Dynamic Pricing: When To Entice Brand Switching And When To
Reward Consumer Loyalty∗†
Yongmin Chen‡
Jason Pearcy§
December 4, 2013
Abstract
This article develops a theory of dynamic pricing in which firms may offer separate prices
to different consumers based on their past purchases. Brand preferences over two periods are
described by a copula admitting various degrees of positive dependence. When commitment
to future prices is infeasible, each firm offers lower prices to its rival’s customers. When firms
can commit to future prices, consumer loyalty is rewarded if preference dependence is low, but
enticing brand switching occurs if preference dependence is high. Our theory provides a unified
treatment of the two pricing policies, and sheds light on observed practices across industries.
JEL Classification: D2, L1.
Keywords: Dynamic pricing, brand switching, consumer loyalty, preference dependence.
∗
We thank Mark Armstrong, Igal Hendel (the Editor), two referees, and seminar participants at Tulane University, University of Arizona, University of Colorado, University of Southern California, University of Toronto, and
participants at the 2007 IIOC in Savannah and the 2007 Econometrics Society Meetings at Duke University for
helpful comments and suggestions. Pearcy acknowledges financial support from the Committee on Research Summer
Fellowship and the Research Enhancement Fund at Tulane University.
†
c
Author Posting. Jason
Pearcy and Yongmin Chen 2010. This is the author’s version of the work. It is posted
here for personal use, not for redistribution. The definitive version was published in RAND Journal of Economics,
Vol. 41, No. 4, Winter 2010, pp. 674–685. http://dx.doi.org/10.1111/j.1756-2171.2010.00116.x
‡
University of Colorado at Boulder and Zhejiang University; [email protected]
§
Montana State University; [email protected]
1
Introduction
In markets with repeat purchases, it is common for a firm to offer different prices to its repeat
customers and customers who switch from a rival. Some industries are popularized by marketing
programs that entice consumers to switch from rivals. For example, long-distance telephone companies often offer lower prices to rivals’ consumers, satellite companies sometimes target cable users
with lower prices, and credit card companies frequently offer lower rates to switching customers.
In some other industries, however, the prevailing practice is for firms to reward consumer loyalty.
Frequent flier programs in the airline industry and frequent stay programs by hotels are familiar
examples.
These pricing practices have received wide attention in economics.1 Chen (1997), Fudenberg
and Tirole (2000), Taylor (2003), and Villas-Boas (1999), among others, have studied pricing practices that pay customers to switch or poach rivals’ customers. A familiar theme of this literature
is that enticing brand switching is the equilibrium outcome when oligopoly firms can price by consumers’ purchase history; the resulting price discrimination reduces industry profits and often, but
not always, benefits consumers. In a different direction, Banerjee and Summers (1987), Caminal
and Matutes (1990), and Caminal and Claici (2007), among others, have studied loyalty-inducing
arrangements, such as frequent flier programs. This second literature assumes that firms can make
certain future price commitments to their repeat customers, which may or may not intensify competition.
While the existing economics literature has provided valuable insights on firms’ dynamic pricing behavior in markets with repeat purchases, some important issues remain to be addressed.
For instance, when commitment to future prices is not possible, Fudenberg and Tirole (2000) find
that firms reward switching consumers if preferences are perfectly dependent (i.e., constant) across
periods, but there is no price discrimination if intertemporal preferences are independent; whereas
when commitment is allowed Caminal and Matutes (1990) find that firms reward consumer loyalty
if preferences are independent across periods. An important unanswered question is whether and
1
Recent economics literature has used various terms to describe these practices, including behavior-based price
discrimination, price discrimination by purchase history, dynamic price discrimination, or dynamic pricing (e.g.,
Armstrong (2006); Fudenberg and Villas-Boas (2006); and Stole (2007)).
1
in what form dynamic price discrimination will arise when, more generally, preferences are neither
independent nor perfectly dependent. Furthermore, the existing studies have taken it as given that
firms either commit to future prices (but do not set prices based on consumers’ past purchases)
or make no future commitment.2 As such, they do not explain why some industries reward brand
switching while some other industries reward customer loyalty. The theory developed in this paper
will connect and extend the literature by considering a more general form of intertemporal preference dependence and by allowing firms to choose endogenously whether to commit to future prices
or to make no such commitment (but retain the ability of ex post price discrimination). This theory
will then shed light on the different pricing practices across industries.
Our basic model follows Fudenberg and Tirole (2000) by considering a two-period duopoly with
horizontal product differentiation, where firms cannot commit to future prices. To allow for a
more general dependence relation of brand preferences across periods, we make use of a copula, a
function that combines or couples univariate marginal distributions to a multivariate distribution.3
For any given marginal distribution of consumer preferences in each period, consumers’ preference
dependence between periods monotonically increase in a single parameter, α ∈ [0, 1], with α = 0 and
α = 1 corresponding to the special cases of independence and complete dependence, respectively.
We find that, at any symmetric equilibrium, positive intertemporal preference dependence (i.e.
