ISSN (Print) 0473-453X

Discussion Paper No. 1201

ISSN (Online) 2435-0982

THE EFFECTS OF PERSONAL DATAMANAGEMENT ON
COMPETITION AND WELFARE

Jiajia Cong
Noriaki Matsushima

March 2023

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

The Effects of Personal Data Management on
Competition and Welfare*
Jiajia Cong,† Noriaki Matsushima‡
March 6, 2023

Abstract
This study examines how consumers’ personal data management affects firms’ competitionin the data collection and data application markets and welfare outcomes. Consumers purchase products from differentiated firms in two markets. Firms compete to collect consumer
data first to predict their preferences in the data application market, where each firm offers
personalized prices to its targeted consumers and a uniform price to untargeted consumers.
Before firms offer prices, their targeted consumers can erase data to become untargeted fora fixed cost. We show that consumers’ privacy management mitigates price competition, reduces firms’ profits, and harms consumer surplus and social welfare in the data application
market; privacy management intensifies competition and improves consumer surplus in thedata collection market. Across these two markets, profits and social welfare decline. Thechange in consumers’ two-market surplus depends on their foresight regarding the outcomesin the data application market, with only forward-looking consumers having a higher surplus.
We extend the model in several directions, including data-enabled product personalization,
privacy costs, data portability, and data ownership, and discuss the implications for privacy
laws.
Keywords: privacy management, data collection, data application, price discrimination, privacy laws
JEL code: D11, D43, D62, L13, L51* We thank Elias Carroni, Chongwoo Choe, Vincenzo Denicolò, Dingwei Gu, Zhuoran Lu, Alireza J. Naghavi, Arina

Nikandrova, Hiroyuki Odagiri, Susumu Sato, Naoki Wakamori, Wen Zhou, and the seminar participants at Competition
Policy Research Center (CPRC) at the Japan Fair Trade Commission (JFTC), Fudan University, ShanghaiTech University, University of Bologna, Workshop on “Digital Platform and Competition” (Kwansei Gakuin University), and Wuhan
University for their valuable discussions and comments. Cong would like to acknowledge the financial support from
NSFC (Projects 72003035, 72192845) and Shanghai Pujiang Program (2019PJC007). We gratefully acknowledge thefinancial support from the Japan Society for the Promotion of Science (JSPS), KAKENHI Grant Numbers JP19H01483,
JP20H05631, JP21H00702, JP21K01452, and JP21K18430. Although Matsushima serves as a member of the CPRC
at the JFTC, the views expressed in this paper are solely ours and should not be attributed to the JFTC. The usual
disclaimer applies.
† Department of Applied Economics, School of Management, Fudan University, jjcong@fudan.edu.cn.
‡ Corresponding author, Institute of Social and Economic Research, Osaka University, nmatsush@iser.osaka-u.ac.jp.

11Introduction

By collecting consumer data and using data analytics, firms can gain a competitive edge in developing data-utilizing markets such as those of healthcare, insurance, and streaming services(Goldfarb and Tucker, 2019, Farboodi et al., 2019, Hagiu and Wright, 2022). As a result, many
firms devote significant resources to collecting and utilizing consumer data, often leveraging it
across multiple industries.
Big tech firms’ entry into the healthcare market is a typical example of data collection anddata application. Google has become a central player in the healthcare industry since it madepublic plans to develop Google Health for health insurers and doctors in 2008 (Ozalp et al.,2022). In 2021, Google acquired Fitbit, which produces smartwatches that collect real-time
health data from millions of customers, for $2.1 billion. Through the Apple Watch, whichcollects a wide range of biometric data, Apple can provide users with health support. Moreover, Amazon allows consumers to communicate with healthcare companies through its smart
speaker, Alexa; the company has completed serial acquisitions of pharmacy and health firms
in recent years and is now offering employees its own virtual healthcare (Farr, 2019).1 In sum,these companies collect customer data through their devices and provide data-utilizing services
to users.
Data-utilized services (e.g., digital healthcare services) not only improve the trade surplusesof such services but also lead to two major concerns: data privacy and personalized pricing.2
Consumer demand for data privacy has sparked a global wave of legislation granting consumers
enhanced control over their data. A typical example of such laws is the European Union’s
General Data Protection Regulation (GDPR). GDPR Article 17, the right to erasure (‘right to be
1 Amazon purchased online pharmacy PillPack in 2018, acquired digital health startup Health Navigator in 2019,and proposed the acquisition of One Medical for approximately $3.9 billion in 2022. In 2022, Amazon proposed theacquisition of iRobot, the maker of the popular Roomba vacuum cleaner, which can enhance its data collection ofcustomers’ homes and private lives.
2 For instance, consumers’ privacy was a major concern in the Google-Fitbit and Amazon-iRobot acquisitions. Trans-

ferring patient data from Ascension to Google Cloud led to an ethical dispute, although Ascension obeyed the Health
Insurance Portability and Accountability Act (HIPAA) (see Schneble et al. (2020) for details).

2

forgotten’), stipulates that consumers can order a firm to erase their data, which it must do
without undue delay, even if the firm had obtained consent from customers to process theirpersonal data. Privacy laws in many other countries also clearly grant consumers the rightto erase their data.3 Although privacy laws secure the right of consumers to manage theirpersonal data, the impact of this right in competitive environments remains unclear and under
debate (Aridor et al., 2022; Ali et al., 2022).
Personalized pricing, where firms use data gathered from personal devices (e.g., smartphones and smartwatches) to offer customized prices to consumers, facilitates consumer surplus exploitation (Wagner and Eidenmüller, 2019).4 Several studies, such as Shiller (2020),
Dubé and Misra (2022), and Smith et al. (2022), have quantitatively confirmed the effectiveness of such exploitation.5 We therefore explore the effects of consumers’ personal data
management to escape personalized pricing, which arises under privacy laws, on profits andwelfare.
This study considers a duopoly model in which consumers purchase products in two independent markets: market B (e.g., wearable devices) is for data collection, and market A(e.g., healthcare and insurance) is for data application. Consumers’ product preferences in thetwo markets are independent. The two firms compete in the two markets and first attempt
to attract consumers in the data collection market with uniform pricing. The personal datacollected uniquely by each firm uncover its customers’ preferences for firms’ products in thedata application market. For example, firms in the healthcare or insurance market can use
the biometric data collected from wristbands to predict a suitable healthcare and insurance
3 In addition to data erasure, consumers can order the firm to transmit their data from the firm to another firm

if technically feasible—so-called data portability (GDPR Article 20). The California Consumer Privacy Act (CCPA) andthe Treasury Laws Amendment (Consumer Data Right) Bill 2019 in Australia also give consumers the right to data
portability. We discuss the effects of data portability in one extension.
4 See Esteves and Resende (2016) and Ezrachi and Stucke (2016) for real-world examples of personalized offers.
5 Although the surplus exploitation concern does not match the standard result in the literature of personalized

pricing in which personalized pricing is worse for firms than is uniform pricing in oligopoly markets (Thisse and Vives,
1988; Shaffer and Zhang, 1995; Zhang, 2011), some studies show that personalized pricing does not always negatively
affect firms (e.g., Shaffer and Zhang, 2002; Choudhary et al., 2005; Matsumura and Matsushima, 2015; Esteves,2022).

3

plan for each consumer. Such a plan can be different from the firm’s standard product. Aftercompetition in market B, each firm in market A offers personalized prices to its targeted customers, for whom the firm has collected data, and a uniform price to untargeted consumers,
for whom the firm does not have data.6 However, before firms decide prices in market A, eachfirm’s targeted customers can erase their data from the firm’s database to become untargeted
consumers at a fixed cost, an option referred to as privacy management.7 Consumers who
erase their data (i.e., opt-out consumers) can escape personalized prices and choose uniform
prices offered by firms. Our benchmark is the scenario in which consumers cannot manage
their privacy.
We find that consumers’ privacy management mitigates price competition, reduces firms’
profits, and negatively affects welfare in market A, the data application market. The explanation of competition mitigation starts with which types of consumers self-select to manage
data. Consumers who strongly prefer one firm in market A choose to erase their data because
they expect to be charged high personalized prices otherwise. Consequently, each firm’s uniform price rises because the price is applied to those opt-out consumers who strongly prefer
the firm. Such increases in uniform prices induce firms to set higher personalized prices for
consumers who do not erase their data, i.e., opt-in consumers.
We sequentially explain the impacts on profits, consumer surplus, and social welfare inmarket A. Firms’ profits in this market decline because their uniform prices cannot efficiently
extract the surplus from opt-out consumers with high reservation values. Opt-in consumers
are worse off, and opt-out consumers are barely better off because they pay an inflated uniform
price and privacy management costs, lowering consumer surplus in market A. The consumerfirm mismatch becomes larger, reducing social welfare, because higher uniform prices help
each firm protect its targeted customers through personalized prices in market A. In addi6 Generally,

consumer data can not only help the firm provide personalized products that generate more trade

surplus but also enable the firm to extract the surplus more efficiently with sophisticated pricing tools. We focus onfirms’ pricing tools in the main model and discuss personalized products in an extension.
7 Consumers can exercise their right to object to their data being used in the data application market under GDPR

Article 21, which is equivalent to data erasure in the model.

4tion, privacy management costs further negatively affect welfare. Interestingly, increasing theprivacy management cost can benefit firms, consumers, and society simultaneously because
fewer consumers would opt out, and the above equilibrium changes would thus become less
impactful.
We show that consumers’ privacy management intensifies competition in market B (i.e.,enhances firms’ incentives to collect data) compared to its absence. The reason for this is thata firm’s profit in market A increases faster with its market share in market B under privacy
management. Specifically, when a firm gains more consumers in market B, its uniform pricein market A becomes higher because it has more consumers who strongly prefer the firm andopt out. This higher uniform price, in turn, diminishes the incentives of marginal consumers toerase their data because the gains from escaping personalized prices are lower. Thus, the firm
can price discriminate against more consumers with a higher willingness to pay and efficiently
extract their surplus in market A. Under privacy management, inflated personalized prices
further facilitate surplus extraction. As price competition intensifies in market B, firms earnlower profits, and consumers benefit. In our model of symmetric firms, the two firms always
split the market equally and generate a fixed consumer-firm match, which leaves the social
welfare of market B unchanged with privacy management.
When combining these two markets, privacy management lowers the two-market profitsand social welfare but enhances total consumer surplus only when consumers are forwardlooking regarding the outcome in market A. Compared to myopic consumers, forward-looking
consumers are more price sensitive in market B because the expected surplus that they willobtain from a firm in market A is positively related to the firm’s market share in market B.
Consumers’ higher price sensitivity further accelerates competition in market B and results inlarger consumer surplus gains.
We consider several factors as extensions, two of which we explain here: data portability(GDPR Article 20) and data ownership. When data portability is available, consumers withweak preferences for both firms make their data available to firms with data portability to trig-

5

ger consumer-by-consumer Bertrand competition on personalized pricing, intensifying competition in market A but mitigating competition in market B because the value of collecting
data plummets. When consumers own data property rights (i.e., opt out by default and opt inby choice), compared to the main model, the equilibrium outcome is better from the welfare
viewpoint, provided that the compensation paid to consumers for data usage is appropriate.
Our study provides the following policy implications. Consumers’ two-market surplus increases only when they have foresight regarding the data application market, and thus, firms
should explain how to use collected data when they gain customers in the data collection
market.8 Both data erasure and data portability result in surplus redistribution, with consumers who manage privacy benefiting. Data erasure destroys surplus simultaneously due toincreased consumer-firm mismatch, while data portability enhances social welfare because it
enables consumers to be targeted and attracted by their preferred firm.
The remainder of this paper is organized as follows. After reviewing the literature in Section
2 and setting up the model in Section 3, we establish the benchmark equilibrium without
privacy management and the equilibrium when privacy management is available in Section
4. Section 5 provides several extensions, such as data-enabled product personalization, data
portability, privacy costs, and ownership of data property rights. Section 6 discusses the policy
implications. Finally, Section 7 concludes the paper.

2

Literature review

Our study contributes to the literature on behavior-based price discrimination (BBPD) withpersonalized pricing, consumer data management, and the role of data in competition.
Our study relates to those studies in which firms collect data (e.g., purchasing history)
and then apply them to pricing in a competitive environment. Caminal and Matutes (1990),
Chen (1997), and Fudenberg and Tirole (2000) are earlier works that consider two-period mod8 In November 2022, Google reached a record $392M privacy settlement with 40 U.S. states over location data,requiring Google to provide consumers with more information and to be more transparent about its data collection
and utilization (https://www.nytimes.com/2022/11/14/technology/google-privacy-settlement.html).

