1+1] with a total mass equal toone. A consumer of type (zk; yk) derives the following indirect utility from buying

the high quality good

UH(zk; yk) = yk � piH � v jzk � xij (1)

where piH is the price of variant i charged by the high quality �rm, jzk � xij is the

5

minimum distance of the location of the ith good from the consumer's ideal good,

and v is the positive associated cost to such inconvenience per unit of distance. If

buying the low quality good then a consumer derives indirect utility

UL(z; y) = h(yk � piL � v jzk � xij) (2)

where piL denotes the price of variant i charged by the low quality �rm.3 We wish

to solve the following two-stage game. In the �rst stage, both companies decide

simultaneously and independently whether to o�er one or two products, that is

nj = f1; 2g for j = H;L where nj is, precisely, the number of products in the marketfor �rm j. For each �rm, the products only di�er in their location in the variety

space, then the low quality �rm products are de�ned by pairs like (xi; h) while the

high quality products are de�ned by pairs like (xi; 1): The provision of one product

entails a �xed cost " which is common for both �rms. In order to focus on �rms'

decision to become multiproduct, we will consider that for each number of products

in the market their possible locations are given in the following way. Products are

located equidistantly from one another. Therefore, when there are two products, one

for each �rm, they are located at the opposite ends of the circumference diameter,

e.g. both products are fully characterized by the following pairs (0; h) and (12; 1).

If there are three, there is one �rm that competes directly with the two products

of the other �rm and the three are equidistant from one another. Finally, if there

are four products, two for each company, we will consider two alternative situations

for equidistant products, either each product is directly competing against the two

products of the rival, the interleaved location, denoted by I; or competition from the

rival �rm is only on one side, the neighbour location, denoted by N: In this way we

will consider two alternative games, the interleaved competition and the neighbour

competition games that di�er only in the exogenous location of the products for

the case where both �rms decide to become multiproduct. In the second stage, and

knowing the �rst-stage choices, companies simultaneously and independently choose

prices. Price discrimination among outlets is not allowed, and therefore the high

quality �rm selects only one price pH regardless of the number of its variants in the

market, and similarly for the low quality �rm which selects pL.

3Ireland (1991) employs this consumer preferences to combine the vertical and the horizontal

dimensions.

6

2.1 The Interleaved Competition Game

Since at the �rst stage �rms only decide on the number of products, i.e. nH � nL;their choice gives rise to four di�erent subgames.

� Suppose that both companies are single-product, i.e. nH = 1 and nL = 1. Thissubgame is denoted by the f1; 1g� subgame; where the �rst component is the highquality �rm's �rst stage choice and the second one that of the low quality �rm. Also

consider, without loss of generality, that products are characterized by the pairs

(0; h) and (12; 1) and then the distance zk is just the distance from consumer k's

ideal to the low quality good located at 0: Given (1) and (2) above we can obtain

the set of consumers indi�erent between purchasing the high quality and the low

quality products:

�y(zk) =2(pH � hpL) + v

2(1� h) � v(1 + h)(1� h) zk (3)

This is a linear and decreasing function of zk, where zk 2 [0; 12] and partitions

total demand C� [y1; y1 + 1] in two groups of consumers. This function yields the

consumer with the highest income that patronizes the low quality �rm as a function

of the distance to this �rm in varietal space terms. Then for a given location zk the

consumers with income lower than �y(zk) will buy the low quality good while those

with greater income will buy the high quality good. Note that we have constructed

the demand in such a way that it is normalized to the area of a square of one unit

length. However, the number of �rms and their location allow us to consider one

of the two symmetric halves of the square. Note that there are several possibilities

depending on the position and slope of �y(zk):4 We will focus on the case where �y(z)

crosses the horizontal sides of the half square in Figure 1.

[insert �gure 1 about here]

This means that the horizontal di�erence, the di�erence in variety, dominates the

vertical di�erence, that of quality, and is referred to as horizontal dominance. For-

mally,��@�y(z)@z

�� is greater than the slope of the half square diagonal, that is, v(1+h)(1�h) > 2:

In order to obtain total demand for the low quality �rm we have to add two ele-

ments. First, all the consumers that are closer than z+ are patronized by the low

4See Neven and Thisse (1990). In fact, and for the ease of the exposition, we look at the linear

demand piece and ensure concavity of pro�ts.

7

quality �rm and second, those consumers that are at a distance not further than z�

and not closer than z+ with an income lower than �y(z) and greater than y1: Note

that z+ is de�ned by the distance that satis�es �y(z+) = y1 + 1; equivalently, z� is

the distance that satis�es �y(z�) = y1: This results in a linear demand as follows,

DL(1; 1) = 2[(y1 + 1� y1)z+ +Z z�

z+�y(z)dz] =

2pH � 2hpL + v � (1� h)v(1 + h)

(4)

Similarly, demand for the high quality company is given by,

DH(1; 1) = 2[((y1 + 1� y1)(z� � z+)�Z z�

z+�y(z)dz) + (y1 + 1� y1)(

1

2� z�)] =

=2hpL � 2pH + vh+ (1� h)

v(1 + h)(5)

These demands are valid as long as z+ � 0 and z� � 12, otherwise, the function

�y(z) would not cross the horizontal sides of the half square.5 Each �rm maximizes

pro�ts, �L = pLDL � " and �H = pHDH � "; resulting in the following equilibriumprices and pro�ts:

p�L(1; 1) =v(2+h)�(1�h)

6h; p�H(1; 1) =

v(1+2h)+(1�h)6

��L(1; 1) =[v(2+h)�(1�h)]2

18vh(1+h)� "; ��H(1; 1) =

[v(1+2h)+(1�h)]218v(1+h)

� "(6)

We need to ensure that the whole market is served. The fact that the poorest

and farthest away consumer purchases (which imposes an upper bound), along with

the condition that z� is smaller than 1=2, results in the following interval for the

transport cost parameter,

2(1� h)(2 + 3y1)(2 + h)

< v <(1� h)2 + 12hy1(2 + h)(1 + 2h)

(7)

In fact, the lower bound on v is more demanding that the above condition for hori-

zontal dominance.

� Suppose that the high quality �rm is single-product while the low quality one

becomes multi-product, i.e. nH = 1 and nL = 2. This subgame is denoted by

5This is equivalent to saying that pL belongs to the interval [p0L; p

00L]; where p

0L is the price for

which �y(z = 0) = y1 and p00L is the price for which �y(z = 1=2) = y1 + 1; and that pH belongs to

the interval [p0H ; p00H ]; where p

0H is the price for which �y(z = 1=2) = y1 + 1 and p

00H is the price for

which �y(z = 0) = y1:

8

the f1; 2g � subgame. The three products are characterized by the pairs (0; h);(13; 1) and (2

3; h): Then, the one unit square is partitioned in three equal area parts.

The two parts around product (13; 1) in which the high quality and the low quality

�rms compete for consumers, and a third part where consumers are equally shared

between the two low quality variants. Then it is important to note that the distance

from the farthest consumer to the high quality �rm is 12; while for the low quality

�rm the farthest consumer is at most at distance 13: (See Figure 2).


