ISSN (Print) 0473-453X

Discussion Paper No. 1179

ISSN (Online) 2435-0982

BEHAVIOR-BASED PRICE DISCRIMINATION
IN THE DOMESTIC AND INTERNATIONALMIXED DUOPOLY

Suzuka Okuyama

July 2022

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanBehavior-based Price Discrimination in the Domestic
and International Mixed duopoly ∗Suzuka Okuyama †Institute of Social and Economic Research, Osaka University

Abstract
This study investigates mixed markets in which a social welfare-maximizing public
ﬁrm and a private ﬁrm engage in behavior-based price discrimination (BBPD).
Total of two cases are considered: one where domestic shareholders completely
own the private ﬁrm and one where foreign shareholders completely own it. In
the domestic mixed duopoly, BBPD is irrelevant from the viewpoint of social
welfare. This is because poaching does not occur. In the international mixed
duopoly, BBPD improves domestic social welfare, as it allows the public ﬁrm tolower its poaching price. In both cases, privatization is more undesirable under
BBPD than uniform pricing.

JEL classiﬁcation: D43, H42,L13
Keywords : Behavior-based price discrimination, Mixed oligopoly,
Foreign ﬁrms, Privatization

1

Introduction

In the telecommunication industry, a ﬁrm that is fully or partially owned by the governments provides mobile services, and oﬀers diﬀerent prices depending on consumers’ past
purchases; this is called behavior-based price discrimination (BBPD). 1 Orange and Docomo oﬀer discounts on cell phone bills to new consumers. Telkomsel oﬀers discounts to∗

The author would like to thank Junichiro Ishida, Tatsuro Iwaisako, Koichi Futagami, and Noriaki
Matsushima for very helpful comments and suggestions. This research did not receive any speciﬁcgrant from funding agencies in the public, commercial, or not-for-proﬁt sectors.
†
Corresponding author. Suzuka Okuyama, Institute of Social and Economic Research, Osaka University, 6-1, Mihogaoka, Ibaraki, Osaka, 567-0047, JAPAN. E-mail: okuyama@iser.osaka-u.ac.jp.
1
In mobile phone markets, consumers can choose between pre- or post-paid plans. When ﬁrms
contract post-paid plans with rivals’ consumers, not only the private ﬁrms but also those that the
governments fully or partially own oﬀer discounts.

1

consumers with new numbers. Telenor oﬀers new consumers a two-month trial period.
On the other hand, BSNL does not oﬀer new consumers discounts.
Recently, government ownership in these markets has become issue. In Japan, NTT,a partially privatized company, made Docomo its subsidiary in 2021. In India, theworld’s second-largest cell phone market, BSNL, which the Indian government fully
owns, is expected to be privatized. The impacts of nationalization and privatization on
welfare in these markets have not been fully discussed. Therefore, this study investigates
whether the presence of a social welfare-maximizing public ﬁrm improves welfare in themarket where ﬁrms engage in price discrimination.
This study provides analyses of BBPD in mixed duopoly markets. We follow thebrand preference approach based on Hotelling models (Fudenberg and Tirole, 2000).
We assume that the public ﬁrm maximizes domestic social welfare and the private
ﬁrm maximizes its proﬁt. In a domestic (resp. international) mixed duopoly, thepublic ﬁrm competes with a private ﬁrm completely owned by domestic (resp. foreign)
shareholders.2 Consumers’ preferences are private information. In the ﬁrst period, ﬁrms
oﬀer uniform prices. In the second, ﬁrms oﬀer diﬀerent prices based on the consumers’purchase histories. This study investigates the impacts of BBPD on domestic social
welfare and discusses privatization.
BBPD is found to be irrelevant from the viewpoint of social welfare in the domestic mixed duopoly, in contrast to the result in Fudenberg and Tirole (2000), whichshow that BBPD is detrimental to social welfare in a domestic pure duopoly. In
Fudenberg and Tirole (2000), two proﬁt-maximizing private ﬁrms poach their rival’s
consumers by oﬀering discounts, and poaching induces socially ineﬃcient switching inthe second period. However, in our model, the public ﬁrm does not oﬀer a discount to
its rival’s consumers, and poaching does not occur in the second period since the public
ﬁrm minimizes the sum of disutilities of taste mismatch in each turf.
BBPD is beneﬁcial to domestic social welfare in the international mixed duopoly.
The public ﬁrm has an incentive to lower its prices to reduce the outﬂow to foreign
shareholders in the international mixed duopoly. By lowering its prices, it faces anincrease in the sum of disutilities of taste mismatch if ﬁrms do not engae in BBPD.
However, BBPD allows the public ﬁrm to lower its poaching price without increasing
the sum of disutilities of taste mismatch in the private ﬁrm’s turf in the second period. Therefore, BBPD lowers equilibrium prices and reduces the outﬂow to foreign
shareholders, although BBPD increases the total discounted sum of disutilities of taste
mismatch in the international mixed duopoly.
Privatization is more undesirable under BBPD than uniform pricing. 3 Welfare loss
2

Some compete with domestic private ﬁrms (e.g., Docomo) and others compete with foreign private
ﬁrms (Orange, Telenor, Telia, Telkomsel, BSNL).
3
Recently, the literature on privatization policies has become richer. Matsumura and Shimizu(2010) examine sequential privatization. Matsumura and Okamura (2015) and Haraguchi et al. (2018)
discuss the relationship between competition and the optimal privatization policy. Lin and Matsumura

2

from privatization is found to be larger under BBPD than uniform pricing in the domestic and international mixed duopoly. In the domestic mixed duopoly, privatization
reduces domestic social welfare under BBPD, although it is irrelevant from the viewpoint of domestic social welfare under uniform pricing. In the international mixed
duopoly, privatization reduces domestic social welfare under both uniform pricing and
BBPD. However, the welfare loss from privatization is larger under BBPD than underuniform pricing. These results indicate that, from the viewpoint of domestic social welfare, the subsidiary acquisition of Docomo improves social welfare and privatization of
BSNL is more detrimental to the market of the post-paid plan than that of the pre-paid
plan.
The remainder of this paper is organized as follows. Section 2 provides a reviewof the literature. Section 3 describes the model setup. The eﬀects of BBPD on ﬁrms’
proﬁt, consumer surplus, and social welfare in the domestic mixed duopoly are examinedin Section 4. Section 5 examines the eﬀects of the foreign mixed duopoly. Section 6
discusses the welfare implications of BBPD. Finally, concluding remarks arepresented
in Section 7.

