Loyal Consumers and Bargain Hunters: Price
Competition with Asymmetric Information
Lester M.K. Kwong∗
Created: February 12, 2002
This Version: April 15, 2003
COMMENTS WELCOME
Abstract
This paper considers a homogeneous good Bertrand market with
asymmetric information. Consumers differ by a unidimensional type
space as well as grouped exogenously as being either a loyal consumer
or a bargain hunter. We find that under such an asymmetric information environment, the equilibrium will be supported by mixed strategies. Furthermore, we find that firms will compete via nonlinear price
schedules bearing similarities to a monopolist to the extent that a “flattening” out effect occurs. More importantly, competition occurs in the
reservation utility of the marginal consumer.
∗
Ph.D. program in economics, University of British Columbia. 997-1873 East Mall,
Vancouver, British Columbia, Canada V6T 1Z1. Email: mkit@interchange.ubc.ca. I
would like to thank Sukanta Bhattacharya, Patrick Francois, Thomas Ross, Kazutaka
Takechi, Markus von Wartburg, Okan Yilankaya, participants in the IO study group and
in the microeconomics lunch seminar, at the University of British Columbia, and especially
Gorkem Celik and Guofu Tan for helpful comments and suggestions. I am also grateful
to the A.D. Scott Fellowship in Economics for financial support.
1
Introduction
The theory of monopoly pricing and the practice of price discrimination has
received a lot of attention from economists because of their welfare implications. While inefficiencies arise due to conflicts of interest between the
firm and consumers, there are methods of minimizing such inefficiencies.
For example, if the willingness to pay by each consumer in the market is
observable, then a monopolist may implement a perfect price discriminating
policy to extract all consumer surplus. Under such a pricing scheme, all inefficiencies are eliminated. However, if the willingness to pay is unobservable,
nonlinear pricing schemes may still be implemented for the same purpose of
consumer surplus extraction. One way of modelling such environments is to
allow consumers to differ in their willingness to pay usually represented by
a unidimensional type space.1 Then, a firm, given some prior belief over the
probability distribution of this taste parameter may construct a nonlinear
pricing scheme to discriminate between different types of consumers.
Such pricing strategies are, therefore, means for extracting information
from consumers regarding their tastes. Furthermore, the implementation of
such nonlinear pricing schemes require little more than the usual assumptions of microeconomic theory. The growth of informational economics and
mechanism design in the past couple of decades provide us with insights into
the construction of such pricing mechanisms. In particular, with the aid of
the revelation principal, the focus of the modeler on the space of strategies,
in which the optimal mechanism lies within, is reduced. This allows for
much simplifications from a theoretical point of view.
The study of nonlinear pricing to monopoly markets has been studied
extensively by Mussa and Rosen (1978), Goldman, Leland, Sibley (1984),
and by Maskin and Riley (1984), to name a few. And the extension into
oligopolistic markets have been studied by Oren, Smith and Wilson (1983,
1984), Ivaldi and Martimort (1994), Stole (1995), Hamilton and Thisse
(1997), and Rochet and Stole (2002). A conclusion one may draw from
such extensions is that a theory for competing mechanisms brings about
complications absent in monopoly markets. For example, a pricing mechanism for one firm may include messages, from the consumer, regarding the
mechanism its opponent is offering them, in addition to the revelation of
their private information. As a result, the structure of such mechanisms will
1
It is for theoretical simplicity that a unidimensional type space is considered. Extensions beyond the unidimensional case has been studied by Armstrong (1996), Armstrong
and Rochet (1999), Laffont, Maskin and Rochet (1987), McAfee and McMillan (1988),
and Rochet and Choné (1998).
1
have an inherent recursive nature. One way to over come such issues, as
the authors above have noted, is by introducing differentiated products into
the market.2 Therefore, by introducing a spatially competitive oligopolistic
setting, a simplification of the problem is attained.
On the other hand, Mandy (1992) considered a pure homogeneous good
Bertrand game where firms compete via nonlinear price schedules. The
equilibrium characterized in his study suggests that when firms may enter
freely into the market, then firms’ profits are driven to zero. However,
the characterization of the equilibrium is incomplete which leads to the
possibility that consumers may be segmented, in terms of which firm they
purchase from, by types. Alternatively, when entry is restricted, firms may
earn positive profits. The profitability of firms is solely determined by the
structure of costs. However, much like the previous case, the exact price
schedules offered by each firm may not be characterized.
Consequently, some general questions that may be of interest in environments of price competition with asymmetric information is whether the
introduction of a second firm into the market disrupts the pricing behavior of
a monopolist. From the mechanism design literature, it is known that there
exists an incentive compatible direct mechanism if there is a single principal.
However, when the number of principals are increased, the existence of such
a mechanism is, generally, unknown.3 In other words, will there be incentives for the second firm to charge, for example, a uniform price so as to
disrupt the incentive compatible direct mechanism a monopolist may offer?
Heuristically, such a strategy is feasible so long as there exists a large number
of consumers on the low end compensated well enough by the uniform price.
But then, one may predict the incumbent’s mechanism to be suboptimal if
such segmentation of the market is to be expected. Alternatively, one may
propose the question regarding the distribution of consumers, in terms of
types, among the firms in the market. For example, will one firm specialize in serving all high end consumers while another firm serves the low end
consumers? More specifically, will consumers be partitioned according to
their types in equilibrium? Such questions, in our opinion, are of particular
interest as competing mechanisms are often an observed phenomenon.
In attempts to provide a partial answer to the questions above, we ad2
This is with the exception of Oren, Smith and Wilson (1983, 1984) who took a different
route and modelled nonlinear pricing in terms of a Cournot model.
3
This issue was addressed in Epstein and Peters (1999) in which they examined the
revelation principal with a language capable of describing mechanisms. In view of their
findings, the model developed in this paper will be one in which the mechanisms a seller
may offer is restricted to those absent of this language.
2
dress the issue of competing mechanisms from a different approach. We
adopt a structure, similar to the framework from previous studies such as
Rosenthal (1980) and Varian (1980), where each firm has access into two
different markets; a captive market and a competitive market. On the assumption that firms cannot distinguish between consumers from the two
markets, the benefits from this assumed structure is obvious; the existence
of the captive markets will restrict the amount of competition in the competitive market. In other words, a zero profit equilibrium, from undercutting,
will not result.
This paper purports to address the question of nonlinear price competition in the framework described above. The remainder of the paper is as
follows. Section 2 will set up the basic framework. We then analyze this
framework under a duopoly setting in Section 3. In Section 4, we analyze
some extensions to the model. More specifically, we consider the case when
the size of the captive market goes to zero and when we allow for more firms
in the market by considering a generalized n-firm oligopoly market. Finally,
some concluding remarks will follow in Section 5.
2
The Basic Framework
Assume a continuum of consumers in the unit interval [0, 1] with
unit density.
Associated with each consumer is a taste parameter θ ∈ θ, θ ≡ Θ ⊂ R+
which is drawn from a continuously differentiable distribution function F (θ)
with density f (θ). Nature draws a θ independently for each individual according to F (θ) and the realization of θ is private information for each
consumer. Furthermore, consumers are assumed to fall into one of the two
groups; loyal consumers and bargain hunters. A loyal consumer is here defined as one whose objective is to consume the product from their current
firm, not necessarily the one with the lowest price. We assume loyal consumers are distributed evenly among all the firms in the market. In contrast,
a bargain hunter is one who seeks to consume the product at the lowest available price. Consequently, a bargain hunter is well informed regarding the
prices of all firms in the market.4
At the outset, this assumption may seem restrictive, but we may interpret a loyal consumer to have very high search costs whereas a bargain
hunter does not. Consequently, the distribution of prices among all firms is
4
Alternatively, one may think of loyal consumers as uninformed consumers while bargain hunters as informed ones. Under this interpretation, an informed consumer is well
informed about the prices in the market while uninformed ones are not.
