Market Segmentation With Nonlinear Pricing

Silvia Sonderegger

Abstract

This paper studies the e¤ect of captive consumers in a competitive model of nonlinear pric-
ing. We focus on the bene…ts and drawbacks of allowing what we call market segmentation,
namely, a situation where the price-quality menu o¤ered to captive consumers can di¤er from
that o¤ered to consumers that are exposed to competition. We …nd that the e¤ect of market
segmentation depends on the relationship between the range of consumer preferences found in
captive markets and that found in competitive markets. When the range of consumer prefer-
ences in captive markets is “wide”, segmentation is quality and (aggregate) welfare reducing,
while the opposite holds when the range of consumer preferences in captive markets is “narrow”.
Segmentation always harms captive consumers, while it always bene…ts consumers located in
competitive markets.
JEL Classi…cations: D43, L1.

1 Introduction

The use of second-degree (or indirect) price discrimination by …rms competing in oligopolistic
settings is a phenomenon that has been increasingly recognized and documented.1 This paper
contributes to the recent body of theoretical literature on this subject, by studying competition
within a horizontally di¤erentiated duopoly.2 Di¤erently from previous studies, we assume that
Department of Economics, School of Economics, Finance and Management, University of Bristol, 12 Priory
Road, Bristol BS8 1TN, United Kingdom. E-mail: S.Sonderegger@bristol.ac.uk. A previous version of this paper
was entitled “Nonlinear Pricing, Multimarket Duopolists and Third-Degree Price Discrimination”. I thank the editor
and two anonymous referees for providing very useful comments and suggestions. I am also indebted to Jean-Charles
Rochet, Lars Stole, John Sutton and seminar participants at the London School of Economics and the University of
Bristol. All errors are my own.
1
The growing number of empirical papers documenting the existence of second-degree price discrimination in
competitive environments include Shepard (1991), Borenstein (1991) and Busse and Rysman (2005).
2
The theoretical literature on this topic includes Spulber (1989), Stole (1995), Rochet and Stole (2002), Armstrong
and Vickers (2001).

1
consumers vary in their degree of captivity to a given …rm, as well as in their unobservable valu-
ation for quality. The presence of captive consumers plays an important role in many industries,
such as for instance the telecommunication industry or the market for industrial gas (Armstrong
and Vickers 1993). The purpose of this paper is that of identifying how the presence of captive
consumers conditions the nature of the competitive equilibrium when di¤erentiated …rms compete
in price-quality menus. This has important policy implications, by informing the debate of whether
market segmentation (where …rms discriminate between captive and non-captive markets) should
or should not be allowed. This is often a controversial issue. Armstrong and Vickers (1993) for
instance cite the example of the United Kingdom’s industrial gas market, where, prior to 1988
British Gas (BG) was free to set prices to its customers without constraints. Customers without
an alternative source of energy complained that they were being charged more than the less cap-
tive consumers. Finally, a Monopolies and Mergers Commission Report removed BG’s freedom to
discriminate between more and less captive consumers.
A detailed description of the model is provided in section 2, while section 3 establishes some use-
ful preliminaries. An important feature of our framework is that a consumer’s marginal valuation
for quality is inversely related to the distance between the consumer’s ideal product-space location
and that of the …rm from which he purchases.3 This assumption has also been made elsewhere in
the literature, e.g. Spulber (1989), Stole (1995) and Hamilton and Thisse (1997). To see what this
assumption entails, consider for instance a theatre, selling tickets for opera performances. Here,
the distance between the consumer and the …rm captures the consumer’s appreciation of opera as
a genre –with a lower distance representing greater appreciation. Take two consumers: a serious
opera lover and one who is less of a connoisseur. Our assumption implies that, for lousy opera
companies, the di¤erence in the maximum price that the connoisseur and the non-connoisseur are
willing to pay to see the performance is relatively small. For high quality opera companies, how-
ever, the di¤erence between the two consumers is much more apparent. The price that a serious
opera lover is willing to pay to see, say, Placido Domingo is much higher than that of someone who
has a more tepid attitude towards opera. An implication of this is that consumers who purchase
higher quality goods also have stronger brand preferences. This is consistent with the empirical
observation that, in several markets, consumers who purchase higher qualities are more brand-loyal
than those who purchase lower qualities, and are therefore less likely to switch supplier to take
3
Although we concentrate on quality provision, the model could equivalently be used to model situations where
quality is homogenous, and the relevant contractible variable is quantity rather than quality. See also Rochet and
Stole (2002, footnote 5) on this point.

2
advantage of small price di¤erences.4
We assume that each consumer belongs to one of three categories: some only have access to one
…rm, some only have access to the other, and some have access to both – i.e., they belong to the
“competitive market”. When facing a given …rm, consumers therefore vary along two dimensions.
First, they vary with respect to their marginal valuations for quality of the product sold by the
…rm. Second, they vary in their captivity status – they are either entirely captive to the …rm, or
they have costless access to both …rms. Captive consumers may be located in speci…c geographical
markets (as e.g. in Rosenthal 1980), or they may have higher search costs than other consumers
(as e.g. in Salop and Stiglitz 1977). Alternatively, they may be repeat-purchase consumers who
are now locked-in (as e.g. in Klemperer 1987).
We allow the range of consumer valuations found in the competitive and the captive markets to
di¤er. While in the competitive market the lowest-valution consumer for each …rm has valuation
v 1, in the …rm’s captive market the lowest-valuation consumer has valuation v s, for some
v > 1 and s 1.5 The parameter s describes the range of consumer preferences found in captive
markets relative to the range of preferences found in competitive markets.
We compare two cases. In the …rst case (presented in section 4), market segmentation is allowed,
namely …rms are allowed to discriminate between consumers located in their captive market and
those located in the competitive market. In the second case (presented in section 5), market
segmentation is not allowed, so consumers located in captive markets and those located in the
competitive market must be o¤ered the same price-quality menu of contracts.
We identify two e¤ects of disallowing segmentation: a competition e¤ ect and a discontinu-
ity e¤ ect. The discontinuity e¤ect arises because, when segmentation is not allowed, consumers
from di¤erent markets (captive/competitive) are lumped together. Although within each mar-
ket consumers are distributed uniformly (and therefore continuously), when consumers from the
two markets are lumped together the resulting density function exhibits a discontinuity. Keeping
everything else equal, this lowers the quality o¤ered to consumers with su¢ ciently low valuations.
The competition e¤ect arises because competition in the competitive market has the by-product
e¤ect of softening the e¢ ciency/informational rents trade-o¤ …rms face when selecting the quality
of low-valuation captive consumers. Keeping everything else equal, this raises the quality o¤ered
4
This is for instance well documented within the car market, as shown in Goldberg (1995), Berry, Levinsohn
and Pakes (1995), Feenstra and Levinsohn (1995). Indeed, Verboven (1996) calls this feature a “stylized fact of this
market”.
5
We assume that, in both competitive and captive markets, the highest-valuation consumer for each …rm has
valuation v.