In order to construct the demand for the low quality �rm we �rst obtain the set

of consumers indi�erent between purchasing the high quality and the low quality

products in the �rst competing part, namely:

y(zk) =3(pH � hpL) + v

3(1� h) � v(1 + h)(1� h) zk (8)

This is a linear and decreasing function of zk, where zk 2 [0; 13 ]: From y(z) above,

horizontal dominance occurs if v(1+h)(1�h) > 3: The second competing part is symmetric

to the �rst one and �nally all consumers in the third part are patronized by the low

quality �rm. Therefore, demand for the low quality �rm is obtained as follows

DL(1; 2) = 2[(y1 + 1� y1)z+ +Z z�

z+�y(z)dz] + (y1 + 1� y1)

1

3= (9)

=6pH � 6hpL + v(3 + h)� 3(1� h)

3v(1 + h)

where now z+ denotes the distance that satis�es y(z+) = y1+1 and z� is the distance

that satis�es y(z�) = y1: Similarly, demand for the high quality �rm is given by

DH(1; 2) = 2[(y1 + 1� y1)(z+ � z�)�Z z�

z+y(z)dz + (y1 + 1� y1)(

1

3� z�)] =

=6hpL � 6pH + 2vh+ 3(1� h)

3v(1 + h)(10)

These demands are valid as long as z+ � 0 and z� � 13: Pro�t maximization by

each �rm results in the following equilibrium prices and pro�ts:

p�L(1; 2) =2v(3+2h)�3(1�h)

18h; p�H(1; 2) =

v(3+5h)+3(1�h)18

��L(1; 2) =[2v(3+2h)�3(1�h)]2

162vh(1+h)� 2"; ��H(1; 2) =

[v(3+5h)+3(1�h)]2162v(1+h)

� "(11)

9

As before, we must guarantee that the poorest and farthest away consumer pur-

chases. It must also be the case that z� be smaller than 1=3: Furthermore, the

"less favoured" consumer might be located in the non-competing part between low

quality variants. Altogether this results in the following interval for the transport

cost parameter,

6(1� h)(2 + 3y1)3 + h

< v < minf3 ((1� h)2 + 12hy1)

(2 + h)(3 + 5h);3(1� h+ 6hy1)

6 + 7hg (12)

where again the lower bound on v is more demanding that the above condition for

horizontal dominance.

� Suppose now that the high quality �rm is multiproduct while the low quality

is single-product, i.e. nH = 2 and nL = 1. This subgame is denoted by the f2; 1g �subgame. The three products are characterized by the pairs (0; h); (1

3; 1) and (2

3; 1):

This con�guration is parallel to the previous one, where the non-competing part is

patronized by the high quality �rm. The equilibrium prices and pro�ts are given by

p�L(2; 1) =v(5+3h)�3(1�h)

18h; p�H(2; 1) =

2v(2+3h)+3(1�h)18

��L(2; 1) =[v(5+3h)�3(1�h)]2

162vh(1+h)� "; ��H(2; 1) =

[2v(2+3h)+3(1�h)]2162v(1+h)

� 2"(13)

The corresponding interval for the transport cost parameter is now given by

maxf6(1�h)(1�3y1)1+3h

; 6(1�h)(2+3y1)5+3h

g < v < minf3((1�h)2+12hy1)(1+2h)(5+3h)

; 3(6y1�(1�h))7+6h

g (14)

� Suppose now that both �rms are multiproduct, i.e. nH = 2 and nL = 2, whichis denoted by the f2; 2g�subgame. The four products are characterized by the pairs(0; h); (1

4; 1); (1

2; h); and (3

4; 1): Note that the number of variants and their position

partitions the unit square in four identical competing parts - see Figure 3.a.

[insert �gure 3a about here]

The set of consumers indi�erent between purchasing the high quality and the

low quality products in the �rst competing part is given by:

y0(zk) =4(pH � hpL) + v

4(1� h) � v(1 + h)(1� h) zk (15)

10

a decreasing function of zk, where zk 2 [0; 14 ]: The condition for horizontal dominanceis v(1+h)

(1�h) > 4:

The equilibrium prices and pro�ts are the following,

p�L(2; 2) =v(2+h)�2(1�h)

12h; p�H(2; 2) =

v(1+2h)+2(1�h)12

��L(2; 2) =[v(2+h)�2(1�h)]2

36vh(1+h)� 2" ; ��H(2; 2) =

[v(1+2h)+2(1�h)]236v(1+h)

� 2"(16)

The transport cost parameter must verify that

4(1� h)(2 + 3y1)(2 + h)

< v <2[(1� h)2 + 12hy1](1 + 2h)(2 + h)

(17)

2.2 The Neighbour Competition Game

As noted above, we are going to consider an alternative location for the case where

both �rms are multiproduct. Thus the neighbour competition game only di�ers

from the interleaved competition game in the f2; 2g� subgame; where now the fourproducts are characterized by the pairs (0; h); (1

4; 1); (1

2; 1); and (3

4; h): In order to

distinguish both subgames we will denote the one corresponding to the neighbour

competition case as f2; 2; ng � subgame: As before the unit square is partitionedin four parts but now there are two competing parts and two parts in which all

consumers are patronized by one and a di�erent �rm - see Figure 3.b.


The set of consumers indi�erent between purchasing the high quality and the low

quality products in the �rst competing part is, as before, y0(zk): The equilibrium

prices and pro�ts read,

p�L(2; 2; n) =v(7+5h)�4(1�h)

24h; p�H(2; 2; n) =

v(5+7h)+4(1�h)24

��L(2; 2; n) =[v(7+5h)�4(1�h)]2

288vh(1+h)� 2"; ��H(2; 2; n) =

[v(5+7h)+4(1�h)]2288v(1+h)

� 2"(18)

The transport cost parameter must verify that

4(1� h)(2 + 3y1)(2 + h)

< v <4[(1� h+ 6hy1]

(7 + 8h)(19)

11

3 Characterization of the Equilibrium

In this section we are going to present the �rs-stage Nash equilibrium where �rms

choose the number of variants to o�er. We begin by establishing the restrictions to

be veri�ed by the parameters in the model. Then, we will analyze the �rms' incen-

tives to become multiproduct for each of the two games. Finally, the equilibrium

analysis is provided.

Restrictions on the Parameters

Before computing the Nash equilibrium in the choice of the number of variants we

must bear in mind some restrictions. Firstly, we want to ensure that the restriction

for horizontal dominance is satis�ed in the four subgames; it must be the case

that v(1+h)(1�h) > 4 or v > 4(1�h)

(1+h). Also, we have just seen that for each subgame the

transport cost parameter must belong to a certain interval: Then, the comparison

and computations are valid for the following interval, which captures the intersection

of the �ve intervals and the condition for horizontal dominance shown above,

6(1� h)(2 + 3y1)3 + h

< v < minf(1� h)2 + 12hy1

(2 + h)(1 + 2h);3(1� h+ 6hy1)

6 + 7hg (20)

Secondly, we must check that equilibrium prices are positive; this is indeed the

case. Thirdly, there is a bound on the size of " to ensure non-negative pro�ts. The

bounds on " in the interleaved competition game is theminf [v(2+h)�2(1�h)]272vh(1+h)

; [v(1+2h)+2(1�h)]2

72v(1+h)g �

"i; where the second term is the smallest for v su�ciently large.6 The bounds on "

the neighbour competition game is the minf [v(7+5h)�4(1�h)]2576vh(1+h)

; [v(5+7h)+4(1�h)]2

576v(1+h)g � "n,

where the second term is the smallest for v su�ciently large.7

6It can be easily checked that [v(2+h)�2(1�h)]2

72vh(1+h) < [v(1+2h)+2(1�h)]272v(1+h) for

v 2 [ 4+2(1+h)(2h�3ph)

4+7h+4h2 ; 4+2(1+h)(2h+3ph)

4+7h+4h2 ]: The opposite holds for values of v outside this interval.