3
known to bargain hunters but not to loyal consumers. We further assume
consumers only purchase from one firm. As a result, consumers are unable
to split their total consumption between the two firms in the market. This,
again, although restrictive, greatly simplifies the analysis to follow.5
Let α ∈ [0, 1] denote the proportion of consumers who are bargain
hunters and 1 − α denote the proportion of consumers who are loyal. We
assume that the distribution of θ, F (θ), as well as the proportion of loyal
consumers and bargain hunters, given by α, is common knowledge to all
firms in the market. However, the actual type of each consumer remains
unknown to the individual firms.
Due to the nature of the asymmetric information, firms in the market will
have incentives to price nonlinearly. Firms in the market are homogeneous
and produce with a constant marginal cost of c. Denote Ti (q) as the price
schedule offered by firm i where q denotes quantity. Firms choose Ti (q)
simultaneously after which consumers make their purchase decision.
Utility for the consumer depends on his taste parameter, θ, as well as
his total consumption of the good q.6 Let u : Θ × R+ → R be the utility
function for a consumer. We assume that the utility function satisfies the
properties:
∂u (θ, q)
∂u (θ, q)
∂ 2 u(θ, q)
∂ 2 u (θ, q)
> 0;
> 0;
<
0;
>0
∂θ
∂q
∂q 2
∂θ∂q
for all θ ∈ Θ and q ∈ R+ .
3
A Duopoly Market
Given the assumption of a homogeneous good market and price competition,
this implies that we essentially have a Bertrand price competition game. As
in the case of the monopoly, a consumer maximizes utility by choice of her
consumption level. However, because there are two firms in the market, the
distinction between bargain hunters and loyal consumers is not trivial.7
5
An alternative approach is to interpret each firm as offering a class of groups differing
in quality and each consumer simply demands one good at most. Under this interpretation,
the assumption seems justified.
6
Note that the utility of a consumer only depends on the taste parameter but not on
being a bargain hunter or a loyal consumer. The additional utility from being a bargain
hunter is the cost savings from consumption which we assume to be equal to search costs.
As a result, consumers are loyal or bargain hunters by nature and have no “real” strategic
advantages over loyal consumers.
7
The reason for this distinction to be trivial in the case of the monopoly is that a
monopolist has no incentives to differentiate between the two groups of consumers. In
4
With the assumption that α < 1, each firm in the market is guaranteed
(1 − α)/2 number of consumers each period.8 The remaining α share is
where competition between the two firms takes place. Given the objective
of a loyal consumer, we may obtain a consumption rule as a function of θ
by her optimization problem. We define qiL (θ) as:
qiL (θ) ≡ arg max u (θ, q) − Ti (q)
q
(1)
where L denotes a loyal consumer and i represents the firm the loyal consumer is loyal to. A bargain hunter’s consumption decision, on the other
hand, involves two stages. Aside from the decision on how much to consume,
he must also choose from which firm to consume. Therefore, a typical θ-type
bargain hunter’s optimization problem may be expressed as:
max max u (θ, q) − T1 (q) , max u (θ, q) − T2 (q)
(2)
q
q
This optimization problem may be solved by a two stage process. More
specifically, for any firm i ∈ {1, 2} in the market, a bargain hunter will have
an optimal consumption rule. We denote this as qi (θ) and define it as:
qi (θ) = arg max u (θ, q) − Ti (q)
q
(3)
Clearly then, the reduced form optimization problem, with Eq. 3, may be
written as:
max {u (θ, qi (θ)) − Ti (qi (θ))}
(4)
i∈{1,2}
The solution to this optimization problem, Eq. 4, may not be unique but
a solution necessarily exists, for all θ ∈ Θ. So, given the two price schedules
offered by the two firms, we may define each firm’s respective segment of
bargain hunters by Θi . More formally:
Θi = θ ∈ Θ | max {u (θ, qj (θ)) − Tj (qj (θ))} = i
(5)
j∈{1,2}
fact, by definition, a bargain hunter is a loyal consumer if there only exists one firm in the
market.
8
While each firm is guaranteed this fraction of consumers each period, identifying the
consumer as being loyal or as a bargain hunter is not possible. As a result, the same price
must be offered to all consumers who purchase from the same firm.
5
u 6
u(θ, q2 (θ)) − T2 (q2 (θ))
HH
HH
j
HH
Y
H
H
u(θ, q1 (θ)) − T1 (q1 (θ))
-
|
{z
Θ1
}|
Θ1,2
{z
Θ2
}
θ
Figure 1: Θ1 and Θ2 given T1 and T2
A graphical representation of the partition of Θ into Θ1 and Θ2 is given
in Figure 1. While the figure shows a nice simple partition of Θ into Θ1 and
Θ2 , ignoring Θ1,2 with zero measure, in general this need not be the case.
Furthermore, the utility level in terms of θ is shown to be smooth, which,
again, may not necessarily be the case. This may especially be true if the
optimal price schedules by the two firms are discontinuous. Furthermore,
Θ1 ∩ Θ2 ≡ Θ1,2 may have positive measure but Θ1 ∪ Θ2 = Θ. We may interpret Θ1,2 as the θ-type bargain hunters who are indifferent to consuming
from either firm. Let Θi,−j = Θ \ Θj be the θ-type bargain hunters who
strictly prefers consuming from firm i. Note that by construction, the subsets of Θ into each firm’s respective market segment is solely determined by
the price schedules offered by each firm. Consequently, an equilibrium may
be defined by this characterization. Therefore, given two price schedules T1
and T2 , we may define a bargain hunter’s optimal consumption rule as:
q1 (θ), if θ ∈ Θ1
B
q (θ) =
(6)
q2 (θ), if θ ∈ Θ2
Claim 1 Given T1 and T2 and the induced partition of Θ:
1. for all θ ∈ Θi , q B (θ) = qiL (θ).
Furthermore, if for some i ∈ {1, 2};
2. Θi = ∅, then for all q B (θ) > 0, Tj < Ti with j 6= i.
6
3. Θi ∈
/ {∅, Θ} for all i ∈ {1, 2}, then there exists some q, q 0 ∈ R+ such
that T1 (q) ≥ T2 (q) and T1 (q 0 ) ≤ T2 (q 0 ) with at least one inequality
strict.
Proof: See appendix.
The above claim states that for all θ ∈ Θi 6= ∅, all θ-type consumers,
regardless of being a bargain hunter or a loyal consumer, will choose to
consume the same quantity level from firm i. Furthermore, we note that the
price schedule offered by firm i must be uniformly higher than that of firm
j’s over the set of quantity levels the bargain hunters are consuming at if
Θi = ∅. This further drives the result that if both Θi and Θj are nonempty
and not equal to Θ, then there must exist at least two quantity levels for
which T1 (q) ≥ T2 (q) and T1 (q 0 ) ≤ T2 (q 0 ) with at least one inequality strict.
This idea that the choice set for a bargain hunter is larger than that
of the loyal consumer is important for the analysis to follows. This implies
that if a firm offering Ti as their price schedule does not capture the θ ∈ Θj
bargain hunters, then simply replicating Tj over the quantity levels such
θ-type bargain hunters are consuming will not change q B (θ) for all θ ∈ Θi .
Consequently, Claim 1 allows us to examine deviations when analyzing an
equilibrium in a simple and systematic way.
Define C P as the set of all possible piecewise continuous price schedules.
