Discussion Paper No. 869

SHOULD FIRMS EMPLOYPERSONALIZED PRICING?

Toshihiro Matsumura
Noriaki Matsushima

March 2013

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Should firms employ personalized pricing?∗
Toshihiro Matsumura†Institute of Social Science, the University of Tokyo
Noriaki Matsushima‡
Institute of Social and Economic Research, Osaka University

Abstract
The recent developments in information technology (IT) have enabled firms
to employ personalized pricing. Should all firms employ personalized pricing
even though the adaptation costs of such pricing strategies are not high? This
paper theoretically demonstrates a situation in which all firms do not always
employ personalized pricing even though the fixed costs to do so is zero. The
model is based on those of Thisse and Vives (1988) and Shaffer and Zhang(2002). Our model incorporates the fact that firms engage in marginal costreducing activities after they decide whether to employ personalized pricing. As
employing personalized pricing induces the rival firm to engage more in reducing
its costs, to mitigate the cost-reducing activities of firms, the less-efficient firmshould not employ personalized pricing. Our main result indicates that such firms
should take into account their relative competitive positions and technologicalenvironments. When firms are small, they would need to reconsider whether to
employ personalized pricing.
JEL codes: M31, L13, D43
Keywords: personalized pricing, heterogeneity, cost reductions, game theory∗

We would like to thank seminar participants at Nagoya University and Shinshu University for
their helpful comments. The authors gratefully acknowledge financial support from Grant-in-Aid fromthe Japanese Ministry of Education, Science and Culture. Needless to say, we are responsible for any
remaining errors.
†
Toshihiro Matsumura, Institute of Social Science, the University of Tokyo, Hongo 7-3-1, Bunkyo,
Tokyo 113-0033, Japan.
‡
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6879-8571. E-mail:nmatsush@iser.osaka-u.ac.jp

1

Introduction

The recent developments in information technology (IT) have enabled firms to employ
personalized pricing (Arora et al., 2008 and references therein). For instance, by using
profiles developed by information collected through loyalty cards, Tesco learns a variety of facts about what and when its customers buy. Tesco’s Clubcard operation uses
these data to send personalized coupons and other offers to every Clubcard householdin each quarter, a huge operation that accounts for more than 6% of the U.K.’s annual
postal volume (Raju and Zhang, 2010, p.131-2).1 However, some companies have not
employed such personalized pricing strategies even though they potentially could. A
simple explanation for this asymmetry concerning personalized pricing is that personalization is expensive and requires data and expensive software to implement (Aroraet al., 2008). Although this explanation is plausible, it raises the following question:
Should all firms employ personalized pricing even though the adaptation costs of suchpricing strategies are not high? This is a reasonable question because the recent rapid
development of IT will allow firms to inexpensively employ such a pricing policy in thenear future. The purpose of this paper is to provide a different, yet simple rationale
behind why some firms do not employ personalized pricing even though personalization
is inexpensive.
The model setting is as follows. Two firms compete in a differentiated productmarket. Each firm determines whether to employ personalized pricing. If a firm employs personalized pricing, it will be able to distinguish among customers and offer
them personalized prices; otherwise, it offers a standard non-discriminatory price toall customers. The model setting is based on those of Thisse and Vives (1988) and1

Recently, grocers like Safeway and Kroger also began offering different methods to determine
individualized prices (The New York Times (August 10,2012)). Hoping to improve razor-thin profit
margins, they are creating specific offers and prices, based on shoppers’ behaviors, which could encourage them to spend more.

1

Shaffer and Zhang (2002). Our model also incorporates the fact that firms engage inmarginal cost-reducing activities. More concretely, we set a model in which the firms
engage in these activities after they determine their pricing policies. In the retailing
context, for instance, retailers need to construct electric devices to distribute appropriate promotional coupons through frequent-shopper programs based on the analysesof household purchase records. Furthermore, to use such devices, these retailers alsoneed new cash registers, computers for gathering the records of customer purchases,
applications for analyzing the records, data analysts, customer service offices for theirfrequent shopper programs, and so on. For instance, issuing promotional coupons foreach customer generates an additional per-customer cost. If the retailer creates a device to incorporate promotional coupons into every customer’s card, it would save thisadditional per-customer cost despite possible additional running costs. Introducing
such a new device is an example of a cost-reducing activity by retailers. Introducing
personalized pricing is time consuming, requires much time to make the pricing policy
work well, and thus is a long-term decision, even though the cost of developing personalized pricing is low. We therefore believe that firm should make their pricing policy
decisions before they engage in cost-reducing activities.
Even if a firm does not employ personalized pricing, it still engages in cost-reducing
activity.2 One would think that a firm can keep its R&D costs down by not employing
personalized pricing. Although we first discuss the case in which the effort costs under2

This case is similar to oligopoly models with cost reductions where firms engage in cost-reducing
activities before they set their prices or quantities, which have been intensively discussed in theliterature of industrial organization (e.g., Brander and Spencer, 1983). Recently, researchers haveinvestigated the relationship between firm asymmetry and R&D activities as discussed in this paper(Cohen and Klepper, 1996; Matsumura and Matsushima, 2010, Ishida et al., 2011). We believe thatour paper can make a new contribution in this context. Although most studies in this literature focus
on how R&D affects equilibrium pricing levels, our paper sheds light on how each firm’s decision on
pricing policy (whether to employ personalized pricing) affects the strategic R&D behavior of firms,
and how the degree of firm asymmetry matters.