Note that the lower bound of the interval is always smaller than the condition for horizontal

dominance. Besides, for v > 85 then

[v(2+h)�2(1�h)]272vh(1+h) > [v(1+2h)+2(1�h)]2

72v(1+h) regardless of the value of

h:7Proceeding as before, [v(7+5h)�4(1�h)]

2

576vh(1+h) < [v(5+7h)+4(1�h)]2576v(1+h) for

v 2 [ 4(7+10h+7h2�12(1+h)

ph)

49+94h+49h2 ; 4(7+10h+7h2+12(1+h)

ph))

49+94h+49h2 ]: The opposite holds for values of v outside

this interval. Note that the lower bound of the interval is always smaller than the condition for

horizontal dominance. Besides, for v > 1 then [v(7+5h)�4(1�h)]2576vh(1+h) > [v(5+7h)+4(1�h)]2

576v(1+h) regardless of

the value of h:

12

Firms' Incentives to Become Multiproduct

We are now ready to provide the �rms' unilateral incentives to become multi-

product. The high quality �rm will o�ers two variants rather than one, provided

that there is one low quality variant if ��H(2; 1)� ��H(1; 1) > 0; that is, ifv(7 + 12h) + 6(1� h)

162(1 + h)> " (21)

Denote the l.h.s. by rH1; although it is a function on both v and h we will not

make this dependence explicit to save on notation. It is straightforward that this

function is positive and de�nes a threshold value for " such that for 0 < " < rH1 the

unilateral incentive is positive while for rH1 < " it is negative.

Similarly, the low quality �rm will o�er two variants than one, provided that

there is one high quality variant if ��L(1; 2)� ��L(1; 1) > 0; that is, ifv(12 + 7h)� 6(1� h)

162(1 + h)> " (22)

Denote the l.h.s. by rL1: It can be checked that this function is positive as long

as the horizontal dominance condition is veri�ed.

These two thresholds, rH1 and rL1, are common for the two games under con-

sideration. However, the decision to become multiproduct once the rival is actually

multiproduct has to be distinguished for each game. Consider �rst the interleaved

competition game.

The interleaved competition game

The high quality �rm will o�er two variants rather than one, provided that there

are two low quality variants if ��H(2; 2)� ��H(1; 2) > 0; that is, if18(1� h)2 + 12vh(1� h)� v2(9 + 24h+ 14h2)

324v(1 + h)> " (23)

Denote the l.h.s. by riH2: The numerator is a concave parabola in v and hence will

be positive for values between the roots. It can be checked that this is not possible

by just comparing with the condition for horizontal dominance (which is indeed a

milder condition than the lower bound on v in eq. 20): Therefore, riH2 is negative

and the high-quality �rm never introduces a second variant provided the low-quality

�rm o�ers two.

Finally, the low quality �rm will o�er two variants rather than one, provided

that there are two high quality variants if ��L(2; 2)� ��L(2; 1) > 0; that is, if18(1� h)2 � 12v(1� h)� v2(14 + 24h+ 9h2)

324vh(1 + h)> " (24)

13

Denote the l.h.s. by riL2: A similar reasoning can be applied to conclude that the

low-quality �rm never introduces a second variant provided the high-quality �rm

o�ers two variants.

The neighbour competition game

The high quality �rm will supply two variants rather than one, provided that

there are two low quality variants if ��H(2; 2; n)� ��H(1; 2) > 0; that is, if

(3 + h)(24(1� h) + v(27 + 41h))2592(1 + h)

> " (25)

Denote the l.h.s. by rnH2. Note that rnH2 is positive and that the high-quality �rm

will introduce a second variant provided the low-quality �rm o�ers two variants for

small enough values of ".

Finally, the low quality �rm will supply two variants rather than one, provided

that there are two high quality variants if ��L(2; 2; n)� ��L(2; 1) > 0; that is, if

(1 + 3h)(�24(1� h) + v(41 + 27h))2592h(1 + h)

> " (26)

Denote the l.h.s. by rnL2: Note that the condition on v that ensures that rnL2 is

positive is milder than that for horizontal dominance. Then the low-quality �rm

will introduce a second variant provided the high-quality �rm o�ers two variants for

small enough values of ".

Equilibrium Analysis

The interleaved competition game

The �rst stage equilibrium can be fully characterized by establishing the ranking

of the four thresholds on " and "i: We know from above that both riH2 and riL2 are

negative and it also follows that rH1 is greater than rL1 for v 2 (0; 125 ); the oppositeotherwise. This implies that depending on whether v 2 (0; 12

5); it is possible to �nd

an " such that 0 < rL1 < " < rH1 concluding that the unique Nash equilibrium can

be (2; 1): For v > 125; this will never happen. However, the following general result

will hold since riH2 and riL2 are negative.

Proposition 1 The choice of two variants by both �rms is never an equilibrium.

In order to complete the characterization of the equilibrium there remains to �nd

if there is a region for the parameter space v � h such that the bound "i is smallerthan the thresholds rH1 and rL1: Figure 4 identi�es all the possible situations.

14


In regions I, VIII and IX it follows that "i is smaller than both thresholds. In par-

ticular, for region I,we have that 0 < "i < rL1 < rH1 with "i = [v(2+h)�2(1�h)]2

72vh(1+h); while

for region VIII, 0 < "i < rL1 < rH1 with "i = [v(1+2h)+2(1�h)]2

72v(1+h); and, �nally, for region

IX, 0 < "i < rH1 < rL1 with "i = [v(1+2h)+2(1�h)]2

72v(1+h): Similarly, in regions II and VII,

"i belongs to the interval (rL1; rH1). For regions III and IV, "i is greater than both

thresholds with rL1 < rH1; while for region V "i is greater than both thresholds with

rH1 < rL1: Finally, for region VI, "i belongs to the interval (rH1; rL1). Regions with

the same colour have the same pattern of equilibria as a function of ": We provide

in the Appendix the corresponding equilibria for each case. However in drawing

Figure 4 we have ignored that v must satisfy a set of restrictions that depend on

both h and y1: Speci�cally, the lower bound on v is stronger than the horizontal

dominance condition. Besides, the interval exists for h high enough, which means

that there is an upper bound on the maximum (possible) degree of vertical di�er-

entiation. Therefore, depending on the particular level of y1 the relevant region for

v � h changes. For expositional reasons we will present the complete characteriza-tion for y1 = 1: Then take y1 = 1; the transport cost parameter must belong to

the interval: 30(1�h)3+h

< v < minf 1+10h+h2

(2+h)(1+2h); 3(1+5h)6+7h

g and such an interval exists forh 2 (0:8327; 1) - see Figure 5.8


Proposition 2 Consider 30(1�h)3+h

< v < minf 1+10h+h2

(2+h)(1+2h); 3(1+5h)6+7h

g and h 2 (0:8327; 1):It follows that:

a) For 30(1�h)3+h

< v < �i"L�L1 then 0 < "i < rL1 < rH1 which implies that for all

" 2 (0; "i) there are two Nash Equilibria f(2; 1); (1; 2)g.b) For maxf30(1�h)

3+h;�i

"L�L1g < v < �i"L�H1 then 0 < rL1 < "

i < rH1.

b.1) If 0 < " < rL1 < "i < rH1; then there are two Nash Equilibria f(2; 1); (1; 2)g;

b.2) if 0 < rL1 < " < "i < rH1; then the Nash Equilibrium is (1; 2);

c) For maxf30(1�h)3+h

;�i"L�H1g < v < minf 1+10h+h2

(2+h)(1+2h); 3(1+5h)6+7h

g then 0 < rL1 < rH1 <"i.