α > 0) is necessary and sufficient for firms to engage in dynamic pricing in favor of switching
consumers, and the enticement for brand switching increases monotonically in α. Our analysis
further reveals how preference dependence matters for equilibrium profits and consumer surplus if
the marginal distribution in each period is uniform: (i) industry profit varies non-monotonically
with α, increasing when α is small but decreasing when α is large; and (ii) compared to the
benchmark where price discrimination by purchase history is not possible, and in contrast to the
familiar theme in the literature, the practice of enticing brand switching increases industry profits,
except when α is at least 0.8973.4
2
An exception is Fudenberg and Tirole (2000), who also consider the case where firms can offer both short-term and
long-term contracts, but again under the assumption that intertemporal preferences are either perfectly dependent
or independent.
3
Copulas have been developed in the statistics literature (Joe, 1997; Nelson, 1999), and they provide a convenient
way of studying scale-free measures of dependence between random variables.
4
We have also obtained qualitatively similar results if the marginal distribution is a beta distribution, Beta(2,2).
2
We also study an extended model in which each firm can, in period 1, choose to commit to
a future price for its repeat customers. The nature of the industry equilibrium again depends
crucially on intertemporal preference dependence. The industry equilibrium is such that it rewards
consumer loyalty if α is small but entices brand switching if α is large.
Our results shed light on the different forms of dynamic pricing in different industries. In the
airline industry, for instance, a consumer’s preference for different airlines may change substantially between two trips, depending on the availability and schedule of flights to possibly different
destinations. Our theory then suggests that the industry equilibrium will involve loyalty programs
that make consumers less willing to switch between airlines. On the other hand, for long-distance
telephone carriers, it seems that a consumer’s preference is unlikely to change from one period to
another. In addition, there could be switching costs which generate (or reinforce) a strong positive
preference dependence across periods. In this case, the industry equilibrium is more likely to involve
paying consumers to switch.
The rest of the paper is organized as follows. Section 2 sets up the basic model, Section 3
characterizes its equilibrium, Section 4 studies the extended model that allows commitment, and
Section 5 concludes. Proofs are contained in the appendix while more technical derivations and
numerical computations are included in a technical appendix available on the authors’ websites.
2
The Basic Model
Two firms, A and B, produce a differentiated product at constant marginal cost c. A unit mass of
consumers are continuously distributed on interval θ ∈ [0, 1]. The brands produced by A and B are
located at θ = 0 and θ = 1, respectively. There are two periods, t = 1, 2. Each consumer demands
at most one unit of the product each period. In period t, a consumer of type θt purchasing from A
or B at price p receives utility uA (θt ) = v − θt − p or uB (θt ) = v − (1 − θt ) − p, respectively, where
v is sufficiently large so that in equilibrium all consumers will purchase. Firms and consumers have
a common discount factor δ ∈ (0, 1]. Each firm maximizes its discounted sum of expected profit,
and each consumer maximizes her discounted sum of expected utility.
A consumer’s preference type, θt , is the realization of a random variable with joint distribution
3
function H(θ1 , θ2 ) and marginal distribution function F (·) that admits density f (·). The consumer
learns her θt in the beginning of each period. A key departure of our paper from the literature is
that we allow a more general relationship of consumer preferences across periods by using a copula.
For our purpose, we use copula Family 11 from Joe (1997, page 148):
H(θ1 , θ2 ) ≡ α min{F (θ1 ) , F (θ2 )} + (1 − α)F (θ1 )F (θ2 ),
(1)
where α ∈ [0, 1]. The (positive) dependence between θ1 and θ2 increases monotonically in α.5 The
following assumption is made throughout the paper:
Assumption 1. Consumer preferences (θ1 , θ2 ) follow the joint distribution H(θ1 , θ2 ). The marginal
distribution of preferences in each period is F (θt ) with density f (θt ), t = 1, 2. Furthermore, f (·)
is differentiable, symmetric about 21 , and
f (θ)
1−F (θ)
strictly increases in θ.
The properties of f and F assumed in Assumption 1 are similar to those in Fudenberg and Tirole
(2000) and ensure the existence of a unique equilibrium for certain cases.
In each period, firms choose their prices simultaneously. For j = {A, B}, a strategy of firm j
specifies p1j in t = 1 and prices (p2j , p̃2j ) in t = 2 based on consumers’ previous purchases, where
p2j and p̃2j are prices for new and repeat consumers. We require that strategies be sequentially
rational. Thus, in equilibrium, these strategies must induce a Nash equilibrium at any secondperiod subgame as well as a Nash equilibrium in period 1.6 Since firms are ex ante symmetric, only
(pure strategy) symmetric equilibrium are considered.
Each firm’s demand is determined by the location or type of the consumer indifferent between
the two firms’ products in the two periods. R and S define the indifferent consumers in period 2
conditional on their first-period choice:
R≡
1 + p2B − p̃2A
,
2
S≡
5
1 + p̃2B − p2A
.