2the two pricing policies are the same in Section 4.1, we also discuss how the main
result in Section 4.1 changes if the costs of cost-reducing activities under the two
pricing policies differ in Section 4.2.
We show that when the ex ante cost difference between the firms is large, the lessefficient firm does not employ a personalized pricing policy even though the fixed costto employ the policy is zero, while the more efficient firm employs the personalized
pricing policy. The key point is strategic interaction between the less-efficient firm’s
pricing policy and its rival’s cost-reducing activity. In fact, this asymmetric result does
not hold when the firms do not engage in cost-reducing activities.3
We explain the intuition behind why personalized pricing is not always adopted.
Let us first consider the case in which no firm employs a personalized pricing policy,
that is, two firms adopt a standard uniform pricing policy. As in the standard pricecompetition with differentiated products, a firm’s cost reduction leads to a reductionin its price. Through the strategic interaction between the firms (due to the strategic
complementarity of prices), the price reduction induces a lower price in the rival firm,
which reduces the profit of the cost-reducing firm. Thus, under a standard uniform
pricing policy, each firm has a smaller incentive to carry out cost-reducing activities
than if the cost reduction had no strategic effect on the rival’s pricing.
Let us next consider the case in which both firms employ a personalized pricing
policy. The competition in each customer market is similar to that of the standard
Bertrand competition. Only the firm with a cost advantage to the rival supplies foreach market, and then the price is equal to the rival’s marginal cost for the market it
supplies. Although a cost reduction by a firm reduces its rival’s prices in the rival’s
target markets, the cost reduction does not affect the cost-reducing firm’s prices in its
3

In the context of marketing, Moorthy (2005) focuses on retail price changes motivated by costchanges.

3

target markets. This suggests that a unilateral cost reduction will be more profitable
under a personalized pricing policy than under a uniform pricing policy. Note that thelarger the firm’s market share, the greater incentive it has for cost-reducing activity
under a personalized pricing policy.
Let us finally suppose that one of the firms (firm 2) unilaterally deviates froma personalized pricing policy and employs uniform pricing. Then, firm 2’s uniform
price is now dependent on the cost of its rival (firm 1), and thus, firm 1’s prices inthe target markets are also dependent on its own cost. A reduction in firm 1’s cost
reduces firm 1’s prices through the reaction of firm 2, in contrast to the situation where
both firms employ personalized pricing. Thus, firm 1 has a weaker incentive for cost
reduction when firm 2 employs a uniform pricing policy, which mitigates competition
between them and increases the profit of firm 2. As we explained, a firm with a larger
market share (firm 1) has a stronger incentive for cost-reducing activity. Moreover,the negative effect of firm 1’s cost reduction through the reaction by firm 2 increases
with the market share of firm 1 because a price decrease caused by price competition,
ceteris paribus, harms the larger firm (firm 1) more. These suggest that a unilateraldeviation from a personalized pricing policy by firm 2 reduces firm 1’s cost-reducing
activity more when firm 1’s market share is larger. If the market share of firm 1 issufficiently large, then at some point, firm 2 actually prefers adopting a uniform pricing
policy in order to dampen the incentive of firm 1 with a larger market share to engage
in cost-reducing activities. This is why the less-efficient firm is motivated to adopt
uniform pricing rather than personalized pricing.
We review related research and discuss the relation between these papers and ours.
Recent studies emphasize that the promise of personalization enables firms to estimate
their customers’ valuations and thus implement finer price discrimination (Shaffer andZhang, 1995, 2000, 2002; Bester and Petrakis, 1996; Villas-Boas, 1999; Fudenberg and4

Tirole, 2000; Chen and Iyer, 2002; Desai and Purohit, 2004; Bhaskar and To, 2004;
Liu and Serfes, 2004; Taylor, 2004; Liu and Zhang, 2006; Dogan et al., 2010; Shulman
and Geng, 2012).4 Among the existing literature, some papers have already investigated why asymmetry concerning the employment of personalized pricing occurs andhave presented interesting and plausible answers (Shaffer and Zhang, 2002; Choudhary
et al., 2005; Ghose and Huang, 2009). Shaffer and Zhang (2002) introduce firm heterogeneity into a spatial competition model with price discrimination and show that
personalized pricing does not always lead to a prisoner’s dilemma situation; that is,a firm in an advantageous position can benefit from personalized pricing. They also
show that only the advantageous firm employs personalized pricing when the per customer cost of offering a personalized price is moderate. However, they do not explicitly
discuss whether each firm commits to employing personalized pricing. Moreover, ourmodel incorporates the effort choices of firms to improve their production environments. Choudhary et al. (2005) examine a vertically differentiated duopoly and showthat the higher-quality firm can be worse off with personalized pricing. The driving
force of their main result is as follows. Personalized pricing employed by a firm motivates its rival to improve its product quality. The effect of the quality improvement
depends on who employs it. When the rival is the high-quality firm, that is, when
the low-quality firm employs personalized pricing, the quality improvement by the rival (the high quality firm) mitigates competition because the quality differentiation
between the two firms increases. The converse also holds true. When the rival is the
low-quality firm, that is, when the high-quality firm employs personalized pricing, thequality improvement by the rival accelerates competition because the quality differentiation between the two firms decreases.5 In our model, quality improvements (which
4

Arora et al. (2008) and Zhang (2009) provide excellent surveys of the discussion of personalized
pricing.
5Moreover, when the high-quality firm employs personalized pricing to escape the intense compe-

5

have qualitatively similar effects with cost reductions) by both the high-quality and the
low-quality firms accelerate competition, which yields the contrasting result in which
the higher-quality firm is always better off with personalized pricing. This property isquite different from that of Choudhary et al. (2005). Ghose and Huang (2009) also
incorporate the efforts of firms to improve product qualities into duopoly competition
with personalized pricing.6 Shaffer and Zhang (2002) show that a prisoner’s dilemma
situation does not arise even when the firms are ex ante symmetric, but they do not
explicitly discuss whether each firm commits to employing personalized pricing.
The paper proceeds as follows. The next section outlines the basic environment.
Section 3 shows the benchmark result. Section 4 elaborates on the basic environmentand displays the main result. Finally, Section 5 concludes the paper.