8For greater values of y1 the set for h widens and the same happens for the subset in v�h thatappears at equilibrium.

15

c.1) If 0 < " < rL1 < rH1 < "i; then there are two Nash Equilibria f(2; 1); (1; 2)g;

c.2) if 0 < rL1 < " < rH1 < "i; then the Nash Equilibrium is (1; 2);

c.3) if 0 < rL1 < rH1 < " < "i; then the Nash Equilibrium is (1; 1):

where the precise expression of �i"L�L1 is given in the Appendix and displayed in

Figures 4 and 5: Note that part a) in the above proposition corresponds to a subset

of region I in Figure 4, part b) to region II and part c) to region III. The above

proposition identi�es that small enough values of " yield multiple equilibria in which

only one �rm supplies two variants, whereas both �rms establish only one variant

for high enough values of ":This occurs regardless of the values of (v; h): However,

for intermediate levels of " the �rm that opens two variants is either the high quality

�rm if v is small enough or the low quality �rm if v is large enough.

The neighbour competition game

We proceed in the same way as above. The �rst stage equilibrium can be fully

characterized by establishing the ranking of the four thresholds with " and "n. To

establish this ranking; we prove in the Appendix that the parameter space v � hcan be partitioned into ten di�erent regions9 as shown in Figure 6. Consider now

y1 = 1; the following result holds (See Figure 7).

[insert �gures 6 and 7 about here]

Proposition 3 Consider 30(1�h)3+h

< v < minf 1+10h+h2

(2+h)(1+2h); 3(1+5h)6+7h

g and h 2 (0:8327; 1):It follows that:

a) For 30(1�h)3+h

< v < �H2�L2 then 0 < rnL2 < r

nH2 < rL1 < rH1 < "

n

a.1) If 0 < " < rnL2 < rnH2 < rL1 < rH1 < "

n; then the Nash Equilibrium is (2; 2):

a.2) if 0 < rnL2 < " < rnH2 < rL1 < rH1 < "

n; then the Nash Equilibrium is (2; 1);

a.3) If 0 < rnL2 < rnH2 < " < rL1 < rH1 < "n; then there are two Nash Equilibria

f(2; 1); (1; 2)g;a.4) If 0 < rnL2 < r

nH2 < rL1 < " < rH1 < "


a.5) If 0 < rnL2 < rnH2 < rL1 < rH1 < " < "


b) For �H2�L2 < v < minf 1+10h+h2

(2+h)(1+2h); 3(1+5h)6+7h

g then 0 < rnH2 < rnL2 < rL1 < rH1

< "n.

b.1) If 0 < " < rnH2 < rnL2 < rL1 < rH1 < "


9A set of propositions, one for each of these regions is given in the Appendix .

16

b.2) if 0 < rnH2 < " < rnL2 < rL1 < rH1 < "


b.3) If 0 < rnH2 < rnL2 < " < rL1 < rH1 < "n; then there are two Nash Equilibria

f(2; 1); (1; 2)g;b.4) If 0 < rnH2 < r

nL2 < rL1 < " < rH1 < "


b.5) If 0 < rnH2 < rnL2 < rL1 < rH1 < " < "


Note that part a) in the above proposition corresponds to a subset of region II in

Figure 6 and similarly, part b) in region III. In contrast with the interleaved compe-

tition game: �rst, both �rms o�ering two variants may be an equilibrium; second,

for low enough v multiproduction of one �rm is not guaranteed now. Asymmetric

equilibria arise noting that (1; 2) shows up for v su�ciently large.

4 Concluding remarks

This paper has looked at �rms' incentives to become multiproduct in a setting both

with horizontal and vertical di�erentiation. We have shown how transport costs,

quality and �xed costs per variant interact to characterize the di�erent equilibria.

The foregoing analysis, in a situation of horizontal dominance, has considered ex-

ogenous quality, given location choices and consumers' heterogeneity both in income

and location. It has been shown that both �rms will not o�er two variants in the

interleaved con�guration. On the other hand, this is the only equilibrium in the

neighbour con�guration, provided that the �xed costs per variant are negligible.

Most interestingly, asymmetric equilibria may arise. Broadly speaking, and other

things equal, an equilibrium where only the low-quality �rm supplies two variants

is more likely to occur for large enough transport costs.

Before concluding we would like to mention some preliminary �ndings which

in fact suggest how to complete the picture. Suppose that both the high and the

low quality �rms were to choose one variant and that it were located at the same

point - a 'matching' strategy. We have checked that �rms obtain higher pro�ts

when each �rm's variant is located equidistantly on the circumference. We have

also worked out the two-stage game with interleaved competition under vertical

dominance. An interesting result is that the con�guration where only the low-quality

�rm becomes multiproduct is not an equilibrium. This points out the relevance of

the type of dominance. Directions of future research are to complete the analysis

of vertical dominance and to look in more detail at the issue of location choice in

17

order to establish whether multiproduct �rms �nd it optimal to di�erentiate in both

dimensions.

References

[1] Anderson, S.P., A. de Palma and J.F. Thisse (1992), Discrete Choice

Theory of Product Di�erentiation, The MIT Press.

[2] Brander, J. and J. Eaton (1984), "Product Line Rivalry", American Eco-

nomic Review, 74, 323-334.

[3] Canoy, M. and M. Peitz (1997), "The Di�erentiation Triangle", Journal of

Industrial Economics, 45, 305-324.

[4] Champsaur, P. and J. Rochet (1989), "Multiproduct Duopolists", Econo-

metrica, 57,533-557.

[5] De Fraja, G. (1993), "Relaxing Spatial Competition through Product Line

Choice (or viceversa)", Regional Science and Urban Economics, 23, 461-486.

[6] De Fraja, G. (1996), "Product Line Competition in Vertically Di�erentiated

Products", International Journal of Industrial Organization, 14, 389-414.

[7] Dobson, P.W. and M. Waterson (1996), "Product Range and Inter�rm

Competition", Journal of Economics and Management Strategy, 35, 317-341.

[8] Dos Santos Ferreira, R. and J.F. Thisse (1996), "Horizontal and Vertical

Di�erentiation: The Launhardt Model", International Journal of Industrial

Organization, 14, 485-506.

[9] Gilbert, R. and C. Matutes (1993), "Product Line Rivalry and Brand

Di�erentiation", Journal of Industrial Economics, 41, 223-240.

[10] Ireland, N. (1987), Product Di�erentiation and Non-Price Competition, Basil

Blackwell.

[11] Ireland, N. (1991), "A Product Di�erentiation Model of Bus Deregulation",

Journal of Transport Economics and Policy, 25, 153-162.

18

[12] Irmen, A. and J. F. Thisse (1998), "Competition in Multi-characteristics

Spaces: Hotelling Was Almost Right", Journal of Economic Theory, 78, 76-102.

[13] Klemperer, P. (1992), "Equilibrium Product Lines: Competing Head-to-

Head May Be Less Competitive", American Economic Review, 82, 740-755.

[14] Ma~nez, J. and M. Waterson (2001), "Multiproduct Firms and Product

Di�erentiation: A Survey", Warwick Economic Research Papers no 594.

[15] Mart��nez-Giralt, X. and D.J. Neven (1988), "Can Price Competition

Dominate Market Segmentation?", Journal of Industrial Economics, 36, 431-

442.

[16] Neven, D.J. and J.F. Thisse (1990), "On Quality and Variety Compe-

tition", in J.J. Gabszewicz, J.F. Richard and L.A. Wolsey (eds.), Economic

Decision Making: Games, Econometrics and Optimization. Contributions in

honour of Jacques H. Dr�eze, NorthHolland, 175-199.