2

Related literature

Our study relates to two strands of literature. Firstly, it is related to a strand that
deals with BBPD. Since the seminal works by Chen (1997), Villas-Boas (1999), andFudenberg and Tirole (2000), numerous studies have investigated BBPD. 4 These threestudies and Choe and Matsushima (2021) demonstrate that BBPD is detrimental toﬁrms unless they are suﬃciently asymmetric since it intensiﬁes competition. Carroni(2016), Jing (2017), and Rhee and Thomadsen (2017) introduce asymmetric competitive advantage and indicate that BBPD can be beneﬁcial to ﬁrms. We assume thatﬁrms maximize diﬀerent objective functions and show that BBPD is detrimental toﬁrms. The reason is that the pricing of the public ﬁrm limits the private ﬁrm’s price inthe second period.
This paper is also related to the strand of literature on mixed oligopoly. In particular, this paper has strong connections to (i) spatial competition and (ii) international
competition.
(2018a) discuss optimal privatization policy in the Stackelberg competition. Sato and Matsumura(2019a, 2019b) investigate the impact of the shadow cost of public funds on the optimal privatization
policy. Kim (2018, 2019), Haraguchi and Matsumura (2020b, 2020c), and Liu et al. (2021) discuss
optimal privatization policies by introducing pollution, corporate social responsibility, asymmetric private ﬁrms, commitment, and corporate taxation policies, respectively. See Matsumura and Okumura(2013), Matsumura and Okumura (2017) for privatization neutrality theorem.
4Chen and Pearcy (2010) discuss the intertemporal dependence of brand preference. Liu and Shuai(2013, 2016) investigate price discrimination in a multi-dimensional model. Esteves and Reggiani(2014) and Zhang et al. (2019) introduce elastic demand.

3

The literature includes two types of spatial competition: price competition underuniform pricing and spatial price discrimination.
In the ﬁrst type model, Cremer et al. (1991) examine a mixed duopoly with horizontal diﬀerentiation in a Hotelling model. They show that the equilibrium location is
optimal, and ﬁrst-best is achieved in the domestic mixed duopoly, whereas maximal differentiation occurs in a domestic pure duopoly. This diﬀerence occurs because the public
ﬁrm minimizes the transport cost. Matsumura and Matsushima (2004) introduce costreducing activities into the model in Cremer et al. (1991). Kitahara and Matsumura(2013) and Matsumura and Tomaru (2015) extend this model by incorporating elastic
demand and shadow cost of public ﬁnding, respectively.
In the second type model, Matsushima and Matsumura (2003) investigate a mixed
oligopoly by using the spatial price discrimination model with Cournot competition. 5
Heywood and Ye (2009b), Ebina et al. (2009), Ye and Wu (2015), and Heywood et al.
(2021) extend Matsushima and Matsumura (2003) by incorporating, price competition,
additional transportation costs, two-ﬁrm mergers, and an upstream mixed oligopoly,
respectively. Matsushima and Matsumura (2006) and Heywood and Ye (2009a) analyze
spatial price discrimination models of mixed markets with foreign private ﬁrms.
In our model, it is assumed that ﬁrms oﬀer diﬀerent prices based on the consumers’purchase histories. Poaching does not occur and ﬁrst-best is achieved in the domestic
mixed duopoly, whereas poaching induces socially ineﬃcient switching in the domestic
pure duopoly as shown in Fudenberg and Tirole (2000). The reason for this diﬀerenceis that the public ﬁrm minimizes the sum of disutilities of taste mismatch.
Since Fjell and Pal (1996) and Pal and White (1998), many studies have discussed
the international mixed oligopoly.6 Fjell and Pal (1996) suggest that the public ﬁrm’s
price is below its marginal cost as the domestic social welfare does not include foreign
ﬁrms’ proﬁts in the international mixed oligopoly. We demonstrate that BBPD allows
the public ﬁrm to further reduce outﬂows to foreign shareholders.

5

In spatial price discrimination models, ﬁrms oﬀer diﬀerent prices based on consumers’ location.
Matsumura (2003) and Matsumura and Tomaru (2012) investigate Stackelberg competition in
the international mixed oligopoly. Matsumura et al. (2009), and Haraguchi and Matsumura (2014)
discuss the international mixed oligopoly with diﬀerentiated products. Lin and Matsumura (2012),
Cato and Matsumura (2012), Cato and Matsumura (2015), and Lee et al. (2018) examine the eﬀect ofpartial privatization, developed by Matsumura (1998) on social welfare. Lyu and Shuai (2017) discusssubcontracting in the international mixed duopoly by using the Hotelling model. Lin and Matsumura(2018b) discuss the privatization neutrality theorem when domestic and foreign investors own theprivate ﬁrm. Matsumura and Matsushima (2012) indicate that the privatization of international airports can improve welfare. Haraguchi and Matsumura (2020a) discuss technology transfer from foreign
enterprises.
6

4

3

Preliminary

We extend Fudenberg and Tirole (2000) by considering mixed duopoly markets. Twoﬁrms, A and B, compete in two periods. Let Firm A be a public ﬁrm, which maximizes
domestic social welfare, and Firm B be a private ﬁrm, which maximizes its proﬁt. The
ﬁrms produce goods at a constant marginal cost, c, and simultaneously set their prices
in each period.
Consumers uniformly distribute over [θ, θ] (θ = −θ, θ < 0). The parameter θ
represents how much consumers prefer goods B over goods A. Consumers’ preferences
are constant over time. Consumers buy one unit of product from Firm A or Firm B
or do not buy at all. The per-period utilities for a consumer indexed by θ purchasingfrom A and B at price p are uA = v − 2θ − p and uB = v + 2θ − p, respectively. We
assume that v is suﬃciently large so that all consumers purchase in equilibrium. Firms
and consumers discount their future by the common factor δ ∈ [0, 1).