6els with third-degree price discrimination based on consumers’ purchase history (so-called
behavior-based price discrimination (BBPD)). Moreover, Choe et al. (2018), Choe et al. (2022),
and Laussel and Resende (2022) incorporate data-enabled personalized pricing into the BBPD
framework.9 Among those studies, our work is closely related to Chen et al. (2022), who
consider two-stage duopoly models in which only one of the firms can apply consumer datacollected in a market to another market to offer personalized prices.10 However, they do notinvestigate how consumers’ data management affects competition in the two markets, which isthe focus of our study.
We contribute to the literature on consumer data management, particularly discussions on
the interaction between consumer data management and firms’ pricing strategy. Belleflamme
and Vergote (2016) and Koh et al. (2017) show that consumers’ data management choices function as signals of their willingness to pay and that the monopolist adjusts discriminatory prices
against two consumer groups (those who manage data and those who do not) accordingly.11
While we share this signaling effect, our study is embedded in a competitive environment
where each firm’s discriminatory prices face competition.12 Therefore, whether data management consumers can cause pricing externalities for other consumers and the magnitude ofsuch externalities can differ.
There are three recent papers on consumer data management. First, in a Hotelling linear city model, Ali et al. (2022) assume that duopoly firms collect personal data only fromconsumers’ revelation of preferences and show that consumers can strategically disclose their
9 Choe et al. (2018) show that personalized pricing intensifies competition in the first period and that the resulting

outcomes are asymmetric. Choe et al. (2022) incorporate data-sharing agreements between competing firms into Choe
et al. (2018). Laussel and Resende (2022) incorporate endogenous product differentiation and product personalization
into Choe et al. (2018) and show a contrasting result to that of Choe et al. (2018).
10 Herresthal et al.

(2022) consider two-stage oligopoly models in which one of the firms can apply customer datacollected in a market to an insurance market and show that the firm’s data application benefits consumers.
11 Using the BBPD framework, Acquisti and Varian (2005) and Conitzer et al.

(2012) consider consumers’ identity

management to escape expected high prices in monopoly models.
12 Montes et al. (2019) and Valletti and Wu (2020) consider consumer data management to escape personalized prices,with the former focusing on the data broker’s optimal data sales strategy and the latter focusing on the endogenizationof profiling technology quality.

7

preferences to amplify firms’ competition and benefit.13 The result is similar to our analysis ofdata portability. One of our contributions is that we consider the interaction between consumer
data management and firms’ incentives to collect data and demonstrate that data management
substantially enhances such incentives.
Anderson et al. (2022) consider a two-stage game with targeted discounting in stage two,
assuming that consumers can escape target discounts before knowing their preferences forfirms’ products, meaning that consumers’ denial does not signal their preferences, different
from our game. They show that the refusal of personalized discounts can benefit firms andconsumers by mitigating discount competition and lowering uniform prices.
Using the two-period BBPD framework with consumers’ heterogeneous willingness to pay,
Ke and Sudhir (2022) study how consumers’ data management affects competition among ex
ante homogeneous firms. Their findings, contrary to ours, show that consumers who strongly
prefer a firm opt in because, in their model setting, these consumers can reap greater benefits from personalized products, but the firm cannot efficiently extract the surplus due touncertainty about their valuations. In their model, consumer surplus increases under privacy
management. We reach diverging welfare results due to various modeling differences, including
market structure, product differentiation, and the firm’s ability to expand and distribute the“pie” with the help of data.
The final section discusses recent research on the role of data and categorizes them into
two categories: competitive edge from data accumulation (Farboodi et al., 2019; Hagiu andWright, 2022; Prüfer and Schottmüller, 2021; Cordorelli and Padilla, 2022; de Cornière andTaylor, 2023) and consumer data markets (Choi et al., 2019; Bergemann and Bonatti, 2019;
Argenziano and Bonatti, 2021; Ichihashi, 2021; Acemoglu et al., 2022; Bergemann et al.,2022). However, the interaction between consumer data management and firms’ competition
is beyond the scope of these papers.
13 Ichihashi (2020) discusses the interaction between information disclosure by a consumer and product recommen-

dation by a seller in buyer-seller models.

8

3

Model

Consider two markets, A and B. Market B is data rich, and firms can collect substantial consumer data in this market. Market A is lucrative, and firms can predict consumers’ preferencesin the product space of this market using their data from market B. Our model is best understood with the help of a concrete example. Thus, we refer to market B as the market for
wearable devices, such as Fitbit wristbands and Apple watches, and market A as the market
for healthcare and insurance. Needless to say, our model applies to other markets where data
are collected in one market and applied in another market.
Two firms serve each market—A1 and A2 in market A and B1 and B2 in market B—wherefirms A1 and B1 are two subsidiaries of firm 1 and firms A2 and B2 are two subsidiaries of firm
2. The two markets have the same group of consumers, and their mass is normalized to one.
In each market, a consumer demands one unit of product. We normalize the marginal cost of
production to zero and treat prices as profit margins.
In market B, consumers are uniformly distributed on [0, 1]. Firms B1 and B2 are located at
0 and 1, respectively, and compete on uniform prices. Firm Bi ’s price is βi (i = 1, 2). Consumery ∈ [0, 1] obtains utility u = vB − ty − β1 from firm B1 and utility u = vB − t(1 − y) − β2 from firm B2 .
We assume that the value of vB is large such that the market is fully covered. If a consumerpurchases from firm Bi , then the firm can collect consumer data through the consumer’s usageof its product. We assume that consumers must consent to data collection in the market B touse products like wristbands and smartwatches properly.
In market A, firms A1 and A2 are located at 0 and 1, respectively. If a consumer is locatedat y ∈ [0, 1] in market B, then her location x in market A is a random variable on [0, 1] that
follows a uniform distribution. In other words, consumers’ product preferences in the twomarkets are independent of each other because the products serve different purposes (Matutes
and Regibeau, 1988; Armstrong and Vickers, 2010). Consumers privately know their exact
locations in market A, which can be interpreted as the healthcare or insurance plan that9

perfectly matches their needs.14 Firm Ai obtains all of firm Bi ’s collected consumer data.
This data transfer means that if a consumer has purchased from firm Bi , then firm Ai can
uncover her exact realized location x in market A and target her with the help of consumer
data. However, firm Aj (j ̸= i) knows only that her location x is uniformly distributed on [0, 1].
Firm Ai charges personalized prices pi (x) to its targeted consumers and a uniform price αi tountargeted consumers.
Before firm Bi transfers consumer data to firm Ai , knowing her preferences in market A,
each consumer can choose whether to erase her data from firm Bi ’s datasets or object to firm
Bi ’s transfer of her data to firm Ai , which is called privacy management. We assume that thecost of privacy management is ε ≥ 0 for every consumer. This formulation is consistent withexisting U.S. privacy laws that give the property rights over data to the entities that collectthem, and consumers opt in by default and can manage their privacy (Economides and Lianos,2021).
If a consumer does not erase her data, then she continues to be a targeted consumer of firm
Ai . Firm Ai ’s targeted consumer has two choices: receiving utility u = vA − txi − pi (x) from firm
Ai or receiving utility u = vA − txj − αj from firm Aj , where xi = 1 − xj . If she erases her data, shecannot be targeted by either firm and thus has two choices: obtaining utility u = vA − txi − αifrom Ai or obtaining utility u = vA − txj − αj from Aj . We maintain the assumption that thevalue of vA is large such that the market is fully covered.
The whole game proceeds as follows. The two firms in market B simultaneously decide theiruniform prices, and consumers make their purchase decisions in market B after observing
the prices. After the purchase decisions in market B, consumers recognize their product preferences over A1 and A2 in market A, and firms collect customer data through the consumer
14 Healthcare or insurance plans are commonly believed to differ substantially (Gaynor and Vogt, 2000; Gaynor and

Town, 2011; Biglaiser and Ma, 2003). For instance, healthcare can differ in terms of lists of approved physicians,
diagnostic testing, real-time monitoring, and high-tech services, while insurance plans may vary in terms of their coverage, care (hospital) providers, reimbursement policies, and other characteristics of the plan provider. The Hotellingmodel is widely used to model differentiation in healthcare and insurance plans (Gal-or, 1997; Ellis, 1998, Biglaiser
and Ma, 2003; Olivella and Vera-Hernández, 2007; Chen et al., 2022).

10

usage of the products in market B. Then, consumers simultaneously decide whether to erase
their data. In market A, the two firms simultaneously post publicly observable uniform prices.
Thereafter, firm Ai offers private personalized prices to its targeted consumers. After observing
all available offers, consumers make purchasing decisions in market A. The sequential timing of price offers in market A is standard in the literature on personalized pricing (Thisse andVives, 1988; Shaffer and Zhang, 2002; Choe et al., 2018), and reflects the flexibility in choosing
personalized prices and allows us to solve for the subgame perfect Nash equilibrium in purestrategies.
We consider two cases regarding consumer expectations of the outcome in market A when
they make purchase decisions in market B. In the myopic case, consumers do not expect
their subsequent choices in the adjacent market and base their decisions solely on utility inmarket B. In the forward-looking case, consumers correctly recognize their expected surplusesin market A and make decisions based on their total surpluses in markets A and B.
We make several simplifying assumptions in the main model and relax them in Section 5.
First, consumer data in the main model are used only for price discrimination in market A (i.e.,
to divide the “pie”) and not to enlarge the pie by creating data-enhanced products. Moreover,consumers do not incur any privacy costs, even if they do not manage their personal data.
These assumptions are relaxed in Section 5.1. Second, in addition to data erasure, we consider
data portability as an extra option for privacy management in Section 5.2. Third, market size
is fixed in the main model, regardless of privacy policies. Privacy-sensitive consumers may
leave the market if privacy management is absent and return when it is available. This caseis discussed in Section 5.3. Finally, Section 5.4 examines an alternative privacy setting where
consumers own data property rights so that they opt out by default and opt in by choice.
Three additional extensions are collected in the online appendix, including some consumers
not purchasing in market B, consumers in market A following a nonuniform distribution, andconsumers in market B choosing whether to consent to data collection.15
15 Our insights are robust when some consumers purchase nothing in market B and when consumers’ preference x in11

4

Analysis

4.1

Benchmark: No privacy management

We start with the benchmark in which consumers cannot erase their data from firm Bi ’s
datasets. In this case, if a consumer is firm Bi ’s targeted consumer, then she must be firm
Ai ’s targeted consumer. Suppose that firm B1 wins consumers on [0, δ] and that firm B2 winsconsumers on [δ, 1] in market B. As shown in Figure 1(a), at any point x ∈ [0, 1] in market A, asegment, δ, of consumers is targeted by firm A1 , and the remaining 1 − δ segment is targetedby firm A2 . In other words, at any point x in market A, firm Ai ’s targeted consumers are firm
Aj ’s untargeted consumers.

1

1
pay αn1

A2 ’s targeted consumers
δ

pay pn2 (x)

δ

A1

δ

B1

pay αn2

pay pn1 (x)A1 ’s targeted consumers

A2

A1

B2

B1

(a) Targeted consumers without privacy management

1/4

3/4
δ

A2
B2

(b) Equilibrium market outcome

Figure 1: Market structure and equilibrium without privacy management
market A follows another distribution, which includes the uniform distribution as a special case. These two extensions
are contained in Sections 1 and 2 of the online appendix, respectively. Moreover, Section 3 of the online appendix
discusses two other privacy management settings, ex ante and sequential privacy management, in which firms need
consumers’ consent to collect their data in market B. In ex ante privacy management, consumers cannot erase theirdata in market A and thus need to decide whether to agree to data collection and application before collection begins.
Consumers in sequential privacy management decide whether to consent to data collection in market B and thendecide whether to consent to data application in market A after data collection. We find that ex ante and sequential
privacy management leads to lower consumer surplus compared to the setting where firms do not request consent tocollect consumer data.