[17] Shaked, A. and J. Sutton (1990), "Multiproduct Firms and Market Struc-

ture", Rand Journal of Economics, 21, 45-62.

A Proofs

A.1 Equilibrium Characterization for the Interleaved Game

Ranking of Thresholds and Bounds on ".

As indicated in the text, both riH2 and riL2 are negative and rH1 is greater than rL1

for v 2 (0; 125): Thus the ranking on thresholds is completely speci�ed. Also note that

"i � minf [v(2+h)�2(1�h)]272vh(1+h)

; [v(1+2h)+2(1�h)]2

72v(1+h)g where [v(2+h)�2(1�h)]2

72vh(1+h)< [v(1+2h)+2(1�h)]2

72v(1+h)for

v 2 [4+2(1+h)(2h�3ph)

4+7h+4h2; 4+2(1+h)(2h+3

ph)

4+7h+4h2]: It follows that 4+2(1+h)(2h�3

ph)

4+7h+4h2< 4(1�h)

(1+h); and

4+2(1+h)(2h+3ph)

4+7h+4h2< 12

5for all h 2 (0; 1): Denote expression 4+2(1+h)(2h+3

ph)

4+7h+4h2by �i(h):

It is an increasing function on h; where �i(0) = 1; �i(1) = 85and �i(h) crosses

4(1�h)(1+h)

at h = 0:4385: Then we can consider three cases: i) For 4(1�h)(1+h)

< v < �i(h)

and h 2 (0:4385; 1) it follows that rL1 < rH1 and "i = [v(2+h)�2(1�h)]2

72vh(1+h); ii) for

maxf4(1�h)(1+h)

; �i(h)g < v < 125; and h 2 (1

4; 1) then rL1 < rH1 and "

i = [v(1+2h)+2(1�h)]272v(1+h)

;

and �nally, iii) for maxf4(1�h)(1+h)

; 125g < v, then rH1 < rL1 and "i = [v(1+2h)+2(1�h)]2

72v(1+h):

19

� i) Consider 4(1�h)(1+h)

< v < �i(h) and h 2 (0:4385; 1):

i.a) The di�erence [v(2+h)�2(1�h)]2

72vh(1+h)� rH1 is positive if 36(1�h)

2�12(6+5h)(1�h)v+(36+8h�39h2)v2648hv(1+h)

>

0 or equivalently if v =2 [6(1�h)(6+5h�2p(13+16h)h)

36+8h�39h2 ;6(1�h)(6+5h+2

p(13+16h)h

36+8h�39h2 ]:

It happens that6(1�h)(6+5h�2

p(13+16h)h)

36+8h�39h2 < 4(1�h)(1+h)

and that6(1�h)(6+5h+2

p(13+16h)h

36+8h�39h2 <

�i(h) for all h 2 (0; 1): Then, we conclude that for 4(1�h)(1+h)

< v <6(1�h)(6+5h+2

p(13+16h)h

36+8h+39h2

we have that [v(2+h)�2(1�h)]272vh(1+h)

< rH1 while for6(1�h)(6+5h+2

p(13+16h)h

36+8h+39h2< v < �i(h);

then rL1 < rH1 <[v(2+h)�2(1�h)]2

72vh(1+h)and the ranking is completed. Denote expression

6(1�h)(6+5h+2p(13+16h)h

36+8h�39h2 by �i"L�H1(h) where �

i"L�H1(0) = 1, �i

"L�H1(1) = 0 and

crosses 4(1�h)(1+h)

at h = 0:458:

i.b) Now [v(2+h)�2(1�h)]272vh(1+h)

> rL1 if36(1�h)2�12(6+h)(1�h)v+(36�12h�19h2)v2

648hv(1+h)> 0 or equiv-

alently if v =2 [6(1�h)(6+h�2p(6+5h)h)

36�12h�19h2 ;6(1�h)(6+h+2

p(6+5h)h)

36�12h�19h2 ]:As before6(1�h)(6+h�2

p(6+5h)h)

36�12h�19h2 <

4(1�h)(1+h)

and6(1�h)(6+h+2

p(6+5h)h)

36�12h�19h2 < �i(h) for all h 2 (0; 1): Then for 4(1�h)(1+h)

< v <

6(1�h)(6+h+2p(6+5h)h)

36�12h�19h2 we have that [v(2+h)�2(1�h)]2

72vh(1+h)< rL1; the opposite for

6(1�h)(6+h+2p(6+5h)h)

36�12h�19h2 <

v. Denote expression6(1�h)(6+h+2

p(6+5h)h)

36�12h�19h2 by �i"L�L1(h) where �

i"L�L1(0) = 1,

�i"L�L1(1) = 0 and crosses 4(1�h)

(1+h)at h = 0:522: It is also important to note that

�i"L�L1(h) < �

i"L�H1(h) for all h:

Therefore, combining the discussion presented in i.a) and i.b) we are able to

construct regions I, II and III in Figure 4. Region I is de�ned by 4(1�h)(1+h)

< v <

�i"L�L1(h) and h 2 (0:522; 1) and it happens that "i =

[v(2+h)�2(1�h)]272vh(1+h)

< rL1 < rH1:

Region II is de�ned by maxf4(1�h)(1+h)

;�i"L�L1(h)g < v < �i

"L�H1(h) and h 2 (0:458; 1)and it happens that rL1 < "

i = [v(2+h)�2(1�h)]272vh(1+h)

< rH1: Finally, region III is de�ned

by maxf4(1�h)(1+h)

;�i"L�H1(h)g < v < �i(h) and h 2 (0:4385; 1) and it happens that

rL1 < rH1 < "i = [v(2+h)�2(1�h)]2

72vh(1+h):

� ii) Consider maxf4(1�h)(1+h)

; �i(h)g < v < 125; and h 2 (1

4; 1):

The di�erence with part i) is that "i = [v(1+2h)+2(1�h)]272v(1+h)

: Then,

ii.a) [v(1+2h)+2(1�h)]2

72v(1+h)> rH1 if

36(1�h)2+12(1+6h)(1�h)v�(19+12h�36h2)v2648v(1+h)

> 0.

It is straightforward to see that the above expression is positive regardless of the

value of v if h > 0:912 and for v < 6(1�h)2p5+6h�(1+6h) if h 2 (0; 0:912). Denote expression

6(1�h)2p5+6h�(1+6h) by �

i"H�H1(h): It is de�ned for h 2 (0; 0:912); �i

"H�H1(0) = 1:728 and

crosses 4(1�h)(1+h)

at h = 0:389:

20

ii.b) [v(1+2h)+2(1�h)]2

72v(1+h)> rL1 if

36(1�h)2+12(5+6h)(1�h)v�(39�8h�36h2)v2648v(1+h)

> 0.