4
4.1

Domestic mixed duopoly
Uniform pricing

We consider the benchmark case where BBPD is not feasible. The ﬁrms cannot identify
the ﬁrst-period decisions of consumers, or BBPD is illegal. The two-period model
reduces to two replications of the static equilibrium. We solve the static model.
Let θU denote the consumer who is indiﬀerent between choosing Firm A and Firm
B. Consumers in [θ, θU ] choose Firm A and consumers in [θU , θ] choose Firm B.
Firm A and Firm B oﬀer pA and pB to consumers, respectively. Then, θU satisﬁesUUv − θ2 − pA = v + θ2 − pB . Solving it for θU yields
θ U = pB − pA .

(1)

Per-period domestic social welfare under uniform pricing is the sum of per-period
consumer surplus and per-period proﬁts of ﬁrms and is given by
1 2 1
1swU = (v − c)(θ − θ) + θ + θ2 − (θU )2 .
4
4
2

(2)

Firm A chooses pA to maximize (2), and Firm B chooses pB to maximize its proﬁt.
The proﬁt function of Firm B isπB = (pB − c)(θ − θU ).

(3)

pB − pA = 0,c + θ + pA − 2pB = 0.

(4)
(5)

The ﬁrst-order conditions are

5The ﬁrst-order conditions yield p∗A = p∗B = c + θ and θU ∗ = 0.
Hence, the total discounted consumer surplus, total discounted proﬁts, and total
discounted social welfare in equilibrium under uniform pricing in the domestic mixed
duopoly are
1 2 1
2CS U = (1 + δ)[(v − c)(θ − θ) + θ + θ2 − 2θ ],
4
4
2
U
U
ΠA = ΠB = (1 + δ)θ ,1 2 1
SW U = (1 + δ)[(v − c)(θ − θ) + θ + θ2 ].
4
4

4.2

(6)
(7)
(8)

BBPD

We now assume that BBPD is feasible. In the ﬁrst period, each ﬁrm sets a uniform
price. In the second, they oﬀer diﬀerent prices to consumers who have diﬀerent purchasehistories after they observe consumers’ ﬁrst-period behaviors. We derive the subgameperfect Nash equilibrium by backward induction.

The second period
Let θ∗ denote the consumer who is indiﬀerent between choosing Firm A and Firm B in
the ﬁrst period. Consumers in [θ, θ∗ ] chose Firm A and consumers in [θ∗ , θ] chose Firm
B in the ﬁrst period. In the second period, Firm A and Firm B oﬀer α and β to theirpast consumers and oﬀer α̂ and β̂ to the rival’s past consumers, respectively. Consumer
θ on [θ, θ∗ ] continues to purchase from Firm A if v − 2θ − α ≥ v + 2θ − β̂, otherwise,
they switche to Firm B. Consumer θ on [θ∗ , θ] continues to purchase from Firm B ifv − 2θ − α̂ ≤ v + 2θ − β, otherwise, they switche to Firm A.
Considering the possibility of corner solutions, we obtain the locations of the indifferent consumers on [θ, θ∗ ] and [θ∗ , θ], θA and θB , respectively.
{
θA =
{
θB =

β̂ − α,
θ∗ ,

if
if

β̂ − α ≤ θ∗ ,β̂ − α ≥ θ∗ ,

(9)

θ∗ ,
β − α̂,if
if

β − α̂ ≤ θ∗ ,β − α̂ ≥ θ∗ .

(10)Domestic social welfare in the second period is given by
1 2 1
1
1 2
1 2
sw2D = (v − c)(θ − θ) + θ + θ2 + θ∗2 − θA
− θB
,
4
4
2
2
2
6

(11)

where the superscript “D” stands for price discrimination. The second-period proﬁt of
ﬁrm B is given byD
πB2= (β − c)(θ − θB ) + (β̂ − c)(θ∗ − θA ).

(12)

Firm A chooses α and α̂ to maximize (11), and Firm B chooses β and β̂ to maximize(12). By taking the parametric conditions in (9) and (10) into account, we solve themaximization problems, leading to{(13)β̂ − θ∗ ,if θ ≤ θ∗ ≤ 0,
α=∗β̂,
if 0 ≤ θ ≤ θ,
(14)
{
β,
if θ ≤ θ∗ ≤ 0,
(15)
α̂ =
∗
∗
β−θ ,
if 0 ≤ θ ≤ θ,(16)∗if α ≤ c − θ∗ ,(17)
α + θ ,∗β̂ = 1c+θ α+,
if α ≥ c − θ∗ ,
(18)
2
2

c+θ
1
α̂ +
,
β= 2
2

α̂ + θ∗ ,ifα̂ ≤ c + θ − 2θ∗ ,

(19)

if

α̂ ≥ c + θ − 2θ∗ .