12

Firm Ai sets personalized prices pi (x) for targeted consumers and a uniform price, αi , foruntargeted consumers. The equilibrium prices are the same as those in Thisse and Vives(1988):
α1n = α2n = t/2,
pn1 (x) =
2t(3/4 − x)

if x ≤ 3/4,


0

if x ≥ 3/4,

pn2 (x) =

(1)


0

2t(x − 1/4)

if x ≤ 1/4,

(2)

if x ≥ 1/4,

where superscript n indicates no privacy management. Firm A1 wins consumers to the leftof the blue lines in Figure 1(b). Concretely, firm A1 wins its targeted consumers on [0, 3/4]
and wins the rival’s targeted consumers on [0, 1/4]. Firm A2 wins the remaining consumers.
nnTherefore, the two firms’ profits are πA= t(2 + 7δ)/16 and πA= t(9 − 7δ)/16, which increase
12

with their market shares in market B.
We now focus on the equilibrium analysis of market B. First, we derive the ex ante expectedsurplus of firm B1 ’s consumers in market A, E[CSB1 ], as follows:
Z 3/4
E[CSB1 ] =0(vA − pn1 (x) − tx)dx +

Z 1
3/4

(vA − α2n − t(1 − x))dx = vA − t.

Similarly, we have that E[CSB2 ] = vA − t. The indifferent consumer δ in market B is determinedbyvB − β1 − tδ + gE[CSB1 ] = vB − β2 − t(1 − δ) + gE[CSB2 ].

(3)

Parameter g ∈ {0, 1} specifies the extent of consumers’ foresight regarding the expected outcome
in market A, in which g = 0 indicates that consumers are myopic, while g = 1 indicates that
consumers are forward looking. The indifferent consumer is δ = (t + β2 − β1 )/(2t), as in thennstandard Hotelling model. The profits of firms B1 and B2 are πB= β1 δ and πB= β2 (1 − δ),12respectively. Firms 1 and 2 decide their uniform prices βi to maximize two-market profits:nnnnΠn1 = πA
+ πB
and Πn2 = πA
+ πB
, respectively. The equilibrium uniform prices in market
1
1
2
2

B are β1n = β2n = 9t/16, implying that the indifferent consumer is δ n = 1/2. We have that
n
n
n
n
πB
= πB
= 9t/32 and πA= πA= 11t/32. The equilibrium profits of firms 1 and 2 are
1212Πn1 = Πn2 = 5t/8.

13

4.2

Equilibrium under privacy management

Consumers in this section can manage privacy by erasing their data from firm Bi ’s datasets.
We start by characterizing consumers’ data erasure strategies. Consumers make privacy management decisions based on anticipated prices in market A, i.e., α1a , α2a , pa1 (x), and pa2 (x), where
superscript a indicates anticipation. Moreover, we focus on the case in which firms in market
A use pure strategies in uniform pricing. Lemma 1 characterizes what kinds of consumers
choose to erase their data.
Lemma 1. Given consumers’ price anticipations in market A, firm B1 ’s consumers erase data if
and only if their locations in market A are smaller than x̃1 , and firm B2 ’s consumers erase data
if and only if their locations in market A are larger than x̃2 , in whichx̃1 ≡ε1 αa − α1aε
1 α2a − α1a+−and x̃2 ≡ + 2
+ .
2
2t
2t
2
2t
2t

(4)

Lemma 1 states that consumers who match well with a firm in market A erase data. Figure2(a) shows these opt-out consumers, who face very high personalized prices if they do not erase
their data.16 As an example, let us consider a consumer of firm B1 . If she erases her data,then she receives utility vA − tx − α1a − ε or utility vA − t(1 − x) − α2a − ε in market A. Otherwise,
she receives utility vA − tx − pa1 (x) or utility vA − t(1 − x) − α2a . When her location x is smaller
than x̄1 ≡ 1/2 + α2a /(2t), in which x̄1 is indifferent between firm A1 ’s zero personalized price andfirm A2 ’s uniform price, firm A1 charges the optimal personalized price pa1 (x) = α2a + t(1 − 2x),
equalizing vA −tx−pa1 (x) and vA −t(1−x)−α2a , and wins over the consumer. Therefore, consumerx ∈ [0, x̄1 ] does not erase her data if and only if
vA − tx − pa1 (x) ≥ max{vA − tx − α1a , vA − t(1 − x) − α2a } − ε,which is equivalent to x > x̃1 . When the consumer’s location x is larger than x̄1 (i.e., she
16 Several recent empirical studies on consumer privacy endorse Lemma 1. Chen and Gal (2021) use experiments tofind that consumers have greater concerns over their privacy when they perceive that they do not receive fair value
created from their personal information and are less willing to disclose their data. Through experiments, Lin (2022)
finds that consumers’ instrumental preference for privacy, which is endogenous to how the firm utilizes consumer
data to generate targeting outcomes, significantly affects consumers’ data sharing with the firm.

14

strongly prefers A2 ’s product), she purchases from firm A2 under price α2 and does not erase
data because erasing data does not generate any benefit but brings costs ε.

x̄2

1

x̃2

1

A2 ’s
consumers haveerased data

A2 ’s targeted
consumers

x̄2∗

x̃2∗

pay
α∗1

pay p∗2 (x)pay α∗2

1
2

δ
A1 ’s
consumers haveerased data

A1

A1 ’s targeted
consumers

x̃1

x̂
δ

B1

pay α∗1

x̄1

A2

A1

B2

B1

(a) Targeted consumers after privacy management

pay
α∗2

pay p∗1 (x)

x̃1∗

1/2

x̄1∗

A2
B2

(b) Equilibrium market outcome

Figure 2: Market structure and equilibrium under privacy management

As shown in Figure 2(a), after privacy management, firm Ai ’s untargeted consumers arethose who have purchased from firm Bj and those who have purchased from firm Bi andhave erased their data (i.e., opt-out consumers). Firm Ai ’s targeted consumers are firm Bi ’s
consumers who choose to opt in.
We now analyze the equilibrium prices in market A after consumers engage in privacy management. Suppose that two firms’ uniform prices are α1 and α2 . Firm A1 ’s opt-out consumers−α1purchase from firm A1 if and only if x < x̂ ≡ 12 + α22t
. Consumers’ price anticipations should

be correct in equilibrium due to rational expectations, implying that inequality x̃1 < x̂ < x̃2should hold in equilibrium.
We formulate the objectives of firms A1 and A2 to derive their optimal uniform prices. Firm
A1 wins the rival’s targeted consumers x < max{x̄2 , 0} with uniform price α1 , in which x̄2 ≡1/2 − α1 /(2t). When α1 ≥ t, firm A1 forgoes poaching the rival’s targeted consumers. Therefore,15

firm A1 ’s profit from its uniform price α1 is



 
1 α1

α1 (1 − δ)−+ δ x̃122t
α δ x̃1

1when α1 ≤ twhen α1 ≥ t and x̂ ≥ x̃1 .

Given the determined x̃1 , firm A1 ’s local optimal price for the second case of the above profit
function is α1 such that x̂ = x̃1 , which violates the definition of x̃1 in (4) because x̃1 = x̂ − ε/(2t).
Therefore, only the first case can be sustainable in equilibrium. Similarly, firm A2 ’s profit fromits uniform price α2 is


  
1 α2

α2 δ
−
+ (1 − δ)(1 − x̃2 )when α2 ≤ t
22tα (1 − δ)(1 − x̃ )when α2 ≥ t and x̂ ≤ x̃2 .
22Only the first case can be sustainable in equilibrium.
t
1δ
Given x̃1 and x̃2 , the two firms’ optimal uniform prices are α1 = 2t + tx̃
1−δ and α2 = 2 +t(1−x̃2 )(1−δ).
δ

Since consumers have rational expectations (i.e., α1a = α1 and α2a = α2 ), based on(4), we have that
α1∗ =

t
t
+ δ(t − ε), α2∗ = + (1 − δ)(t − ε),
2
2

(5)

(1 − δ)(t − ε)δ(t − ε), x̃∗2 = 1 −
.
t
t

(6)

and
x̃∗1 =

Given any δ ∈ [0, 1], we need to make the equilibrium cutoffs (in Figure 2(b)) lie between zeroand one, which requires t/2 ≤ ε ≤ t, and ensure that no firm has an incentive to deviate, which
requires assumption 1.
Assumption 1. We assume that√

3t/2 ≤ ε ≤ t.

Under this assumption, the share of consumers erasing data is less than 7% on the equilibrium
path.17 Aridor et al. (2022) empirically find that the GDPR decreases total recorded online
searches by 10.7%, while Goldberg et al. (2022) find a reduction of 11.7% in website pageviews
among EU users post-GDPR. When ε approaches its upper bound t, the number of opt-out
consumers approaches zero, and the equilibrium under privacy management converges to that
under no privacy management.
17 The equilibrium δ ∗ is equal to 1/2, as we show later.

16

In comparison to uniform prices under no privacy management in (1), uniform prices under
privacy management in (5) increase. The intuition behind this result is as follows. Firm Ai uses
its uniform price to serve its own opt-out consumers, who strongly prefer the firm, and to poachthe rival’s targeted consumers. Its uniform price needs to compete with the rival’s uniform priceαj for the first batch of consumers and compete with the rival’s personalized prices pj (x) for thesecond batch. The rival’s uniform price αj is higher than the flexible personalized price pj (x) for
marginal consumers, implying that opt-out consumers are less price elastic. Therefore, opt-out
consumers’ high willingness to pay and low price elasticity make firm Ai a “fat cat” in uniform
price competition (Fudenberg and Tirole, 1984).
Each firm’s personalized price competes with the rival’s uniform price, implying that p∗1 (x) =α2∗ + t(1 − 2x) and p∗2 (x) = α1∗ + t(2x − 1). Therefore, firms’ personalized prices also increase under
privacy management.
Proposition 1. When consumers can manage their privacy, firms in the data application market charge higher uniform and personalized prices compared to the benchmark without privacy
management. Moreover, a firm’s price increases are larger(i) when the privacy management cost ε declines or(ii) when the firm’s market share in the data collection market expands.
Compared to uniform prices under no privacy management, the terms δ(t−ε) and (1−δ)(t−ε)
in (5) represent the increases in uniform prices due to opt-out consumers. A lower ε leads
to larger uniform price increases because more consumers opt out, as in (6), and the firm
becomes more concerned about opt-out consumers who strongly prefer the firm when deciding
its uniform price. A larger δ leads to a higher α1∗ and a lower α2∗ because firm A1 focuses
more on its opt-out consumers, whose reservation values are high, and firm A2 focuses moreon its rival’s opt-in consumers and charges a lower uniform price to poach these consumers.
The effect of δ on prices becomes stronger as the privacy management cost decreases because
consumers are more likely to opt out.
17Consumers’ incentives to opt out are directly impacted by δ and ε. A larger δ leads to a higher
α1∗ and a lower α2∗ , which implies lower personalized prices p∗1 (x) = α2∗ + t(1 − 2x). Therefore, firm
A1 ’s consumers have weaker incentives to opt out because the benefits of escaping personalized
prices decline. Formally, x̃∗1 in (6) decreases with δ. Conversely, a higher δ leads to higherp∗2 (x) = α1∗ + t(2x − 1) and a lower α2∗ , increasing the incentives of firm A2 ’s consumers to opt
out. Therefore, x̃∗2 in (6) decreases with δ. As before, a decrease in ε strengthens the effects ofδ on x̃∗1 and x̃∗2 .
Corollary 1. As the value of δ increases,(i) α1∗ and p∗2 (x) increase, and α2∗ and p∗1 (x) decrease;(ii) firm A1 consumers’ incentives to opt out weaken, and firm A2 consumers’ incentives to opt
out strengthen (i.e., x̃∗1 and x̃∗2 decrease); and(iii) the impacts of δ in (i) and (ii) become stronger as the value of ε decreases.
A firm cannot win its targeted consumers who strongly prefer the rival’s product, even if
its personalized price is zero, as shown in Figure 2(b). Specifically, firm A1 loses its targeted
consumers on [x̄∗1 , 1], and firm A2 loses its targeted consumers on [0, x̄∗2 ]. In equilibrium, weand x̄∗2 = 41 − δ(t−ε)
obtain x̄∗1 = 34 + (1−δ)(t−ε)
2t2t .
Opt-out consumers in market A pay strictly lower uniform prices than personalized opt-in
prices under no privacy management. However, they cause negative externalities on other optin consumers through price and mismatch channels. Compared to no privacy management,
opt-in consumers either pay higher personalized prices or pay the rival firm’s higher uniform
prices under privacy management. Furthermore, higher uniform prices help firms protect theirtargeted consumers by personalized pricing, allowing firms to retain lukewarm consumers who
would switch if privacy management were not feasible. For instance, firm A1 keeps its targeted
consumers on [3/4, x̄∗1 ] only under privacy management.
Corollary 2. Opt-out consumers in market A pay strictly lower prices than under no privacy

18

management but cause negative externalities on opt-in consumers in market A by increasing
their prices or making them incur higher mismatch costs.
Finally, the equilibrium profits of firms A1 and A2 in market A are as follows:
πA1 = α1∗ [δ x̃∗1 + (1 − δ)x̄∗2 ] + δ

Z x̄∗1x̃∗
1p∗1 (x)dx,

πA2 = α2∗ [(1 − δ)(1 − x̃∗2 ) + δ(1 − x̄∗1 )] + (1 − δ)Z x̃∗2x̄∗
2p∗2 (x)dx.