2
(2)
Our formulation generalizes the model in Fudenberg and Tirole (2000), where either α = 1 or α = 0. A similar
copula is used in Nalebuff (2004) and Armstrong and Vickers (2009) in modeling the correlation of consumer values
for two products. Chen and Riordian (2008) use a more general copula approach to product differentiation.
6
Although firms have imperfect information about consumers’ types, each firm’s aggregate demand is well defined,
as shown in the next section. Given each firm’s demand, the game can be solved using the solution concept of
subgame perfect Nash equilibrium.
4
Conditional on purchasing A in period 1, A is purchased again if θ2 ≤ R, otherwise B is purchased.
Similarly S defines the indifferent consumer in period 2 conditional on an initial purchase of B.
θ∗ defines the indifferent consumer in period 1. Equating the discounted sum of expected utilities
from purchasing A or B satisfies
Z
∗
S
2θ = 1 + p1B − p1A − δ(p2B − p̃2B ) − 2δ
G(θ2 |θ∗ )dθ2
(3)
R
where G is the conditional distribution of θ2 given θ1 .
The first-period demand for Firms A and B are simply F (θ∗ ) and 1 − F (θ∗ ), respectively. In
the second period, each firm serves repeat/loyal consumers and new/switching consumers. For
instance, Firm A’s demand from repeat consumers in period 2 is H(θ∗ , R) and F (S) − H(θ∗ , S) is
their demand from new consumers. Firm profits are determined by multiplying the relevant price
minus marginal cost by the demand.
The benchmark case where each firm can only charge a single price to all consumers in each
period is used as a comparison throughout the paper. Standard derivations reveal that the equilibrium prices, profits and aggregate consumer surplus (marked with superscript µ) in each period
are
pµj
3
=c+
1
f
1
2
πjµ
=
1
2f
1
2
V
µ
1
=v−c−
−2
f (1/2)
Z
1
2
θdF (θ) .
(4)
0
Equilibrium and the Effects of Preference Dependence
This section studies the equilibrium properties of the basic model where firms cannot commit to
future prices and how these properties relate to the preference dependence parameter, α. In the
second period, Firm j chooses p̃2j and p2j to maximize second period profits. The possible second
5
period prices are:
p̃2A =
p2A =
p̃2B
p2B



c +
2
f (R)
F (R) +


c + 2 F (R)
f (R)



c + 2 F (S) −
α
1−α
if θ∗ ≤ R
,
if
θ∗
αF (θ∗ )
1−(1−α)F (θ∗ )
f (S)
(5a)
≥R
if θ∗ ≤ S
,


∗
c + 2 F (S)
if θ ≥ S
f (S)



c + 2 1−F (S)
if θ∗ ≤ S
f (S)
,
=


1
∗
c + 2
if θ ≥ S
f (S) 1−α − F (S)



c + 2 1−F (R)
if θ∗ ≤ R
f (R)
.
=

∗)
F
(θ

2
∗
c +
if θ ≥ R
f (R) α+(1−α)F (θ∗ ) − F (R)
(5b)
(5c)
(5d)
Equilibrium pricing strategies are determined by first assuming a certain relationship between R,
S and θ∗ , and then solving the respective first-order conditions.7 We now state and prove a general
result concerning the direction of price discrimination.
Proposition 1. At any symmetric equilibrium, p̃n2 ≥ pn2 ,8 where the strict inequality holds for all
α > 0; and the amount of price discrimination in favor of switching consumers, p̃n2 − pn2 , increases
monotonically in α.
Therefore, if commitment to a future price is not possible, in equilibrium firms will reward new
(i.e. switching) customers whenever intertemporal preferences are positively dependent.9 If α = 0,
then p̃n2 = pn2 and there is no price discrimination. As α increases, there are more rewards to
consumers who switch suppliers in the second period. Fudenberg and Tirole (2000) find that, for
α = 1, firms will poach rivals’ customers in equilibrium. Our result generalizes their finding to any
α > 0, and reveals how the degree of price discrimination in the market varies with the dependence
of consumers’ intertemporal brand preferences.10
7
The (local) second-order conditions are satisfied by Assumption 1.
Equilibrium prices are denoted with superscript n indicating the equilibrium with no price commitment.
9
Note that Proposition 1 does not establish equilibrium existence which requires strategies to be sequentially
rational and globally optimal.
10
Interestingly, the dependence parameter α has a similar effect on the degree of price discrimination as the
8
6
With p̃n2 ≥ pn2 , the period-2 symmetric prices and firm profits are
p̃n2 = c +
π2n =
2F (R)
2 [1 − (1 + α) F (R)]
, pn2 = c +
.
f (R)
(1 + α) f (R)
(6)
[1 − (1 + α) F (R)]2 + [(1 + α) F (R)]2
(1 + α) f (R)
(7)
and we therefore have the following.