3
to these consumers.
Which of these two e¤ects dominates depends crucially on the value of s. In particular, the
results obtained when captive markets are “narrow” (s < 1=2) are radically di¤erent from those
obtained when they are “wide” (s > 1=2). For wide captive markets the competition e¤ect al-
ways dominates. Hence, forbidding segmentation raises the the qualities o¤ered to a whole set
of consumers. These results are in contrast with the …ndings of Spulber (1989), Stole (1995) and
Hamilton and Thisse (1997), where competition simply shifts rents, but does not a¤ect quality
allocations. We show that, with captive markets, this may no longer be true.
If captive markets are su¢ ciently narrow, though, the competition e¤ect disappears altogether,
so disallowing segmentation only brings about the discontinuity e¤ect. Hence, in this case forbid-
ding segmentation lowers the qualities o¤ered to a whole set of types – namely, consumers with
su¢ ciently low valuations.
Section 6 compares the quality allocations obtained under the segmentation and the no-
segmentation régimes. In section 7, we characterize their welfare properties. Again, these change
in narrow and wide captive markets. For narrow captive markets, aggregate welfare is higher when
segmentation is allowed, while the opposite occurs for wide captive markets. Moreover, segmen-
tation always harms captive consumers, while it bene…ts those located in the competitive market.
Section 8 concludes. All the proofs that are not in the main text can be found in the appendix.

2 The model

There are two …rms, denoted as …rm L and …rm R, positioned at the left and right extremities
of a Hotelling line of length 1, and a continuum of consumers of total mass one. A consumer’s
preferences are determined by his ideal product-space location on the line (his location, in short),
which is his private information. Let z be the (absolute value of the) distance between a consumer’s
ideal product-space location and the …rm’s location. Similarly to Spulber (1989), Stole (1995) and
Hamilton and Thisse (1997), we assume that the utility of a consumer who purchases quality q at
price p is equal to:
u(z; p; q) = (v z) q p: (1)

for some v 2 R+ . In what follows, we refer to (v z) as the consumer’s valuation. If a consumer

does not consume the good at all, his utility is equal to 0. In order to simplify the analysis, we
impose the following restriction on v:

Assumption A1 v 3.

4
 This assumption is su¢ cient to ensure that at equilibrium all consumers are served by one
of the two …rms. As made clear by (1), our setting di¤ers from textbook models of horizontal
di¤erentiation. Take for instance a …rm selling a product of quality q at price p. Consider the
“standard” textbook approach (for simplicity, let the marginal “transportation cost” be unity).
The utility of consuming the good for a consumer located at distance z from the …rm is vq p z.
Hence, the di¤erence in utility between a consumer located at distance z0 > 0 and a consumer
located at distance z1 > z0 from the …rm is z1 z0 , independent of quality q. By contrast, in our
model this di¤erence is q(z1 z0 ), increasing in q. A good of zero quality is equally worthless for
both consumers. As quality increases, the utility obtained by the consumer at distance z0 grows
faster than that obtained by the consumer located at distance z1 .
In addition to having di¤erent valuations, consumers also di¤er in their degrees of captivity to
the …rms. In order to keep the analysis simple, we consider an extreme case: some consumers are
entirely captive to …rm L (they belong to …rm L’s captive market), some consumers are entirely
captive to …rm R (they belong to R’ s captive market), and some consumers have access to both
…rms (they belong to the competitive market). Firms can observe if a consumer is captive or if
he is exposed to competition. Whether they can utilize this information when making o¤ers to
consumers, though, depends on whether segmentation is or is not allowed. If it is not, …rms must
allow all consumers to select their favorite choice from the same menu of price-quality o¤ers. By
contrast, if segmentation is allowed then …rms can o¤er di¤erent deals to captive consumers com-
pared to consumers who are exposed to competition.6 We assume that consumers are equally split
between the three markets. This implies that the mass of consumers in each market (competitive
market, L’s captive market and R’s captive market) is equal to 1=3. We also make the following
assumption.

Assumption A2 In the competitive market, the consumers’ ideal product-space locations are
uniformly distributed on [0; 1]. In L’s captive market, they are uniformly distributed on
[0; s], while in R’s captive market they are uniformly distributed on [1 s; 1], for some
s 2 (0; 1].

Assumption A2 makes clear that the range of consumer valuations found in the competitive
and the captive markets may di¤er. Consider for instance …rm L (the case for …rm R is analogous).
The ideal product-space locations of its captive consumers belong to [0; s]. By contrast, those of
6
This is presumably what British Gas did before captive consumers successfully …led a complaint with the
Monopolies and Mergers Commission, as reported by Armstrong and Vickers (1993).

5
consumers exposed to competition belong to [0; 1]. Note that, at equilibrium, …rm L does not
serve all the consumers who are exposed to competition, but only those whose ideal product-
space locations belong to [0; 1=2]. When s < (>) 1=2, this is larger (narrower) than the range of
product-space locations of the consumers located in L’s captive market.
We assume that the two …rms are perfectly symmetric, with quadratic costs of production.
The pro…t from selling quality q at price p is then p q 2 =2.
Conditional on a consumer with valuation v z purchasing quality q at price p, joint surplus
from trade is equal to S(q; z) = (v z) q q 2 =2: The full information …rst-best allocation that
maximizes S(q; z) is given by q F B (z) = v z. This is the benchmark against which we evaluate
the e¢ ciency of quality allocations that result under asymmetric information.7

The timing of the game is as follows. At t = 1, the …rms simultaneously make price-quality
o¤ers; at t = 2, consumers located in the competitive market choose which …rm to consume from
(if any) and which quality to purchase, while consumers located in the captive markets decide
whether or not to purchase from the unique …rm they have access to, and (if yes) which quality to
purchase; at t = 3 market transactions take place; …nally, at t = 4 payo¤s are realized.
Throughout the analysis we concentrate on symmetric, pure strategy, deterministic equilibria.
We also concentrate on equilibria where …rms only o¤er those qualities that they actually sell.8 In
what follows we therefore use the term equilibrium with these conditions left implicit.9

3 Preliminaries

Before we turn to the analysis, it is instructive to establish some preliminary results. Let zi
indicate the distance between a consumer’s ideal product-space location and that of …rm i = L; R.
The precise value of zi is a consumer’s private information. From the revelation principle, our
search for the optimal contract can be restricted to direct revelation mechanisms that are incentive
compatible. The following lemma characterizes the conditions that must be satis…ed in order to
have incentive compatibility. For any given mechanism o¤ered by …rm i, we indicate the indirect
7
Note that since the value of s determines the distribution of consumer valuations in the captive markets, it also
a¤ects the maximum surplus achievable. However, our analysis studies the e¤ects of di¤erent régimes (discrimina-
tory/non discriminatory) keeping s …xed, so this is not an issue here.
8
This restriction is introduced to avoid “out of equilibrium” o¤ers that may complicate things, without really
adding to the results.
9
D’Aspremont, Gabzewicz and Thisse (1979) show that, in the linear-cost speci…cation of the Hotelling model, a
pure-strategy price equilibrium may fail to exist when …rms are located su¢ ciently close to the centre of the segment.
This occurs because demand functions are discontinuous. For some price con…gurations, all consumers in …rm i’s
turf switch to …rm j for a small reduction in pj . In our framework, this potential problem is ruled out, since …rms
are located at the extremities of the Hotelling segment.

6
utility of a consumer located at distance zi from …rm i who truthfully reveals it as ui (zi ).10
Lemma 1: The following conditions are necessary and su¢ cient for incentive compatibility:

(IC.1) u0i (zi ) = qi (zi )

(IC.2) qi (zi ) is non-increasing in zi :

Proof: The proof of lemma 1 is standard and is therefore omitted. See for instance Fudenberg
and Tirole (1991, Chapter 7).