2

Basic environment

Before we discuss the main concern of our paper, we explain the model without costreducing activities. In Section 4, we explicitly discuss our main concern.
Consider a linear city along the unit interval [0, 1], where firm 1 is located at 0
and firm 2 is located at 1. Consumers are uniformly distributed along the interval,
and the density of the consumer distribution is 1. Each consumer buys exactly one
unit of the good, which can be produced by either firm 1 or firm 2. A consumer
locating at x ∈ [0, 1] incurs a transportation cost of tx (t(1 − x)) when purchasing a
product from firm 1 (firm 2), where t is a positive constant. Each consumer derives
a surplus from consumption (gross of price and transportation costs) equal to v. We
tition through the additional investment by the low-quality firm, an additional quality improvement
is needed for the high-quality firm. The additional investment cost is high because the marginal investment cost of the high-quality firm is high. This additional cost is not compensated by the gainfrom personalized pricing.
6
Product customization is also a means to discriminate between customers. Alptekinoğlu andCorbett (2008) discuss product customization in duopoly competition.

6

assume that v is so large that every consumer consumes one unit of the product. We
think that the model reflects the spatial nature of competition, which is applicableto many markets including competition among retailers in cities, manufacturers withdifferentiated products, and so on.
The firms produce homogeneous goods with the constant marginal cost ci > 0(i = 1, 2). We assume that t > |c2 − c1 | to ensure that the quantity supplied by thefirms is always positive. Let pi (x) denote the price of firm i (i = 1, 2), where x is a pointon the interval [0, 1]. That is, the prices set by the firms depend on point x ∈ [0, 1].
Given that firms 1 and 2 set p1 (x) and p2 (x), respectively, consumer x buys a goodfrom firm 1 if and only if p1 (x) + tx ≤ p2 (x) + t(1 − x); note that if p1 (x) + tx =p2 (x) + t(1 − x), the consumer randomly chooses one of the goods.
Each firm has two options to set its price pi (x). The first is uniform pricing (“UP”).
When firm 1 (firm 2) employs UP, its price strategy is given by p1 (x) = p1 (p2 (x) = p2 ),where pi is a constant value set by firm i (i = 1, 2). We interpret pi as firm i’s regular
price which is common for all consumers. The second is a targeted pricing (“TP”).
When firm i employs TP, it modifies its price at each point x even after setting its
regular price pi . We interpret the modification of the price at each point x as promotion
strategies of firm i. Note that we assume that information about the location of eachconsumer is available to the firms. Firms can obtain such information from a numberof sources, such as through repeated past transactions with the customers (Liu andSerfes, 2004).7
The game runs as follows. First, each firm simultaneously determines its pricing
policy, UP or TP. Second, after observing the rival’s pricing policy, each firm simultaneously sets its regular price. If firm i has chosen UP, it is its final price. Third, if firm i
7

Liu and Serfes (2004) and Shaffer and Zhang (2000, 2002) provide more detailed discussions on
this issue.

7has employed TP, it modifies its price at each point x. If the two firms have employed
TP, they simultaneously modify their prices at each point x. The solution concept
is the subgame perfect Nash equilibrium. We therefore solve the game by backward
induction.
The timing structure in our model follows that in the related papers (Banks andMoorthy, 1999; Liu and Serfes, 2004; Rao, 1991; Shaffer and Zhang, 1995, 2002; Thisse
and Vives, 1988; Zhang, 2009). These papers assume that firms choose their promotional strategies after they have chosen their regular prices. This assumption reflects
the common view that a firm’s choice of regular price can be adjusted slower than its
choice of promotional offers such as targeted coupons. In addition, if both decisions
are made simultaneously, no pure strategy equilibrium exists in the subgames where
only one firm employs TP.

3

Benchmark analysis without cost-reducing activities

There are four cases depending on the pricing strategies set by the firms. We discusseach of the four cases.
UP-UP: When both firms employ UP, the indifferent consumer, x̂nn , is given as
p1 + tx̂nn = p2 + t(1 − x̂nn ) → x̂nn =t + p2 − p1.
2t

The profits are given by
π1 = (p1 − c1 )x̂nn , π2 = (p2 − c2 )(1 − x̂nn ).
The first-order conditions lead topi =3t − c1 + c2
(3t − ci + cj )23t + 2ci + cj
, x̂nn =, πi =
.
3
6t
18t
8

UP-TP: Consider the case in which firm 1 employs UP and firm 2 employs TP. We
can apply the discussion of this case to that in which firm 2 employs UP and firm 1
employs TP. When only firm 1 employs UP, the timing is as follows. First, firm 1 sets
its price. Second, observing this price, firm 2 sets its prices.
Given that firm 1 sets its price p1 , firm 2 takes the demand of consumer x if andonly if
p1 + tx > p2 (x) + t(1 − x) → p2 (x) < p1 + t(2x − 1).
If the right-hand side in the latter inequality is larger than the marginal cost of firm
2, c2 , it sets p2 (x) at p1 + t(2x − 1) − ε to take the demand of consumer x, where ε issufficiently small; we omit this value. Otherwise, it sets p2 (x) at c2 . The optimal priceof firm 2 at point x is given by½
p2 (x) =p1 + t(2x − 1) if c2 < p1 + t(2x − 1),c2otherwise.