Corollary 1. At any symmetric equilibrium, industry prices and firm profits in period 2 decrease
as α increases.
As α increases, consumers are more likely to be captive to their first-period supplier, which
might suggest that a higher α would enable firms to raise prices and earn higher second-period
profits. However, a higher α motivates each firm to lower its poaching price, in order to induce
the rival’s customers to switch. This in turn motivates each firm to lower prices for their repeat
customers. As a result, a higher α intensifies competition in period 2, resulting in lower equilibrium
prices and profits.
We can also generally compare the period-2 prices and profits under dynamic pricing to our
benchmark when dynamic pricing is not possible. We find pµ > p̃n2 > pn2 and because each firm’s
period-2 output at any symmetric equilibrium is 21 , we have:
Corollary 2. Suppose α > 0. Then at any symmetric equilibrium, industry prices and firm profits
in the second period are lower with dynamic pricing than without.
Since total output is fixed in our model, overall social surplus is lower under dynamic pricing
due to preference mismatch caused by excessive switching. The comparison of consumer welfare
with and without dynamic pricing in the second period is complicated by the fact that, under
dynamic pricing, consumers pay lower prices but also incur utility loss due to purchasing from a
less preferred brand. Nevertheless, consumer surplus in period 2 must be higher under dynamic
pricing, using the following revealed-preference argument: any consumer who switches could have
purchased the product from their previous supplier for p̃n2 ; and since pµ > p̃n2 , such a consumer
parameter of switching costs in Chen (1997), where price discrimination vanishes when the average switching cost
approaches zero, and price discrimination increases in the average switching cost.
7
must be better off than when the price is pµ without dynamic pricing. Furthermore, pµ > p̃n2 also
implies that any consumer who does not switch must be better off under dynamic pricing. We
therefore have:
Corollary 3. Suppose α > 0. Then second-period consumer surplus is higher with dynamic pricing
than without, but dynamic pricing leads to lower overall social surplus.
3.1
Uniform Distribution
To solve the equilibrium for the entire model explicitly and to gain additional insights, we assume
F (θ) = θ. The candidate symmetric equilibrium prices for the two periods are
pn1 = c + 1 + δ
α(33 + 29α − 29α2 − 9α3 )
,
9(1 + α)3
p̃n2 = c +
(3 + α)
,
3(1 + α)
pn2 = c +
(3 − α)
.
3(1 + α)
(8)
We have verified that these are prices of a subgame perfect Nash equilibrium of the entire game if
α = 0 or α ≥ 0.151. However for 0 < α < 0.151 and δ = 1, a firm can make a profitable global
deviation and hence no pure strategy equilibrium exists.11 We thus assume α ∈ [0.151, 1] for the
rest of this section.
As α increases, prices for all consumers in period 2 decrease, whereas prices in period 1 increase
for α ≤ 0.4661 but decrease for α > 0.4661. Each firm’s sales to repeat and new customers in period
2 are q̃2n =
3+α
12
and q2n =
3−α
12 .
Thus, higher α leads to fewer switching consumers in equilibrium.
Since prices are higher in period 1 than in period 2, profits are higher in period 1 than in period
2 as well. Second period profits decrease as α increases, while first-period profits increase in α for
α ≤ 0.4661 but decrease in α for α > 0.4661. The discounted sum of profits are non-monotonic in
α, increasing for α < 0.3451 but decreasing for α > 0.3451.
It is interesting that equilibrium profits vary non-monotonically with α. Intuitively, price discrimination affects firm profits in two ways. On the one hand, a firm with more consumers in period
1 will have more incentive to raise its price for its repeat customers. This makes demand in period
1 less elastic, enabling firms to raise their period 1 price. On the other hand, past purchases par11
We assume δ = 1 for convenience. This assumption is relaxed in the technical appendix which includes a
derivation of the equilibrium conditions.
8
tially reveal consumers’ brand preferences, motivating firms to lower prices in period 2 to compete
for customers who have higher expected preferences for the rivals’ brand. When α is higher, the
second-period prices for switching consumers are lower; this reduces profits in period 2, but can
have the indirect effect of making consumers either more or less price sensitive in period 1. As a
result, equilibrium profit in period 1 is non-monotonic in α, first increasing and then decreasing;
whereas equilibrium profit in period 2 monotonically decreases in α. When α is relatively small,
the effect on the first-period profits dominates, so that discounted sum of profits increases in α.
However, when α exceeds some critical value, profits decrease in α, initially because the effect on
the second-period profits starts to dominate, and later because the first-period profits also decrease
in α.
Compared to the benchmark, the equilibrium price with price discrimination is higher for all
consumers in period 1 but lower for all consumers in period 2, the same as in Fudenberg and Tirole
(2000). Firm profits with price discrimination are always higher in period 1 but lower in period 2
and each firm’s discounted sum of profits is higher when α < 0.8973 but lower when α > 0.8973.