More formally:
C P = {T : R+ → R+ | T is piecewise continuous}
(7)
Since C P may be interpreted as the set of all possible tariffs offered by each
firm, given our restrictions, it is essentially the set of pure strategies for
each firm. Alternatively, a mixed strategy
R is given by a measurable function
∆i : C P → [0, 1] with the property that T ∈C P ∆i (x) dx = 1 and ∆i (x) ≥ 0
for all x ∈ C P . We seek a Nash equilibrium in the price schedules offered
by each firm.9
Given T1 and T2 , each firm’s market segment of the bargain hunters is
given by the partition of Θ, expected profits for each firm may be expressed
as:
Z
Z
α
L
πi = α
(Ti (θ) − cqi (θ))f (θ)dθ +
(Ti (θ) − cqiL (θ))f (θ)dθ
2
Θi,−j
Θi,j
Z
1−α
(Ti (θ) − cqiL (θ))f (θ)dθ
(8)
+
2
Θ
9
Note that we have not restricted our attention to purely nonlinear price schedules as
the class of functions C P does not eliminate the uniform price as a possibility.
7
where Ti (θ) ≡ Ti (qiL (θ)). In equilibrium, each firm’s choice of Ti maximizes
Eq. 8.
Since for all Ti ∈ C P , qiL (θ) is well defined, we therefore, let C P to be the
set of all possible price schedules a firm may offer evaluated at the induced
demand of its loyal consumers, qiL (θ). More specifically:
C P = {T ◦ q L : Θ → R+ | T ∈ C P and q L = arg max u(θ, q) − T (q)}
q
(9)
It is clear that there exists a one to one mapping between C P and C P . Define
T to be the set of monetary transfer rules of all incentive compatible direct
mechanisms. Then:
T = T : Θ → R+ | for all θ, θ0 ∈ Θ, U (θ, θ) ≥ U θ, θ0
(10)
where U (θ, θ0 ) ≡ u(θ, q(θ0 )) − T (θ0 ). It is clear that T ⊂ C P . Note that all
incentive compatible direct mechanisms are well defined.10 In fact, for all
T ∈ T , they must differ only in one dimension. More importantly, they differ
by a degree of a constant given by the constant of integration.11 This, may
be interpreted as the reservation utility for the marginal consumer which we
define to be u
b. We characterize our first result in the following proposition.
Proposition 1 Given α ∈ (0, 1), there exists no pure strategy equilibrium.
Proof: See appendix.
In establishing this result, we first considered the idea that the price
schedules offered by the two firms will not, roughly speaking, intersect, in
equilibrium. In particular, we rely on two specific deviations in our proof
of this proposition. More specifically, we consider a deviation for firm i by
replicating firm j’s tariff over the quantity levels for which bargain hunters
purchase from firm j. From Claim 1, we know that if θ ∈ Θj,−i then such
θ-type bargain hunters purchase from firm j. Furthermore, Ti (qjL (θ)) >
Tj (qjL (θ)) for all θ ∈ Θj,−i . So in equilibrium, the profits obtained from the
proportion of loyal consumers who are consuming such quantity levels must
be strictly greater than those obtained by replicating such portions of the
price schedule.
In addition, we consider a deviation by firm i, and firm j by replicating
Tj and Ti , respectively, over all q ∈ R+ . If both deviations are jointly
unfeasible then it cannot be the case that profits for a firm from selling only
10
11
See, for example, Maskin and Riley (1984).
See, for example, Laffont (1989).
8
to θ-type loyal consumers is strictly greater than replicating its competitors
tariff over such quantity levels.
This, therefore, suggests that it must be the case Θi ∈ {∅, Θ} in any
pure strategy equilibrium. In other words, no equilibrium where bargain
hunters are segmented between the two firms, in pure strategies, may be
supported. This further allows us to consider equilibria where such segmentation does not occur. We may further deduce that with the existence of
a group loyal consumers, profits must be bounded above zero. Therefore,
motives to undercut an opponent’s price schedule, uniformly, by some arbitrary always exist. As a result, an equilibrium where Θi = Θj = Θ occurs
is not rationalizable.
This result, in essence, allows us to reduce the set of pure strategies from
C P to T since equilibrium behavior necessarily implies an incentive compatible direct mechanism. In other words, an equilibrium in pure strategies
necessarily implies that Ti ◦ qiL , Tj ◦ qjL ∈ T and that competition is for all of
the θ-type individuals. As a result, the individual firm must offer a contract
as prescribed by the monopolist.12
This further implies that in the duopoly case, all competition must occur
in u
b since any functional changes in the optimal tariff will not be incentive
compatible and will, therefore, be suboptimal. As a result, the proof for the
nonexistence of a pure strategy equilibrium follows nicely from the discontinuities in the profit function. Because undercutting an opponent’s tariff,
uniformly, by an arbitrarily small amount will cause jumps in the profit
function due to monopolization of the bargain hunting group of consumers,
profits will be driven to zero. However, in such a process of undercutting,
the inability to differentiate between bargain hunters and loyal consumers
essentially eliminates all monopoly power. Consequently, the loss of all
monopoly power is not an equilibrium in this model, since an outside option
with strictly positive profits exists. This corresponds to the case when a
firm forgoes competition for bargain hunters and simply monopolizes loyal
consumers. Due to Proposition 1, equilibrium, in so far as it exists, will be
in mixed strategies.
In specifying a mixed strategy equilibrium, one essentially has to consider
all strategies within the set C P , the set of all piecewise continuous price
schedules, which makes characterization cumbersome. Therefore, we provide
the following lemma to simplify this process.
12
This is by the assumption that a direct incentive compatible mechanism is the optimal
mechanism for a monopolist in this environment.
9
Lemma 1 In any mixed strategy equilibrium, if S(G1 ), S(G2 ) ⊂ C P are the
equilibrium supports for firms 1 and 2, then no two pure strategies Tsi , Tkj ∈
{S(G1 ) ∪ S(G2 )} are such that Θs,k is a countable set, for s, k = 1, 2 except
when Θs,k = ∅. Furthermore, Θs,k is a convex set.
Proof: See Appendix.
The implications of Lemma 1 is that firms will not randomize with
price schedules such that the resulting distribution of bargain hunters is
segmented. This follows from the proof that Θs,k is convex. In other words,
whenever two price schedules, T s and T k , intersect, then Θs ⊂ Θk or vice
versa. In addition, we may establish the following corollary.
Corollary 1 In any mixed strategy equilibrium, T ∈ T restricted to the
b ⊂ Θ such that for all θ ∈ Θ,
b T (θ) − cq(θ) ≥ 0. Furthermore,
domain Θ
b and T (Θ \ Θ)
b = c · q(Θ \ Θ).
b
Θs,k ≡ {Θ \ Θ},
Proof: To establish this corollary, consider, first, the strategies such that
for all T k , T s ∈ S(Gi ), i = 1, 2, with Θk,s = ∅, they lie within the space C P .
Define the set of such strategies as S and assume T ∈ S but T ∈
/ T . Since
such strategies are uniformly above or below one another, sup{S} = T is well
defined.13 Let T M be the optimal price schedule offered by the monopolist.
Then, it is clear that for all T ∈ S, T ≤ T M for all θ ∈ Θ. Then, if T ∈
/ T,
expected profits can be improved by simply charging T M so T ∈ T . It is
then clear that for every T ∈
/ T but in S a deviation to some T 0 ∈ T exists
by the idea that Θk,s = ∅ for all T ∈ S.14
Now consider a strategy T = T M − k such that ∃θ ∈ Θ for which
b =
T (θ) − cq(θ) < 0. It is evident that a deviation to offering T 0 where T 0 (Θ)
0
b
b
b
T (Θ) and T (Θ \ Θ) = cq(Θ \ Θ) is profitable since incentive compatibility
b is preserved and zero profits are earned by all θ ∈ {Θ \ Θ}
b
for all θ ∈ Θ
15
types.