The indifferent consumer x̂nd is given by
c2 = p1 + t(2x − 1) → x̂nd =t + c2 − p1.
2t

Anticipating the prices of firm 2, firm 1 sets its price, p1 . The profit of firm 1 is givenby
π1 = (p1 − c1 )x̂nd =(p1 − c1 )(t + c2 − p1 ).
2t

The first-order condition leads top1 =t + c1 + c2
t + c2 − c1
(t + c2 − c1 )2, x̂nd =, π1 =.
2
4t
8t

The profit of firm 2 is given byZ 1(p1 + t(2m − 1) − c2 )dm =

π2 =
x̂nd

9

(3t + c1 − c2 )2.
16t

Conversely, under the case in which firm 2 employs UP and firm 1 employs TP, theprofits of the firms are given by(3t + c2 − c1 )2(t + c1 − c2 )2
π1 =, π2 =.
16t
8t
TP-TP: When both firms employ TP, the lowest price of firm i for each consumer
is ci . Firm 1 takes the demand of consumer x if and only if
p1 (x) + tx < c2 + t(1 − x) → p1 (x) < c2 + t(1 − 2x).
If the right-hand side in the latter inequality is larger than the marginal cost of firm
1, c1 , it sets p1 (x) at c2 + t(1 − 2x) − ε to take the demand of consumer x, where ε issufficiently small; we omit this value. Otherwise, it sets p1 (x) at c1 . The optimal priceof firm 1 at point x is given by½
p1 (x) =c2 + t(1 − 2x) if c1 < c2 + t(1 − 2x),c1
otherwise.

A similar argument is applied to the prices of firm 2.
½
p2 (x) =c1 + t(2x − 1) if c2 < c1 + t(2x − 1),c2otherwise.

The indifferent consumer x̂dd is given by
c1 + tx = c2 + t(1 − x) → x̂dd =t + c2 − c1.
2t

The profits of the firms are given by
Z x̂dd(t + c2 − c1 )2
,
4t
0
Z 1
(t + c1 − c2 )2(c1 + t(2m − 1) − c2 )dm =
π2 =
.
4t
x̂dd
π1 =

(c2 + t(1 − 2m) − c1 )dm =10Table 1: Payoff matrix when the marginal costs are exogenous
1/2

UP

TP

(3t + c1 − c2 )218t

UP(3t + c2 − c1 )218t(3t + c1 − c2 )216t(t + c2 − c1 )28t(t + c1 − c2 )28t

TP(3t + c2 − c1 )2
16t(t + c1 − c2 )24t(t + c2 − c1 )24t

Note: In each cell, the lower left and the upper right items are the payoffs of firms
1 and 2, respectively.

Payoff matrix: From the above discussion, we obtain the payoff matrix in Table 1.
From the payoff matrix in Table 1, we obtain the following lemma.
Lemma 1. For any c1 and c2 , employing TP dominates employing UP.
This property follows the results discussed in Thisse and Vives (1988) and Zhang(2009). Note that as in Shaffer and Zhang (2002), if c2 − c1 is large, firm 1’s profit in
the case in which both firms employ TP is larger than that in which no firm does.
Before we explain the key point of the main result presented in the next section, we
review the intuition why both firms employ a personalized pricing policy without costreducing activities, which is shown by Thisse and Vives (1988) and Shaffer and Zhang(2002). Personalized pricing allows a firm to price-discriminate among customers. The
firm employing a personalized pricing policy can give special discounts to targetedcustomers without changing the prices for other customers whose loyalty for the firm ishigh. On the contrary, under uniform pricing policy, in order to get targeted customers,
the firm employing this pricing policy must provide discounts for all customers including
those who would buy without discounts. Thus, firms have incentives to employ a11personalized pricing policy.
Because the effect of price discrimination does not depend on the assumptions that
total demand is inelastic and/or all consumers have the same demand, we believe that
Lemma 1 holds under broader situations than what we discuss in this paper.

4

Cost reduction

The result in the previous section changes when we take into account the cost-reducing
activities of firms. Note that we can derive a qualitatively similar result even though thedifference between the products is in quality and the firms engage in quality-improving
activities. We discuss it after we show the main result.
We consider the following game. First, each firm decides its pricing policy. Second,observing these pricing policies, each firm engages in its cost-reducing efforts. Third,
each firm simultaneously sets its regular price. If firm i has chosen UP, it is its final
price. Finally, if firm i has employed TP, it modifies its price at each point x. If thetwo firms have employed TP, they simultaneously modify their prices at each point x.
Under this scenario, the above payoff matrix represents the outcomes after the third
stage in the four subgames that follow the first stage.
We have implicitly assumed that the pricing policy decision is time consuming
and a long-term one; in other words, it has an investment nature. As mentioned in the
Introduction, firms need to spend enough time to construct the IT system for TP. Then,after the construction of the IT system, firms employing the IT system improve theefficiencies of their operations and services that are suitable for the system. Of course,
even though a firm does not employ TP, it engages in marginal cost reducing activity
after it determines not to employ TP. Furthermore, because the timing structure isalso employed in Choudhary et al. (2005), our result in this section is comparable to12that in Choudhary et al. (2005).
Following the pricing policy employed in the first stage, each firm engages in its
efforts to reduce the operation cost. If the firm employs TP, the cost-reducing activities
include activities like the stabilization of computer systems for TP and efforts to learn
how to use these systems efficiency. If the firm employs UP, the cost-reducing activities
are related to the improvement of its daily activities. Under this interpretation, one
may think that employing TP requires much effort to make the operations of the firm
work well. To clarify the logic behind our main result, we first assume that the effort
costs of firm i to reduce ci are common in the two pricing policies. We also discuss acase in which the effort costs under the two pricing policies differ after we investigate
the symmetric case.