Thus, when α is sufficiently high, the ability of firms to engage in dynamic price discrimination
lowers their equilibrium profits, as in Fudenberg and Tirole (2000) for the special case of α = 1;12
but when α is not so high, dynamic price discrimination actually raises equilibrium profits.
Compared to the benchmark, price discrimination lowers consumer surplus in period 1 but
increases consumer surplus in period 2. The discounted sum of consumer surplus is higher with
price discrimination when α > 0.9452 but lower otherwise. For α ∈ (0.8973, 0.9452), both industry
profits and consumer surplus are lower under discriminatory pricing, due to the deadweight loss
caused by preference mismatch.
Summarizing our findings under an uniform marginal distribution, we have:
Proposition 2. Assume F (θ) = θ and α ∈ [0.151, 1], then an equilibrium exists. The equilibrium
price in period 1 first increases and then decreases in α, while equilibrium prices in period 2 always
decrease in α. Industry profit first increases and then decreases in α, while aggregate consumer
surplus first decreases and then increases in α. Compared to the benchmark, firm profits are higher
12
In fact, other papers in the literature on dynamic pricing, including Chen (1997), Villas-Boas (1999), and Taylor
(2003), have similarly found that dynamic price discrimination tends to intensify competition and reduce firm profits.
9
if α < 0.8973 but lower if α > 0.8973; and consumer surplus is lower if α < 0.9452 but higher if
α > 0.9452.
4
The Extended Model: Allowing Commitment
We now assume that in period 1 each firm can commit to a future price for its period 2 repeat
customers. Specifically, we enlarge each firm’s strategy space so that in period 1, firm j chooses
among {(p1j , N ) , (p1j , p̃2j )}, where (p1j , N ) means that j does not make any commitment to future
prices, and (p1j , p̃2j ) means that j commits to charging p̃2j in period 2 to consumers who purchase
from it in period 1. Hence, if j’s choice in period 1 is (p1j , N ), in period 2 it can charge p2j and
p̃2j for new and repeat consumers respectively. If j’s choice in period 1 is (p1j , p̃2j ), in period 2 it
only chooses price p2j for new consumers.13 As before, firms make their choices simultaneously in
each period.
If both firms choose a no-commitment strategy in period 1, (p1j , N ), the outcome is the
no-commitment equilibrium studied in Section 3. If both firms choose a commitment strategy,
(p1j , p̃2j ), in period 1, then we have a commitment equilibrium. Our analysis of the extended
model is facilitated by the following result.
Lemma 1. Assume that both firms’ strategies satisfy sequential rationality. Then, a firm’s payoff
under any no-commitment strategy is weakly lower than its payoff under a commitment strategy
given any strategy of the other firm.
Lemma 1 allows us to just focus on commitment strategies in the analysis of the extended model.14
In what follows, we analyze the extended model assuming that both firms choose (p1j , p̃2j ) in period
1.
The pricing strategy for a symmetric commitment equilibrium, (pc1 , p̃c2 , pc2 ), depends on the
relationship between R, θ∗ and S. As before in Section 3, only two situations can be consistent
13
Note that choosing (p1j , p̃2j ) in period 1 is the same as offering a long term contract with no breach penalty.
Caminal and Matutes (1990) study a similar commitment model for the case of α = 0. Fudenberg and Tirole (2000)
allow firms to offer both short-term and long-term contracts, for the cases of α = 1 and α = 0. In our model, a firm
either offers a short-term or a long-term contract, and there is no breach penalty.
14
If we assume that the extended game has a unique equilibrium, then Lemma 1 implies that there is no loss of
generality to focus on commitment strategies.
10
with a symmetric equilibrium: either S ≤ θ∗ ≤ R which occurs if and only if p̃2j ≤ p2j , or
S ≥ θ∗ ≥ R which occurs if and only if p̃2j ≥ p2j . Define E = p̃c2 − pc2 so when E ≤ 0, firms reward
loyalty; when E ≥ 0, firms entice brand switching.
4.1
Equilibrium Rewarding Loyalty
With E ≤ 0, the expressions for the candidate second-period symmetric equilibrium prices are:
pE≤0
2
2(1 − F (R))
=c+
,
f (R)
p̃E≤0
2
2(1 − F (R)) f (R)2 + (1 − F (R))f 0 (R)
.
=c−
f (R)3
(9)
From Assumption 1, p̃E≤0
< c. Thus for the symmetric equilibrium rewarding repeat customers,
2
the second-period price for repeat customers is below marginal cost. This is consistent with the
findings of Caminal and Matutes (1990) and Fudenberg and Tirole (2000) where the special case of
α = 0 is considered and may explain the popularity of loyalty programs by airlines and hotels that
effectively price below marginal costs when they reward loyal customers with a free airline ticket
or a free hotel stay.15
Each firm’s corresponding sales to repeat and new customers in period 2 are q̃2E≤0 =
and q2E≤0 =
(1−α)(1−F (R))
.