Firm A1 ’s profit increases with δ, and firm A2 ’s profit increases with 1 − δ. Their profits increase
with privacy management cost ε under Assumption 1.
Lemma 2. As ε decreases from its upper bound t, firm A1 ’s profit increases faster with δ ∈ [1/2, 1],
and firm A2 ’s profit increases faster with (1 − δ) ∈ [1/2, 1], implying that compared to no privacy
management, privacy management makes a firm’s profit in the data application market increase
faster with its market share in the data collection market.
Lemma 2 says that as ε declines from t, more consumers erase data in market A, resulting
in firm Ai ’s profit increasing with its consumer base faster. The intuition behind Lemma 2 is asfollows. Based on Corollary 1(iii), as the value of ε decreases, an increase in δ further enlarges
α1∗ , which makes the surplus extraction with uniform price more efficient. At the same time, alarger δ further diminishes the range of A1 ’s opt-out consumers (i.e., [0, x̃∗1 ]), and thus, A1 canprice discriminate against more consumers with a high willingness to pay with personalized
prices. These two effects increase the profitability of A1 . However, as the value of ε decreases,
an increase in δ diminishes the range in which A1 serves A2 ’s targeted consumers (i.e., x̄∗2 ) moreand decreases A1 ’s personalized prices p∗1 (x) more. These two effects diminish the profitabilityof A1 . We can check that the former two positive effects dominate the latter effects, i.e., that
∂ 2 πA1 /∂δ∂ε < 0 holds if δ ≥ 1/2 and ε is not small.
Now, we turn our attention to the equilibrium analysis of market B. First, we derive the ex

19

ante expected surplus of firm Bi ’s consumer in market A, E[CSBi ], as follows:
Z x̃1
E[CSB1 ]=0=
E[CSB2 ]=(vA − α1∗ − tx − ε)dx +x̃1(vA − p∗1 (x) − tx)dx +

Z 1
x̄1

(vA − α2∗ − t(1 − x))dx(1 − δ)2 ε2vA − (1 + δ − δ 2 )t + (1 − δ)(2δ − 1)ε +
,
t
Z x̄2
Z x̃2
Z 1
∗
∗
(vA − α1 − tx)dx +(vA − p2 (x) − t(1 − x))dx +(vA − α2∗ − t(1 − x) − ε)dx
0

=

Z x̄1

x̄2

x̃2

δ 2 ε2.
vA − (1 + δ − δ 2 )t − δ(2δ − 1)ε +t

The difference between E[CSB1 ] and E[CSB2 ] is18
E[CSB1 ] − E[CSB2 ] =(t − ε)ε(2δ − 1)1
> 0 if and only if δ > .
t
2

(7)

The firm with a larger market share in market B provides a higher expected consumer surplus
to its customers in market A.
The indifferent consumer δ in market B is determined the same way as in (3), which leads to2 −β1 )
δ = 21 + 2(tt(β2 −g(t−ε)ε) . Demand in market B is more price elastic if consumers are forward looking(i.e., g = 1). Firms’ profits in market B are πB1 = β1 δ and πB2 = β2 (1 − δ). Firms 1 and 2 decide
their uniform prices βi to maximize two-market profits: Π1 = πA1 + πB1 and Π2 = πA2 + πB2 . The
equilibrium uniform prices in market B are
β1∗ = β2∗ =(2t + ε)(4t − ε) g(t − ε)ε
−
,
16t
t

(8)

∗
∗
implying that δ ∗ = 1/2. The equilibrium profits of firms B1 and B2 are πB= πB= (2t + ε)(4t −1
2ε)/(32t) − g(t − ε)ε/(2t).
The equilibrium outcomes in market A are determined by replacing δ with 1/2. Marginal
consumers who are indifferent between erasing and keeping their data arex̃∗1 =1ε1ε− , x̃∗2 = + .
2 2t
2 2t

(9)

18 It is straightforward to check that E[CSB1 ] increases with firm B1 ’s market share δ and that E[CSB2 ] increases

with firm B2 ’s market share 1 − δ under Assumption 1. If firm B1 earns a larger market share (i.e., a larger δ value),then firm A1 ’s uniform price α∗1 increases and the rival’s α∗2 decreases, which leads to a lower p∗1 (x). As a result, it
is ex ante less likely for B1 ’s consumer to opt out because x̃∗1 decreases and is more likely to enjoy the lower p∗1 (x) inmarket A, leading to a higher E[CSB1 ].

20

We have that x̄∗1 = 1 − ε/(4t) and x̄∗2 = ε/(4t). The equilibrium prices in market A are
α1∗ = α2∗ = t − ε/2.
p∗1 (x) =
2t(1 − x) − ε/2when x̃∗1 < x < x̄∗1 ,
0when x̄∗1 ≤ x,p∗2 (x) =

(10)


0

2tx − ε/2when x ≤ x̄∗2 ,(11)

when x̄∗2 < x < x̃∗2 .

∗∗The equilibrium profits of firms A1 and A2 are πA= πA= (12t2 +3ε2 −4tε)/(32t). The equilibrium
1
2

profits of firms 1 and 2 are Π∗1 = Π∗2 = (10t2 + ε2 − tε)/(16t) − g(t − ε)ε/(2t). Figure 2(b) shows theequilibrium outcomes in the two markets.
Compared with its counterpart without privacy management, equilibrium prices in market
B decrease when privacy management is available because firm Ai ’s profit grows faster with
Bi ’s market share (Lemma 2). Specifically, as a firm’s market share in market B becomeslarger, the firm’s uniform price in market A becomes higher because it faces a larger number of
opt-out consumers with a higher willingness to pay (Corollary 1(i)). This higher uniform pricediminishes the incentives of marginal consumers to erase their data because the gains fromescaping personalized prices are lower (Corollary 1(ii)). Thus, the firm can price discriminate
against more consumers with a high willingness to pay and efficiently extract their surplus. The
inflated personalized prices in market A under privacy management further facilitate surplus
extraction. Therefore, firms in market B compete more aggressively to obtain a larger market
share.
Proposition 2. Compared to no privacy management, when consumers can manage their privacy, price competition in market B intensifies and becomes more severe as consumers’ foresight
regarding market A improves (i.e., g = 1).
The intuition behind why price competition in market B further intensifies when consumers
care more about their anticipated surplus in market A (g = 1) is as follows. Consumers rationally expect that the firm with a larger market share in market B provides a higher expectedconsumer surplus in market A (see (7)). When consumers are forward looking, firm Bi with alarger market share is more likely to win consumers, leading to more fierce price competition21

in market B. In contrast, when consumers are myopic (i.e., g = 0) or do not manage theirprivacy (i.e., ε = t), the utility difference between E[CSB1 ] and E[CSB2 ] vanishes in consumers’purchase decisions in market B.
Corollary 3. As privacy management cost ε increases, equilibrium prices under privacy management decrease in market A and increase in market B.
As ε increases, fewer consumers choose to erase their data in the equilibrium, leading tointensified uniform price competition in market A and hence lower personalized prices. Basedon Lemma 2, a higher privacy management cost leads to a lower marginal gain in market A
from increasing customers in market B. Thus, a higher privacy management cost mitigates
uniform price competition in market B. Since we can interpret no privacy management as thepolar case in which no consumer erases data because of prohibitively high costs (i.e., ε = t),Corollary 3 immediately implies the price changes under privacy management in both markets(Proposition 1 and Proposition 2).
Proposition 3. When consumers can manage their privacy, all firms earn lower profits than theywould under no privacy management, and their profits increase with privacy management costε.
Interestingly, all firms’ profits under privacy management increase with privacy management cost ε, despite that it is a deadweight loss.19 Three forces lead to this result. First, as
ε increases, fewer consumers opt out, allowing firms to price discriminate against more consumers with a high willingness to pay. Second, firm Ai attracts more of the rival’s targeted
consumers with αi∗ and loses more of its targeted consumers. The gain outweighs the loss
because the price αi∗ charged to poached consumers is higher than p∗i (x) for its lost distanttargeted consumers. These two forces bring firms higher profits in market A. Third, based
19 This result helps explain why Facebook and Google deliberately increase consumers’ costs to delete cookies (https://www.cnil.fr/en/cookies-cnil-fines-google-total-150-million-euros-and-facebook-60-million-euros-non-compliance).
In January 2022, CNIL, the Data Protection Authority for France, fined Google and Facebook 150 million and 60
million euros, respectively, for “not providing an equivalent solution (button or other) enabling the Internet user toeasily refuse the deposit of these cookies”.

22

on Corollary 3, firms compete less aggressively and obtain higher profits in market B as εincreases.

4.3

Impacts of privacy management on welfare

We compare consumer surplus in the two markets under privacy management with that underno privacy management. The consumer surplus without privacy management in market A isn∗
CSA
= vA − t, which declines to CSA= vA − (5t2 − ε2 )/(4t) under privacy management because of

the weakened price competition (Proposition 1). In more detail, opt-in consumers become worse
off (Corollary 2). The surplus of opt-out consumers remains the same as in the benchmark
because some opt-out consumers are better off by avoiding high personalized prices, whileothers are worse off by paying inflated uniform prices and incurring privacy management costs.
n
The consumer surplus in market B without privacy management is CSB= vB −13t/16, which∗= vB − (12t2 + 2tε − ε2 )/(16t) + g(t − ε)ε/t under privacy management because
increases to CSB∗
and total
the price competition intensifies (Proposition 2). Additionally, consumer surplus CSB∗∗
consumer surplus CSA+ CSB
increase with the degree of consumer foresight g. Concretely, if
n
n
∗
∗
+ CSB
< CSA
+ CSB
consumers are myopic about the outcome in market A (g = 0), then CSA
nn∗∗holds. These two
+ CSB
> CSA
+ CSB
holds, and if they are forward looking (g = 1), then CSA

inequalities suggest that providing consumers with information about how their data will be
used benefits them by intensifying competition.20
Social welfare in market A without privacy management is SWAn = vA − 5t/16 and declines to
SWA∗ = vA −(8t2 −7ε2 +4tε)/(16t) under privacy management because more consumers mismatchwith the less preferred firm. The costs of privacy management, a deadweight loss, further
diminish social welfare. Social welfare in market B is always equal to vB − t/4, regardlessof privacy management. Therefore, total social welfare in the two markets decreases under
privacy management.
20 An experimental study by Lin (2022) indicates that consumers are able to engage in strategic reasoning when mak-ing data-sharing decisions and that their beliefs are accurate to the first order only when the information environmentis transparent.

23

Proposition 4. Compared to no privacy management, when consumers can manage their privacy, the total social welfare in the two markets declines; the total consumer surplus declines if
consumers are myopic about the outcome in market A and increases if they are forward looking.
Specifically,(i) opt-out consumers’ surplus in market A does not change, and opt-in consumers in market
A are worse off, leading to lower consumer surplus in market A, and consumer surplus inmarket B increases;(ii) social welfare declines in market A and does not change in market B.
Propositions 3 and 4 conclude that compared to no privacy management, firms’ profits,
total consumer surplus, and total social welfare can all be negatively affected under privacy
management if consumers are myopic about the outcome in market A. In this case, only some
opt-out consumers are better off. The resulting changes in firms’ competition in both markets
harm other agents and society. In other words, privacy management leads to simultaneous
surplus redistribution and destruction.
Corollary 4. Under privacy management, as cost ε increases,(i) consumer surplus increases in market A and decreases in market B, and the total consumer
surplus of the two markets increases (decreases) if consumers are myopic (forward looking),and(ii) social welfare in market A increases, and the total social welfare of the two markets increases.
Corollary 4 states how privacy management cost ε affects welfare. A higher ε directly negatively affects welfare because it is a deadweight loss. In addition, it impacts welfare by affecting
consumers’ decision to opt out and subsequent market competition. In market B, a higher
ε hurts consumer surplus by mitigating competition (Corollary 3). In market A, a higher ε24reduces the number of opt-out consumers and intensifies price competition, enhancing consumer surplus; social welfare improves as well due to less consumer-firm mismatch.
Consumers’ foresight influences how ε affects consumer surpluses in the two markets.
When consumers are forward looking, an increase in ε diminishes the difference between
E[CSB1 ] and E[CSB2 ] (see (7)), which mitigates price competition in market B because obtaining
market share yields a smaller advantage in attracting consumers. Therefore, when consumers
are forward looking, they lose more surplus in market B as ε increases.
In fact, as ε approaches its upper bound t, the number of opt-out consumers converges
to zero, and the welfare level under privacy management approaches that under no privacy
management.