Given the structure of the mixed strategy equilibrium, we may, therefore,
simply restrict our attention to strategies that satisfy Corollary 1. In other
words, we may simply consider price schedules T ∈ T with the “flattening”
13
By this we mean the highest price schedule offered in equilibrium.
This follows since for all T, T 0 ∈ T , |T − T 0 | = k, for all θ ∈ Θ, since maps within T
differs only by a constant of integration.
15
b is preserved since the price schedule is
Note that incentive compatibility for all θ ∈ Θ
“flattened” out for lower θ-types at a higher price than along the unrestricted map T ∈ T .
b
Furthermore, we do not claim that incentive compatibility is preserved over θ ∈ {Θ \ Θ}
types. However, their behavior is irrelevant as zero profits are earned from such θ-types
anyways.
14
10
out property along θ-types where T (θ) − cq(θ) ≤ 0. This is depicted in
Figure 2.
T (q(θ))
6
T (q(θ))
@
@
R
6
u
b
?
c
b
q(θ)
q(θ)
-
q(θ)
Figure 2: The “flattening” out property
Let Gi (b
ui ) be firm i’s equilibrium distribution function over u
bi with
density gi (b
ui ) and S (Gi ) denote the support of this distribution.16 Let
πiM (b
ui ) be monopoly profits from offering u
bi to the marginal consumer.
Then we may establish S (Gi ) in the following lemma.
Lemma 2 In equilibrium, S (Gi ) = [0, u] for i ∈ {1, 2} where u satisfies the
equality:
1+α
1−α
M
πi (u) =
πiM (0)
(11)
2
2
where:
πiM (b
u)
Z
=
Θ
(T (θ) − cq L (θ) − u
b)I(T (θ) − cq(θ) − u
b ≥ 0)f (θ)dθ
(12)
and I(·) is the indicator function.
Proof: See Appendix.
The idea behind lemma 2 is that a firm will not wish to exhaust all
profits since there is always an outside option of not pursuing bargain hunters
because they may exercise monopoly power over loyal consumers. Therefore,
16
Essentially, a firm’s mixed strategy is ∆i (Ti ) where Ti ◦ qiL ∈ T . But given that for all
T, T 0 ∈ T , they only differ by a constant, we abuse notation and simply refer to a mixed
strategy as Gi (b
ui ) to imply the randomization given by ∆i .
11
the region over which firms will randomize their price schedule must not
yield profits lower than this outside option. The objective of the firm, given
that it offers an incentive compatible price schedule, faces the following
maximization problem:17
Z u
max
πis Gj (b
ui ) + πif (1 − Gj (b
ui )) gi (b
ui ) dui
(13)
gi (b
ui ) 0
subject to:
Z
0
u
gi (b
ui ) dui = 1; gi (b
ui ) ≥ 0 for all u
bi ∈ S (Gi )
πis (b
ui ) Gj (b
ui ) + πif (b
ui ) (1 − Gj (b
ui )) ≥ π for all u
bi ∈ S (Gi )
(14)
(15)
where πis and πif represent firm i’s profits when it successfully offers the
lowest nonlinear tariff and when it does not, respectively. More specifically,
πis and πif as:
1+α M
πi (b
ui )
(16)
πis (b
ui ) =
2
1−α M
πi (b
ui )
(17)
πif (b
ui ) =
2
The constraints in Eq. 14 are the boundary conditions of the equilibrium density function and those in Eq. 15 may be interpreted as the firm’s
profit maximizing constraint. Note that π is the minimum profit the firm
is guaranteed, which is equal to the expected profits from serving only the
loyal group of consumers. In other words, π ≡ πif (0).
The solution to the optimization problem, Eq. 13, is obtained trivially
since we may exploit the indifference condition of any two pure strategies
within S (Gi ). By construction, equilibrium strategies will be symmetric,
and in particular, in mixed strategies.
Proposition 2 In equilibrium, both firms will randomize their price schedules according to the distribution function:
M
1−α
πi (0) − πiM (b
ui )
2
Gi (b
ui ) =
(18)
απiM (b
ui )
for i ∈ {1, 2}.
17
We may express the optimization problem in this way since we have already restricted
our attention to incentive compatible price schedules. Therefore, a more formal way to
express this is with the condition that Ti ◦ qiL ∈ T over θ ∈ Θ such that T (θ) − cq(θ) ≥ 0
and Ti = c · qi otherwise.
12
Proof: See Appendix.
The interpretation of the equilibrium distribution, Eq. 18, is quite simple. In equilibrium, the frequency at which competition for the bargain
hunters, with changes in u
bi , is determined by the gains of successfully capturing the α share of the market and by the losses due to increases in u
bi in
the remaining (1 − α)/2 share of the market in which the firm has monopoly
power.
Corollary 2 The equilibrium in Proposition 2 is the unique equilibrium of
the game.
Proof: This is immediate given Corollary 1.
From the analysis above, we derived a symmetric mixed strategy equilibrium in nonlinear price schedules. Some interesting properties of this
equilibrium suggests that the price schedules offered by both firms will be
pseudo-incentive compatible, in terms of a direct mechanism, over the whole
of Θ.18 Consequently, the existence of the loyal group of consumers is sufficient in disciplining the equilibrium from segmentation. Furthermore, due
to the degree of freedom in the price schedules given by u
b, this is where
competition for consumers exists. In essence, the problem considered here
reduces down to a single dimension.
4
Extensions
In this section of the paper, we consider some basic extensions that may be
of interest in this context. More specifically, we will focus on a generalization
to an arbitrary n number of firms as well as examining the equilibrium of the
limiting game when α approaches one. This corresponds to a game where
there exists no loyal consumers. We begin by analyzing the case when α
approaches one.
4.1
When α Approaches One
The question regarding the equilibrium derived in Proposition 2 when α
approaches one is one of particular interest. This corresponds to a game
where a captive market, or no loyal consumers, exist. Correspondingly, this
18
By this, we mean that incentive compatibility holds, in terms of a direct mechanism,
over θ-type consumers in which positive rent is extracted.
13
will be a game of pure Bertrand competition without a positive outside
option.
Note that by taking the limit of the definition of the equilibrium support,
given by Lemma 2, one finds that:
Z
π M (u) = (T (θ) − cq L (θ) − u
b)I(T (θ) − cq(θ) − u
b ≥ 0)f (θ)dθ = 0
Θ
Furthermore, taking the limit of the equilibrium distribution function given
by Eq. 18, we find that:
lim Gi (b
ui ) = 0
α→1
This implies that as α approaches one, all the probabilities are transferred to
the upper-bound of the equilibrium support. This is intuitively plausible. As
the number of loyal consumers decreases, the gains from competition for the
bargain hunters increase. Put in a different way, the loss from competition
due to the loss of monopolization over loyal consumers, decreases. As a
result, firms compete more rigorously by placing higher probabilities on
the upper-bound of the equilibrium support of the distribution function.
In the limit, the flattening out process, as suggested by the mixed strategy
equilibrium developed in the preceding section implies that price is flattened
out to marginal cost for all θ ∈ Θ. In fact, this is also the unique equilibrium
of the limiting game. We state this formally in the following proposition.
Proposition 3 If α = 1, then the unique equilibrium is that for all i ∈
{1, 2}, Ti = c · q.
Proof: See Appendix.