2
F (R)+α(1−F (R))
2
Since F (R) ≥ 1/2 when E ≤ 0, in equilibrium there are always more
loyal customers than switching customers (q̃2E≤0 > q2E≤0 for α > 0). Importantly, however, even
though firms reward loyalty through price commitments, switching still occurs in equilibrium there is no complete lock-in by a loyalty-rewarding program.
In order to solve the equilibrium explicitly for the entire game and to gain additional insights,
we again consider F (θ) = θ. We set δ = 1 and find that a symmetric equilibrium exists with E ≤ 0
when α ≤ 0.6245. Equilibrium prices in periods 1 and 2 for each firm are:
p1E≤0 = c + 1 + δα + δ/3,
1
p̃E≤0
=c− ,
2
3
1
pE≤0
=c+ .
2
3
(10)
Compared to the benchmark, dynamic pricing raises the first-period equilibrium price but lowers
E≤0
In practice, there is often the constraint that price is not negative. Thus, when c is small, we could have p̃2j
= 0,
or firms reward repeat customers with a free good. The equilibrium in this case will be qualitatively similar to what
we have characterized, although some details will differ. For convenience we do not formally consider such corner
cases.
15
11
the second-period prices, similarly as in the basic model without commitment. Additionally, period 1 prices are higher with commitment than without, whereas period 2 prices are lower with
commitment than without and overall consumer surplus is higher with commitment. Hence, even
though commitment results in firms rewarding loyalty in equilibrium, the market nevertheless becomes more competitive in period 2 under commitment, resulting in lower firm profits in period
2, which also leads to lower overall profits. This is consistent with the findings in Caminal and
Matutes (1990) and in Fudenberg and Tirole (2000), extending their insights to settings with general
preference dependence relations.
Summarizing our findings for the extended model with E ≤ 0, we have:
Lemma 2. (i) At any symmetric commitment equilibrium that rewards consumer loyalty, firms
commit to below-cost pricing for their loyal customers in period 2. (ii) Assume F (θ) = θ. Then,
(pE≤0
, p̃E≤0
, pE≤0
) given in equation (10) are equilibrium prices of the extended model if and only
1
2
2
if α ≤ 0.6245. At this equilibrium, firms charge lower prices to repeat customers. Compared to
dynamic pricing without commitment, fewer consumers switch suppliers, overall firm profits are
lower, and overall consumer surplus is higher.
4.2
Equilibrium Enticing Switching
With E ≥ 0, the candidate second-period prices at any symmetric equilibrium are:
p̃E≥0
2
pE≥0
2
2 (1 + α)f (R)2 (3αF (R) − (1 − F (R))) − f 0 (R)(αF (R) − (1 − F (R)))2
=c+
(1 + α)2 f (R)3
2(1 − (1 + α)F (R))
=c+
.
(1 + α)f (R)
(11)
(12)
When α = 0, E = p̃2E≥0 − p2E≥0 < 0, which implies that an equilibrium with E ≥ 0 does not exist
if α is sufficiently close to zero.16
16
When α = 0 firms offer the same prices to both new and repeat customers in period 2 if no commitment is
possible (Proposition 1). With commitment, a firm has the incentive to commit to a lower period-2 price to attract
consumers in the first period. Consequently, the equilibrium will have the property of E < 0 instead of E > 0. The
same is true if α is sufficiently small.
12
To solve the equilibrium explicitly, we again assume F (θ) = θ. Then:
(3 − 5α + 57α2 + 49α3 − 80α4 − 24α5 )
,
(3 + 5α)2 (1 + α)2
(3α2 + 6α − 1)
(1 + 4α − α2 )
=c+
, pE≥0
=c+
.
2
(1 + α)(3 + 5α)
(1 + α)(3 + 5α)
pE≥0
=c+1+δ
1
p̃E≥0
2
(13)
Setting δ = 1, we find that a symmetric equilibrium with E ≥ 0, given by (13), exists when
α ≥ 0.7163. Similar to the cases where E ≤ 0 with commitment and when no commitment to
future prices can be made, the symmetric equilibrium first-period price is higher and the secondperiod prices are lower than the benchmark price pµ .
Each firm’s corresponding equilibrium sales to repeat and new customers in period 2 are q̃2E≥0 =
α2 +6α+5
12+20α
> q̃2n and q2E≥0 =
1+4α−α2
12+20α
< q2n . Thus, the ability of firms to commit to the future price
for their repeat customers reduces the number of switching consumers in equilibrium. Furthermore,
equilibrium prices are lower for all consumers with commitment than without, which implies that
firm profits are lower and consumer surplus is higher at the commitment equilibrium than at the
no-commitment equilibrium.
Summarizing our findings for the extended model with E ≥ 0:
Lemma 3. (i) There can be no symmetric equilibrium with E ≥ 0 if α is close to zero. (ii)
When F (θ) = θ, pE≥0
, p̃E≥0
, p2E≥0 given in equation (13) are equilibrium prices of the extended
1
2
model if and only if α ≥ 0.7163. At this equilibrium, firms charge lower prices for new customers.