Condition (IC.1) is the …rst order condition for local incentive compatibility, while condition
(IC.2) is the second-order condition. Together, these two conditions ensure global incentive com-
patibility. From (IC.1), we see that in order to preserve incentive compatibility, …rms must o¤er
higher rents to high valuation (i.e., low zi ) consumers. These rents are increasing in the quality
o¤ered to low valuation consumers. This is the familiar e¢ ciency/informational rents trade-o¤ …rst
identi…ed by Mussa and Rosen (1978). As is standard in the literature – see for instance Bolton
and Dewatripont (2005), p.85 –in what follows we will …rst characterize each …rm’s optimization
problem ignoring the monotonicity constraint (IC.2), and then reinstate monotonicity whenever
the solution of the unconstrained problem fails to satisfy it. Denote …rm i’s contractual o¤er when
a consumer declares his distance from the …rm’s location to be zb as (pi (b
z ); qi (b
z )). Conditional
on a consumer’s truthfully declaring his distance to be zi , …rm i’s pro…t when contracting with
this consumer is given by pi (zi ) qi (zi )2 =2. Substituting for pi (zi ) = (v zi ) qi (zi ) ui (zi ) this
becomes (v zi ) qi (zi ) ui (zi ) qi (zi )2 =2: We now turn to the analysis.

4 Market segmentation is allowed

We start o¤ by investigating the nature of the equilibrium in the case where market segmentation
is allowed. Each …rm can discriminate between consumers located in its captive market, and
those located in the competitive market. This is especially easy if the two markets are divided
geographically. Even when this is not the case, though, …rms may nonetheless discriminate between
captive and non-captive consumers by tailoring their o¤ers to consumers’characteristics. This is a
10
Consumers di¤er along two dimensions, namely their distance zi from the …rm, and also their market location
(captive/competitive). However, conditional on purchase from …rm i = L; R, the preferences of a consumer located
at distance zi from …rm i are independent of the market (captive/competitive) to which the consumer belongs.
Accordingly, when segmentation is not allowed our search for optimal contracts may be restricted to environments
where all consumers with the same zi who purchase from the same …rm are o¤ered the same contract. [When
segmentation is allowed, …rms are free to condition their contractual o¤er on the market (captive/competitive)
to which a consumer belongs, so incentive compatibility along this dimension is not an issue.] This point is also
discussed in section 5.

7
widespread practice, which has become increasingly common. As argued by Liu and Serfes (2004),
“recent advances in information technology and software tools (..) have taken price discrimination
to a new level”. These authors cite a series of examples of the so-called practice of dynamic pricing,
where consumers pay di¤erent prices according to their demographics, purchasing history and so
on. Sha¤er and Zhang (2000) provide examples of …rms targeting promotions at consumers with
greater exposure to rivals.
Note that although …rms can discriminate between captive and non-captive consumers, within
each market a consumer’s valuation is his private information.11 The situation is therefore one
where …rms use a combination of direct and indirect price discrimination, o¤ering nonlinear price
schedules that are conditional on the market (captive/competitive) where the consumer is located.

Captive market. In its captive market, each …rm is a monopolist. The distance between the
consumer with lowest valuation and the …rm is s, while that between the consumer with the highest
valuation and the …rm is 0. Since the total mass of consumers located in each captive market is
1=3, consumer density is equal to 1=3s. Firm i = L; R’s programme is to maximize
Z s !
1 qi (zi )2
(v zi ) qi (zi ) ui (zi ) dzi (2)
0 3s 2

subject to incentive compatibility (IC.1) and participation: ui (zi ) 0 for all zi 2 [0; s]. The
problem faced by the …rm is a standard screening problem, à la Mussa and Rosen (1978). It is
straightforward to show that the optimal contract satis…es: ui (s) = 0 and

qi (zi ) = v 2zi : (3)

Competitive market. In the competitive market, the distance between the consumer with lowest
valuation and the …rm is 1, while that between the consumer with the highest valuation and the
…rm is 0. Firm i = L; R’s programme is to maximize
Z !
1
qi (zi )2
mi (zi ) (v zi ) qi (zi ) ui (zi ) dzi (4)
0 2

subject to incentive compatibility (IC.1), where mi (zi ) indicates the density of consumers with
distance zi purchasing from …rm i. Let Bi (zi ) denote the utility which a consumer located at
distance zi from …rm i can obtain by purchasing from …rm i = fL; Rg i. This is the consumer’s
reservation utility when contracting with …rm i. A consumer will purchase from …rm i only if
ui (zi ) Bi (zi ). What is the value of Bi (zi )? Since the total length of the Hotelling segment
11
If this was not the case, the …rms would be able to operate …rst-degree price discrimination.

8
is equal to 1, zi = 1 z i for i = L; R and i = fL; Rg ni. A consumer with a high valuation
when dealing with L also has a low valuation when dealing with …rm R, and vice-versa.12 We can
therefore write
Bi (zi ) = max (0, u i (z i )) = max (0, u i (1 zi )) : (5)

Hence, Bi0 (zi ) is either equal to 0 or it is equal to u0 i (1 zi ). From lemma 1, u0 i (1 zi ) =

q i (1 zi ) 0, where q i (1 zi ) denotes the product quality which a consumer located at
distance z i =1 zi is o¤ered when contracting with …rm i. This implies that Bi0 (zi ) 0, and
brings us to the following lemma.

Lemma 2: Given …rm i’s contract, a consumer located in the competitive market buys from …rm
i if and only if zi zim , where zim satis…es: ui (zim ) = Bi (zim ), and ui (zi ) > (respectively:; <)
Bi (zi ) for all zi < (respectively; >) zim .

Proof: Consider …rm i’s problem. To attract a consumer with distance zi located in the com-
petitive market, it must o¤er at least Bi (zi ). Since the competitive market is fully covered, we
must also have qi (zi ) > 0. From lemma 1, this implies that at equilibrium the utility of con-
sumers located in the competitive market purchasing from …rm i is strictly decreasing in zi . Given
Bi0 (zi ) 0, this implies that the consumers’ participation constraint in the competitive market
binds at a single point.
Within the competitive market the farthest consumer served by …rm i is located at distance
zim from the …rm. Substituting for u0i (zi ) from (IC.1), we can write
Z zim
ui (zi ) qi (x)dx = Bi (zim ): (6)
zi

Each …rm i’s problem consists of selecting a marginal type zim and quality allocations qi (zi ) to
maximize the …rm’s expected payo¤ subject to incentive compatibility, taking the rival’s contrac-
tual o¤ers as given.13 By analogy with Spulber (1989), Stole (1995) and Hamilton and Thisse
(1997), it is easy to show that, at equilibrium, the qualities o¤ered to consumers in the competi-
tive market are identical to those o¤ered to their counterparts in the captive markets. Intuitively,
for …rm i the only di¤erence between the captive and the competitive market arises because, in
the latter, the consumer’s reservation utility may be di¤erent from zero. Since Bi0 (zi ) is positive,
however, this does not a¤ect the …rm’s trade-o¤ between e¢ ciency and informational rents when
12
Note that since the relationship between zL and zR is one-to-one, the value of zi , i = L; R provides a full
characterization of a consumer’s preferences for i = L; R and i = fL; Rg i.
13
In what follows, the expression “…rm i’s marginal type” will be used to indicate the distance zim of the farthest
consumer served by …rm i in the competitive market.