We now turn our attention by examining a more general case of a n-firm
oligopoly type market.
4.2
n Number of Firms
In extending the model to include n number of firms, note that each firm
captures (1−α)/n number of loyal consumers. The objective of the loyal consumer remains unchanged but the choice set of a bargain hunter increases.
We maintain our notation and denote qi (θ) as the induced demand for a
θ-type bargain hunter facing the price schedule Ti (q) from firm i. Similarly,
we may define a partition of Θ given the set of price schedules offered by the
n firms, {Ti }ni=1 . Let N = {1, 2, . . . , n} be the set of firms in the market.
Then:
Θi = θ ∈ Θ | max{u(θ, qj (θ)) − Tj (qj (θ))} = i
(19)
j∈N
14
Similarly, a bargain hunter’s optimal consumption rule is, therefore:
q B (θ) = qi (θ), if θ ∈ Θi
(20)
We now state the equivalence of Claim 1 under this more general environment.
Claim 2 Given {Ti }ni=1 and the induced partition of Θ:
1. for all θ ∈ Θi , q B (θ) = qiL (θ).
Furthermore, if for some i ∈ N ;
2. Θi = ∅, then for all q B (θ) > 0, Ti > min{Tj }nj=1 .
3. Θi ∈
/ {∅, Θ} for all i ∈ N , then there exists a vector {qi }i∈N ∈ Rn+ ,
such that Ti (q) ≤ min{Tj }j6=i for all i ∈ N , with at least one inequality
strict.
Proof : The proof is analogous to that of Claim 1 and is thus omitted here
for brevities.
j
Define Θi to be the θ-type bargain hunters who are indifferent to purchasing from exactly j number of firms with firm i being one of them. More
formally:
Θji = {θ ∈ Θi | ∃!N 0 ⊂ N with #{N 0 } = j and θ ∈ ∩s∈N 0 Θs }
(21)
Then profits for firm i may be expressed as:
πi =
Z
n
X
α
j=1
j
1
(Ti (θ)−cqiL (θ))f (θ)dθ+
j
Θi
−α
n
Z
(Ti (θ)−cqiL (θ))f (θ)dθ (22)
Θ
The remaining analysis in the case of n firms is similar to that of the
duopoly and we summarize the results in the following proposition.
Proposition 4 For n < ∞, the equilibrium may be characterized by the
following four properties.
1. No firm will employ a pure strategy.
2. Firms will randomize over strategies as prescribed by Corollary 1.
15
3. The support of the mixed strategy is given by S(Gi ) = [0, u], for all
i ∈ N , where u solves:
1 − (n − 1)α
1−α
πiM (u) =
πiM (0)
n
n
where πiM (u) as in Eq. 12.
4. Each firm randomizes according to the distribution function:
Gi (b
ui ) =
1−α
n
! 1
πiM (0) − πiM (b
ui ) n−1
απiM (b
ui )
Proof: See Appendix.
As can be seen, one may construct a similar equilibrium, to the case
of the duopoly, taking on a n-firm oligopoly type market. As the number
of firms increases, the proportion of loyal consumers each firm will capture
will decrease. This is evident if one fixes the market demand to the unit
interval and distributes (1 − α)/n number of loyal consumers to each firm.
Consequently, since the support of the equilibrium distribution, S (Gi ), is
determined by this allocation of loyal consumers to each firm, this will inevitably change the equilibrium support for each firm’s strategies.
An interesting question that one may ask, at this conjuncture, is what
will happen when n approaches infinity. It is intuitively clear that as n
approaches infinity, the number of loyal consumers each firm has approaches
zero. Similar to the case when α approaches one, by taking the limit of the
equilibrium support, we find that the upper bound implies:
πiM (u) = 0
This is consistent to the case when α approaches one as both cases suggest
that each firm has no loyal consumers. Similarly, by taking the limit of the
equilibrium distribution function, we find that:
lim Gi (b
ui ) = 1
n→∞
This implies that as n increases, the probabilities each firm place on the
lower-bound of their equilibrium support increases. This contrasts the case
when α approaches one since the probabilities, there, are transferred to
the upper-bound. While the interpretation of α approaching one and n
approaching infinity is similar, (i.e., the number of loyal consumers each
16
firm has approaches zero,) the difference lies in the gains and the losses
due to competition. As mentioned above, as α approaches one, The gains
from competition for bargain hunters increases. On the other hand, as n
approaches infinity, the number of loyal consumers each firm has decreases.
However, the probability of being able to offer the lowest price schedule also
decreases. Consequently, the probability of offering something in the upperbound of the equilibrium support goes down and the probability of offering
something in the lower-bound increases.
5
Conclusion
This paper has derived an equilibrium in a homogeneous good duopoly market with asymmetric information. Consumers are assumed to have private
information regarding their type, which corresponds to the realization of
some θ ∈ Θ, as well as being classified into two groups: loyal consumers
and bargain hunters. The equilibrium derived is a symmetric mixed strategy equilibrium where firms randomize their price schedules in hopes of
capturing all of the bargain hunters. An important factor in deriving this
equilibrium is the assumption that firms must offer the same schedule to
both groups of consumers.19 This may be translated into the scenario in
which the two groups are indistinguishable when transactions take place in
the market.
Furthermore, it has been shown that both firms will offer a price schedule
which partially coincides to that of an incentive compatible direct mechanism through a “flattening” process. Clearly then, introduction of entrants
into an environment considered in this paper will not highly disrupt the
incentive compatibility of mechanisms offered.
While we have only analyzed the equilibrium of a static game, one may
interpret the mixed strategy as the randomization of the price schedules each
firm will offer in each of an infinitely, or finitely, repeated game. However,
one explicit assumption must be made, namely, firms will not collude in the
extended game.
Another important point to note is that the assumption of the exogenous grouping of consumers given by α, suggests an equilibrium where only
positive profits are sustained. As shown above, without such an assumption,
19
See, for example, Rosenthal (1980) and Varian (1980). Alternatively, if the two groups
of consumers are distinguishable by the firm, then the equilibrium may be trivially determined as the optimal nonlinear price schedule, in terms of a monopolist’s problem, is
offered to loyal consumers while pure Bertrand competition for the bargain hunters will
occur (i.e., T = c · q for all θ ∈ Θ).
17
or if one takes α to be equal to 1, then we essentially derive the outcome
of a Bertrand game where a zero profit, with price equalling marginal cost,
equilibrium occurs.
To extend this model to capture the notion of an arbitrary n number
of firms, we derived a similar equilibrium. Previous studies such as Varian
(1980) and Rosenthal (1980) have shown a similar equilibrium in markets
with n number of firms in the absence of asymmetric information with consumers differing by types. The difference in the results lie in the pricing
for the goods. In our case, equilibrium was in a price schedule which is
incentive compatible, in the direct mechanism sense, where in the others,
only a uniform price is considered. The general results of this model seem
to be consistent with the studies cited here. The crucial assumption behind
such results though, lies in the inability of firms to differentiate between the
captive market and the competitive market. This, in essence forces them to
offer the same price between the two markets.
18
A
Appendix
We provide the proofs of our results presented in the paper.
Proof of Claim 1:
1. Suppose that for some θ ∈ Θi , q B (θ) 6= qiL (θ). This implies that there
exists some other bundle q 0 = q B (θ) or q 0 = qiL (θ) with the associated
transfer Ti (q 0 ) such that it yields higher utility for either the loyal consumer
or the bargain hunter. Note that the associated transfer must be from firm
i since θ ∈ Θi . Clearly then, such a choice of q 0 is available for both loyal
consumers and bargain hunters so if one finds it optimal to deviate so must
the other type which contradicts the assumption that there exists some
θ ∈ Θi such that q B (θ) 6= qiL (θ).