(20)These best-response functions are depicted in Figure 1.7 The best-response function
for Firm A in its turf, which is given by (13) when θ ≤ θ∗ ≤ 0 and (14) when 0 ≤ θ∗ ≤ θ,
is depicted in Figures 1 (a) and (b), respectively. The best-response function for Firm
A in Firm B’s turf, which is given by (15) when θ ≤ θ∗ ≤ 0 and (16) when 0 ≤ θ∗ ≤ θ,
is depicted in Figures 1 (c) and (d), respectively. It is clear that corner solutions areachieved in Figures 1 (a) and (d) and interior solutions are achieved in Figures 1 (b)and (c).
The best-response function for Firm B in Firm A’s turf, which are given by (17) and(18), is depicted in Figures 1 (a) and (b). That for Firm B’s turf, which are given by(19) and (20), is depicted in Figures 1 (c) and (d). Since the private ﬁrm does not set
its price lower than its marginal cost, Firm B’s best-response functions have horizontal
parts in Figure 1.8
Figure 1 (d) depicts equilibrium under the condition c+θ−2θ∗ > 0. The ﬁgure under the conditionc + θ − 2θ∗ ≤ 0 is omitted since equilibrium prices under the condition c + θ − 2θ∗ ≤ 0 is the same asunder the condition c + θ − 2θ∗ > 0.
8
Following Fudenberg and Tirole (2000), we assume that a proﬁt-maximizing ﬁrm does not charge
prices below the marginal cost.
77

( Figure 1 about here )The best-response functions of the public ﬁrm can be interpreted as follows. Firm
A’s best-response functions have slopes of 1 and vertical intercepts of 0 in Figures 1 (b)and (c). The public ﬁrm chooses its price equal to that of the private ﬁrm when θ∗ ≥ 0
in the public ﬁrm’s turf and when θ∗ ≤ 0 in the private ﬁrm’s turf, respectively. Firm
A’s best-response functions have intercepts of θ∗ in Figures 1 (a) and (d). The public
ﬁrm lowers its price, α, so that θA is equal to θ∗ when θ∗ ≤ 0 in the public ﬁrm’s turf,
and increases its price, α̂, so that θB is equal to θ∗ when θ∗ ≥ 0 in the private ﬁrm’s turf
given the private ﬁrm’s prices. This is because the sum of disutilities of taste mismatchis minimized if the public ﬁrm supplies consumers in [θ, 0] and the private ﬁrm supplies
consumers in [0, θ].
The best-response functions of the private ﬁrm can be interpreted as follows. Firm
B’s best-response function in Firm A’s turf has a slope of one-half if α ≥ c − θ∗ and ahorizontal part if α ≤ c − θ∗ as shown in Figures 1 (a) and (b). The private ﬁrm attacks
the public ﬁrm’s turf by lowering its price, β̂, but not by lowering prices below marginal
cost, c. That in the Firm B’s turf has a slope of one if α̂ ≥ c + θ − 2θ∗ and a slope of
one-half and a horizontal part if α̂ ≤ c + θ − 2θ∗ as shown in Figures 1 (c) and (d). The
private ﬁrm captures the entire market if the public ﬁrm’s price, α̂ is suﬃciently high.
Figure 1 indicatess that there is a unique equilibrium represented by EA in (a) and(b), and EB in (c), respectively. However, there are multiple equilibria in (d). We
assume that a social welfare-maximizing ﬁrm puts a little more value on the consumer
surplus than on the producer surplus. Firm A lowers its price as much as possible andthe equilibrium is then determined at EB in Figure 1 (d).
Thus, equilibrium prices in the second period are
α = c − θ∗ , β̂ = c, α̂ = β = c + θ,
α = β̂ = c + θ∗ , α̂ = c + θ − 2θ∗ , β = c + θ − θ∗ ,if θ ≤ θ∗ ≤ 0,
if 0 ≤ θ∗ ≤ θ.

(21)
(22)

Since θA = θ∗ and θB = 0 when θ ≤ θ∗ ≤ 0 and θA = 0, θB = θ∗ when 0 ≤ θ∗ ≤ θ, Firm
A supplies consumers [θ, 0] and Firm B supplies [0, θ] for all θ∗ . We have
1 2 1
sw2D = (v − c)(θ − θ) + θ + θ2 , for all θ ≤ θ∗ ≤ θ,
4
4
{ 2
θ ,
if θ ≤ θ∗ ≤ 0,D
πB2=
θ∗2 + (θ − θ∗ )2 , if 0 ≤ θ∗ ≤ θ.

(23)
(24)
(25)

Equation (23) shows that the sum of disutilities of taste the mismatch is minimized forall θ∗ .

8

The ﬁrst period
We obtain the indiﬀerent consumers in the ﬁrst period. In the case of θ ≤ θ∗ ≤ 0, theconsumer, θ∗ , is indiﬀerent between choosing Firm A in the ﬁrst period at price a andthen choosing Firm A in the second period at price α, or choosing Firm B in the ﬁrst
period at price b and then choosing Firm A at price α̂. Thus, θ∗ satisﬁes
θ∗
θ∗
θ∗
θ∗
− a + δ(v −− α) = v +− b + δ(v −− α̂).
v−
2
2
2
2
Substituting the second-period equilibrium prices and solving it for θ∗ yieldsθ∗ =b − a + δθ
,
1−δ

if

θ ≤ θ∗ ≤ 0.

(26)

(27)

In the same way, in the case of 0 ≤ θ∗ ≤ θ, the consumer, θ∗ , is indiﬀerent between
choosing Firm A in the ﬁrst period at a, and then choosing Firm B in the second periodat β̂, or choosing Firm B in the ﬁrst period at b, and then choosing Firm B at β. Thus,θ∗ satisﬁes
θ∗
θ∗
θ∗
θ∗
v−
− a + δ(v +− β̂) = v +− b + δ(v +
− β).
(28)
2
2
2
2
Substituting the second-period equilibrium prices and solving it for θ∗ yieldsθ∗ =b − a + δθ,
1 + 2δ

if

0 ≤ θ∗ ≤ θ.

(29)

Domestic social welfare in the ﬁrst period is given by
[
1 2 1
1 ]
sw1D = (v − c)(θ − θ) + θ + θ2 − θ∗2 ,4
4
2
and the ﬁrst-period proﬁt of Firm B is given byD
πB1= (b − c)(θ − θ∗ ).