5

Extensions and discussions

5.1

Data-enabled product personalization and privacy costs

Now, we assume that firms can provide data-enabled personalized products and that consumers incur a privacy cost from their data being exploited. Concretely, if firm Ai ’s consumer
does not erase her data, then firm Ai can utilize data analytics to offer her a personalized
product that has a higher matching value for her than Ai ’s standard product. We parameterize
this by ∆v > 0 such that the consumer derives intrinsic value vA + ∆v from the personalized
product. However, the consumer incurs a privacy cost c ≥ 0 from her data being exploited inmarket A. If the consumer erases data, then she cannot enjoy the personalized product but
saves the privacy costs. We define ω = ∆v − c as the net benefit of product personalization andadopt the following assumptions: max{−3t, 5t − 2ε − 2vA } ≤ ω ≤ 2ε − t and t̂ ≤ ε ≤ t.21 We canestablish equilibria when consumers cannot manage their privacy (the benchmark) and when
they can. Figure 3 shows the equilibrium outcomes when consumers are myopic and forward
21 If ω is very large and consumers can retain part of the benefit under endogenous prices, then no consumer will

erase her data. In this case, all of firm Ai ’s targeted consumers always purchase from the firm, implying the largest
consumer-firm mismatch.

25

looking.

11αn1 = t−ω2pn2 (x) = 2t(x − 14 ) + ω2

ε−ω
4t

t+ε
2t

p∗2 (x) = 2tx − ε−ω
2

α∗1

1
2

t−

(ε+ω)
2

p∗1 (x) = 2t(1 − x) − ε−ω
2

α∗2

1
2

pn1 (x) = 2t( 34 − x) + ω2

A1

t−ω
4t

1
2

α∗1 =

αn2 = t−ω
2

3t+ω
4t

B1

t−
A2

A1

B2

B1

(a) Equilibrium without data erasure

(ε+ω)
2

t−ε
2t

α∗2 =

1
2

A2
1 − ε−ω
4t
B2(b) Equilibrium with data erasure

Figure 3: Equilibrium under production personalization and privacy costs

Compared to the benchmark, firm Ai earns a lower profit under privacy management when
benefit ω is large (i.e., ω > (t − 3ε)/6). The intuition behind this finding is as follows. As ω
rises, Ai ’s personalized prices without privacy management pni (x) increase and apply to moreconsumers because the rival’s consumer poaching becomes more difficult, resulting in higher
profits for the firm. In contrast, under privacy management, firm Ai ’s uniform price αi∗ decreases with ω. The firm’s profit loss in inefficient surplus extraction from opt-out consumers
rises with ω. As in the main model, firms in market B always earn lower profits under privacy
management due to intensified competition.
Proposition 5. Compared to no privacy management, when consumers can manage their privacy,(i) firms in market A earn lower profits if and only if the net benefit of product personalizationis relatively large (i.e., ω > (t − 3ε)/6), and firms in market B always earn lower profits, and(ii) the welfare comparison results are identical to Proposition 4, except that opt-out consumers
become strictly better off now.
26

In the equilibrium under privacy management, as ω increases, firm Ai has to lower its
uniform price to poach the rival’s targeted consumers, but its personalized prices increase andapply to more consumers. As a result, firm Ai ’s equilibrium profit increases with ω as long∗as it is not small (i.e., ω > 2t/3 − ε). This relationship between ω and πA
implies that a largericustomer base generates higher profits in market A, thus accelerating competition in market
B. Ultimately, a higher ω negatively affects firm i’s total profits.

5.2

Data portability

We discuss the case where consumers can use data erasure and data portability in privacy
management. Specifically, data portability means that firm Bi ’s targeted consumers can ask
the firm to transfer their data to the rival firm at no cost.22 Due to analytical difficulty, we
assume that consumers are myopic about the outcome in market A. ex post of no privacy
management is the same as that in Section 4.1.
Firm A1 ’s consumer x chooses from four privacy management options to minimize her anticipated total cost (including price, transportation costs, and data erasure costs if applicable).
First, if the consumer erases data and opts out, then her total cost is either α1a + tx + ε or
α2a + t(1 − x) + ε, as shown by the solid green lines in Figure 4(a). Second, if consumer x chooses
data portability and does not erase her data in A1 ’s datasets, then she is targeted by both
firms. Her total cost is t(1 − x) when x ≤ 1/2 and tx when x ≥ 1/2 (dashed blue lines). Third, if
consumer x chooses data portability and erases her data in A1 ’s datasets, then she is targeted
only by firm A2 and incurs total cost α1a + tx + ε (upward-sloping green line). Fourth, if she does
not manage privacy, then her total cost is α2a + t(1 − x) (dotted red line).
In the equilibrium of market A, firm A1 ’s consumers on [0, x̃1 ] opt out by erasing data; consumers on [x̃1 , x̄1 ] choose data portability and do not erase their data in firm A1 ’s datasets; andconsumers on [x̄1 , 1] do not manage privacy and purchase from firm A2 . Firm A2 ’s consumers
22 According to privacy laws, the responsibility of data portability falls mainly on the firm, and consumers do notincur high costs. For example, the guidelines in the GDPR state that “the overall system implementation costs should
not be charged to the data subjects”. The CCPA clearly requires no charge to consumers for data portability. The
results do not change if we assume that data portability entails a small cost for consumers.

27total
cost

t+ε
6t

1

αa2 + tαa1 + tx + ε

5t+2ε
6t

2t−ε
α∗∗
1 = 3 pay t(1 − 2x) pay t(2x − 1)

αa2 + t(1 − x) + ε

α∗∗
2

1
2

t
αa1 + εα∗∗1t(1 − x)αa2 + t(1 − x)

tx

A1
x̃1

1/2

2t−ε
pay t(1 − 2x) pay t(2x − 1) α∗∗
2 = 3

x̄1

1

x

t−2ε
6t

1/2
1/2

5t−ε
6t

B1

(a) Total costs

A2
B2(b) Equilibrium outcome

Figure 4: Total costs and equilibrium outcome

adopt a similar privacy management strategy. Figure 4(b) shows the equilibrium outcome.
Compared to no privacy management, equilibrium uniform prices in market A increase
because of opt-out consumers. As a result, consumers who do not manage their privacy become
worse off. Firm Ai ’s consumers who have chosen data portability benefit from fierce consumerby-consumer price competition and are not affected by inflated uniform prices. Firms in market
A earn lower profits than them under no privacy management. In addition to the inability to
price discriminate against nearby opt-out consumers, fierce competition for consumers using
data portability further reduces profits. Price competition in market B significantly mitigatesas the value of targeted consumers in market A diminishes substantially. As a result, firms
in market B obtain higher profits. In total, firms 1 and 2 earn higher profits under privacy
management.
Consumer surplus in market A increases compared to that under no privacy management.
However, consumer surplus in market B declines dramatically due to the mitigated price competition in this market, leading to a lower total consumer surplus. When data portability isavailable, consumers purchase from their preferred firm in both markets, implying that social
28

welfare is maximized.
Proposition 6. Suppose that consumers have access to data erasure and data portability inprivacy management. Compared to no privacy management,(i) uniform prices increase in market A due to opt-out consumers, but personalized price competition intensifies;(ii) firms in market A earn lower profits, and those in market B earn higher profits, resulting inhigher profits for firms 1 and 2;(iii) consumer surplus in market A increases, and that in market B and total consumer surplus
decrease; and(iv) social welfare in both markets achieves the maximum value.

5.3

Heterogeneous consumer privacy types and market failureThis section extends the main model by incorporating heterogeneous consumer privacy concerns. We focus on a scenario in which privacy-sensitive consumers may exit the market if theycannot manage their privacy, leading to market failure. Privacy management fixes such failure
by attracting these consumers back to the market, thus producing a new type of “good”. The
return of these consumers, however, weakens price competition in the data application market
and expands negative externalities on (privacy-insensitive) opt-in consumers, aggravating the“bad” and “ugly” aspects of privacy management.
To capture privacy types, we label a consumer in market A as (x, θ), where x is her locationand θ ∈ {0, c} indicates her privacy cost, which is her private information. Privacy cost θ accrues
in market A if her data are transferred and exploited in this market. With probability r ∈ [0, 1],
the consumer is a privacy-sensitive type with privacy cost θ = c; with probability 1 − r, theconsumer is a privacy-insensitive type without privacy cost θ = 0.
Before firm Bi transfers consumer data to firm Ai , anticipating the data transfer and privacy
cost θ, its consumers can decide whether to exit market A. If a consumer exits market A, then29

her utility in this market becomes zero. Conversely, if she stays and does not erase her data,then firm Ai knows her (x, θ), and firm Aj (j ̸= i) knows only the distribution of x and θ. Then,
she is firm Ai ’s targeted consumer and has two choices: obtain utility u = vA − txi − pi (x) − θfrom firm Ai or obtain utility u = vA − txj − αj − θ from firm Aj . If she stays and erases her data,then the consumer does not incur any privacy costs and can escape targeting by either firm.
The game proceeds as in the main model except for the privacy management stage, in which
all consumers simultaneously decide whether to exit the market and erase their data. To
neatly illustrate our idea, we assume that privacy-sensitive consumers’ privacy cost is very
high: c > vA . Moreover, when c is very high, the option of data portability is no longer relevant
because consumers never choose it.

□ Benchmark: No privacy management and market failure

Our first observation is thatall privacy-sensitive consumers choose to exit market A because they receive negative utility
from staying in the market due to the high privacy cost c. As shown in the left panel of
Figure 5, firms A1 and A2 lose rδ and r(1 − δ) shares of privacy-sensitive consumers, causing“market failure.” These firms have only privacy-insensitive consumers left in the market. The
equilibrium analysis is the same as that in Section 4.1. However, equilibrium profits, consumer
surplus, and social welfare all decline due to exiting consumers. The larger r and vA are, themore significant the welfare losses.

□ Equilibrium under privacy management

Our analysis begins with the observation thatwhen privacy management is available, all privacy-sensitive consumers stay in market A anderase their data at the cost of ε. Privacy-insensitive consumers’ privacy management strategy
is the same as that in Lemma 1. Figure 5(b) shows the distribution of opt-in consumers andthose consumers who have erased their data. Compared to the main model (Figure 2(a)), both
firms in market A now have larger numbers of untargeted consumers due to opt-out privacysensitive consumers.
Here, opt-out privacy-insensitive consumers mitigate price competition in market A and
30

1

1

δ
θ = c consumers stay and erase data
A1 ’s θ = 0
consumers
erase data

(1 − r)δ of θ = 0 consumers stayδ

B1

A2 ’s θ = 0
consumers
erase data

θ = c consumers stay and erase datar(1 − δ) of θ = c consumers exitrδ of θ = c consumers exit

A1x̃2
A2 ’s targeted
consumers (θ = 0)

(1 − r)(1 − δ) of θ = 0 consumers stay
δ

x̄2

A2

A1

B2

B1

(a) Consumers’ distribution without privacy management

A1 ’s targeted
consumers (θ = 0)
x̃1

x̂

x̄1

δ

A2
B2

(b) Consumers’ distribution under privacy management

Figure 5: Distribution of consumers without and with privacy management

bring negative externalities to other consumers, consistent with Corollary 2. Moreover, equilibrium prices αi∗ and p∗i (x) increase with r, suggesting that having a larger portion of privacysensitive consumers who erase their data further mitigates price competition in market A.
The competition to obtain privacy-sensitive consumers, which involves uniform versus uniform prices, is less severe than that for acquiring opt-in privacy-insensitive consumers, which
requires uniform versus personalized prices. Thus, price competition becomes less aggressive
as the number of privacy-sensitive consumers increases (i.e., a larger r value). Furthermore, x̄∗1
increases with r, and x̄∗2 decreases with r. These relations imply that a larger r generates moreconsumer-firm mismatch in market A. As a result, the return of privacy-sensitive consumers
yields negative externalities for privacy-insensitive consumers by increasing prices and making
them incur higher mismatch costs.
We find that compared to no privacy management, firm Bi always earns lower profits under privacy management because price competition in market B intensifies as the benefits ofwinning consumers increase in market A. Consistent with the main model, forward-looking
consumers motivate firms to compete more aggressively in market B. Conversely, firms in31

market A earn higher profits under privacy management, except when r is extremely small,leading to firms 1 and 2 earning higher profits under privacy management, except when r issmall.
Proposition 7. Suppose that consumers have heterogeneous privacy sensitivity. Compared tono privacy management, when privacy management is available,(i) market failure due to privacy-sensitive consumers exiting the market is fixed,(ii) the return of privacy-sensitive consumers and the opting out of privacy-insensitive consumers mitigate price competition in market A, and(iii) price competition intensifies in market B.
We now discuss the welfare implications of Proposition 7. Privacy management enhances
consumer surplus by attracting privacy-sensitive consumers back to market A. However, thereturn of these consumers further mitigates price competition and negatively affects consumer
surplus in market A. Therefore, when the former effect dominates the latter, that is, when
the share of privacy-sensitive consumers is larger than a threshold level, privacy management
improves consumer surplus in market A. Proposition 7(iii) implies that privacy management
increases consumer surplus in market B. Similarly, when the share of privacy-sensitive consumers is larger than a threshold level, privacy management improves social welfare in market
A. However, the availability of privacy management does not change social welfare in market
B.