9
selecting quality. Conditional on type zim purchasing from …rm i, all consumers located at distance
smaller than zim from …rm i also necessarily purchase from that …rm.14 The trade-o¤ involved
in o¤ering higher or lower quality is then the same as that faced by a monopoly. Hence, in this
environment, competition shifts rents away from …rms to consumers, but does not a¤ect quality
allocations (and aggregate welfare). As will become clear below, this is in contrast with the case
where segmentation is not allowed.
In any symmetric equilibrium, the marginal consumer in the competitive market is located mid-
way between the two …rms. Hence, zLm = zR
m = 1=2. The full characterization of the equilibrium of

the game is found by substituting for zim = 1=2 in the …rm’s …rst order condition for zim and then
imposing symmetry in the two …rms’ contractual o¤ers. This yields u(1=2) = (v 1)(v 2)=2.
The following proposition summarizes our results.

Proposition 1 (Characterization of equilibrium contract when segmentation is allowed.) When

market segmentation is allowed, the equilibrium quality allocations in both the captive and the
competitive market are the same, and are given by (3). In the competitive market, ui (1=2) =
(v 1)(v 2)=2, i = L; R:

5 Market segmentation is not allowed

We now turn to the case in which market segmentation is not allowed. Consumers located in the
competitive and the captive markets must be o¤ered the same menu of price-quality contracts.
Conditional on purchase from …rm i = L; R, the preferences of a consumer located at distance
zi from …rm i are independent of the market (captive/competitive) to which a consumer belongs.
Accordingly, in our search for the optimal contracts, we consider direct revelation mechanisms of
the form fqi (zi ); pi (zi )g.15
Since s 1, for each …rm the farthest available consumer is located in the competitive market,
at distance 1 from the …rm. Firm i’s problem is to maximize
Z !
1
qi (zi )2
mi (ui (zi ) ; zi ) (v zi )qi (zi ) ui (zi ) dzi (7)
0 2

subject to incentive compatibility (IC.1) and a participation constraint.

From Lemma 2, we know that consumers located in the competitive market purchase from …rm
i only if their distance zi from the …rm is zim . By contrast, captive consumer have no choice but
14
Note that this would not necessarily be true if Bi0 (zi ) was negative.
15
This is without loss of generality given that we restrict attention to mechanisms that do not involve randomization
– this point is also made by Rochet and Stole (2002), pp.282–283.

10
purchase from …rm i. The density of consumers located at distance zi from …rm i who purchase
from …rm i is therefore equal to:

if zim < s 8
>
> (1 + s) =3s for zi 2 [0; zim ]
>
<
mi (zi ) = 1=3s for zi 2 (zim ; s] (8)
>
>
>
: 0 for z > s
i

if zim > s 8
>
> (1 + s) =3s for zi 2 [0; s]
>
<
mi (zi ) = 1=3 for zi 2 (s; zim ] : (9)
>
>
>
: 0 for z > z m
i i

In both (8) and (9) the density mi (zi ) has a downward discontinuity (at zi = zim and zi = s,
respectively). To see why, consider for instance (8) –the case of (9) is analogous. Here, purchasing
consumers with distance zi > zim from …rm i may only be located in the captive market –if they
were located in the competitive market, they would not be purchasing from i. In what follows, we
refer to these consumers as being “unambiguously captive”. By contrast, purchasing consumers
with distance zi 2 [0; zim ] may be either located in the captive market or in the competitive
market.16 Clearly, the latter have a greater density than unambiguously captive consumers, and,
as a result, we have a downward discontinuity. As will become clear below, this discontinuity plays
an important role in our analysis.
In a symmetric equilibrium, zLm = zR
m = 1=2. When s < 1=2 (i.e., the distribution of preferences

in the captive market is “su¢ ciently narrow”) at equilibrium each …rm’s marginal type is also the
farthest consumer being served. This is however no longer true when s > 1=2 (i.e., the distribution
of preferences in the captive market is “su¢ ciently wide”). The distinction between these two
cases turns out to be crucial for characterizing the equilibrium contract. The case where s < 1=2 is
qualitatively equivalent to a situation in which the captive market is entirely absent. In contrast,
the case where s > 1=2 possesses novel features, that derive from the presence of a captive market.
As a shorthand, in what follows we indicate s < 1=2 as the narrow captive markets case, and
s > 1=2 as the wide captive markets case.
16
When purchasing consumers located at a distance zi from a …rm may originate both from the …rm’s captive
market and from the competitive market, the density mi (zi ) is given by the sum of 1=3s (the density of consumers
in the captive market) and 1=3 (the density of consumers in the competitive market). This yields (1 + s) =3s.

11
5.1 “Wide” captive markets: s > 1=2

In this section we characterize the properties of the optimal contract whenever s > 1=2. In this
case, the farthest consumer served by each …rm at equilibrium is located at distance s from the
…rm. The marginal density of consumer distances being faced by each …rm is given by (8). To solve
for the equilibrium contracts, we derive each …rm i’s optimal mechanism conditional on zim < s.17
Ignoring monotonicity concerns, each …rm i solves (7) subject to (IC.1). Rearranging (6) for any
zi we can write
Z zi
ui (zi ) = Bi (zim ) qi (x)dx: (10)
zim

Substituting for ui (zi ) into the …rm’s objective function, we see that the …rm’s problem can
equivalently be expressed as one of selecting quality allocations and the marginal type zim < s to
solve
Z Z !
s
1 qi (zi )2 zi
1
max m (v zi ) qi (zi ) + qi (x)dx + I(zi ) dzi (11)
qi (zi ), zi 0 3 2 zim s
Z s
1 1
Bi (zim ) + I(zi ) dzi
0 3 s

where I(zi ) is given by 8

< 1 for zi zim
I(zi ) = : (12)
: 0 for zi > zim

subject to the captive consumers’participation constraint18

ui (zi ) 0 8zi 2 [0; s] : (13)

Since the reservation utility of all captive consumers is equal to 0, the necessary and su¢ cient
condition for (13) to hold is that ui (s) 0 or, equivalently, that
Z s
qi (x)dx Bi (zim ) 0. (14)
zim

After integration by parts, the Lagrangian for the problem can be written as

Z s
1 qi (zi )2 1
qi (zi ) v 2zi + s (1 3 ) (1 I(zi )) 2
+ I(zi ) dzi (15)
0 3 s
1
Bi (zim ) (zim + 1 3 )
3
17
At equilibrium, zim = 1=2 < s. Hence, although the conditions we derive below do not guarantee overall
optimality, they need to hold true at equilibrium.
18
Participation in the competitive market is determined by the choice of zim .

12
where 0 is the Lagrange multiplier for constraint (14). The unconstrained quality allocations
–derived by ignoring condition (IC.2) –are19

for zi 2 [0; zim ]

qi (zi ) = v 2zi (16)

for zi 2 (zim ; s]
qi (zi ) = v 2zi + s (1 3 ): (17)

Consumers located at distance zi 2 (zim ; s] are unambiguously captive, while those located at
distance zi 2 [0; zim ] from the …rm may be either located in the captive or the competitive market.
For these latter consumers the unconstrained quality allocations that emerge when segmentation is
not allowed are identical to those that would emerge if segmentation was allowed. For ambiguously
captive consumers, the two quality allocations di¤er by s (1 3 ). As shown in the appendix, in
our framework this value is always positive. Hence, disallowing segmentation induces the …rm
to o¤er higher qualities to low-valuation consumers than it would otherwise (i.e., if segmentation
was allowed). In order to fully appreciate the di¤erent e¤ects at play when selecting the optimal
contract, it is instructive to analyze the benchmark case where captive markets are entirely absent,
yet the density of zi is given by (8), as it is here. This is done in the following section.