4.1

Symmetric effort costs

We first consider the case in which the effort costs to decrease the marginal cost do
not depend on the decision of pricing policy.
We now investigate how c1 and c2 are determined by the firms. Assume that theex ante marginal costs of firms are c1 = c − e1 and c2 = c + d − e2 , where c and d are
positive constants and ei is firm i’s effort to reduce its marginal cost (i = 1, 2). From
above, we have that firm 1 is more efficient than firm 2. I(ei ) is the effort cost of firm i.
We assume that I(ei ) = γe2i , where γ is a positive constant. For simplicity, we assume
that the effort costs under the two pricing policies are the same. This assumption
clarifies the effects of pricing policy on the efforts and prices in the second and third
stages. To guarantee that the second-order conditions for cost-reducing activities are
satisfied, we assume that tγ ≥ 1.
We discuss the four cases in the main text: (1) both firms employ TP; (2) only firm
1 employs TP; (3) only firm 2 employs TP; and (4) no firm employs TP.
13

Both firms employ TP: The profits in the second stage are given as
π1 =(t + d − e2 + e1 )2(t − d − e1 + e2 )2− γe21 , π2 =− γe22 .
4t
4t

The first-order conditions yield the following reaction functions of the firms:
e1 (e2 ) =t + d − e2t − d − e1
, e2 (e1 ) =.
4tγ − 14tγ − 1

Solving these equations, we have2tγ + 2dγ − 1
2tγ − 2dγ − 1, e2 =,
4γ(2tγ − 1)4γ(2tγ − 1)(4tγ − 1)(2tγ − 2dγ − 1)2(4tγ − 1)(2tγ + 2dγ − 1)2
π1 =
,
π
=
.
2
16γ(2tγ − 1)2
16γ(2tγ − 1)2e1 =Only firm 1 employs TP: The profits in the second stage are given asπ1 =(3t + d − e2 + e1 )2(t − d − e1 + e2 )2− γe21 , π2 =− γe22 .
16t
8t

The first-order conditions yield the following reaction functions of the firms:
e1 (e2 ) =3t + d − e2t − d − e1
, e2 (e1 ) =.
16tγ − 18tγ − 1

Solving these equations, we have
6tγ + 2dγ − 1
4tγ − 4dγ − 1, e2 =,
2γ(16tγ − 3)2γ(16tγ − 3)(16tγ − 1)(6tγ + 2dγ − 1)2(8tγ − 1)(4tγ − 4dγ − 1)2
π1 =, π2 =.
4γ(16tγ − 3)24γ(16tγ − 3)2e1 =Only firm 2 employs TP: The profits in the second stage are given asπ1 =(3t − d − e1 + e2 )2(t + d − e2 + e1 )2− γe21 , π2 =− γe22 .
8t
16t

The first-order conditions yield the following reaction functions of the firms:
e1 (e2 ) =t + d − e23t − d − e1
, e2 (e1 ) =.
8tγ − 116tγ − 114

Table 2: Payoff matrix in the full game
1/2

UP

TP

(18tγ − 1)(9tγ − 3dγ − 1)236γ(9tγ − 1)2(16tγ − 1)(6tγ − 2dγ − 1)24γ(16tγ − 3)2UP(18tγ − 1)(9tγ + 3dγ − 1)236γ(9tγ − 1)2(8tγ − 1)(4tγ + 4dγ − 1)24γ(16tγ − 3)2(8tγ − 1)(4tγ − 4dγ − 1)24γ(16tγ − 3)2(4tγ − 1)(2tγ − 2dγ − 1)2
16γ(2tγ − 1)2TP(16tγ − 1)(6tγ + 2dγ − 1)24γ(16tγ − 3)2(4tγ − 1)(2tγ + 2dγ − 1)2
16γ(2tγ − 1)2

Solving these equations, we have4tγ + 4dγ − 1
6tγ − 2dγ − 1, e2 =,
2γ(16tγ − 3)2γ(16tγ − 3)(8tγ − 1)(4tγ + 4dγ − 1)2(16tγ − 3)(6tγ − 2dγ − 1)2
π1 =
,
π
=
.
2
4γ(16tγ − 3)24γ(16tγ − 3)2e1 =No firm employs TP: The profits in the second stage are given asπ1 =(3t + d − e2 + e1 )2(3t − d − e1 + e2 )2− γe21 , π2 =− γe22 .
18t
18t

The first-order conditions yield the following reaction functions of the firms:
e1 (e2 ) =3t − d − e1
3t + d − e2
, e2 (e1 ) =.
18tγ − 118tγ − 1

Solving these equations, we have
9tγ − 3dγ − 1
9tγ + 3dγ − 1, e2 =,
6γ(9tγ − 1)6γ(9tγ − 1)(18tγ − 1)(9tγ + 3dγ − 1)2(18tγ − 1)(9tγ − 3dγ − 1)2
π1 =
,
π
=
.
2
36γ(9tγ − 1)236γ(9tγ − 1)2

e1 =Payoff matrix: From the above discussion, we obtain the payoff matrix in Table 2.
15

d˜
d¯

3/4

f (tγ)