(30)

(31)

Firm A chooses a to maximize SW D = sw1D + δsw2D and ﬁrm B chooses b to maximize
DDΠD
B = πB1 + δπB2 .
The diﬀerentiation of the above objective functions yields
a = b + δθ,for all θ ≤ θ∗ ≤ θ,c + (1 − 2δ)θ
1


 a+
,
if θ ≤ θ∗ ≤ 0,
2
2
b=
2


 1 − 2δ a + (1 + 2δ)c + (1 + δ + 2δ )θ , if 0 ≤ θ∗ ≤ θ.
2
2
Solving these best-response functions, we have the following proposition.
9

(32)
(33)
(34)

Proposition 1 Equilibrium prices in the domestic mixed duopoly are given bya = α̂ = β = c + θ, α = β̂ = c, b = c + (1 − δ)θ.
Thus, we havep∗A = a = α̂ > α, p∗B = β ≥ b > β̂, θ∗ = θA = θB = 0.
Proof. See Appendix A.
Proposition 1 reveals the public ﬁrm does not oﬀer a discount to its rival’s consumers,
and both ﬁrms do not poach their rival’s consumers in the domestic mixed duopoly. 9
Equations (27) and (32) indicates that it is optimal that Firm A oﬀers its price so thatθ∗ is 0 in the ﬁrst period. Given θ∗ , Firm A oﬀers its prices to minimize the sum of
disutilities in the second period. Hence, poaching, which induces socially ineﬃcientswitching, does not occur.
Total discounted consumer surplus, total discounted proﬁts, and total discounted
domestic social welfare in equilibrium under BBPD in the domestic mixed duopoly are
1 2 1
2CS D = (1 + δ)[(v − c)(θ − θ) + θ + θ2 ] − 2θ ,
4
4
2
D
D
ΠA = Π B = θ ,1 2 1
SW D = (1 + δ)[(v − c)(θ − θ) + θ + θ2 ].
4
4

(35)
(36)
(37)

Proposition 2 Comparing the outcome under uniform pricing, (6), (7), and (8), we
have
U
CS U ≤ CS D , ΠUi ≥ ΠD
= SW D , (i = A, B).
i , SWThus, BBPD does not aﬀect domestic social welfare since poaching does not occur inthe domestic mixed duopoly.
In the domestic mixed duopoly, BBPD is beneﬁcial to consumers, detrimental to ﬁrms,
and neutral to domestic social welfare. Fudenberg and Tirole (2000) consider two symmetric proﬁt-maximizing ﬁrms and demonstrate that BBPD is beneﬁcial to consumers
and detrimental to ﬁrms and social welfare. Our result indicates that BBPD does notharm domestic social welfare, unlike Fudenberg and Tirole (2000).
Domestic social welfare in the ﬁrst period equals in both cases since the ﬁrst-period
diﬀerential consumers are located at 0. However, the second-period domestic social
welfare in Fudenberg and Tirole (2000) is smaller than our model. This is because two
ﬁrms poach their rival’s consumers in Fudenberg and Tirole (2000), whereas poachingdoes not occur, and no consumers switch in our model.
9

Poaching occurs if the public ﬁrm engages in uniform pricing and the private ﬁrm engages in
BBPD, or the former maximizes the weighted sum of social welfare and its proﬁt.

10

5International mixed duopoly

We next discuss a market where a public ﬁrm competes with a foreign private ﬁrm
owned by foreign shareholders.

5.1

Uniform pricing

We consider the case where BBPD is not feasible. Since the foreign private ﬁrm’s proﬁt
ﬂows out to foreign owners, per-period domestic social welfare under uniform pricing
in the international mixed duopoly is the sum of per-period consumer surplus and the
per-period proﬁt of Firm A and is given by
1 2 1
1
swU f = (v − c)(θ − θ) + θ + θ2 − (θ̃U )2 − (pfB − c)(θ − θ̃U ),
4
4
2

(38)

where the superscript “f ” represents for the international mixed duopoly and θ̃U isgiven by pfB − pfA . Solving as in the previous section, we obtain pfA∗ = c, pfB∗ = c + 2θ ,and θ̃U ∗ = 2θ . Thus, we have
1 2 1
3 2
CS U f = SW U f = (1 + δ)[(v − c)(θ − θ) + θ + θ2 ] − (1 + δ) θ ,448UfΠA = 0,
1 2
ΠUBf = (1 + δ) θ .
4

(39)
(40)
(41)

The public ﬁrm’s price in the international mixed duopoly is equal to its marginal
cost and lower than that in the domestic mixed duopoly when ﬁrms engage in uniform
pricing. This is because that the public ﬁrm’s lower price increases the sum of disutilitiesof taste mismatch but decreases the outﬂow to foreign shareholders. The former eﬀect
is smaller than the latter eﬀect when pfA∗ ≥ c and vice versa.

5.2

BBPD

We now assume that BBPD is feasible.

The second period
Let θ∗f denote the ﬁrst-period indiﬀerent consumer in the international mixed duopoly.
Firm A and Firm B oﬀer αf and β f to their past consumers and oﬀer α̂f and β̂ f to therival’s past consumers, respectively, in the second period. As in the previous section,

11

f
f
we obtain the locations of the indiﬀerent consumers on [θ, θ∗f ] and [θ∗f , θ], θAand θB,
respectively.

{
f
θA
=

β̂ f − αf ,
θ∗f ,

if
if

β̂ f − αf ≤ θ∗f ,β̂ f − αf ≥ θ∗f ,

(42)

θ∗f ,
β f − α̂f ,ifif

β f − α̂f ≤ θ∗f ,β f − α̂f ≥ θ∗f .

(43)

{
f
=
θB

In the case of the international mixed duopoly, domestic social welfare in the second
period under BBPD is given by
1 2 1
1
1 f 2 1 f 2
sw2Df = (v − c)(θ − θ) + θ + θ2 + (θ∗f )2 − (θA) − (θB )
4
4
2
2
2
f
f
f
f
∗f
−(β − c)(θ − θB ) − (β̂ − c)(θ − θA ).

(44)

Firm A chooses αf and α̂f to maximize (44) and ﬁrm B chooses β f and β̂ f to maximize
Df
f
f
πB2
= (β f − c)(θ − θB
) + (β̂ f − c)(θ∗f − θA
).