2. Suppose Θi = ∅ but that for some θ ∈ Θj with q B (θ) > 0, Tj > Ti . Then a
deviation to consuming from firm i exists which contradicts the assumption
that Θi = ∅.
3. If Θi , Θj ∈
/ {∅, Θ} then some bargain hunters consume from firm 1 and
some from firm 2. If there does not exist some q, q 0 ∈ R+ such that T1 (q) ≥
T2 (q) and T1 (q 0 ) ≤ T2 (q 0 ) with at least one inequality strict, then from part
2, this implies that for some i ∈ {1, 2}, Θi = ∅ or that Θi = Θj , which
contradicts the assumption that Θi ∈
/ {∅, Θ}.
Proof of Proposition 1: To establish this proposition, we begin by first
showing that the two firms will not, in equilibrium, offer two price schedules
T1 and T2 such that Θi ∈
/ {Θ, ∅} for i = 1, 2.
Suppose given Ti and Tj , Θi , Θj ∈
/ {∅, Θ}. Then firm i’s profits may be
written as:
Z
Z
α
L
πi = α
(Ti (θ) − cqi (θ))f (θ)dθ +
(Ti (θ) − cqiL (θ))f (θ)dθ
2
Θi,−j
Θi,j
Z
1−α
+
(Ti (θ) − cqiL (θ))f (θ)dθ
(23)
2
Θ
Clearly then, if the pair (Ti , Tj ) is an equilibrium, then firm i will not find
it profitable to deviate by replicating Tj over the set qjB (θ) > 0. Therefore,
we derive the inequalities that for all i ∈ {1, 2}:
Z
Z
1−α
1
L
(Ti (θ) − cqi (θ))f (θ)dθ ≥
(Tj (θ) − cqjL (θ))f (θ)dθ (24)
2
2 Θj,−i
Θj,−i
Alternatively, we may consider a deviation by firm i by replicating Tj over
all q ∈ R+ . Such a deviation is unprofitable if and only if:
Z
1
πi ≥
(Tj (θ) − cqjL (θ))f (θ)dθ
(25)
2 Θ
19
Eq. 25 must also hold for firm j 6= i if the pair (Ti , Tj ) is, indeed, an
equilibrium. Therefore, we have:
Z
Z
L
(Ti (θ) − cqi (θ))f (θ)dθ +
(Tj (θ) − cqjL (θ))f (θ)dθ
Θi,−j
Θj,−i
Z
Z
L
−
(Ti (θ) − cqi (θ))f (θ)dθ −
(Tj (θ) − cqjL (θ))f (θ)dθ ≥ 0 (26)
Θj,−i
Θi,−j
Clearly, this cannot be true given Eq. 24 and thus, such a partition of
Θ is not possible. Therefore, if an equilibrium exists, for all i ∈ {1, 2},
Θi ∈ {∅, Θ}.
Therefore, if a pure strategy equilibrium exists, then Θi ∈ {Θ, ∅} for
i = 1, 2. Suppose that Θi = Θj = Θ. Then Ti = Tj and profits are such that
πi = πj . As a result, if such an equilibrium exists, it must be unprofitable
for one firm to undercut by > 0. This amounts to the condition:
Z
α
(Ti (qiB (θ)) − cqiB (θ))f (θ)dθ > 0
(27)
≥
2 Θ
which, by assumption, is not possible since α > 0 and maybe arbitrarily
chosen.
Therefore, suppose that given Ti and Tj , Θi = ∅ and Θj = Θ. Then
all bargain hunters purchase from firm j. Clearly then profit maximization
requires that Ti ◦ qiL ∈ T . Similarly, profit maximization requires that
Tj ◦ qjL = Ti − ◦ qiL ∈ T for some > 0.
The above implies that all competition between firms 1 and 2 must
occur in u
b1 and u
b2 . Suppose these are equilibrium values and, without
loss of generality, assume that u
b1 < u
b2 .20 By continuity, ∃ > 0 such
that u
b1 < u
b2 − and that profits for firm 2 increase. Clearly then, a
unilateral deviation for firm 2 exists, violating the notion of an equilibrium.
Therefore, suppose that u
b1 = u
b2 . Then each firm captures exactly half of
the market. Again, by continuity, ∃ > 0 such that for firm i that increases
u
bi to u0i = u
bi + will capture the whole of the bargain hunters. This gain
in profits is clearly greater than the loss due to the change so long as
profits, given u
bi , are not zero. In the case that πi (b
ui ) = 0, this contradicts
profit maximization since offering a contract with u
bi = 0 yields strictly
positive profits. This may be accomplished by forgoing competition for
bargain hunters and by monopolizing loyal consumers. Therefore, unilateral
20
Roughly speaking, one could imagine that u
bi is so low such that for some θ ∈ Θ,
negative profits are earned. Consequently, we consider price schedules which flatten out
over θ ∈ Θ whenever T (θ) − c · q(θ) < 0 along the total cost curve.
20
deviations exists for all values of u
b1 and u
b2 proving the nonexistence of a
pure strategy equilibrium. Alternatively, one may think of a Bertrand price
competition game with a strictly positive outside option.
i
Proof of Lemma 1: First note that if Θs,k is countable, then Ts ∈ S(Gs )
and Tkj ∈ S(Gk ) intersects at least once at unique points over the relevant
domain.21 The proof of this, then, is analogous to that of Proposition 1. By
definition of a mixed strategy equilibrium, all pure strategies in the support
will yield the same level of expected profits. Therefore, consider any two
price schedules offered by firms s, k = 1, 2 such that Θs 6= Θk ∈
/ {Θ, ∅}.
j
j
i
i
Define Θs (Ts , Tk ), with s 6= k Ts ∈ S(Gs ) and Tk ∈ S(Gk ), to be the
θ ∈ Θ type bargain hunters that will purchase from firm s given Tsi and Tkj .
Θs,−k (Tsi , Tkj ) and Θs,k (Tsi , Tkj ) are similarly defined. Therefore, consider the
strategy Tsi for firm s. Expected profits, given Gk is:
Z
i
πs (Tsi , Tkj )gk (Tkj )dTkj
(28)
E(πs |Ts , Gk ) =
Tkj ∈S(Gk )
A similar expression may be derived for another pure strategy Tsl ∈ S(Gs ).
Then, first consider the case when Tsi and Tsl are such that Θi (Tsi , Tsl ) ∈
/
{Θ, ∅}.22 We can define an ordering over a partition, indexed by Λ, of Θ such
that either Θλs (Tsi , Tkj ) ⊆ {Θλs (Tsl , Tkj ) ∪ ∅} or {Θλs (Tsi , Tkj ) ∪ ∅} ⊇ Θλs (Tsl , Tkj )
S
for all Tkj ∈ S(Gk ) and λ∈Λ Θλs = Θ. Then consider the pure strategy Tes
where:
(
Tsl if Θλs (Tsi , Tkj ) ⊆ {Θλs (Tsl , Tkj ) ∪ ∅}
(29)
Tes =
Tsi if {Θλs (Tsi , Tkj ) ∪ ∅} ⊇ Θλs (Tsl , Tkj )
It is clear that Tes takes the lower envelope of the two functions Tsi and Tsl .
Then, by the logic of the proof of Proposition 1, E(πs |Tsi , Gk ) = E(πs |Tsl , Gk ) ≥
E(πs |Tes , Gk ) provides a contradiction, unless Θλs (Tsi , Tkj ) ⊂ Θλs (Tsl , Tkj ), or
vice versa, for all λ ∈ Λ and for all Tkj ∈ S(Gk ) which further implies that
Tsi and Tsl are uniformly above or below, or partially overlaps, Tkj for all
Tkj ∈ S(Gk ).