2/4
1/4
tγ
0

1

2

3

4

5

6

7

Figure 1: Condition under which firm 2 does not employ TP

We easily find that for firm 1, employing TP dominates employing UP. Given thisproperty, whether firm 2 employs TP depends on the exogenous parameters. We defined˜ ≡ d/t. d˜ denotes the cost difference normalized by per length transport cost, t. We
assume that γd < tγ − 1/2 to ensure that the quantities supplied by the firms are¯ d¯ is the upper
always positive. The inequality is rewritten as d˜ < 1 − 1/(2tγ) ≡ d.
˜bound of d.
A simple comparison leads to the following proposition.
Proposition 1. (i) If d˜ > f (tγ), then only firm 1 employs TP; otherwise, both firms
employ TP, wheref (tγ) ≡hip(2tγ − 1) (4tγ − 1)(128t2 γ 2 − 16tγ + 1) − 2(16tγ − 3) (4tγ − 1)(8tγ − 1)2tγ(512t3 γ 3 − 64t2 γ 2 − 60tγ + 7).

(ii) The inequality d¯ > f (tγ) always holds.
Figure 1 indicates the range over which d¯ > d˜ > f (tγ). Note that as tγ → ∞, bothf (tγ) and d¯ converge to 1.
When both firms employ TP, a cost reduction implemented by firm i does notchange firm j’s prices for the points that firm i supplies. This implies that the cost16

reducing activities of firm i do not accelerate the price competition between the two
firms. That is, the strategic effect of cost-reducing activities is quite different fromthat in the standard price competition model. Under the strategic environment, theexpected market share of a firm is the only factor that affects the incentive of that firm’s
cost-reducing activities. The more efficient the firm, the greater is the output overwhich it can apply the outcomes of its cost-reducing activities, and hence, the greater
are its returns from such activities. Because the converse also holds, the incentive of
the inefficient firm to engage in cost-reducing activities gradually diminishes as the exante cost difference increases.
When the efficient firm employs TP and the inefficient firm employs UP, a cost
reduction implemented by the efficient firm changes the inefficient firm’s price. As in thestandard price competition model, the cost reduction of the efficient firm accelerates theprice competition between them. This pricing strategy of the inefficient firm diminishes
the incentive of the efficient firm to engage in cost-reducing activities. This effect isstronger when the ex ante cost difference is large because the efficient firm obtaining alarger market share has a stronger incentive to engage in cost-reducing activities. Thisimplies that employing UP is beneficial for the inefficient firm when the ex ante costdifference is large. Therefore, when d˜ is large enough, the inefficient firm employs UP.
Employing TP is not always a dominant strategy for firms even though it is costless.
We check the market share of firm 1 under TP-TP and TP-UP when d˜ = f (tγ) (seeFigure 2). This would be helpful to understand the situation in which firm 2 should
employ UP. Firm 1’s market share where d˜ = f (tγ) is smaller when tγ is smaller. This
reflects the fact that the stronger the incentive of firm 1 to reduce its cost, the stronger
is the incentive of firm 2 to mitigate the cost-reducing activity by firm 1. Figure 2 also
indicates that the equilibrium market share of firm 2 is smaller if it chooses UP insteadof TP given that firm 1 chooses TP.
17

x`
1
TP-UP

0.95

TP-TP

0.9
0.85
0.8
2

3

4

5

6

7

tΓ

Figure 2: Market share of firm 1 when d˜ = f (tγ).
Note: “TP-TP” indicates the case where both firms employ TP, and “TP-UP”
indicates the case where only firm 1 employs TP.

We have discussed the case in which firms engage in cost-reducing activities. We
can develop a model where they engage in quality-improving activities, which is mathematically equivalent to the model discussed in this subsection. Consider the following
model. The marginal cost of the firm is c. The product quality of firm 1 is q1 = v+d+e1
and that of firm 2 is q2 = v + e2 , where v is a positive constant and sufficiently large,d(> 0) is the initial quality advantage of firm 1, and ei is firm i’s effort level toraise the quality of its product. Consumer x buys a good from firm 1 if and only if
q1 − p1 (x) − tx ≥ q2 − p2 (x) − t(1 − x); note that if q1 − p1 (x) − tx = q2 − p2 (x) − t(1 − x),the consumer randomly chooses one of the firms. The remaining assumptions are thesame to those of the previous model. In this model, we can derive exactly the same
payoff matrix in Table 2, and thus, we obtain the following result.8
Corollary. Firm 1 ( ex ante higher-quality firm) always adopts TP, while firm 2 ( exante lower-quality firm) adopts UP if d˜ > f (tγ).
8The mathematical procedure to derive the result is available upon request.