Compared to dynamic pricing without commitment, all prices are lower, fewer consumers switch
suppliers, firm profits are lower, and consumer surplus is higher.
Interestingly, although the nature of price discrimination differs for the two different commitment equilibria, they share the common feature that being able to commit to a future price reduces
brand switching, reduces overall firm profits,17 and increases overall consumer surplus. One way
to think of this result is that the ability to make future price commitment adds another dimension
along which firms compete. Consequently, competition is more fierce for the commitment equilib17
Nalebuff (2001) finds that firms will commit to loyalty programs when such commitment is possible, and rewarding
loyalty increases profits. In his setting, firms commit to price discounts and move sequentially, which might explain
why loyalty programs are more profitable (collusive) outcome.
13
rium, hurting firms but benefiting consumers. Since committing to a lower future price for its repeat
customers will increase a firm’s current demand, a firm has more incentive to lower its price for
repeat customers under commitment. But without commitment the general tendency is for firms
to charge lower prices for rivals’ switching customers. The balance of these two considerations
suggests that under commitment there will be fewer switching consumers, even if in equilibrium
firms still price lower to switching consumers.
Combining Lemma 2 and Lemma 3, we have:
Proposition 3. (i) When α is sufficiently small, firms reward consumer loyalty at any symmetric
equilibrium of the extended model. (ii) Assume that F (θ) = θ. Then, if α ≤ 0.6245, there exists a
unique symmetric industry equilibrium, where firms reward consumer loyalty; whereas if α ≥ 0.7163,
there exists a unique symmetric industry equilibrium where firms entice brand switching. Compared
to dynamic pricing without commitment, at either commitment equilibrium fewer consumers switch
suppliers, overall firm profits are lower, and overall consumer surplus is higher.
When α is small, or a consumer’s past purchase does not predict a much stronger preference
for the same brand in the future, firms have strong incentives to commit to a lower price for repeat
customers. This results in an industry equilibrium that rewards consumer loyalty. We think this
may explain why loyalty programs are prevalent in, for instance, the airline industry. A consumer’s
preferred airline is likely to change for different trips that may have different destinations, depending
on the availability and schedule of flights from different airlines. Loyalty programs give consumers
lower prices on repeat purchases and make consumers less willing to switch between different
airlines.
On the other hand, when α is large, or a consumer’s past purchase predicts a significantly
stronger preference for the same brand in the future, it is likely that consumers are unwilling to
switch brands after they have made an initial purchase. This provides the incentive for firms to
target rivals’ customers with lower prices. In this case, the industry equilibrium is characterized
by enticing consumers to switch. We think this may explain the type of dynamic pricing practices
in markets such as those for telephone services, cable or satellite services, and banking or credit
card services, where a new consumer is often charged a lower price (offered a better deal) than a
14
repeat consumer. Although for simplicity we have assumed in our model that consumers’ brand
preferences in period 2 are determined entirely by their intrinsic utility functions, for these markets
it is perhaps more appropriate to consider consumers’ preference distribution in period 2 as at least
partly arising from or incorporating some distribution of consumer switching costs, as in Gehrig and
Stenbacka (2004). Our insights could extend to such settings: having low switching costs is similar
to having a small α, where firms are motivated to reward consumer loyalty in order to prevent
consumers from switching brands; conversely, having significant switching costs is similar to having
a large α, where firms offer better deals to new customers in order to entice brand switching.18
5
Conclusion
This paper has developed a theory of dynamic pricing for competing firms who can charge separate
prices to consumers with different purchase histories. Our analysis shows how the nature of industry
equilibrium depends on a more general intertemporal relationship of consumer preferences and on
the ability of firms to commit to future prices. Our analysis also connects and extends the existing
literature on dynamic pricing which either focuses on special cases of preference dependence or treats
firms’ price commitment decisions as exogenously given. Furthermore, our theoretical predictions
offer explanations of observed pricing practices across industries.
A key feature of our approach is to use a copula to model a consumer’s preference dependence
through time. For analytical tractability, we have assumed a particular copula family that has
a convenient preference dependence parameter (α). In future research it would be interesting to
extend our analysis to other copula functions. One primary advantage of the copula approach
is that the analysis can separate the effects of preference dependence from those of the marginal
distributions. In fact, some of our central results are obtained under the assumption of a general
marginal distribution function, F (·). When the marginal distributions are uniform, additional
results are obtained concerning how firm profits and consumer surplus varies with α and with firms’
ability to commit to future prices. We have also verified that our results hold qualitatively when
18
The marginal distributions of θ in periods 1 and 2 need not be the same, especially when the distribution of θ
in period 2 includes consumer switching costs. Our assumption that they both equal to F (·) is made for analytical
tractability.
15
the marginal distributions are Beta(2,2), suggesting that the insights gained under the uniform
distribution are robust to some variation in the marginal distribution.
Appendix
The proofs of Proposition 2, Lemma 2 and Lemma 3 are in the technical appendix. See Section H.1
of the technical appendix for the proof of Proposition 2, Section J for Lemma 2 and Section K for
Lemma 3.