5.1.1 A benchmark case

The aim of this section is that of separating the direct e¤ect of disallowing segmentation in the
presence of captive markets from the indirect e¤ect, which arises because the presence of captive
markets changes the distribution of consumer preferences faced by the …rms. This indirect e¤ect
can be replicated by simply letting the density of zi be given by (8), which is what we do here. Note
that since we are assuming that captive markets are entirely absent, here the marginal type is also
the farthest consumer being served by each …rm. Hence, in this benchmark the …rm’s marginal
type is s, and zim simply represents the point where the marginal density of zi has a (exogenously
given) discontinuity.
The analysis closely follows that presented in section 4. Each …rm selects quality allocations
to solve:

Z s
1 qi (zi )2 1
max (v zi ) qi (zi ) ui (zi ) + I(zi ) dzi
qi (zi ) 0 3 2 s
19
As discussed below, these allocations actually violate condition (IC.2).

13
subject to incentive compatibility and participation, where I(zi ) is given by (12). It is straightfor-
ward to verify that the optimal quality allocations prescribe:
for zi 2 [0; zim ]
qi (zi )benchmark = v 2zi (18)

for zi 2 (zim ; s]

qi (zi )benchmark = v 2zi zim s: (19)

The quality allocations described in (18) and (19) exhibit a downward discontinuity at zi =
zim . This arises from the discontinuous density of consumers faced by each …rm. Intuitively,
under (8) the standard e¢ ciency-informational rents trade-o¤ bites more severely for low-valuation
consumers (i.e., with distance greater than zim from the …rm) than for higher-valuation ones. When
serving low-valuation consumers, …rms are therefore more willing to sacri…ce quality, in order to
save on the informational rents to be o¤ered to higher-valuation consumers.

5.1.2 Two e¤ects

In the benchmark case analyzed above, captive markets are entirely absent, but the marginal
density of zi is exogenously assumed to satisfy (8). This allows us to distinguish two di¤erent
e¤ects that operate in our framework. The …rst e¤ect arises because, when segmentation is not
allowed, consumers from di¤erent markets (captive/competitive) are lumped together, and the
resulting density function exhibits a discontinuity. We call this e¤ect the discontinuity e¤ ect of
disallowing segmentation. As seen above, this e¤ect pushes the quality o¤ered to unambiguously
captive consumers downwards (leaving the qualities o¤ered to other consumers unchanged).20
Disallowing segmentation also generates another e¤ect – what we call the competition e¤ ect.
Similar to the discontinuity e¤ect, the competition e¤ect only a¤ects the qualities o¤ered to con-
sumers who are unambiguously captive. However, this e¤ect operates in the opposite direction.21
The basic idea behind the competition e¤ect is that the consumer bene…ts generated by competition
in the competitive market generate positive spillovers also for unambiguously captive consumers.
These positive spillovers are especially relevant, since they operate on the quality levels that are
o¤ered to these consumers. As a result, total surplus is also a¤ected. To see how this happens,
consider …rst high-valuation consumers (located at distance smaller than zim from the …rm). Since
20
Recall that here unambiguously captive consumers are those located at distance greater than zim .
21
This can be seen by comparing (16) and (17) with their counterparts derived in section 5.1.1, namely (18) and
(19). While for zi 2 [0; zim ] there is no di¤erence (in both cases quality allocations are identical to those in section
4), for zi 2 (zim ; s] the quality allocations in (17) exceed those in (19).

14
these include consumers belonging to the competitive market, competition between the two …rms
ensures that they obtain a good deal (i.e., yielding high utility) when purchasing the good. Hence,
for these consumers, competition brings a direct utility bene…t. The quality levels they are of-
fered at equilibrium are however the same as those that would be o¤ered by a monopolist, since
(as discussed in section 4) competition does not a¤ect the trade-o¤ faced by …rms when selecting
quality to o¤er these consumers – it only a¤ects the price. What about consumers with lower
valuations? These consumers are unambiguously captive, so for them competition brings no di-
rect bene…t. However, for these consumers competition does nonetheless have an indirect e¤ect.
Intuitively, since through competition high-valuation consumers are receiving high rents anyway,
the qualities o¤ered to low-valuation consumers can be raised without fear that they may attract
higher-valuation types, for whom they are not intended.22 In other words, competition for higher
valuation consumers softens the e¢ ciency/informational rents trade-o¤ faced by the …rm when
selecting the qualities o¤ered to lower-valuation consumers. This induces the …rm to raise the
qualities it o¤ers these consumers. Importantly, the competitive e¤ect arises because the …rm is
constrained to o¤er all consumers the same price/quality menu from which to select, and would
not arise otherwise.
Overall, therefore, quality allocations are determined by the interplay between two e¤ects –
namely the discontinuity e¤ ect and the competition e¤ ect – moving in opposite directions. The
discontinuity e¤ect lowers the qualities o¤ered to low-valuation, unambiguously captive consumers,
while the competition e¤ect pushes them upwards. Under assumption A1 – details can be found
in the appendix – the competition e¤ect is always su¢ ciently strong to ensure that the uncon-
strained quality schedule (given by (16) and (17)) exhibits an upward jump.23 However, since it
violates condition (IC.2), an upward jump in the quality schedule is incompatible with incentive
compatibility. The second-best quality allocation therefore prescribes pooling over some interval
[z0 ; z1 ], where z1 > 1=2 > z0 (since at equilibrium zim = 1=2). The interval [z0 ; z1 ] is derived by
trading o¤ allocative e¢ ciency and rent extraction with respect to the pooling interval as a whole.
Note that at equilibrium the quality allocations o¤ered to [z0 ; s] are above those that would be
o¤ered by a monopolist or if segmentation was allowed. Indeed, some types in (z0 ; s] may even be
o¤ered quality allocations that are above the …rst best level. In this case, disallowing segmentation
22
Recall that, from (IC.1), in our environment the informational rents that must be o¤ered to preserve incentive
compatibility are are always decreasing in zi . Firms must prevent high valuation consumers from understating their
valuations, not vice-versa.
23
The upward jump emerges because the competition e¤ect bites in a discontinuous fashion. It raises the qualities
o¤ered to consumers that are located exclusively in captive markets while it leaves unchanged the qualities o¤ered
to consumers that may be located in both the competitive and the captive market.

15
may generate a new type of distortions, namely upward distortions. The following proposition
summarizes our …ndings. A fuller characterization of the equilibrium conditions can be found in
the appendix.