5.4

Who should own data property rights?

We now discuss how consumers’ personal data management affects market competition andwelfare when they own data property rights. In this section, consumers opt out by default and
opt in by choice, and firms can exploit a consumer’s data only when she opts in. This setting
mirrors the privacy model in GDPR, where firms have to obtain consumers’ consent to collectand process their data. The game proceeds as in the main model, except that the two firms
32

in market A commit to a benefit, b ∈ [0, t], for opt-in consumers and deliver this benefit once
they opt in. In addition, opt-in consumers obtain a net benefit, ω ∈ [0, 2b − t], from productpersonalization in the data application market (as in Section 5.1).23 The equilibrium analysis
follows a similar process as that in the main model.
We compare the equilibrium to the benchmark in Section 5.1 where firms own data andconsumers can opt out with a fixed cost, ε. The comparative results depend on the value ofb. The first case focuses on b = ε, which equalizes consumers’ tradeoff between opting in andopting out, regardless of who owns the data. The second case focuses on the profit-maximizing
b value for firms 1 and 2: b = t when consumers are forward looking (g = 1), and b = (t + ω)/2
when consumers are myopic (g = 0). Table 1 shows the comparative results.
Table 1: Profits and welfare change when consumers own the data
∗
πA
i

∗
πB
i

Π∗1 & Π∗2

∗
CSA

∗
CSB

∗
∗
+ CSB
CSA

SWA∗

SWB∗

b=ε

↓

↑

↑

↑

↓

0

↑

0

b = t (g = 1)

↓

↑

↑

↑

↓

↑

↑

0

b = t+ω
2 (g = 0)

↓

↑

↑

↑

↓

↓ iff

↓ iff

ω < 2ε − tω̂1 < ω < ω̂20

Note: ↑ indicates an increase, ↓ indicates a decrease, 0 indicates no change, and iff indicates if andonly if.

In market A, firms earn a lower profit because they pay benefits to opt-in consumers; consumer surplus increases due to the benefits and saved privacy management cost ε. Firm Bi isless incentivized to collect data with an attractive price when consumers own the data property rights, resulting in higher profits and lower consumer surplus in market B. However, theprice mitigation in market B is not so significant when consumers are forward looking (g = 1),
making total consumer surplus increase, as in the b = t row.
23 Benefit b serves the same role as ε in the equilibrium analysis; it can be a lump-sum payment, a discount, a tie-in

gift, or any other benefit to induce consumers to opt in. The assumption of ω ≤ 2b−t ensures pure strategy equilibriumin the price competition of market A.

33

Proposition 8. Suppose that consumers own the data property rights. Compared to the scenario
where firms own the data,(i) firms in market A earn a lower profit, and firms in market B earn a higher profit due tomitigated competition, leading to higher profits for firms 1 and 2;(ii) consumer surplus increases in market A and decreases in market B; and(iii) giving data property rights to consumers leads to higher total consumer surplus and social
welfare when these consumers are forward looking.

6

Policy implications

6.1

Rights to data erasure and data portability

We can extend the main model to show that firm Bi has no incentive to voluntarily offer consumers the choice of data erasure.24 To understand this aspect, suppose that firm Bi offers
data erasure. Firm Ai cannot charge high personalized prices to opt-out consumers who havea high willingness to pay and wins fewer of the rival’s targeted consumers because of its inflated uniform price. This finding implies that privacy laws are necessary to ensure consumers’
right to delete their personal data. However, such a right is likely to backfire on consumers
economically and negatively affect society (Proposition 4).
In the following policy discussion on data portability as an additional option, we focus on thecase in which Assumption 1 holds and adopt two benchmarks: one is no privacy management(Section 4.1) and the other is data erasure only (Section 4.2). Similar to Section 5.2, which
requires ε < t/2, we can establish equilibria for forward-looking and myopic consumers whent/2 ≤ ε ≤ t holds. The equilibrium outcome in market A is similar to that in Figure 4(b), but no∗∗consumer chooses to erase data (i.e., x̃∗∗
1 = 0 and x̃2 = 1) because opting out with high cost εand paying the inflated uniform price is no longer the best choice. Instead, a firm’s consumers
with a high willingness to pay choose data portability to be targeted by both firms in market A
24 Due to analytical difficulty, here, we assume that consumers are myopic about the outcome in market A.

34

and pay low personalized prices. Compared to no privacy management and data erasure only,offering consumers the additional choice of data portability improves the profits of firms 1 and
2 and social welfare but reduces total consumer surplus, consistent with Proposition 6.
Proposition 9. Compared to no privacy management, data erasure and data portability change
competition in the data collection and application markets in opposite directions; total consumer
surplus is always negatively affected, while total social welfare is enhanced whenever data
portability is available.
Data erasure weakens competition in the data application market and intensifies competition in the data collection market, and the opposite is true for data portability. Consumers’
surplus loss from competition mitigation in one market outweighs their gains from intensifiedcompetition in the other market. Data erasure negatively affects social welfare due to a larger
consumer-firm mismatch, while data portability improves social welfare because it enables
consumers to be targeted and hence attracted by their preferred firm.

6.2

Banning cross-market data transfer and data walls

Suppose that regulators prohibit firm Bi from transferring consumer data to firm Ai . Alternatively, suppose that firms 1 and 2 commit to setting up an inside data wall to block the data
flow from business B to business A.25 As a result of the ban, consumers no longer need toactively manage their data; instead, they all opt out passively. Firms in market A cannot target
any consumers, implying that the equilibrium prices are the same as those in the classical
Hotelling model: α1∗ = α1∗ = t. Price competition in market B also becomes the Hotelling type:β1∗ = β2∗ = t.
Compared to Section 4.2 where consumers can only erase their data, all firms earn higher
profits under the ban. In market A, the severe competition between uniform prices and per25 For example, in the Google-Fitbit merger, Google assured global Fitbit users that it would not use their health andwellness data for Google ads and that it would separate the Fitbit data from other Google ad data and store Fitbit data
in a separate data silo (https://blog.google/products/devices-services/fitbit-acquisition/).

35

sonalized prices downgrades to competition between uniform prices. Specifically, uniform priceαi∗ = t is higher than that in (10) and the personalized prices in (11) for most opt-in consumers.
The ban also mitigates price competition in market B. As a result, consumer surplus declines
in each market. Since all consumers purchase from their preferred firms in each market, social
welfare achieves the maximum value under the ban.
The same results are obtained when we compare the equilibrium under the ban with thatin Section 4.1, where consumers cannot manage their privacy, or with the scenario in which
consumers have access to data erasure and data portability.
Proposition 10. Suppose that firms 1 and 2 cannot transfer data across markets. Compared tothe case with no such restriction, regardless of whether consumers can manage their privacy,(i) all firms earn higher profits due to weaker price competition in each market, and(ii) while consumer surplus declines in each market, social welfare weakly improves in market
A.

7

Conclusions

We consider a duopoly model in which consumers purchase products in two markets: one fordata collection, and one for data application. The two firms compete in the two markets andfirst attempt to gain consumers in the data collection market. The data collected uniquelyby each firm capture its customers’ preferences for firms’ products in the data application
market. Following competition in the data collection market, each firm in the data application
market offers personalized prices to its targeted customers and a uniform price to untargeted
consumers. Before firms’ pricing decisions, each firm’s targeted customers can erase their datafrom the firm’s database to become untargeted at a fixed cost. Consumers who erase their data
can escape personalized prices and choose uniform prices offered by firms.
We find that consumers who strongly prefer one firm in the data application market choose
to erase their data. These consumers opting out increases firms’ uniform prices, which induces36firms to set higher personalized prices. Nevertheless, firms earn lower profits in the data application market because they cannot extract enough surplus from opt-out consumers who havea high willingness to pay. Higher prices clearly negatively affect consumers. Moreover, due tohigher uniform prices, each firm easily protects its targeted customers through personalized
pricing, resulting in increased consumer-firm mismatches and decreased social welfare. Interestingly, consumer privacy management intensifies competition in the data collection market,
although it diminishes the profitability of data applications. Combining these two markets,
consumer privacy management leads to lower profits and social welfare; consumers benefit
only when they are forward looking. The results are robust to various extensions.
Our study has direct implications for privacy laws. Data erasure and data portability cannegatively affect consumers because the former weakens competition in the data application
market, while the latter dampens competition in the data collection market. However, regarding
social welfare, data portability is more likely to be beneficial for welfare than is data erasure
because the former enables a consumer to be targeted and attracted by her preferred firm,
while data erasure tends to worsen consumer-firm mismatch. Furthermore, banning firms’cross-market data transfer harms consumers because it mitigates competition in both markets.
Giving consumers data property rights can be beneficial to consumers themselves and society
as a whole if they are forward looking.

37

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A

Appendix

A.1

Proof of equilibrium uniform prices in market A

In this section, we show how to determine the conditions under which firm A1 and firm A2x̃12)
and α2∗ = tδ+2t(1−δ)(1−x̃. Firmdo not deviate from the subgame equilibrium α1∗ = t(1−δ)+2tδ2(1−δ)
2δ

A1 has two possible deviations. The first one is increasing α1 above t and does not poach therival’s targeted consumers. The second is reducing α1 to attract firm A2 ’s consumers who haveerased their data (i.e., consumers on [x̃∗2 , 1]).
□ Firm A1 ’s deviation one: α1 ≥ t

Notice that vA − t − α2∗ > 0 always holds based on ourAssumption 1, implying firm A1 cannot become a local monopoly for its consumers on [0, x̃∗1 ].
Given its deviation price α1d , the indifferent consumer is x̂d = 12 +

d
α∗
2 −α1
2t . Firm A1 maximizes its

ddeviation profit from uniform price πA= δα1d x̂d subject to x̂d ≤ x̃∗1 . The optimal deviation price1d= δ(1−δ)(t−ε)[(2δ+1)t+2(1−δ)ε]and the deviation profit is πA. Its equilibrium
is α1d = t(1+2δ)+2(1−δ)ε
2
2t
1
2

∗
profit from the uniform price is πA= (1−δ)(t+2δt−2δε). Then we have8t
1

(1 − δ)
∗
d
πA
− πA
=
1
1




n √4ε2 + t2 − 2ε o−4(t − ε)2 δ 2 − 8ε(t − ε)δ + t2
> 0 if and only if δ < min 1,.
8t
2(t − ε)□ Firm A1 ’s deviation two: x̂d ≥ x̃∗2

To make x̂d ≥ x̃∗2 hold, firm A1 ’s deviation price
α1d ≤ δ + 21 t − (δ + 1)ε. Firm A1 ’s deviation profit from the uniform price isdπA= δα1d x̃∗1 + (1 − δ)α1d (x̄d2 + x̂d − x̃∗2 ) =1−4(1 − δ)(α1d )2 + (1 − δ)[3t(1 + 2δ) − (2 + 6δ)ε]α1d
.
4t


d
Firm A1 maximizes πAsubject to α1d ≤ δ + 12 t − (δ + 1)ε. Based on our Assumption 1, firm A1 ’s
1
optimal deviation is α1d = δ + 12 t − (δ + 1)ε, implying x̂d = x̃∗2 . Such deviation is unprofitablebecause firm A1 reduces its price and does not gain more demand.
Firm A2 has two possible deviations similar to firm A1 . We find firm A2 has no incentive tono√
4ε2 +t2
deviate if and only if δ > max 0, 2t−2(t−ε). In summary, no firm has the incentive to deviate
if and only if