Proof of Proposition 1. For any symmetric equilibrium, θ∗ = 1/2. Then, equation (2) reduces to
R = 21 − E2 , and S = 12 + E2 , where E ≤ 0 if and only if 21 ≤ R and S ≤ 21 ; and E ≥ 0 if
and only if 12 ≥ R and S ≥ 21 . Symmetry of f (·) from Assumption 1 implies f (R) = f (S),
F (R) = 1 − F (S), and F ( 12 ) = 21 . Also, for any symmetric equilibrium, it must be that either
S ≤ 12 ≤ R or S ≥ 12 ≥ R.
α
Next, E < 0 implies S < 12 < R, or F (R) > F (S), and note that 1−α
≥ 0. If E < 0, from
equation (5) we would have
2
α
n
n
E = p̃2 − p2 =
F (R) − F (S) +
>0
f (R)
1−α
which is a contradiction. Therefore, we must have E ≥ 0, or p̃n2 ≥ pn2 and S ≥ 12 ≥ R. From equa- 2
α
tions (5a) and (5b), noticing F 21 = 1/2 and f (R) = f (S), we have E = f (R)
.
F (R) − F (S) + 1+α
n
n
When α = 0, it must be that E = 0 so that p̃2 = p2 ; otherwise we would have E < 0, which is not
possible.
Since F (S) = 1 − F (R) by the symmetry condition, we obtain
2
1
2
E=
2F (R) −
=
(2(1 + α)F (R) − 1) .
f (R)
1+α
f (R)(1 + α)
Implicitly differentiating E with respect to α and noticing R =
1
2
−
E
2,
we obtain:
dE
2f (R)
=
.
2
dα
(1 + α) [3(1 + α)f (R) + f 0 (R) − 2(1 + α)F (R)f 0 (R)]
If f 0 (R) ≥ 0,
dE
dα
0 ⇐⇒ 3(1 + α)f (R)2 + f 0 (R) − 2(1 + α)F (R)f 0 (R) > 0
⇐⇒ 2(1 + α) f (R)2 − f 0 (R)F (R) + f (R)2 (1 + α) + f 0 (R) > 0,
>
which holds since f (R)2 − f 0 (R)F (R) > 0 from Assumption 1. Otherwise if f 0 (R) < 0,
2(1 + α) f (R)2 − f 0 (R)F (R) + f (R)2 (1 + α) + f 0 (R)
1
2
0
− F (R)
+ (1 + α)f (R)2 ,
= 2(1 + α) f (R) + f (R)
2(1 + α)
16
which is again larger than 0 because
tion 1. Therefore
dE
dα
1
2(1+α)
< 1, and f (R)2 + f 0 (R) [1 − F (R)] > 0 from Assump-
> 0, or p̃n2 − pn2 increases monotonically in α.
Proof of Corollary 1. Since f (R)2 − F (R) f 0 (R) > 0 from Assumption 1 and
sition 1, we have
dp̃n2
1 dE
f (R)2 − F (R) f 0 (R)
−
< 0,
=2
dα
2 dα
f (R)2
which, together with E = p̃n2 − pn2 , implies that
the same in a symmetric equilibrium.
dpn
2
dα
< 0.
dπ2n
dα
dE
dα
> 0 from Propo-
< 0 since each firm’s output remains
Proof of Corollary 2. If α > 0, E = p̃n2 − pn2 > 0 from Proposition 1. From (4) and (6) we have:
!
1
F
1
2F
(R)
F
(R)
2
−
> 0,
pµ − p̃n2 =
=2
−
f (R)
f (R)
f 12
f 21
where the inequality holds because R =
1
2
− E2 <
1
2
and because
F (v)
f (v)
is increasing (Assumption 1).
Proof of Lemma 1. Consider any no-commitment strategy by firm k, k = A, B, in which k chooses
(p1k , N ) in period 1 and (p2k , p̃2k ) in period 2. Any strategy by firm j 6= k specifies either (p1j , N )
or (p1j , p̃2j ) in period 1.
If j’s strategy specifies (p1j , N ), then in period 2 j and k charge (p2j , p̃2j ) and (p2k , p̃2k ), and
sequential rationality requires that these second-period prices be
to each other.
best responses
0
0
Now consider a commitment strategy by firm j, where j chooses p1j , p̃2j in period 1 and where
p01j = p1j . If p̃02j = p̃2j , then the second-period prices for the two firms must be the same as when j
chooses (p1j , N ) in period 1, and thus the payoff for j under the commitment strategy is the same
as the payoff under no-commitment for j. Additionally j can commit to some p̃02j 6= p̃2j , which can
potentially result in a higher payoff for j, because p̃02j is chosen to maximize πj instead of only π2j .
If firm k specifies (p1k , p̃2k ) in period 1, a similar argument establishes that a commitment
strategy is weakly preferred for firm j.
17
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