It is straightforward to see that the above expression is positive regardless of the

value of v if h > 0:9356 and for v < 6(1�h)2p16+13h�(5+6h) if h 2 (0; 0:9356). Denote expres-

sion 6(1�h)2p16+13h�(5+6h) by �

i"H�L1(h); it is de�ned for h 2 (0; 0:9356); �i

"H�L1(0) = 2

and crosses 4(1�h)(1+h)

at h = 0:337: Finally, note that �i"H�L1(h) > �

i"H�H1(h) for h <

0:775 the opposite happens for h � 0:775 and that �i"H�L1(0:775) = �

i"H�H1(0:775)

= 125:

Combining ii.a) and ii.b) we are able to construct regions IV, VII and VIII in

Figure 4. Region IV is de�ned by maxf4(1�h)(1+h)

; �i(h)g < v < minf�i"H�H1(h);

125g

and h 2 (0:389; 1) and it happens that rL1 < rH1 < "i = [v(1+2h)+2(1�h)]272v(1+h)

: Region

VII is de�ned by maxf4(1�h)(1+h)

;�i"H�H1(h)g < v < �i

"H�L1(h) and h 2 (0:337; 0:775)and it happens that rL1 < "

i = [v(1+2h)+2(1�h)]272v(1+h)

< rH1: Finally, region VIII is de�ned

by maxf4(1�h)(1+h)

;�i"H�L1(h)g < v < 12

5and h 2 (1

4; 0:775) and it happens that "i =

[v(1+2h)+2(1�h)]272v(1+h)

< rL1 < rH1:

� iii) for maxf4(1�h)(1+h)

; 125g < v.

The di�erence with part ii) is that rH1 < rL1. Then regions V, VI and IX are

de�ned. Region V is de�ned by 125< v < �i

"H�L1(h) and h 2 [0:775; 0:9356] andby 12

5< v for h > 0:9356; and it happens that rH1 < rL1 < "i = [v(1+2h)+2(1�h)]2

72v(1+h):

Region VI is de�ned by �i"H�L1(h) < v < �

i"H�H1(h) for h 2 (0:775; 0:912) and by

�i"H�L1(h)< v for h 2 (0:912; 0:9356) and it follows that rH1 < "i =

[v(1+2h)+2(1�h)]272v(1+h)

<

rL1: Finally, region IX is de�ned by 4(1�h)(1+h)

< v for h 2 (0; 14); by 12

5< v for

h 2 (14; 0:775) and by �i

"H�H1(h) < v for h 2 (0:775; 0:912) and it follows that"i = [v(1+2h)+2(1�h)]2

72v(1+h)< rH1 < rL1:


The nine di�erent regions de�ned above determine �ve di�erent rankings of the

thresholds and "i: The following �ve propositions characterize the �rst stage Nash

Equilibrium for them.

Proposition 4 Consider regions I, VIII and IX as de�ned above. It follows that

0 < "i < rL1 < rH1: Then for all " 2 (0; "i) there are two Nash Equilibria

f(1; 2); (2; 1)g:

Proposition 5 Consider regions II and VII and IX as de�ned above. It follows

that 0 < rL1 < "i < rH1:

21

a) If 0 < " < rL1 < "i < rH1; there are two Nash Equilibria f(1; 2); (2; 1)g:

b) If 0 < rL1 < " < "i < rH1, the Nash Equilibrium is (2; 1):

Proposition 6 Consider regions III and IV as de�ned above. It follows that 0 <

rL1 < rH1 < "i:

a) If 0 < " < rL1 < rH1 < "i; there are two Nash Equilibria f(1; 2); (2; 1)g:

b) If 0 < rL1 < " < rH1 < "i, the Nash Equilibrium is (2; 1):

c) If 0 < rL1 < rH1 < " < "i, the Nash Equilibrium is (1; 1):

Proposition 7 Consider region VI as de�ned above. It follows that 0 < rH1 < "i <

rL1:

a) If 0 < " < rH1 < "i < rL1; there are two Nash Equilibria f(1; 2); (2; 1)g:

b) If 0 < rH1 < " < "i < rL1, the Nash Equilibrium is (1; 2):

Proposition 8 Consider region V as de�ned above. It follows that 0 < rH1 < rL1 <

"i:

a) If 0 < " < rH1 < rL1 < "i; there are two Nash Equilibria f(1; 2); (2; 1)g:

b) If 0 < rH1 < " < rL1 < "i, the Nash Equilibrium is (1; 2):

c) If 0 < rH1 < rL1 < " < "i, the Nash Equilibrium is (1; 1):

The proof is straightforward and follows a parallel reasoning in all of the �ve

propositions. Then we only formally prove the last one. Note that the ranking of

the thresholds and "i is speci�ed once the region on the parameter space v � h isdetermined. For region V, 0 < rH1 < rL1 < "i; also recall that both rH2 and rL2

are negative. Then, we can construct the �rst stage best response functions for all

possible " 2 (0; "i): Consider the high quality �rm's best response. Set nL = 1; thenthe high quality �rm will select nH = 2 i� " < rH1: Set nL = 2; then the high quality

�rm will select nH = 2 i� " < rH2: Similarly for the low quality �rm's best response.

Set nH = 1; then the low quality �rm will select nL = 2 i� " < rL1: Set nH = 2;

then the high quality �rm will select nL = 2 i� " < rL2: Therefore, we conclude that

(2; 2) is never a Nash Equilibrium since rH2 and rL2 are negative. If " < rH1 < rL1;

then both �rms will prefer a second variant rather than one if the rival is selecting

one variant. This is part a) of the proposition and the NE are both f(1; 2); (2; 1)g:If rH1 < " < rL1 then only the low quality �rm will select two variants if the high

22

quality �rm is selecting one; for the high quality �rm nH = 1 is a dominant strategy.

This is part b) and the NE is (1; 2): Finally, for rH1 < rL1 < "; it follows that nH = 1

and nL = 1 are dominant strategies for the high and low quality �rms, respectively.

This is part c) and the NE is (1; 1):

A.2 Equilibrium Characterization for the Neighbour Game

Ranking of Thresholds and Bounds on ":

We �rst will rank the four thresholds on "; rH1 =v(7+12h)+6(1�h)

162(1+h); rL1 =

v(12+7h)�6(1�h)162(1+h)

; rnH2 =(3+h)(24(1�h)+v(27+41h))

2592(1+h)and rnL2 =

(1+3h)(�24(1�h)+v(41+27h))2592h(1+h)

: Then, we will check how

the bound on "; "n � minf [v(7+5h)�4(1�h)]2576vh(1+h)

; [v(5+7h)+4(1�h)]2

576v(1+h)g compares with the thresh-

olds ranking. The ranking will be done for the parameter space v � h where

h 2 (0; 1) and v > 12(1�h)3+h

: We consider this lower bound for convenience since

it permits to avoid situations that will never arise at equilibrium because they will

not satisfy the lower bound restriction on v that ensures that all consumers buy

and that the indi�erent consumer function crosses the horizontal sides of the mar-

ket square. The lower bound for v is taken by setting y1 = 0 in6(1�h)(2+3y1)

3+h; with

4(1�h)(1+h)

< 12(1�h)3+h

< 6(1�h)(2+3y1)3+h

; then it is a milder bound for v than 6(1�h)(2+3y1)3+h

and

stronger than that ensuring horizontal dominance.