18

This is a contrasting result to Choudhary et al. (2005). They examine a verticallydifferentiated duopoly and show that a higher-quality firm can be worse off with TP,
while a lower-quality firm cannot. There are three important differences between ourmodel and theirs. First, in our model horizontal product differentiation exists. Second,
we assume that quality improvement requires fixed cost, while they assume that it raises
a marginal cost. Third, most importantly, in our model, an increase in the productquality in a firm raises the willingness to pay for consumers uniformly, while in themodel of Choudhary et al. (2005), an increase in the product quality in a firm raises
the willingness to pay for consumers unequally, more so for consumers who have ahigher preference for high quality.
Consider a standard vertical differentiation model discussed by them. Supposethat firm 1 is the higher-quality firm and firm 2 is the lower-quality firm. Firm 1(firm 2) supplies the product for consumers whose willingness to pay for a quality
improvement is relatively high (low). Consider the uniform pricing policy. Supposethat firm 1 improves its quality. This quality improvement increases the willingness
to pay for consumers unequally, more so for consumers who have a higher preference
for high quality, and less for marginal consumers. Firm 1 raises the price more than
the increase in willingness to pay for marginal consumers and reaps more profits fromconsumers whose loyalty to firm 1 is high. As a result, competition between two firms
is mitigated. On the contrary, a quality improvement in firm 2’s product increases
the willingness to pay more for marginal consumers over whom firms compete than forthe consumers whose loyalty to firm 2 is high. As a result, competition is accelerated.
Consider the personalized pricing policy. An increase in the quality of each firm’s
product improves the competitive advantage more for consumers whose loyalty to firm
1 is higher. Thus, the effect of an increase in the quality of firm 1 on firm 2’s profit is
weak because the effect on firm 2’s consumers is weak. On the contrary, the effect of an
19increase in the quality of firm 2 on firm 1’s profit is strong because the effect on firm 1’s
consumers is strong. In other words, the strategic effects of quality improvement areasymmetric in both policies. For these reasons, the profit loss of firm 1 by the rival’s
quality improvement is significant for firm 1, and thus, firm 1 adopts UP to dampen
the quality-improvement activity by firm 2.
In our model, quality improvements by both the higher-quality and the lower-quality
firms accelerate competition, which is unlike in the above vertical differentiation model.
In our model, the higher-quality firm is the firm with a larger market share, and thelower-quality firm has a stronger incentive for dampening the quality-improvement
activity by the rival whose market share is large, as is discussed in the cost-reducing
model. Thus, the lower-quality firm adopts UP for this strategic purpose.

4.2

Asymmetric effort costs

We consider the case in which the effort costs to decrease the marginal cost depend on
the decision of pricing policy. We only mention the additional assumption on the modelin the previous subsection. We assume that I(ei ) = αγe2i if firm i employs TP, where
α > 1. That is, the effort cost under TP is larger than that under UP. Employing TP
requires more skills for the firm. This higher requirement level is also applied to theeffort cost under TP. We therefore assume that α > 1. To guarantee that the secondorder conditions for cost-reducing activities are satisfied, we assume that αtγ ≥ 1. The
timing structure of the game is the same as in the previous subsection.
The decision of firm 2 to employ TP depends on the ex ante cost difference between
˜ as depicted in Figure 1. The threshold value of d˜ depends not only on tγ
the firms, d,but also on the difficulty of cost reduction under TP, α. Denote the threshold value
by f (αtγ). We have the following result.
Proposition 2.

(i) Firm 1 always employs TP, and firm 2 does if and only if20f HΑtΓLd
1


d
1

-

d

0.8

0.9
f HΑtΓL

0.6

0.8
0.4
0.7

0.2
1.2

1.4

1.6

1.8

2

0.6

Α

1.2

1.4

1.6

1.8

2

Α

˜ f (αtγ).
Figure 3: Threshold value of d,˜
Note: We fix tγ at 2 for presentation. d¯ is the upper bound of d.

d˜ < f (αtγ). (ii) f (αtγ) is increasing in α.
This means that as cost reduction under TP becomes more difficult, firm 2’s incentive to employ TP increases (see Figure 3). As explained earlier, the key point ofour model is that the employment of TP by firm 2 enhances the incentive of firm 1
for cost-reducing activity given that firm 1 employs TP. An increase in the difficultyof cost reduction (an increase in α) weakens this strategic reaction of firm 1 to firm 2’s
adoption of TP, which enhances the incentive of firm 2 to employ TP.

4.3

Pricing policy and marginal costs

We have assumed that employing TP does not increase the firm’s marginal cost. Instead
we assume that the effort costs to decrease the marginal cost depend on the decisionof pricing policy, we briefly discuss the case in which the decision of pricing policy
increases the ex ante marginal cost. We only mention the additional assumption on
the model in the basic model. Assume that the ex ante marginal cost of firm 1 is
c1 = c + f − e1 if it employs TP otherwise this is c1 = c − e1 . Assume also that theex ante marginal cost of firm 2 is c2 = c + d + f − e2 if it employs TP otherwise thisis c2 = c + d − e2 . Furthermore, for analytical simplicity, we also assume that f < d,21

that is, the marginal cost increment through employing TP is not large. The timing
structure of the game is the same as in the previous subsection. Under the modified
setting, we find that firm 2 does not employ TP for wider parameter ranges because no
employment of TP has additional value for firm 2. This functions as a marginal cost
reducing investment. By no employment of TP, the ex ante marginal cost of firm 2 isc2 = c + d − e2 instead of c2 = c + d + f − e2 . The additional benefit induces firm 2
not to employ TP.

5Conclusion

The recent developments in IT enable firms to employ personalized pricing (Arora etal., 2008 and references therein). Following the recent trend, this paper provides asimple model to investigate whether firms should employ personalized pricing. The
model setting in this paper is based on those of Thisse and Vives (1988) and Shaffer
and Zhang (2002). Our model incorporates the fact that firms engage in cost-reducing
activities after they determine their pricing policies. The main result of this paper isthat when the ex ante cost difference between the firms is large, employing personalized
pricing harms the less-efficient firm even though employing this pricing is costless. This
result does not hold when the firms do not engage in cost-reducing activities. Thatis, when the ex post activities related to cost reductions are important to firms, lessefficient firms do not have to employ personalized pricing. This result has a managerial
implication for firms that consider whether to employ personalized pricing. Our main
result implies that such firms should take into account their relative competitive positions and technological environments. When firms are small and their cost structures
is easily changed by their efforts, they would need to reconsider whether to employ
personalized pricing.