(45)

By taking the parametric conditions in (42) and (43) into account, we solve the publicﬁrm’s maximization problems, leading to{β̂ f − θ∗f , if β̂ f ≥ c + θ∗f ,
(46)
f
α =
f
∗f
c,
if β̂ ≤ c + θ ,
(47)
and

{
f

α̂ =

if β f ≥ c + θ∗f ,

c,
∗f

∗f

β − θ , if β ≤ c + θ .
f

f

(48)
(49)

The private ﬁrm’s best-response function is the same as in (17), (18), (19), and (20).
By comparing (13)-(16) and (46)-(49), it can be seen that the public ﬁrm oﬀers a lower
price in the international mixed duopoly than the domestic mixed duopoly for a given
the private ﬁrm’s prices. The outﬂow to foreign shareholders makes the public ﬁrm
more aggressive in pricing in the international mixed duopoly.
Figure 2 depicts these best-response functions. Figures 2(a) and (b) show equilibrium in the Firm A’s turf. Firm A’s best-response function is given by (46) and (47),and that of Firm B by (17) and (18) in Figures 2(a) and (b). Firm A supplies all∗
consumers in its turf if θ∗ ≤ 0 and supplies consumers on [θ, θ2 ] if θ∗ ≥ 0. The public
ﬁrm supplies the consumers who prefer the private ﬁrm to the public ﬁrm to reduce the12outﬂow when θ∗ is larger than 0. Figures 2(c) and (d) show equilibrium in Firm B’sturf. Firm A’s best-response function is given by (48) and (49), and that of Firm B isgiven by (19) and (20) in Figures 2(c) and (d). Firm B supplies all consumers in its turf
if θ∗ ≥ 2θ and supplies consumers on [ 2θ , θ] if θ∗ ≤ 2θ . The public ﬁrm does not supply
consumers in the private ﬁrm’s turf to prevent an increase in the sum of disutilities oftaste mismatch when θ∗ is suﬃciently large. The results are diﬀerent from the second
period’s equilibrium in the domestic mixed duopoly in that the public ﬁrm can supply
consumers who prefer the private ﬁrm to the public ﬁrm.
( Figure 2 about here )Proceeding in the same way as in Section 4, equilibrium prices in the second periodareθ
αf = c − θ∗f , β̂ f = c, α̂f = c, β f = c + ,2∗fθθ
αf = c, β̂ f = c +, α̂f = c, β f = c + ,22∗fθ
αf = c, β̂ f = c +, α̂f = c + θ − 2θ∗f , β f = c + θ − θ∗f ,2

if θ ≤ θ∗f ≤ 0,
θif 0 ≤ θ∗f ≤ ,2
θ
if ≤ θ∗f ≤ θ.
2

(50)
(51)
(52)

Therefore, we have
1 2 1
3
sw2Df = (v − c)(θ − θ) + θ + θ2 − (θ∗f )2 − (θ − θ∗f )2 ,4
4
8
1 ∗f 2Df∗f 2
πB2 = (θ − θ ) + (θ ) ,
4

(53)
(54)

if 2θ ≤ θ∗f ≤ θ. In this case, Firm A does not poach Firm B’s old consumer sinceff
θB= θ∗f but Firm B poaches that of Firm A since θA< θ∗f .
In Appendix B, we show that only the case of 2θ ≤ θ∗f ≤ θ is possible in equilibrium.

The ﬁrst period
In the case of 2θ ≤ θ∗f ≤ θ, the consumer, θ∗f , is indiﬀerent between choosing Firm A
in the ﬁrst period at price, af , and then choosing Firm B in the second period at price,β̂ f , or choosing Firm B in the ﬁrst period at price, bf , and then choosing Firm B at
price, β f . Thus, θ∗f satisﬁes
θ∗f
θ∗f
θ∗f
θ∗f
f
f
f
− a + δ(v +− β̂ ) = v +− b + δ(v +− β f ).
v−
2
2
2
2
13

(55)

Substituting the second-period equilibrium prices and solving it for θ∗ yieldsθ∗f =2(bf − af + δθ),
2 + 3δθ≤ θ∗f ≤ θ.
2

if

(56)

Domestic social welfare in the ﬁrst period is given by
[
]
1 2 1
1
sw1Df = (v − c)(θ − θ) + θ + θ2 − (θ∗f )2 − (bf − c)(θ − θ∗f ) ,
4
4
2

(57)

and the ﬁrst-period proﬁt of Firm B is given by
Df= (bf − c)(θ − θ∗f ).
πB1

(58)

Firm A chooses af to maximize SW Df = sw1Df + δsw2Df and Firm B chooses bf to
Df
Dfmaximize ΠDf
B = πB1 + δπB2 .
The diﬀerentiation of the above objective functions yields
2(2 + 3δ)c − (4 + δ)δθ + 5δbf − (4 + 11δ)af = 0,2(2 + 3δ)c + (4 + δ 2 )θ + 4(1 − δ)af − 2(4 + δ)bf = 0.

(59)
(60)

Solving (59) and (60) gives equilibrium prices in the ﬁrst period and substituting these
into (52), we can obtain equilibrium prices.
Proposition 3 Equilibrium prices and the ﬁrst-period indiﬀerent consumer in the international mixed duopoly are given by
af = c −δ(6 − δ)
θ,2(7δ + 8)

αf = c,β̂ f = c +bf = c +3δ + 2θ,
7δ + 85δ 2 + 2δ + 8θ,2(7δ + 8)α̂f = c −5δθ,7δ + 8θ∗f =2(3δ + 2)
θ,7δ + 8βf = c +δ+4θ.
7δ + 8Thus, we haveffpfA = αf ≥ af ≥ α̂f , pfB ≥ bf ≥ β f > β̂ f , θA
< θ∗f , θB
= θ∗f .