)
( √
)
√
2t − 4ε2 + t2
4ε2 + t2 − 2εmax 0,≤ δ ≤ min 1,.
2(t − ε)2(t − ε)
(

44

A.2

Proof of Proposition 4

Consumer surplus in market A without privacy management is
1
n
CSA
=

"Z
3/4

2

1
+
2

0

(vA − tx − pn1 (x))dx +

"Z

1/4

0

(vA − tx − α1n )dx +

Z 1
3/4

Z 1
1/4

#
(vA − t(1 − x) − α2n )dx#(vA − t(1 − x) − pn2 (x))dx=vA − t.
Consumer surplus in market A under privacy management is
1
∗
CSA
=

x̃∗
1

"Z

2

0

1
+
2

(vA − tx − α1∗ − ε)dx +

"Z ∗
x̄2

=vA −

0

(vA − tx − α1∗ )dx +Z x̄∗1x̃∗
1Z x̃∗2x̄∗
2(vA − tx − p∗1 (x))dx +

Z 1
x̄∗
1

#
(vA − t(1 − x) − α2∗ )dx(vA − t(1 − x) − p∗2 (x))dx +

Z 1
x̃∗
2

#
(vA − t(1 − x) − α2∗ − ε)dx

5t2 − ε2.
4tn∗holds.
< CSA
It is straightforward to check that CSA

Consumer surplus in market B without and with privacy management is
n
CSB
=

Z 1/2
0

(vB − tx − β1n )dx +∗CSB(g = 0) =

Z 1/2
0

∗
CSB
(g = 1) =

Z 1/2
0

Z 1
1/2

(vB − t(1 − x) − β2n )dx = vB −(vB − tx − β1∗ )dx +(vB − tx − β1∗ )dx +

Z 1
1/2

Z 1
1/2

13
t,
16

(vB − t(1 − x) − β2∗ )dx = vB −12t2 + 2tε − ε2,16t(vB − t(1 − x) − β2∗ )dx = vB −12t2 − 14tε + 15ε2
.
16t

∗
n
∗
∗
n
n
We can check that CSB> CSB
always holds. Moreover, CSA
+ CSB
< CSA
+ CSB
hold wheng = 0 and the reverse holds when g = 1.
5t
Social welfare in market A without privacy management is SWAn = vA − 16
. Social welfare in
2

2

+4tε
market A under privacy management is SWA∗ = vA − 8t −7ε. We have SWA∗ < SWAn . Social
16t

welfare in market B always equals vB − 4t .

45

A.3

Proof for Section 5.1□ Benchmark: no privacy management The analysis is similar to Section 4.1. In market A,
the equilibrium uniform prices are α1n = α2n = (t − ω)/2 and the personalized prices are2t(3/4 − x) + ω/2 if x ≤ 3/4 + ω/(4t)0
if x ≤ 1/4 − ω/(4t)nnp1 (x) =p2 (x) =02t(x − 1/4) + ω/2 if x ≥ 1/4 − ω/(4t).
if x ≥ 3/4 + ω/(4t)In market B, the indifferent consumer is δ = 1+β22 −β1 , which is independent of g because
E[CSB1 ] = E[CSB2 ]. Firm 1 and firm 2 decide their uniform prices to maximize total profitsnnΠ1 = β 1 δ + π A
and Π2 = β2 (1 − δ) + πA
, respectively. The equilibrium uniform prices in market
1
2

B are β1n = β2n = (ω 2 − 10tω + 9t2 )/(16t). The indifferent consumer in market B is δ n = 1/2. Wennnnhave πB= πB= (ω 2 − 10tω + 9t2 )/(32t) and πA= πA= (3ω 2 + 2tω + 11t2 )/(32t). The equilibrium
12
12

profits of firms 1 and 2 are Πn1 = Πn2 = (ω 2 − 2tω + 5t2 )/(8t).
□ Equilibrium under privacy management The analysis is similar to Section 4.2. Given δ,
the equilibrium uniform prices in market A is α1∗ = t(1/2+δ)−δε− ω2 and α2∗ = t(3/2−δ)−(1−δ)ε− ω2 .
The personalized prices are p∗1 (x) = α2∗ + t(1 − 2x) + ω and p∗2 (x) = α1∗ − t(1 − 2x) + ω. The
cutoffs are x̃∗1 = (1 − δ)(t − ε)/t, x̃∗2 = 1 − δ(t − ε)/t, x̄∗1 = ((5 − 2δ)t + ω − 2(1 − δ)ε)/(4t), andx̄∗2 = ((1 − 2δ)t + 2δε − ω)/(4t). It is straightforward to check that x̃∗1 ∈ (0, 1) and x̃∗2 ∈ (0, 1)
always hold. Under our assumption of ω > −3t, x̄∗1 > 0 and x̄∗2 < 1 always hold. We havet−ω∗x̄∗1 < 1 is equivalent to δ > ω+t−2ε2(t−ε) and x̄2 > 0 is equivalent to δ < 2(t−ε) . When ω ≤ 2ε − t,t−2ε+ω
t−ω
2(t−ε) < δ < 2(t−ε) holds for any δ ∈ [0, 1]. We assume ω ≤ 2ε − t holds in the following analysis.

Similar to the main model, firms in market A should not have incentive to deviate from α1∗ and
α2∗ , which requires
max{δ̂2 (ε, ω), δ̃2 (ε, ω), 0} ≤ δ ≤ min{δ̂1 (ε, ω), δ̃1 (ε, ω), 1}.

(12)

The derivation of (12) can be found in the Online Appendix. We find that (12) holds for anyδ ∈ [0, 1] when ε ∈ [t̂(ω, t), t], in which t̂(ω, t) increases with ω and ranges from 0.35t to t.
In market B, the equilibrium uniform prices are β1∗ = β2∗ = ((2t + ε)(4t − ε) − ω(10t − ω))/(16t) −∗∗g(t − ε)ε/t, implying the indifferent consumer is δ ∗ = 1/2.26 The profits of firms are πA
= πA
=
1
2
26 The ex-ante expected surpluf of firm B ’s consumer in market A is E[CSiBi ] + ω/2, where E[CSBi ] is in Section 4.2.

46

∗
∗
(12t2 +3ε2 −4tε+ω(6ε+3ω−4t))/(32t) and πB= πB= ((2t+ε)(4t−ε)−ω(10t−ω))/(32t)−g(t−ε)ε/(2t).
1
2

The equilibrium profits of firms 1 and 2 are Π∗1 = Π∗2 = (10t2 + ε2 − tε + ω(3ε + 2ω − 7t))/(16t) −g(t − ε)ε/(2t). The welfare comparison is similar to the main model and is omitted here.

A.4

Proof for data portability in Section 5.2 and Section 6.1

Consumers’ total costs from different options are shown in Figure 6. The left panel shows thecase of small ε, and the right panel shows the case of large ε.
□ When ε is small

Based on the left panel of Figure 6, consumers on [0, x̃1 ] opt out

by erasing data, consumers on [x̃1 , x̄1 ] choose data portability, and consumers on [x̄1 , 1] do not
αa +ε

αa

manage privacy and opt in. We have x̃1 = 12 − 12t and x̄1 = 21 + 2t2 . Firm B2 ’s targeted consumers
adopt a similar privacy management strategy, in which x̃2 = 21 +

total
cost

total
cost
αa2 + tαa2 + tαa1 + tx + ε

αa
αa
2 +ε
and x̄2 = 21 − 2t1 .
2t

αa1 + tx + ε

αa1 + εαa2 + t(1 − x) + ε

αa2 + t(1 − x)tt

αa2 + t(1 − x) + εtx

αa1 + εt(1 − x)x̃1

1/2

t(1 − x)αa2 + t(1 − x)

tx

x̄1

1

1/2

x

t+αa2
2t

1

x

Figure 6: Total costs and consumers’ privacy management strategy

Firm A1 ’s profit from its uniform price α1 and firm A2 ’s profit from its uniform price α2 areα1 [(1 − δ)x̄2 + δ x̃1 ] when α1 ≤ t − εα2 [δ(1 − x̄1 ) + (1 − δ)(1 − x̃2 )] when α2 ≤ t − εα1 (1 − δ)x̄2

when t − ε ≤ α1 ≤ t, α2 δ(1 − x̄1 )when t − ε ≤ α2 ≤ t.

(1−δ)(t−2ε)
t−ε+δε∗∗∗∗∗∗In the equilibrium of market A, we have α1∗∗ = t−δε
2−δ , α2 =1+δ , x̃1 =2t(2−δ) , x̃2 =t(2+δ)−(1−δ)εt(1−δ)+δε
2t+tδ+2δε∗∗, and x̄∗∗2 = 2t(2−δ) .
2t(1+δ) , x̄1 =
2t(1+δ)47∗∗∗∗
The constraints x̃∗∗1 < 1, 0 < x̃2 , 0 < x̄1 < 1,

∗∗∗∗and 0 < x̄∗∗2 < 1 always hold. The constraints 0 < x̃1 and x̃2 < 1 hold for any δ if and only if

0 < ε < 2t . Firm A1 and firm A2 have no incentive to deviate if and only if max{0, δ2 (t, ε)} ≤ δ ≤min{1, δ1 (t, ε)}, which holds for any δ ∈ [0, 1] when 0.37t ≤ ε < 2t .27 Two firms’ equilibrium profits
in market A are∗∗∗∗πA= α1∗∗ [δ x̃∗∗
1 + (1 − δ)x̄2 ] + δ
1

Z 21
x̃∗∗
1

t(1 − 2x)dx + (1 − δ)

∗∗
∗∗
πA
= α2∗∗ [(1 − δ)(1 − x̃∗∗2 ) + δ(1 − x̄1 )] + δ
2

Z x̄∗∗
1
1
2

Z 21
x̄∗∗
2

t(1 − 2x)dx,t(2x − 1)dx + (1 − δ)Z x̃∗∗
2
1
2

t(2x − 1)dx.

−β1
In market B, the indifferent consumer is δ = 21 + β22t
. Firm i decides βi to maximize its total
2

2, implying
profit Πi = πBi + πAi . The equilibrium uniform prices are β1∗∗ = β2∗∗ = 100t +8tε−11ε108tδ ∗∗ = 12 . The equilibrium outcomes in market A are determined by replacing δ with 12 . The
uniform price αi∗∗ increases compared to no privacy management.
∗∗= vA −
Under data erasure and data portability, consumer surplus in market A is CSA
25t2 −4tε+7ε2, and consumer surplus in market B is
36t

∗∗
CSB
=

Z 1/2
0

(vB − tx − β1∗∗ )dx +

Z 1
1/2

(vB − t(1 − x) − β2∗∗ )dx = vB −127t2 + 8tε − 11ε2
.
108t

n
n
∗∗
∗∗
n
∗∗
n
∗∗
,
+ CSB
< CSA
+ CSB
, CSA
< CSB
, CSB
> CSA
Compared to no privacy management, CSA

SWA∗∗ > SWAn , and SWB∗∗ > SWBn .
This case requires 2t < ε < t. The right panel of Figure 6□ When ε is relatively large

shows the total costs of consumers’ different options. In this case, no consumer opts out by
3
1∗∗
erasing data. Firms’ optimal uniform prices are α1∗∗ = α2∗∗ = 2t , implying x̄∗∗
1 = 4 and x̄2 = 4 . It

is straightforward to check that no firm has incentive to deviate from α1∗∗ and α2∗∗ . Two firms’
equilibrium profits in market A are∗∗πA= α1∗∗ (1 − δ)x̄∗∗
2 +δ
1
∗∗
πA
= α2∗∗ δ(1 − x̄∗∗
1 )+δ
2

Z 21
0

t(1 − 2x)dx + (1 − δ)Z x̄∗∗
1
1
2

Z 21

t(2x − 1)dx + (1 − δ)x̄∗∗2t(1 − 2x)dx =

Z 1
1
2

t(2x − 1)dx =t(3 + δ),16t(4 − δ).
16

Notice that E[CSB1 ] = E[CSB2 ] always holds, implying the indifferent consumer in market B
27 The detailed proof of this condition can be found in Section 5 of the online appendix.