Ranking the four thresholds

a) First, as we have seen before rH1 is greater than rL1 for v 2 (0; 125 ):b) The di�erence rH1 � rnH2 =

24(1�h)2+(31+42h�41h2)v2592(1+h)

is always positive.

c) The di�erence rL1 � rnH2 =�24(1�h)(7+h)+(111�38h�41h2)v

2592(1+h)is positive for v >

24(1�h)(7+h)111�38h�41h2 . Denote expression

24(1�h)(7+h)111�38h�41h2 by �L1�H2(h); which is a decreasing

function of h; with �L1�H2(0) =5637and �L1�H2(1) = 0: Also, it crosses

12(1�h)3+h

at

h = 0:761:

Combining a), b) and c) we distinguish three parts: i) For h 2 (0:761; 1) andv 2 [ 12(1�h)

3+h;�L1�H2(h)] we have that rH1 > rnH2 > rL1; ii) For h 2 (1

3; 1) and

maxf12(1�h)3+h

;�L1�H2(h)g < v < 125we have that rH1 > rL1 > rnH2; and �nally iii)

For h 2 (0; 1) and maxf12(1�h)3+h

; 125g < v we have that rL1 > rH1 > rnH2. Therefore

there remains to situate rL2 in the threshold ranking.

d) The di�erence rnH2 � rnL2 =(1�h)(24(1+6h+h2)�(41+110h+41h2)v)

2592h(1+h)is positive for v <

24(1+6h+h2)41+110h+41h2

: Denote expression 24(1+6h+h2)41+110h+41h2

by �H2�L2(h); which is a decreasing

function of h; with �L1�H2(0) =2441and �L1�H2(1) = 1: Also, it crosses

12(1�h)3+h

at

h = 0:696:

23

e) The di�erence rH1 � rnL2 =24(1+6h�7h2)�(41+38h�111h2)v

2592h(1+h)is positive for all v if

h > 0:803 and for v < 24(1+6h�7h2)41+38h�111h2 if h 2 (0; 0:803): Denote expression

24(1+6h�7h2)41+38h�111h2

by �H1�L2(h): It is only de�ned for h 2 (0; 0:803) and �L1�H2(0) =2441; it crosses

12(1�h)3+h

at h = 0:506:

f) Finally, The di�erence rL1� rnL2 =24(1�h)2�(41�42h�31h2)v

2592h(1+h)is positive for all v if

h > 0:657 and for v < 24(1�h)241�42h�31h2 if h 2 (0; 0:657): Denote expression

24(1�h)241�42h�31h2

by �L1�L2(h): It is only de�ned for h 2 (0; 0:657) and �L1�L2(0) =2441: it crosses

12(1�h)3+h

at h = 0:624:

� Consider part i) together with d), e) and f), and since �H2�L2(h); �H1�L2(h)

and �L1�L2(h) are greater than �L1�H2(h) if h 2 (0:761; 1) then the full

threshold ranking is rH1 > rnH2 > rL1 > rnL2 > 0 for this part. We label

this region as I. That is region I is de�ned by v 2 [ 12(1�h)3+h

;�L1�H2(h)] and

h 2 (0:761; 1).

� Consider part ii) together with d), e) and f). Four regions can be de�ned witha di�erent threshold ranking. First of all note that easy computations yield

the following:

1)�H1�L2(h) > �L1�L2(h) for h 2 (0:642; 1) also �H1�L2(0:642) = �L1�L2(0:642) =125:

2)�L1�L2(h) > �H2�L2(h) for h 2 (0:612; 1):Then we distinguish four regions in part ii):

Region II de�ned by maxf12(1�h)3+h

;�L1�H2(h)g < v < �H2�L2(h) and h 2(0:696; 1) in which happens that rH1 > rL1 > r

nH2 > r

nL2 > 0:

Region III de�ned by maxf12(1�h)3+h

;�H2�L2(h)g < v < minf�L1�L2(h);125g and

h 2 (0:624; 1) where the following ranking holds rH1 > rL1 > rnL2 > rnH2 > 0:Region VI de�ned by maxf12(1�h)

3+h;�L1�L2(h)g < v < �H1�L2(h) and h 2

(0:506; 0:642) where rH1 > rnL2 > rL1 > r

nH2 > 0 holds.

Region VII de�ned by maxf12(1�h)3+h

;�H1�L2(h)g < v < 125and h 2 (1

3; 0:642) in

which rnL2 > rH1 > rL1 > rnH2 > 0 holds.

� Consider part iii) together with d), e) and f). The next three regions can beobtained.

Region IV is de�ned by 125< v < �H1�L2(h) for h 2 (0:642; 0:802) and 12

5< v

for h 2 (0:802; 1) in which the following ranking holds, rL1 > rH1 > rnL2 > rnH2 > 0:

24

Region V is de�ned by �H1�L2(h) < v < �L1�L2(h) for h 2 (0:642; 0:657) and�H1�L2(h) < v for h 2 (0:657; 0:802) in which rL1 > rnL2 > rH1 > rnH2 > 0:Region VIII is de�ned by maxf12(1�h)

3+h; 125;�L1�L2(h)g < v and h 2 (0; 0:657) in

which the following ranking holds rnL2 > rL1 > rH1 > rnH2 > 0:

Ranking the bound on " and the thresholds

First of all recall that "n � minf [v(7+5h)�4(1�h)]2576vh(1+h)

; [v(5+7h)+4(1�h)]2

576v(1+h)g and that the

�rst term is the minimum for v < 4(7+10h+7h2+12(1+h)ph))

49+94h+49h2; noting that for v > 1 the

second term is the minimum for all h:

Then, we �rst compare [v(7+5h)�4(1�h)]2576vh(1+h)

with the three thresholds that are at the

top of the ranking that is, rnL2; rL1 and rH1: Easy computations yield that expression[v(7+5h)�4(1�h)]2

576vh(1+h)is always greater that the three of them by using the horizontal

dominance condition on v. Then we conclude that for v < 4(7+10h+7h2+12(1+h)ph))

49+94h+49h2; it

always happens that "n is greater than the greatest threshold.

Second, we compare [v(5+7h)+4(1�h)]2

576v(1+h)with rH1; r

nL2 and rL1:As above,

[v(5+7h)+4(1�h)]2576v(1+h)

is greater than rH1 for v satisfying the horizontal dominance condition. Next,[v(5+7h)+4(1�h)]2

576v(1+h)> rL2 i�

144(1�h)2+24(2+19h�21h3)v�(82+75h�468h2�441h3)v25184h(1+h)v

> 0; where this

expression is positive for all v if h > 0:417 or if v <12(2+19h�21h3+(1�h)

p2(1+3h)(2+77h+69h2))

82+75h�468h2�441h3

for h 2 (0; 0:417): Denote the latter expression by �n"H�L2(h). It is de�ned for

h 2 (0; 0:417): Note that it equals 2441at h = 0; crosses 12(1�h)

3+hat h = 0:263 and

that �n"H�L2(h) is greater than both �L1�L2(h) and �H1�L2(h) for all h: Then we

conclude that �n"H�L2(h) only crosses region VIII and splits it in two sub-regions.

Finally, [v(5+7h)+4(1�h)]2576v(1+h)

> rL1 i�144(1�h)2+24(23�2h�21h2)v�(159�406h�441h2)v2

5184(1+h)v> 0;

where this expression is positive for all v if h > 0:296 or if v < 12(23�2h�21h2+4(1�h)p43+35h)

159�406h�441h2

for h 2 (0; 0:296): Denote the latter expression by �n"H�L1(h). It is de�ned for

h 2 (0; 0:296): Note that it equals 3:715 at h = 0; crosses 12(1�h)3+h

at h = 0:020

and that �n"H�L1(h) is greater than �L1�L2(h); �H1�L2(h) and �

n"H�L2(h) for all

h: Then we conclude that �n"H�L1(h) only crosses region VIII and splits it in two

sub-regions.

Taking into account the above discussion we know that for all of the regions I to

VII, the full ranking always has "n at the top. However, region VIII is partitioned

in three sub-regions.