We can check that θ∗f satisﬁes 2θ ≤ θ∗f ≤ θ.
Proposition 3 implies that BBPD decreases equilibrium prices of the public ﬁrm,
fa and α̂f , to below its marginal cost. BBPD allows the public ﬁrm to determine its
poaching price, α̂f , by considering only the reduction of the outﬂow without considering
disutilities of taste mismatch in the private ﬁrm’s turf. Consumers in the private ﬁrm’s
turf have lower relative preferences for the public ﬁrm’s goods as θ∗f is larger. As shownin (52), no consumers in the private ﬁrm’s turf switch to the public ﬁrm’s goods even14

if the latter oﬀers α̂f lower than c when 2θ ≤ θ∗f ≤ θ. Therefore, the public ﬁrm canreduce the private ﬁrm’s proﬁt by reducing α̂f without increasing taste mismatch inthis situation. If θ∗f ≤ 2θ , the public ﬁrm does not make α̂f lower than c to prevent anincrease in disutilities of taste mismatch.
This increases an incentive for the public ﬁrm to lower its ﬁrst-period price. The
equation (54) demonstrates that the second-period’s proﬁt of the private ﬁrm is decreasing in θ∗ in the range of 2θ ≤ θ∗ ≤ 45 θ. The public ﬁrm can reduce the second
period’s proﬁt for the private ﬁrm by capturing a larger market share in the ﬁrst period. Therefore, the public ﬁrm’s ﬁrst-period price, af , is lower than c and is decreasingin δ.
Proposition 3 implies that only the private ﬁrm poaches its rival’s consumers becauseff
θA= 12 θ∗f and θB= θ∗f . The public ﬁrm supplies consumers in [θ, 12 θ∗f ] and the private
ﬁrm supplies consumers in [ 12 θ∗f , θ] in the second period as illustrated in Figure 3.
( Figure 3 about here )BBPD decreases the second period’s sum of disutilities of taste mismatch sincefθA< 2θ but increases that of the ﬁrst period since θ∗f > 2θ . As a result, BBPD increases
the total discounted sum of disutilities of taste mismatch, although BBPD decreases
equilibrium prices in the international mixed duopoly.
Total discounted consumer surplus, total discounted proﬁts, and total discounted
domestic social welfare in equilibrium under BBPD in the international mixed duopoly
are
CS Df

=

ΠDf
A

=

ΠDf
B

=

SW Df

=

47δ 3 + 44δ 2 + 36δ + 48 2
1 2 1
(1 + δ)[(v − c)(θ − θ) + θ + θ2 ] −θ ,442(7δ + 8)2
δ(6 − δ)(13δ + 12) 2−
θ ,2(7δ + 8)2
25δ 3 + 62δ 2 + 56δ + 32 2
θ ,2(7δ + 8)2
1 2 1
17δ 3 + 55δ 2 + 54δ + 24 2
(1 + δ)[(v − c)(θ − θ) + θ + θ2 ] −θ .
44(7δ + 8)2

(61)
(62)
(63)
(64)

Proposition 4 Comparing the outcome under uniform pricing, (39), (40), and (41)yieldsUf
CS U f ≤ CS Df , ΠUi f ≥ ΠDf≤ SW Df , (i = A, B).
i , SWThus, BBPD improves domestic social welfare, although it increases the total discounted
sum of disutilities of taste mismatch in the international mixed duopoly.
BBPD is beneﬁcial to consumers in the international mixed duopoly since it increases
the sum of disutilities of taste mismatch but decreases equilibrium prices. BBPD is15

detrimental to the ﬁrms in the international mixed duopoly since it intensiﬁes competition. The public ﬁrm’s proﬁt falls into the red. The private ﬁrm’s proﬁt decreases, although BBPD increases its market share. BBPD is beneﬁcial to domestic social welfare
in the international mixed duopoly. BBPD reduces the outﬂow to foreign shareholders,
although it is detrimental to international social welfare since it increases the sum of
disutilities of taste mismatch. Fudenberg and Tirole (2000) indicate that private ﬁrms
oﬀer discounts to poach their rival’s consumers in the second period. The same is true
for the private ﬁrm in our model. However, the public ﬁrm oﬀers its poaching price toregulate one of the rival’s prices in the second period.

6

Privatization

This section discusses the privatization of the public ﬁrm. We assume that a privatizedpublic ﬁrm maximizes its proﬁt. Domestic social welfare under uniform pricing in thedomestic mixed duopoly is given by (8) and that in a domestic pure duopoly, where there
are two proﬁt-maximizing private ﬁrms completely owned by domestic shareholders, canbe obtained as follows:
1 2 1
SW U p = (1 + δ)[(v − c) + θ + θ2 ],
4
4

(65)

where “p” represents for the pure duopoly. Domestic social welfare under BBPD in thedomestic mixed duopoly is given by (37) and that in the domestic pure duopoly can beobtained as follows:
1 2 1
1 2
SW Dp = (1 + δ)[(v − c) + θ + θ2 ] − δθ .
4
4
9

(66)

We can see that (8) is equal to (65) and (37) is larger than (66).
Proposition 5 In the domestic mixed duopoly, privatization reduces domestic social
welfare under BBPD, whereas it is neutral to it under uniform pricing.
Domestic social welfare under uniform pricing in the international mixed duopoly
is given by (39) and that in an international pure duopoly can be obtained as follows:
1 2 1
2SW U pf = (1 + δ)[(v − c) + θ + θ2 ] − (1 + δ)θ .
4
4

(67)

In the international pure duopoly, a proﬁt-maximizing ﬁrm completely owned by domestic shareholders competes with a proﬁt-maximizing ﬁrm completely owned by foreign
shareholders. Domestic social welfare under BBPD in the international mixed duopoly
is given by (64) and that in the international pure duopoly is equal to (67). We can seethat (39) and (64) are larger than (67) and Proposition 4 indicates that (64) is larger
than (39).
16

Proposition 6 In the international mixed duopoly, privatization reduces domestic social welfare under both uniform pricing and BBPD. The welfare loss from privatizationis larger under BBPD than under uniform pricing.
From Proposition 5 and 6, we can conclude that the existence of the public ﬁrm increases
domestic social welfare more when ﬁrms engage in BBPD than when they engage inuniform pricing. Therefore, privatization is more undesirable under BBPD than uniform
pricing.