Proposition 2 (Characterization of equilibrium contract when segmentation is not allowed and

s > 1=2:) When segmentation is allowed and s > 1=2, pooling always emerges at equilibrium;
that is, at equilibrium < 1=3, and the same quality q is o¤ ered to all types in [z0 ; z1 ], where
z0 < 0:5 < z1 and q satis…es
Z z1
1
( v 2zi + s (1 3 ) (1 I(zi )) q )( + I(zi ))dzi = 0: (20)
z0 s

where I(zi ) is given by (12). The qualities o¤ ered to zi 2 (z1 ; s] and zi 2 [0; z0 ), i=L,R are given
by (17) and (16), respectively. Moreover, ui (1=2) = q(v + 6 3:5) q 2 =2; i = L; R:

5.2 “Narrow” captive markets: s < 1=2

We now consider the case of narrow captive markets. This is rather di¤erent from the case where
captive markets are wide, seen above. In particular, at equilibrium the farthest consumer being
served by each …rm is located in the competitive market (at distance zim from the …rm’s location,
where you may recall that at equilibrium zim = 1=2). This eliminates the competition e¤ect, since
this e¤ect can only bite for those consumers located at distance greater than zim . The trade-o¤
faced by the …rm when selecting quality is therefore the same as in section 5.1.1, the only di¤erence
between the two cases being that here the marginal density of zi is given by (9) instead of (8). To
sum up, therefore, when s < 1=2, the competition e¤ect of disallowing segmentation disappears,
while the discontinuity e¤ect remains. However, this e¤ect takes a di¤erent form from the previous
section, since it alters the qualities o¤ered to consumers in (s; zim ], i.e. consumers unambiguously
located in the competitive market (while in the previous section it operated on the qualities of
consumers unambiguously located in the captive market).
The equilibrium quality allocations satisfy:
for zi 2 [0; s]

qi (zi ) = v 2zi (21)

for zi 2 (s; zim ]

qi (zi ) = v 2zi 1: (22)

16
 In this case, therefore –and in contrast with the case where s > 1=2 –the standard monopoly
result of underprovision of quality always carries through to the duopoly scenario. Note however
that here quality allocation exhibit a downward discontinuity at zi = s. This is a consequence of
the discontinuous density of consumers, described in (9). In contrast with what obtained in section
5.1, here this discontinuity does not violate condition (IC.2).

Proposition 3 (Characterization of equilibrium contract when segmentation is not allowed and

s < 1=2:) When segmentation is allowed and s < 1=2, pooling never emerges at equilibrium, and the
quality allocations are characterized by (21) and (22). Also, ui (1=2) = maxf0; (v 2) (v 5) =2g;
i = L; R:

6 Discussion: e¤ect of segmentation on quality allocations

Comparing the quality allocations that emerge under segmentation with those that emerge when
segmentation is not allowed, we see that the e¤ect of segmentation on quality changes depending
on the dispersion of preferences in the captive market –namely whether s > 1=2 or s < 1=2:
First, consider the case in which the captive market is relatively narrow, namely s < 1=2.
Letting the subscript NS indicate the case where segmentation is disallowed and the subscript S
indicate the case where segmentation is allowed, we have
8
< = q S (zi ) for zi s
i
qiN S (zi ) :
: < q S (z ) for z > s
i i i

Hence, segmentation has the e¤ect of increasing the quality allocated to all zi > s –while leaving
that of zi s unchanged. As discussed in section 5.2, this occurs because the introduction of
segmentation eliminates the discontinuity e¤ect, and therefore makes less stringent the trade-
o¤ between e¢ ciency and informational rents faced by the principal when selecting the quality
allocated to zi > s.
Now consider the case in which the captive market is relatively wide, namely s > 1=2. In that
case the equilibrium quality schedule exhibits pooling for all types in [z0 ; z1 ] –where z0 < 1=2 < z1
–and 8
< = q S (zi ) for zi z0
i
qiN S (zi ) :
: > q S (z ) for z > z
i i i 0

Here, segmentation has the e¤ect of decreasing the quality allocated to all zi > z0 –while leaving
that of zi z0 unchanged. Although segmentation eliminates the discontinuity e¤ect –something

17
that, caeteris paribus, increases quality –it also eliminates the competition e¤ect –something that,
caeteris paribus, decreases quality. As shown in section 5.1, this latter element is dominant. The
overall result is that, when s > 1=2, segmentation has a negative e¤ect on quality, as it strictly
lowers the quality o¤ered to consumers with su¢ ciently low valuations.

7 Welfare comparison between segmented and non-segmented régimes

Proposition 4 (E¤ ect of market segmentation on aggregate welfare.) When s > 1=2 disallowing
market segmentation raises aggregate welfare, while when s < 1=2 it lowers it.

As proposition 4 indicates, a one-to-one correspondence exists between the e¤ect of segmenta-

tion on equilibrium qualities, and its e¤ect on aggregate welfare: segmentation increases welfare
whenever it induces higher quality o¤ers, and reduces it otherwise. This is especially intuitive
when s < 1=2, since in that case disallowing segmentation lowers the quality o¤ered to a subset
of consumers (leaving the quality o¤ered to the rest unchanged). By contrast, when s > 1=2,
disallowing segmentation raises the qualities o¤ered to some consumers. However, this does not
necessarily mean that distortions are reduced. A subset of types may actually end up being o¤ered
qualities that are above their …rst-best levels. Hence, in this case the welfare e¤ect of segmentation
is unclear a priori. On the one hand, when segmentation is not allowed we obtain underprovision
of quality (as we would in a monopoly). On the other hand, when segmentation is allowed, we may
actually end up with quality overprovision for some consumers. Proposition 4 shows that from an
aggregate welfare perspective this latter scenario is always preferable to the former.
An implication of proposition 4 is that, empirically, the welfare e¤ect of segmentation can be
inferred by looking at the range of qualities o¤ered by the …rms. Consider s > 1=2. With
segmentation, the lowest quality o¤ered by each …rm at equilibrium is v 2zi , while without
segmentation is above v 2zi . The highest quality being o¤ered by both …rms is the same with
and without segmentation, and is equal to v. So, in this case, segmentation widens range of
qualities being o¤ered be the …rms. By contrast, when s < 1=2 segmentation narrows the range
of qualities being o¤ered be the …rms.24 Overall, therefore, our results suggest that segmentation
improves welfare whenever it narrows the range of qualities on o¤er, and it lowers welfare otherwise.
We now examine another issue of interest for policy-making when dealing with price segmen-
tation, namely its e¤ect on consumers located in di¤erent markets. The following proposition
24
With segmentation, the lowest quality o¤ered by each …rm is v 1, while without segmentation it is below that
value. The highest quality being o¤ered by both …rms is the same with and without segmentation, and is equal to v.

18
addresses this issue.

Proposition 5 (E¤ ect of market segmentation on consumer welfare). If market segmentation is al-
lowed, then consumers located in the competitive market become better o¤ , while captive consumers
become worse o¤ .

As proposition 5 indicates, segmentation always harms captive consumers, while it makes

consumers located in the competitive market better o¤. This shares some similarities with the
results obtained by the literature on third-degree price discrimination within the context of a
monopoly – such as e.g. Varian (1985) – where discrimination redistributes income away from
“low elasticity” consumers towards “high elasticity” consumers and the producer.
The rationale for the result is simple. First, consider captive consumers. It is easy to see
that they always bene…t from disallowing segmentation. In the competitive market, competition
induces the …rms to relinquish higher rents to consumers. If the …rms cannot discriminate, captive
consumers also bene…t from this. Those with zi 1=2 are o¤ered the same contracts that are also
o¤ered to consumers in the competitive market, and are therefore directly a¤ected by competition
there. Captive consumers with zi > 1=2 (if any) do not enjoy such direct bene…ts, since no
consumers in the competitive market share their low valuations. However, they nonetheless bene…t,
through the competition e¤ect highlighted in section 5.
Now consider consumers located in the competitive market. Disallowing segmentation makes
the …rms more reluctant to compete against each other by raising consumer rents. This is because
those higher rents also need to be o¤ered to captive consumers.25 As a result, without segmenta-
tion, the equilibrium value of u (1=2) is lower than with segmentation, something that hurts the
consumers.26 What about quality allocations? From the incentive compatibility constraint IC.1,
R 1=2
ui (zi ) = u (1=2) + zi qi (x)dx: For a given value of u (1=2), rents in the competitive market are
clearly a¤ected by qualities allocations in that market. Through the discontinuity e¤ect, when cap-
tive markets are narrow, segmentation raises the qualities o¤ered to consumers with zi 2 (s; 1=2].
So here segmentation bene…ts the consumers in the competitive market in two ways: it raises
u (1=2) and it also raises the qualities o¤ered to some of them. When captive markets are wide,
however, segmentation lowers the qualities o¤ered to consumers with zi > z0 . We have two con-
trasting e¤ects. On one hand, segmentation raises u (1=2), while on the other it lowers the qualities
o¤ered to some consumers. Proposition 5 shows that the former e¤ect is always dominant.
25
This shares similarities with example 2 in Galera and Zaratiegui (2006).
26
This is formally proved in the appendix (proof of proposition 5).