Sub-region VIII-c is de�ned by maxf12(1�h)3+h

;�n"H�L1(h)g < v and h 2 (0; 0:296)

in which the full ranking is rnL2 > rL1 > "n > rH1 > rH2 > 0:

Sub-region VIII-b is de�ned by maxf12(1�h)3+h

;�n"H�L2(h)g < v < �n

"H�L1(h) if

25

h 2 (0:020; 0:296) and by maxf12(1�h)3+h

;�n"H�L2(h)g < v if h 2 (0:296; 0:417) in

which the full ranking is rnL2 > "n > rL1 > rH1 > r

nH2 > 0:

Sub-region VIII-a is de�ned by maxf12(1�h)3+h

; 125;�L1�L2(h)g < v < �n

"H�L2(h) if

h 2 (0:263; 0:417) and by maxf12(1�h)3+h

; 125;�L1�L2(h)g < v if h 2 (0:417; 0:657)) in

which the full ranking is "n > rnL2 > rL1 > rH1 > rnH2 > 0:


The ten di�erent regions de�ned above determine ten di�erent rankings of the

thresholds and "n: The following ten propositions characterize the �rst stage Nash

Equilibrium for them.

Proposition 9 Consider region I as de�ned above. It follows that 0 < rnL2 < rL1 <

rnH2 < rH1 < "n:

a) If 0 < " < rnL2 < rL1 < rnH2 < rH1 < "

n; the Nash Equilibrium is (2; 2).

b) If 0 < rnL2 < " < rL1 < rnH2 < rH1 < "

n; the Nash Equilibrium is (2; 1):

c) If 0 < rnL2 < rL1 < " < rnH2 < rH1 < "


d) If 0 < rnL2 < rL1 < rnH2 < " < rH1 < "


e) If 0 < rnL2 < rL1 < rnH2 < rH1 < " < "


Proposition 10 Consider region II as de�ned above. It follows that 0 < rnL2 <

rnH2 < rL1 < rH1 < "n:

a) If 0 < " < rnL2 < rnH2 < rL1 < rH1 < "


b) If 0 < rnL2 < " < rnH2 < rL1 < rH1 < "


c) If 0 < rnL2 < rnH2 < " < rL1 < rH1 < "n; there are two Nash Equilibria

f(1; 2); (2; 1)g:d) If 0 < rnL2 < r

nH2 < rL1 < " < rH1 < "


e) If 0 < rnL2 < rnH2 < rL1 < rH1 < " < "


Proposition 11 Consider region III as de�ned above. It follows that 0 < rnH2 <

rnL2 < rL1 < rH1 < "n:

a) If 0 < " < rnH2 < rnL2 < rL1 < rH1 < "


b) If 0 < rnH2 < " < rnL2 < rL1 < rH1 < "


c) If 0 < rnH2 < rnL2 < " < rL1 < rH1 < "n; there are two Nash Equilibria

f(1; 2); (2; 1)g:d) If 0 < rnH2 < r

nL2 < rL1 < " < rH1 < "


e) If 0 < rnH2 < rnL2 < rL1 < rH1 < " < "


26

Proposition 12 Consider region IV as de�ned above. It follows that 0 < rnH2 <

rnL2 < rH1 < rL1 < "n:

a) If 0 < " < rnH2 < rnL2 < rH1 < rL1 < "


b) If 0 < rnH2 < " < rL2 < rH1 < rL1 < "n; the Nash Equilibrium is (1; 2):

c) If 0 < rnH2 < rnL2 < " < rH1 < rL1 < "n; there are two Nash Equilibria

f(1; 2); (2; 1)g:d) If 0 < rnH2 < r

nL2 < rH1 < " < rL1 < "


e) If 0 < rnH2 < rnL2 < rH1 < rL1 < " < "


Proposition 13 Consider region V as de�ned above. It follows that 0 < rnH2 <

rH1 < rnL2 < rL1 < "

n:

a) If 0 < " < rnH2 < rH1 < rnL2 < rL1 < "


b) If 0 < rnH2 < " < rH1 < rnL2 < rL1 < "


c) If 0 < rnH2 < rH1 < " < rnL2 < rL1 < "


d) If 0 < rnH2 < rH1 < rnL2 < " < rL1 < "


e) If 0 < rnH2 < rH1 < rnL2 < rL1 < " < "


Proposition 14 Consider region VI as de�ned above. It follows that 0 < rnH2 <

rL1 < rnL2 < rH1 < "

n:

a) If 0 < " < rnH2 < rL1 < rnL2 < rH1 < "


b) If 0 < rnH2 < " < rL1 < rnL2 < rH1 < "


c) If 0 < rnH2 < rL1 < " < rnL2 < rH1 < "

n; there is no Nash Equilibrium:

d) If 0 < rnH2 < rL1 < rnL2 < " < rH1 < "


e) If 0 < rnH2 < rL1 < rnL2 < rH1 < " < "


Proposition 15 Consider region VII as de�ned above. It follows that 0 < rnH2 <

rL1 < rH1 < rnL2 < "

n:

a) If 0 < " < rnH2 < rL1 < rH1 < rnL2 < "


b) If 0 < rnH2 < " < rL1 < rH1 < rnL2 < "


c) If 0 < rnH2 < rL1 < " < rH1 < rnL2 < "

n; there is no Nash Equilibrium:

d) If 0 < rnH2 < rL1 < rH1 < " < rnL2 < "


e) If 0 < rnH2 < rL1 < rH1 < rnL2 < " < "


Proposition 16 Consider region VIII-a as de�ned above. It follows that 0 < rnH2 <

rH1 < rL1 < rnL2 < "

n:

a) If 0 < " < rnH2 < rH1 < rL1 < rnL2 < "


27

b) If 0 < rnH2 < " < rH1 < rL1 < rnL2 < "


c) If 0 < rnH2 < rH1 < " < rL1 < rnL2 < "


d) If 0 < rnH2 < rH1 < rL1 < " < rnL2 < "


e) If 0 < rnH2 < rH1 < rL1 < rnL2 < " < "


Proposition 17 Consider region VIII-b as de�ned above. It follows that 0 < rnH2 <

rH1 < rL1 < "n < rnL2:

a) If 0 < " < rnH2 < rH1 < rL1 < "n < rnL2; the Nash Equilibrium is (2; 2).

b) If 0 < rnH2 < " < rH1 < rL1 < "n < rnL2; the Nash Equilibrium is (1; 2):

c) If 0 < rnH2 < rH1 < " < rL1 < "n < rnL2; the Nash Equilibrium is (1; 2):

d) If 0 < rnH2 < rH1 < rL1 < " < "n < rnL2; the Nash Equilibrium is (1; 1):

Proposition 18 Consider region VIII-c as de�ned above. It follows that 0 < rnH2 <

rH1 < "n < rL1 < r

nL2:

a) If 0 < " < rnH2 < rH1 < "n < rL1 < r

nL2; the Nash Equilibrium is (2; 2).

b) If 0 < rnH2 < " < rH1 < "n < rL1 < r

nL2; the Nash Equilibrium is (1; 2):

c) If 0 < rnH2 < rH1 < " < "n < rL1 < r

nL2; the Nash Equilibrium is (1; 2):

Proofs are straightforward once the full ranking of thresholds and "n is estab-

lished. It follows the same lines of reasoning as those in the case of the interleaved

competition game.

28

Figure 1: Consumer allocation for the {1,1}-subgame.

Figure 2: Consumer allocation for the {1,2}-subgame.

Figure 3.a: Consumer allocation for the {2,2}-subgame.

Figure 3.b: Consumer allocation for the {2,2,n}-subgame.

Figure 4: Regions in the v x h space for the interleaved competition game.

Figure 5: Equilibrium regions corresponding to Proposition 2.

Figure 6: Regions in the v x h space for the neighbour competition game.

Figure 7: Equilibrium regions corresponding to Proposition 3.

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