For the remainder of the first part of the Lemma, we are left to show
that if one firm randomizes price schedules, Tsi ∈ S(Gs ), then the other
firm will not randomize with any schedules, Tkj ∈ S(Gk ) such that both
Θs , Θk ∈
/ {Θ, ∅}, for all Tkj ∈ S(Gk ) and for all Tsi ∈ S(Gs ). Again, this
21
By this, we mean over R+ such that qzj (θ) > 0 for z = 1, 2.
Since both Tsi and Tsl are offered by the same firm, we define Θi (Tsi , Tsl ) as the set of
θ ∈ Θ who will prefer Tsi over Tsl if they were both available.
22
21
may be established in a manner similar to that of Proposition 1. Without
loss of generality, suppose Tkj ≥ Tkl over the relevant domain. Then for
any Tsi , Θk (Tkj , Tsi ) ⊆ Θk (Tkl , Tsi ). Consider πs (Tkj , Tsi ) and πs (Tkl , Tsi ) and
assume they are not pointwise equal in Tsi . If there exists a Tsi ∈ S(Gs )
such that πs (Tkj , Tsi ) = πs (Tkl , Tsi ), then consider a deviation to Tbk such
that Tbk = Tkl over Θk (Tkl , Tsi ) and Tbk = Tks elsewhere.23 It follows that
πs (Tbk , Tsi ) > πs (Tkj , Tsi ) = πs (Tkl , Tsi ) at the point Tsi . Furthermore, for all
Tsv ∈ S(Gs ) such that Θk (Tkl , Tsi ) ⊂ Θk (Tkl , Tsv ), πs (Tbk , Tsi ) ≥ πs (Tkl , Tsi )
in Tsi . Conversely, for all Tsv ∈ S(Gs ) such that Θk (Tkl , Tsi ) ⊃ Θk (Tkl , Tsv ),
the inequality πs (Tbk , Tsi ) ≥ πs (Tkj , Tsi ) holds in Tsi .24 Therefore, a deviation
in pure strategy exists. In the case where there does not exist any Tsi ∈
S(G ) such that πs (Tkj , Tsi ) = πs (Tkl , Tsi ), then choose Tsi ∈ S(Gs ) such that
R s j i
|πs (Tk , Ts ) − πs (Tkl , Tsi )|dx, over the relevant domain, is minimized. A
similar argument now completes the proof.
The contradiction derived in the above proof requires both Θl , Θs ∈
/
{Θ, ∅}. Therefore, convexity of Θs,k immediately follows if we allow for
either Θl ∈
/ {Θ, ∅} or Θs ∈
/ {Θ, ∅} since non-convex sets will generate Θk
and Θs otherwise.
Proof of Lemma 2: Suppose u
bi < 0, then the individual rationality constraint will not hold. Therefore, in equilibrium, it must be the case that
u
bi ≥ 0. Suppose u
bi > u. Then the maximum profits a firm receives will
be lower than if it simply monopolizes the loyal group of consumers and
does not compete for bargain hunters.25 This is clearly not profit maximizing behavior, thus establishing an upper-bound on the support of Gi in
equilibrium for all i ∈ {1, 2}.
Proof of Proposition 2: A mixed strategy must yield the same profits
for all pure strategies in its support in order for it to be an equilibrium,.
Therefore, it is without loss of generality that we restrict our attention to
[0, u]. The expected profits for any u
bi ∈ [0, u] are given by:
E (πi | u
bi ) = πis Gj (b
ui ) + πif (1 − Gj (b
ui ))
(30)
By the firm’s individual profit maximizing constraint, Eq. 15, expected
Similarly, if πs (Tkj , Tsi ) and πs (Tkl , Tsi ) are pointwise equal in Tsi , then consider any
S(Gs ).
These inequalities follow since if Θk (Tkl , Tsi ) is an increasing set in Tsi , then Θk (Tbk , Tsi )
is nondecreasing in Tsi and vice versa. Consequently, πs (Tbk , Tsi ) is nondecreasing in all
directions of Tsi if, in fact, Tkl , Tkj ∈ S(Gk ) is, indeed, a mixed strategy equilibrium.
25
Maximum, here, is interpreted as in the case that the firm does offer the lowest price
schedule and captures all of the bargain hunters.
23
Tsi ∈
24
22
profits must be at least equal to π. Therefore:
E (πi | u
bi ) ≥ π
(31)
In equilibrium, this condition must bind with equality since if E (πi | u
bi ) > π,
then firm j may simply reduce u
bj until this condition binds with equality.
Therefore, rearranging Eq. 30 yields the equilibrium distribution for firm j.
Conversely, firm i’s equilibrium distribution function may be determined by
the indifference condition of firm j. The symmetry of this problem makes
the solution trivial.
Note that the nonexistence of an optimal deviation is guaranteed by
Lemma 1. Consequently, off equilibrium strategies may be disregarded.
Proof of Proposition 3: For α = 1, the proof for the nonexistence of
two pure strategies which yield Θ1 6= Θ2 , both with positive measure, in
Proposition 1 still holds. Thus, for all i ∈ {1, 2}, Θi ∈ {∅, Θ}. So if Θi = ∅,
then from Condition 2 of Lemma 1, we know that for all θ ∈ Θj such that
qjB (θ) > 0, Tj < Ti . Furthermore, πi = 0, and πj ≥ 0. In fact, πj = 0 since if
πj > 0, then there exists some > 0 such that firm i may deviate by setting
Ti = Tj − and earn positive profits.
Now, note that if for some θ0 ∈ Θ, qjB (θ0 ) > 0 and that Tj (qjB (θ0 )) −
cqjB (θ0 ) > 0, then there must exist some θ 6= θ0 such that Tj (qjB (θ)) −
cqjB (θ) < 0. Since Tj (qjB (θ)) < Ti (qiB (θ)), by definition, firm j will not
deviate by setting Tj (qjB (θ)) = Ti (qiB (θ)) if the profits from the permutated
price schedule, Tbj , is lower. This is only possible if Tbj (b
qjB (θ)) − cb
qjB (θ) ≤
Tj (qjB (θ)) − cqjB (θ) < 0 where qbjB is the induced demand from Tbj . Note
that Tj (b
qjB (θ)) < Ti (b
qjB (θ)) since otherwise, a type θ individual will switch
and firm j makes positive profits. So given any increases on portions of the
tariff by firm j where negative profits are earned, such θ-types individuals
will always maintain on such portions, a uniform increase over that curve
is always feasible. But that implies that setting Tbj = c · q is also possible
over all q such that for all θ ∈ Θ, Tj (qjB (θ)) − cqjB (θ) < 0. But then, this
is a contradiction since zero profits are earned from such θ-type consumers.
Consequently, there cannot exist any θ-type for which firm j is making
positive profits from and thus, profits must be pointwise equal to zero over
all q for which qjB (θ) > 0. A simple Bertrand argument now completes the
proof and is therefore, omitted.
Proof of Proposition 4:
1. We prove this first by establishing that no firms will employ pure strategies such that, in equilibrium, Θi ∈
/ {Θ, ∅}. We proceed by induction.
23
Suppose given {Ti }i∈N , there exists a firm, j ∈ N such that Θj ∈
/ {∅, Θ}.