19
8 Concluding remarks

To our knowledge, this paper is the …rst to explicitly analyze the e¤ects of market segmentation
– namely, discrimination between captive and non-captive consumers – in a setting where …rms
compete by o¤ering non-linear price-quality contracts. We identify two e¤ects of disallowing seg-
mentation, a discontinuity e¤ect and a competition e¤ect, working in opposite directions. The
overall impact of segmentation on quality allocations – and welfare – depends on the interplay
between these two e¤ects. In turn, this depends on whether the captive market is “narrow” or
“wide”.
In this paper, captive markets are exogenously given. In reality, however, …rms may take actions
that a¤ect their captive consumers base. Future research could analyze the two-stage game where
…rms …rst select their captive base, and then compete by o¤ering price-quality menus.27 This
would also open up the possibility of asymmetric environments/equilibria. For instance, …rst-
movers may invest in creating a large base of captive consumers, in order to obtain a dominant
position (this is what Fudemberg and Tirole 1984 call a “top dog” strategy). Although in the
present paper we concentrate on symmetric environments and symmetric equilibria (primarily for
analytical convenience), the qualitative e¤ects we identify are independent of the precise details of
the model (such as for instance whether there is full symmetry between the …rms). Our analysis
therefore provides a useful building block for further research on the topic.

9 Appendix
Proof of proposition 1
Captive market The problem faced by the principal is a standard screening model, à la Mussa
and Rosen (1978). Firm i’s optimal contractual o¤er to consumers located in the competitive
market solves (2) subject to (IC.1), (IC.2) and participation: ui (zi ) 0 8zi 2 [0; s]. Given (IC.1),
the necessary and su¢ cient condition for participation is that ui (s) 0. Ignoring constraint (IC.2)
for the time being, we substitute for ui (zi ) using (IC.1) and operate integration by parts. The
…rm’s problem can then be written as
Z s
1 qi (zi )2
max qi (zi ) (v 2zi ) dzi :
qi (zi ) 0 3s 2

It is straightforward to show that the unconstrained optimal quality allocations are given by (3)
(and therefore satisfy (IC.2)).
27
For instance, …rms may incur expenditures that make switching costly for at least some of their consumers (as
in Tirole 1988, p. 326).

20
Competitive market Firm i’s optimal contractual o¤er to consumers located in the competitive
market solves (4) subject to (IC.1), (IC.2) and participation: ui (zi ) 0 8zi 2 [0; zim ]. Since ui (zi )
is non-increasing in zi , the necessary and su¢ cient condition for participation is that ui (zim ) 0,
which is true from the de…nition of zim . Ignoring constraint (IC.2) for the time being, we substitute
for ui (zi ) using (6) and operate integration by parts. The …rm’s problem can then be written as
Z zim
1 qi (zi )2 1
maxm qi (zi ) (v 2zi ) dzi Bi (zim ) zim :
qi (zi );zi 0 3 2 3

qi (zim )2
The …rst order condition28 with respect to zim is qi (zim ) (v 2zim ) 2 Bi0 (zim ) zim Bi (zim ) =
0: The unconstrained optimal quality allocations are given by (3) (and therefore satisfy (IC.2)).
Substituting for zim = 1=2; qi (zim ) = v 2zim = v 1 and allowing for symmetry, we obtain
ui (1=2) = (v 1)(v 2)=2, i = L; R:
Proof of proposition 2
As seen in the main text, when s > 1=2 …rm i’s equilibrium contractual o¤er must solve (11),
subject to (13), the captive consumer’s participation constraint, where zim is de…ned by ui (zim ) =
Bi (zim ). Condition (14) is necessary and su¢ cient condition for (13) to hold. Ignore constraint
(IC.2) for the time being. The Lagrangian for the problem is then
Z Z zi !
zim
1+s qi (zi )2
L = max (v zi ) qi (zi ) + qi (x)dx dzi +
qi (zi ), zim 0 3s 2 zim
Z Z zi !
s
1 qi (zi )2 Bi (zim ) m
(v zi 3 s) qi (zi ) + qi (x)dx dzi (zi + 1 3 )
zim 3s 2 zim 3

where 0 is the Lagrange multiplier for constraint (13). After integration by parts, this becomes
(15). The FOC for zim is
2 2
qi (kim ) qi+ (zim )
1+s
3s (v 2kim )qi (kim ) 2
1
3s (v 2zim + s 3s )qi+ (zim ) 2
(23)
Bi0 (zim ) m Bi (kim )
3 (zi +1 3 ) 3 =0

where qi+ (kim ) = lim"!0 qi (zim + ").29 The …rst order condition with respect to is

ui (s) = 0: (24)

With interior solutions, the unconstrained equilibrium quality allocations are given by (16) and
(17). These violate (IC.2) whenever < 1=3. In that case, the second-best quality schedule must
prescribe pooling over some range [z0 ; z1 ], where 0 z0 < 1=2 < z1 s. As shown e.g. by La¤ont
and Martimort (2002, chapter 3), the optimal pooling procedure must satisfy (20). There are three
28
It is straightforward to show that the second order condition for zim is always satis…ed under (IC.2).
29
The distinction between qi (zim ) and lim"!0 qi (zim + ") is meaningful because, as shown below, in the absence of
pooling the optimal quality schedule may exhibit a jump at zim .