7

Conclusion

We extend Fudenberg and Tirole (2000) by considering a mixed duopoly to investigate
mixed markets in which the social welfare-maximizing public ﬁrm and the private ﬁrm
engage in behavior-based price discrimination.
We ﬁnd that BBPD is irrelevant from the viewpoint of domestic social welfare when
the public ﬁrm competes with the domestic private ﬁrm. Since the public ﬁrm chooses
its prices to minimize the sum of disutilities of taste mismatch in each turf, poachingdoes not occur in the second period.
BBPD is also found to improves domestic social welfare when the public ﬁrm competes with the foreign private ﬁrm. The former chooses its prices by considering minimization of the sum of disutilities of taste mismatch and the reduction of the outﬂow
to foreign investors in the international mixed duopoly. However, it lowers its poaching
price by considering only the reduction of the outﬂow to foreign investors in the second
period when ﬁrms engage in BBPD. Although BBPD increases the total discounted sumof disutilities of taste mismatch, it reduces the outﬂow to foreign investors by lowering
equilibrium prices.
Privatization worsens domestic social welfare in both domestic and international
mixed duopoly when ﬁrms engage in BBPD. Compared to the case of uniform pricing,
BBPD is more desirable from the viewpoint of domestic social welfare.
Introducing long-term contracts, in which ﬁrms promise to supply their goods atthe same prices in the ﬁrst and second periods, remains for future research.

17

(

)

(

)

(

)

(

)

Figure 1: The second period’s equilibrium in the domestic mixed duopoly

18

(

)

(

)

Figure 2: The second period’s equilibrium in the international mixed duopoly

19

Figure 3: The second period’s market in the international mixed duopoly

20Appendix
Appendix A Equilibrium prices, a and b

Figure 4: The ﬁrst period’s equilibrium in the domestic mixed duopoly
In this appendix, we obtains ﬁrst-period equilibrium prices, a and b, in the domestic
mixed duopoly. The best-response functions, (32), (33), and (34) can be depicted asin Figure 4. Figure 4(a) shows equilibrium when δ ≤ 12 and Figure 4(b) illustrates
it when δ ≥ 21 . Equation (32) shows that the Firm A’s best-response function has apositive slope and its b-intercept is negative for all θ ≤ θ∗ ≤ θ. Equation (33) indicatessthat Firm B’s best-response function has a positive slope when θ ≤ θ∗ ≤ 0, which islocated below Firm A’s best-response function. Equation (34) shows that Firm B’s
best-response is located above Firm A’s best-response function. It has a positive slope
if δ ≤ 12 as shown in Figure 4(a) and negative slope if δ ≥ 21 as shown in Figure 4(b)when 0 ≤ θ∗ ≤ θ.
There are two local optimal solutions, which are represented by E1 and E2 in each
Figure 4(a) and Figure 4(b). Firm B earns the following proﬁts, respectively.
]2[
12
a − c + (1 − 2δ)θ + δθ ,4(1 − δ)]2
1[2
ΠD
a − c + (1 − δ)θ + δθ ,
B =
4

ΠD
B =

if

θ ≤ θ∗ ≤ 0,

(A1)

if

0 ≤ θ∗ ≤ θ.

(A2)

when c + θ ≤ a ≤ c + (1 + δ)θ. We ﬁnd that (A1) is larger than (A2) in the range ofc + θ ≤ a ≤ c + (1 + δ)θ. Hence, Firm B can earn more proﬁt on the solid line than thedashd lines in the ﬁgures. Therefore, equilibrium is given by E1 in Figure 4 and theﬁrst-period’s equilibrium prices are given by a = c + θ and b = c + (1 − δ)θ.

21

Appendix B Indiﬀerent consumer, θ∗f
In the case of θ ≤ θ∗f ≤ 0, the ﬁrst-period indiﬀerent consumer can be derived as
θ∗f =

b−a
.
1−δ

(A3)

The objective functions of Firm A and Firm B are
[
]
1
1 2 1
SW Df = (v − c)(θ − θ) + θ + θ2 − (θ∗f )2 − (bf − c)(θ − θ∗f )4
4
2[
1 2 1 ]
+δ (v − c)(θ − θ) − θ + θ2 ,
8
4
1 2
f
∗f
ΠDf
B = (b − c)(θ − θ ) + δ θ .
4

(A4)

(A5)

First-order conditions are(1 − δ)c + δbf − af = 0,

(A4)

c + (1 − δ)θ + a − 2b = 0.

(A6)

f

f

Solving these equations yields af = c + δ(1−δ)θ and bf = c + 1−δθ. Hence, we obtain
2−δ
2−δ
1−δ
∗f
∗f
θ = 2−δ θ, which violates θ ≤ θ ≤ 0.
In the case of 0 ≤ θ∗f ≤ 2θ , the ﬁrst-period indiﬀerent consumer can be derived asθ∗f =2(b − a)
.
2−δ

(A7)

The objective functions of Firm A and Firm B are
]
[
1 2 1
1
SW Df = (v − c)(θ − θ) + θ + θ2 − (θ∗f )2 − (bf − c)(θ − θ∗f )
4
4
2
[
]
1 2 1
1
+δ (v − c)(θ − θ) − θ + θ2 + (θ∗f )2 ,
8
4
8
(
f
∗f
ΠDf
B = (b − c)(θ − θ ) + δ

(A8)

2)

(θ∗f )2 θ
+
4
4

.

(A9)

2(2 − δ)c + δbf − (4 − δ)af = 0,

(A10)

(2 − δ)2 θ + 2(2 − δ)c + 4(1 − δ)af − 2(4 − 3δ)bf = 0.

(A11)First-order conditions areδ(2−δ)
Solving these equations yields af = c + 2(8−5δ)θ and bf = c + (2−δ)(4−δ)θ. Hence, we2(8−5δ)obtain θ∗f = 2(2−δ)θ, which violates 0 ≤ θ∗f ≤ 2θ .
8−5δ
22

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