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25

Appendix (not for publication)
We explain the calculation to derive Corollary in Section 4.1. There are four cases
depending on the pricing strategies set by the firms. We discuss each of the four cases.
UP-UP

When both firms employ UP, the indifferent consumer, x̂nn , is given as

q1 − p1 − tx̂nn = q2 − p2 − t(1 − x̂nn ) → x̂nn =t + q1 − q2 + p2 − p1.
2t

The profits are given by
π1 = (p1 − c)x̂nn , π2 = (p2 − c)(1 − x̂nn ).
The first-order conditions lead topi =3t + q1 − q2
(3t + qi − qj )2
3t + 3c + qi − qj
, x̂nn =, πi =.
36t
18t

By substituting q1 = v + d + e1 and q2 = v + e2 into πi , we have
π1 =(3t + d + e1 − e2 )2(3t − d − e1 + e2 )2, π2 =.
18t
18t

The profits are equivalent with those under ‘No firm employs TP’ in Section 4.1 except
the effort costs, γe2i .
UP-TP

Consider the case in which firm 1 employs UP and firm 2 employs TP. We

can apply the discussion of this case to that in which firm 2 employs UP and firm 1
employs TP. When only firm 1 employs UP, the timing is as follows. First, firm 1 sets
its price. Second, observing this price, firm 2 sets its prices.
Given that firm 1 sets its price p1 , firm 2 takes the demand of consumer x if andonly ifq1 − p1 − tx < q2 − p2 (x) − t(1 − x) → p2 (x) < q2 − q1 + p1 + t(2x − 1).
1

If the right-hand side in the latter inequality is larger than the marginal cost of firm 2,
c2 , it sets p2 (x) at q2 − q1 + p1 + t(2x − 1) − ε to take the demand of consumer x, where
ε is sufficiently small; we omit this value. Otherwise, it sets p2 (x) at c2 . The optimal
price of firm 2 at point x is given by½
q2 − q1 + p1 + t(2x − 1) if c < q2 − q1 + p1 + t(2x − 1),p2 (x) =cotherwise.
The indifferent consumer x̂nd is given byc = q2 − q1 + p1 + t(2x − 1) → x̂nd =t + c + q1 − q2 − p1.
2t

Anticipating the prices of firm 2, firm 1 sets its price, p1 . The profit of firm 1 is givenby
π1 = (p1 − c)x̂nd =(p1 − c)(t + c + q1 − q2 − p1 ).
2t

The first-order condition leads top1 =t + 2c + q1 − q2
t + q1 − q2
(t + q1 − q2 )2, x̂nd =, π1 =.
2
4t
8t

The profit of firm 2 is given by
Z 1
π2 =

(q2 − q1 + p1 + t(2m − 1) − c)dm =x̂nd(3t + q2 − q1 )2.
16t

By substituting q1 = v + d + e1 and q2 = v + e2 into πi , we have
π1 =(t + d + e1 − e2 )2(3t + e2 − d − e1 )2, π2 =.
8t
16t

The profits are equivalent with those under ‘Only firm 2 employs TP’ in Section 4.1
except the effort costs, γe2i .
Conversely, under the case in which firm 2 employs UP and firm 1 employs TP, theprofits of the firms are given byπ1 =(3t + q1 − q2 )2(t + q2 − q1 )2, π2 =.
16t
8t
2

By substituting q1 = v + d + e1 and q2 = v + e2 into πi , we have
(3t + d + e1 − e2 )2(t + e2 − d − e1 )2
π1 =, π2 =.
16t
8t
The profits are equivalent with those under ‘Only firm 1 employs TP’ in Section 4.1
except the effort costs, γe2i .
TP-TPWhen both firms employ TP, the lowest price of firm i for each consumer

is c. Firm 1 takes the demand of consumer x if and only if
q1 − p1 (x) − tx > q2 − c − t(1 − x) → p1 (x) < q1 − q2 + c + t(1 − 2x).
If the right-hand side in the latter inequality is larger than the marginal cost of firm 1,c, it sets p1 (x) at q1 − q2 + c + t(1 − 2x) − ε to take the demand of consumer x, where
ε is sufficiently small; we omit this value. Otherwise, it sets p1 (x) at c. The optimal
price of firm 1 at point x is given by½
p1 (x) =q1 − q2 + c + t(1 − 2x) if q2 < q1 + t(1 − 2x),cotherwise.

A similar argument is applied to the prices of firm 2.
½
p2 (x) =q2 − q1 + c + t(2x − 1) if q1 < q2 + t(2x − 1),cotherwise.

The indifferent consumer x̂dd is given by
q1 − c − tx = q2 − c − t(1 − x) → x̂dd =t + q1 − q2.
2t

The profits of the firms are given by
Z x̂dd(t + q1 − q2 )2
,
4t
0
Z 1
(t + q2 − q1 )2(q2 − q1 + c + t(2m − 1) − c)dm =
π2 =
.
4t
x̂dd
π1 =

(q1 − q2 + c + t(1 − 2m) − c)dm =3

By substituting q1 = v + d + e1 and q2 = v + e2 into πi , we have(t + d + e1 − e2 )2(t + e2 − d − e1 )2
π1 =, π2 =.
4t
4t
The profits are equivalent with those under ‘Both firms employ TP’ in Section 4.1
except the effort costs, γe2i .

4