Then there must exist Θ−j ≡ Θ \ Θj such that for all θ ∈ Θ−j , the bargain
hunters will not purchase from firm j. Therefore, for all θ ∈ Θ−j , the firm
will only get loyal hunters. Clearly, no deviations exist for firm j if and only
if:
(Z
)
Z
L
L
(Tj (θ) − cqj (θ))f (θ)dθ ≥ max
(Ti (θ) − cqi (θ))f (θ)dθ (32)
i∈N,i6=j
Θ−j
Θ−j
Similarly, if there exists a firm s such that Θ−j ⊆ Θ−s , then Tj = Ts over
θ ∈ Θ−j by the above inequality. So, without loss of generality, assume
there are 1 ≤ m ≤ n − 1 firms using Tj and n − m firms using TB where TB
captures all the bargain hunters over Θ−j .26 Define:
Z
πi (Θ−j ) =
(Ti (θ) − cqiL (θ))f (θ)dθ
Θ−j
Then the following inequalities must hold:
1−α
α
α
πB (Θ−j ) ≥
(πj (Θ−j ) − πB (Θ−j )) ≥
πB (Θ−j ) (33)
n−m
n
n−m+1
From Proposition 1 we have already shown that this cannot hold for
n = 2. Therefore, suppose there does not exist a m such that 1 ≤ m ≤ n − 1
such that Eq. 33 is true. This necessarily implies that either:
α
1−α
πB (Θ−j ) <
(πj (Θ−j ) − πB (Θ−j ))
n−m
n
(34)
α
1−α
πB (Θ−j ) >
(πj (Θ−j ) − πB (Θ−j ))
n−m+1
n
(35)
is true or else:
is true. Then, consider the case of n + 1. Note that if Eq. 34 is true, then
there exists a m such that:
m<n−
πB (Θ−j )
αn
1 − α πj (Θ−j ) − πB (Θ−j )
(36)
Then if no deviations for the case of n + 1 is possible, then the inequality:
α
1−α
πB (Θ−j ) ≥
(πj (Θ−j ) − πB (Θ−j ))
n−m+1
n+1
26
b −j ⊆ Θ−j
This is without loss of generality as we may simply focus on a subset Θ
where this statement is true.
24
must hold. But that implies that:
πB (Θ−j )
α
>n
1 − α πj (Θ−j ) − πB (Θ−j )
which would imply m < 0 for Eq. 36 to hold. Therefore, suppose Eq. 35 is
true. Then this implies that there exists a m such that:
m>n+1−
πB (Θ−j )
αn
1 − α πj (Θ−j ) − πB (Θ−j )
(37)
is true. If no deviations for the case of n + 1 is possible, then the inequality:
n + 2 − (n + 1)
πB (Θ−j )
αn
≥m
1 − α πj (Θ−j ) − πB (Θ−j )
is true. This implies that:
1>
πB (Θ−j )
α
1 − α πj (Θ−j ) − πB (Θ−j )
But then this further implies that m > n from Eq. 35 which is not possible
since by assumption, 1 < m < n − 1. Therefore, Eq. 33 for the case of n + 1
cannot hold. Consequently, by the principal of mathematical induction,
Θj ∈
/ {∅, Θ} cannot occur if {Ti }ni=1 are equilibrium price schedules offered
by the n firms.
This suggests that for all i ∈ N , Ti ◦qiL ∈ T . A simple Bertrand argument
completes the proof.
2. The proof of Lemma 1 can be made invariant to the number of firms,
j
provided n < ∞, simply by replacing Tkj by T−s
and Gk by G−k to denote a
vector of price schedules offered by all n firms less firm s, and the vector of
distribution functions by all n firms less firm s, respectively. Consequently,
Corollary 1 follows immediately for the n firm case.
3 and 4. The proofs of Lemma 2 and Proposition 2 can easily accommodate
n firms, thus, the proof is omitted.
25
References
[1] Armstrong, Mark (1996), “Multiproduct Nonlinear Pricing.” Econometrica, 64, 51-75.
[2] Armstrong, Mark and Jean-Charles Rochet (1999), “Multi-dimensional
Screening: A User’s Guide.” European Economic Review, 43, 959-979.
[3] Epstein, Larry G. and Michael Peters (1999), “A Revelation Principal
for Competing Mechanisms.” Journal of Economic Theory, 99, 119-160.
[4] Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their
Applications. 2nd Ed. New York: John Wily & Sons, Inc.
[5] Goldman, M. Barry, Hayne E. Leland, and David S. Sibley (1984),
“Optimal Nonuniform Prices.” Review of Economic Studies, 51, 30519.
[6] Guesnerie, Roger and Jean-Jacques Laffont (1984), “A Complete Solution to a Class of Principal-Agent Problems with an Application to
the Control of a Self-Managed Firm.” Journal of Public Economics, 25,
329-69.
[7] Hamilton, Jonathan H. and Jacques-François Thisse (1997), “Nonlinear
Pricing in Spatial Oligopoly.” Economic Design, 2, 379-97.
[8] Ivaldi, M. and David Martimort (1994), “Competition Under Nonlinear
Pricing.” Annales d’Economie et de Statistique, 34, 71-114.
[9] Kaplansky, Irving (1977), Set Theory and Metric Spaces. 2nd Ed. New
York: Chelsea Publishing Company.
[10] Kwong, Lester M.K. (2003), “Coupons and “Everyday Low Prices”:
Price Competition with Multiple Pricing Instruments.” Canadian Journal of Economics, forthcoming.
[11] Laffont, Jean-Jacques (1989), The Economics of Uncertainty and Information. Cambridge: MIT Press.
[12] Laffont, Jean-Jacques, Eric Maskin, and Jean-Charles Rochet (1987),
“Optimal Nonlinear Pricing with Two-Dimensional Characteristics,” in
Information, Incentives and Economic Mechanisms, ed. by Theodore
Groves, Roy Radner, and Stanley Reiter. Minneapolis: University of
Minnesota Press.
26
[13] Laffont, Jean-Jacques and David Martimort (2002), The Theory of Incentives: The Principal-Agent Model. New Jersey: Princeton University
Press.
[14] Mandy, David M. (1992), “Nonlinear Bertrand Competition.” Econometrica, 60, 1293-1330.
[15] Maskin, Eric and John Riley (1984), “Monopoly with Incomplete Information.” Rand Journal of Economics, 15, 171-96.
[16] McAfee, R. Preston and John McMillan (1988), “Multidimensional Incentive Compatibility and Mechanism Design.” Journal of Economic
Theory, 46, 335-54.
[17] Mussa, M., and S. Rosen (1978), “Monopoly and Product Quality.”
Journal of Economic Theory, 18, 301-17.
[18] Oren, Shmuel, Stephen Smith and Robert Wilson (1983), “Competitive
Nonlinear Tariffs.” Journal of Economic Theory, 29, 49-71.
[19] Oren, Shmuel, Stephen Smith and Robert Wilson (1984), “Pricing a
Product Line.” Journal of Business, 57, S73-S100.
[20] Stole, Lars (1995), “Nonlinear Pricing and Oligopoly.” Journal of Economics and Management Strategy, 4, 529-62.
[21] Rochet, Jean-Charles and Philippe Choné (1998), “Ironing, Sweeping,
and Multidimensional Screening.” Econometrica, 66, 783-826.
[22] Rochet, Jean-Charles and Lars Stole (2002), “Nonlinear Pricing with
Random Participation.” The Review of Economic Studies, 69, 277-311.
[23] Rosenthal, Robert W. (1980), “A Model in Which an Increase in the
Number of Sellers Leads to a Higher Price.” Econometrica, 48, 1575-79.
[24] Tirole, Jean. The Theory of Industrial Organization. 1988, Cambridge:
The MIT Press.
[25] Varian, Hal R. (1980), “A Model of Sales.” American Economic Review,
70, 651-59.
27