21
possible scenarios: (i) pooling binds only on the left: q = v 2z0 for some 0 z0 < 1=2, and
q < v 2z1 +s(1 3 ) for all s z1 > 1=2; (ii) pooling binds only on the right: q = v 2z1 +s(1 3 )
for some s z1 > 1=2, and q > v 2z0 for all 0 z0 < 1=2; (iii) pooling binds on both sides:
q =v 2z0 for some 0 z0 < 1=2, and q = v 2z1 + s(1 3 ) for some s z1 > 1=2. Note
however that since z1 1=2, 0 and s 1, q v follows. Hence, at equilibrium, case (ii)
cannot arise; we must have q = v 2z0 for some 0 z0 < 1=2.
We now show that under A1 no equilibria without pooling can exist. This is because zim = 1=2 is
inconsistent with 1=3. Substituting for qi (zim ), qi+ (zim ) (from (16) and (17)), zim = 1=2 and
allowing for Bi (zim ) = ui (zim ) (for symmetry), (23) can be written as
1+s 1
(v 1)2 maxf(v 1+s 3s ); 0g2 3(v 1)(1=2 ) ui (1=2) = 0 (25)
2s 2s
Rs
where ui (1=2) = 1=2 maxf(v 2zi + s 3s ); 0gdzi .30 We consider three cases
Case 1: < (v s)=3s, so that v 2zi + s 3s > 0 for all zi s.
In this case, (25) can be rewritten as 1+s
2s (v
1
1)2 2s (v 1 + s 3s )2 3(v 1)(1=2 )
Rs
1=2 (v 2zi + s 3s )dzi = 0: Under A1 the values of that solve this equation are inconsistent
with (v s)=3s > 1=3.
Case 2: There exists a zb satisfying s > zb > 0:5 such that v 2b
z+s 3s = 0. Notice that
a necessary condition for this to hold is that < (v 1+ s)=3s.31 In this case, (25) becomes
1+s 1
R zb
2s (v 1)2 2s (v 1+s 3s )2 3(v 1)(1=2 ) 1=2 (v 1+s 3s )dzi = 0: Under A1
the values of that solve this equation are inconsistent with (v 1 + s)=3s > 1=3.
1+s
Case 3: (v 1 + s)=3s, so that ui (0:5) = 0. In this case, (25) becomes 2s (v 1)2
3(v 1)(1=2 ) = 0. Under A1 the values of that solve this equation are inconsistent with
(v 1 + s)=3s.
This proves that under A1 there are no equilibria without pooling. Hence, at equilibrium < 1=3,
and pooling emerges over some interval [z0 ; z1 ] . The quality allocations in (z0 ; s] are therefore
above those that would be o¤ered by a monopolist. Finally, the equilibrium value of ui (1=2),
i = L; R is found by imposing symmetry on (23).
Proof of proposition 3
When s < 1=2 …rm i’s equilibrium contractual o¤er must satisfy
Z s Z zim
1+s qi (zi )2 1 qi (zi )2
maxm (v zi ) qi (zi ) ui (zi ) dzi + (v zi ) qi (zi ) ui (zi ) dki
qi (zi );zi 0 3s 2 s 3 2

subject to (IC.1) and (IC.2).32 Ignore constraint (IC.2) for the time being. Substituting for ui (zi )
30
Rs
More precisely, ui (1=2) = ui (s) + 1=2
maxf(v 2zi + s 3s ); 0gdzi : However, from (24), when 1=3 > 0
then necessarily we must have ui (s) = 0.
31
This ensures that v 2zi + s 3s > 0 for zi = 1=2:
32
The captive consumer’s participation constraint, (13), is here automatically satis…ed, since ui (s) > ui (zim ) =
Bi (zim ) 0.

22
from (10), and after integration by parts, the problem becomes
Z s
1+s qi (zi )2
maxm (v 2zi ) qi (zi ) dzi +
qi (zi );zi 0 3s 2
Z zm
i 1 qi (zi )2 Bi (zim ) (1 + zim )
(v 2zi 1) qi (zi ) dki :
s 3 2 3

The unconstrained equilibrium quality allocations satisfy (21) and (22).33 The …rst order condi-
tion34 with respect to zim is

qi (zim )2
qi (zim )(v 2zim 1) Bi0 (zim ) (1 + zim ) Bi (zim ) = 0: (26)
2
qi (zim )2
Allowing for symmetry, we can rewrite (26) as qi (zim )(v 2zim 1) 2 qi (zim ) (1 +
zim ) ui (zim ) = 0. Substituting for qi (zim ) = v 2zim 1, zim = 1=2 we obtain ui (1=2) =
(v 2) (v 5) =2. If (v 2) (v 5) =2 > 0, this is the equilibrium value of ui (1=2). Otherwise,
ui (1=2) = 0.35

Proof of proposition 4
Part 1): When s > 1=2, aggregate welfare is higher when segmentation is not allowed.
(i) First, consider zi 2 [z1 ; s] (if any). For these types, the equilibrium quality allocation without
segmentation is qiN S (zi ) = v 2zi + s 3s , where < 1=3, while that with segmentation is given
by qiS (zi ) = v 2zi < qiN S (zi ). The di¤erence between surplus without segmentation and that
with segmentation is

qiN S (zi )2 qiS (zi )2

(v zi ) qiN S (zi ) (v zi ) qiS (zi ) + (27)
2 2
Substituting for qiN S (zi ) and qiS (zi ) in (27) we obtain 21 s (1 3 ) ( s + 2zi + 3s ) > 0, where the
last inequality follows from s 1, z1 1=2, 0. (ii) Second, consider zi 2 (z0 ; z1 ). For these
types qiN S (zi ) =q =v 2z0 , while qiS (zi ) =v 2zi < q. Substituting for these values in (27),
we obtain 2z0 (zi z0 ) > 0. (iii) Finally, for zi 2 [0; z0 ] (if any), the equilibrium quality allocation
with and without segmentation are equal. Surplus is therefore also equal.
Part 2): When s < 1=2, aggregate welfare is higher when market segmentation is
allowed.
From inspection of (21),(22) and (3), qiF B (zi ) qiS (zi ) qiN S (zi ) (the last inequality being strict
for some zi ). Hence, aggregate welfare is higher under segmentation.
Proof of proposition 5
33
It is easy to verify that under A1 these allocations are strictly positive and satisfy the monotonicity constraint
(IC.2).
34
It is easily checked that the second order condition for zim is always satis…ed.
35
Note that, when Bi (zi ) = 0 for all zi 1=2, it cannot be optimal for …rm i to set zim < 1=2: If zim < 1=2, the
qi (zim )2
lhs of (26) becomes qi (zim )(v 2zim 1) 2
> 0. Hence, the market is fully covered.

23
Part 1): Consumers located in the competitive market are better o¤ when segmen-
tation is allowed.
i) s > 1=2. Without segmentation utility is
( R z0
uN S (1=2) + q(1=2 z0 ) + zi (v 2x) dx if zi < z0
uN S (z i) =
uN S (1=2) + q(1=2 zi ) if 1=2 zi z0

where 1=2 zi since we are considering consumers in the competitive market. From proposition
2, uN S (1=2) = q(v + 6 3:5) 0:5q 2 ; and q = v 2z0 . In contrast, with segmentation utility is
Z 1=2
uS (zi ) = uS (1=2) + (v 2x) dx (28)
zi

where (from proposition 1) uS (1=2) = (v 1)(v 2)=2. Clearly, uN S (zi ) uS (zi ) is greatest for
zi < z0 , where it is equal to
Z 1=2
uN S (1=2) uS (1=2) + q(1=2 z0 ) (v 2x) dx. (29)
z0

Substituting for q = v 2z0 , (29) becomes 2v + 6z0 + 6v 12z0 vz0 0:75, which is negative
for all consistent with pooling (namely 1=3 > 0) and all z0 0.36
ii) s < 1=2. Without segmentation, utility is
( R 1=2
uN S (1=2) + zi (v 2x)dx if zi < s
uN S (zi ) = R 1=2 R zi :
uN S (1=2) + s (v 2x 1)dx s (v 2x)dx if zi s

where, from proposition 3, uN S (1=2) = maxf0; (v 2) (v 5) =2g. With segmentation, utility is

given by (28). Since uN S (1=2) uS (1=2) = maxf (v 1)(v 2)=2; 1 v=2g < 0 under A1,
uN S (z i) uS (z i) < 0 follows.
Part 2): Captive consumers are better o¤ when segmentation is not allowed
This follows because disallowing segmentation (i) (weakly) raises the quality levels o¤ered to all
captive consumers (and strictly raises the qualities o¤ered to some of them) and (ii) weakly in-
creases u(s). Under (IC.1), both (i) and (ii) raise the captive consumers’welfare.

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