48

is independent of g. The analysis in market B is the same as in the case of small ε. The∗∗
equilibrium uniform prices are β1∗∗ = β2∗∗ = 15t= 12 .
16 , implying δWhen 2t < ε < t, we have two benchmarks to compare. In benchmark one (with superscriptn), consumers cannot manage their privacy (Section 4.1). In the other benchmark (with superscript ∗), consumers can only use data erasure in privacy management (Section 4.2). We have
n
∗
∗∗
∗∗
n
∗
n
∗
the following ranking about firms’ profits: πA
> πA
> πA
, πB
> πB
> πB
, and Π∗∗i > Πi > Πi .
i
i
i
i
i
i
n
In market A, consumer surplus without privacy management is CSA= vA − t; consumer
2

2

−ε
∗
surplus under data erasure is CSA= vA − 5t 4t; consumer surplus under data erasure anddata portability is
1
∗∗
CSA
=

"Z 1
2

2
1
+
2

0

(vA − tx − t(1 − 2x))dx +

"Z

=vA −

x̄∗∗
2

0

(vA − tx − α1∗∗ )dx +Z x̄∗∗
1
1
2

Z 12
x̄∗∗
2

(vA − t(1 − x) − t(2x − 1))dx +(vA − tx − t(1 − 2x))dx +

Z 1
1
2

Z 1
x̄∗∗
1

#
(vA − t(1 − x) − α2∗∗ )dx#(vA − t(1 − x) − t(2x − 1))dx

11
t.
16

∗
n
∗∗
It is straightforward to check that CSA
< CSA
< CSA
holds.
n= vB − 13
In market B, consumer surplus without privacy management is CSB
16 t; consumer
22∗= vB − 12t +2tε−εsurplus under data erasure is CSB; consumer surplus under data erasure16t

and data portability is
∗∗
CSB
=

Z 1/2
0

(vB − tx − β1∗∗ )dx +

Z 1
1/2

(vB − t(1 − x) − β2∗∗ )dx = vB −

19
t.
16

∗
n
∗∗
n
n
∗∗
∗∗
We can check that CSB
> CSB
> CSB
holds. Moreover, CSA
+ CSB
> CSA
+ CSB
always holds,
∗
∗
∗∗
∗∗
and CSA
+ CSB
> CSA
+ CSB
holds under our assumption of ε >

√

3
2 t.

5t
In market A, social welfare without privacy management is SWAn = vA − 16
; social welfare
2

2

+4tε
under data erasure is SWA∗ = vA − 8t −7ε; social welfare under data erasure and data
16t

portability is SWA∗∗ = vA − 4t . We have SWA∗ < SWAn < SWA∗∗ . Social welfare in market B always
equals vB − 4t .

49

A.5

Proof for Section 5.3□ Benchmark: no privacy management

The equilibrium prices in market A are the same asnn
in Section 4.1. Given δ, two firms’ profits are πA= t(1−r)(2+7δ)/16 and πA= t(1−r)(9−7δ)/16.
1
2

The equilibrium uniform prices in market B are β1n = β2n = t(9 + 7r)/16, implying the indifferentnnnnconsumer in market B is δ n = 1/2. We have πB= πB= t(9+7r)/32 and πA= πA= 11t(1−r)/32.
1
2
1
2The equilibrium profits of firms 1 and 2 are Πn1 = Πn2 = t(5 − r)/8. It is straightforward tonncalculate that consumer surplus is CSA= (1 − r)(vA − t) and CSB= vB − (13 + 7r)t/16, and social

welfare is SWAn = (1 − r)(16vA − 5t)/16 and SWBn = vB − t/4.
□ Equilibrium under privacy management
uniform prices areα1 [(1 − r)(1 − δ)x̄2 + (1 − r)δ x̃1 + rx̂]
α1 [(1 − r)δ x̃1 + rx̂]The profits of firms A1 and A2 from theirwhen α1 ≤ t,when α1 ≥ t and x̂ ≥ x̃1 .

α2 [(1 − r)δ(1 − x̄1 ) + (1 − r)(1 − δ)(1 − x̃2 ) + r(1 − x̂)]

when α2 ≤ t,
α2 [(1 − r)(1 − δ)(1 − x̃2 ) + r(1 − x̂)]

when α2 ≥ t and x̂ ≤ x̃2 .

The definitions of x̃i , x̄i , and x̂ are the same as in Section 4.2. Still, in the two cases of Ai ’s
profit function, only the first case can sustain in the equilibrium. Given δ, the equilibriumoutcomes in market A are−δ(r − 1)(ε(r − 2) + 2t) + ε(r − 1)r + 2rt + tε(r − 1)(δ(r − 2) + 2) + t(2δ(r − 1) + 3)
, α2∗ =
,
r
+
2
r+2


ε 2δ r2 − 3r + 2 − r2 + 2r − 4 + t(4δ(r − 1) − r + 4)
2δ(r − 1)(ε(r − 2) + 2t) − (r − 4)(εr + t)x̃∗1 =, x̃∗2 =,2(r + 2)t2(r + 2)tε(r − 1)(δ(r − 2) + 2) + t(2δ(r − 1) + r + 5)
(r − 1)(δε(r − 2) + 2δt − εr − t)x̄∗1 =, x̄∗2 =,2(r + 2)t2(r + 2)t
(2δ − 1)ε r2 − 3r + 2 + t(4δ(r − 1) − r + 4)x̂∗ =.
2(r + 2)tα1∗ =We find that x̄∗i ∈ (0, 1), x̂∗ ∈ (0, 1), x̃∗1 < 1, and x̃∗2 > 0 always hold when ε > t/2. The x̃∗1 > 0noεr 2 −2εr+4ε+rt−4t
holds when t/2 < ε < 3t/4 or 3t/4 < ε < t and 0 < δ < min 1, 2εr. The x̃∗2 < 1 holds
2 −6εr+4ε+4rt−4t

50

n
o
2
−4εr+3rtwhen t/2 < ε < 3t/4 or max 0, 2εr2 εr< δ < 1. Two firms’ equilibrium profits are−6εr+4ε+4rt−4t∗= α1∗ [(1 − r)(1 − δ)x̄∗2 + (1 − r)δ x̃∗1 + rx̂∗ ] + (1 − r)δ
πA
1

Z x̄∗1x̃∗
1p∗1 (x)dx,∗πA= α2∗ [(1 − r)δ(1 − x̄∗1 ) + (1 − r)(1 − δ)(1 − x̃∗2 ) + r(1 − x̂∗ )] + (1 − r)(1 − δ)2Z x̃∗2x̄∗
2p∗2 (x)dx.

Now we focus on market B. Firm Bi ’s privacy-insensitive consumers’ expected surplus inmarket A are
θ=0
E[CSB
]=
1

θ=0
E[CSB
]=
2

Z x̃∗10

Z x̄∗20(vA − α1∗ − tx − ε)dx +(vA − α1∗ − tx)dx +Z x̄∗1Z x̃∗2
x̄∗
2

x̃∗
1

(vA − p∗1 (x) − tx)dx +

Z 1
x̄∗
1

(vA − p∗2 (x) − t(1 − x))dx +(vA − α2∗ − t(1 − x))dx,

Z 1
x̃∗
2

(vA − α2∗ − t(1 − x) − ε)dx.

Firm Bi ’s privacy-sensitive consumers’ expected surplus in market A are
θ=c
θ=c
E[CSB
] = E[CSB
]=
1
2

Z x̂∗
0

(vA − α1∗ − tx − ε)dx +

Z 1
x̂∗

(vA − α2∗ − t(1 − x) − ε)dx.

The indifferent consumer δ in market B is determined by
θ=0
θ=c(1 − r)(vB − β1 − tδ + gE[CSB]) + r(vB − β1 − tδ + gE[CSB
])
1
1
θ=0
θ=c
=(1 − r)(vB − β2 − t(1 − δ) + gE[CSB]) + r(vB − β2 − t(1 − δ) + gE[CSB
]),
2
2
2

2

2

1 −β2 −t)+ε g(r−2)(r−1) +2εg(r−1) t. Two firms’ profit in market B are πB1 =which implies δ = (r+2)t(β2ε2 g(r−2)(r−1)2 +4εg(r−1)2 t−2(r+2)t2

β1 δ and πB2 = β2 (1 − δ). Firm 1 and firm 2 decide their uniform prices to maximize the total
profit Π1 = πB1 + πA1 and Π2 = πB2 + πA2 .
The equilibrium uniform prices in market B areε2 (r − 1) −16g r2 − 3r + 2 + r3 + 9r + 2 + 4ε(r − 1)t −8g(r − 1) + r2 + 2r − 1 + 4 r2 + 7r + 4 t2∗∗.
β1 = β2 =16(r + 2)t∗
The indifferent consumer is δ ∗ = 12 . The equilibrium profits of firm B1 and firm B2 are πB= β1∗ /2
1∗
and πB= β2∗ /2.
2

The equilibrium outcomes in market A are determined by replacing δ with 12 . We have(1−r)εεε∗x̃∗1 = 21 − 2t
, x̃∗2 = 12 + 2t
, x̄∗1 = 1 − (1−r)ε4t , and x̄2 =4t . The equilibrium prices in market A are

α1∗ = α2∗ = t − 12 (1 − r)ε. The equilibrium personalized prices are2t(1 − x) − 1 (1 − r)ε when x̃∗ < x < x̄∗
0
1
1
2
∗
∗
p1 (x) =p2 (x) =0
2tx − 1 (1 − r)εwhen x ≥ x̄∗1
2
51

when x ≤ x̄∗2
when x̄∗2 < x < x̃∗2 .

∗∗The equilibrium profits of firm A1 and firm A2 are πA
= πA
=
1
2

ε2 (r 3 −3r 2 −r+3)+4ε(r 2 −1)t+4(r+3)t2.
32t

The equilibrium profits of firm 1 and firm 2 areε2 (r − 1)2Π∗1 = Π∗2 =
−8g(r − 2) + r2 + r + 2 + 2ε(r − 1)t −8g(r − 1) + 2r2 + 5r + 1 + 4 r2 + 6r + 5 t2.
16(r + 2)t

Consumer surplus in market A is
1−r
∗
CSA
=

"Z ∗
x̄2

2

0

(vA − tx − α1∗ )dx +

"Z

x̃∗
1

Z x̃∗2x̄∗
2(vA − t(1 − x) − p∗2 (x))dx +Z x̄∗1Z 1

1−r+(vA − tx − α1∗ − ε)dx +(vA − tx − p∗1 (x))dx +
2
0
x̃∗
1
"Z
#
Z 1
1/2
∗
∗
+r
(vA − tx − α1 )dx +(vA − t(1 − x) − α2 )dx
0

x̃∗
2

#
(vA − t(1 − x) − α2∗ − ε)dx

Z 1
x̄∗
1

#
(vA − t(1 − x) − α2∗ )dx

1/2

2

=vA +

5t
ε (1 − r)− rε − .
4t4Consumer surplus in market B is
∗
CSB
=

Z 1/2
0

(vB − tx − β1∗ )dx +

Z 1
1/2

(vB − t(1 − x) − β2∗ )dxε(r − 1) −8g(r − 1) + r2 + 2r − 1
r2 + 8r + 6 t
ε2 (r − 1) −16g r2 − 3r + 2 + r3 + 9r + 2−−.
= vB −
16(r + 2)t4(r + 2)4(r + 2)Social welfare in market A is SWA∗ = vA +ε2 (r 3 −3r 2 −5r+7)+4εt(r 2 −4r−1)+4t2 (r−2).
16tSocial welfare

in market B is equal to vB − 4t . The comparisons of firms’ profits, consumer surplus, and social
welfare are easy to derive and are omitted here.

A.6

Proofs of Section 5.4

The equilibrium analysis in market A is the same as in Section 5.1 after replacing the privacy
management cost ε with the opt-in benefit b. In the equilibrium of market A, we have α1∗ =t(1/2 + δ) − δb − ω2 , α2∗ = t(3/2 − δ) − (1 − δ)b − ω2 , x̃∗1 = (t−b)(1−δ), x̃∗2 = 1 − δ(t−b), x̄∗1 = 5t−2b−2(t−b)δ+ω,tt
4t
and x̄∗2 = t−2(t−b)δ−ω. Two firms’ personalized prices are
4t




t(5/2 − δ) − (1 − δ)b + ω − 2tx if x̃∗ < x < x̄∗
0
1
1
2
∗
∗
p1 (x) =p2 (x) =0t(δ − 1/2) − δb + ω + 2tx
if x ≥ x̄∗1
2

52

if x ≤ x̄∗2
if x̄∗2 < x < x̃∗2 .

Two firms’ equilibrium profits in market A are∗πA=α1∗ [δ x̃∗1 + (1 − δ)x̄∗2 ] + δ
1

Z x̄∗1x̃∗
1p∗1 (x)dx − bδ(1 − x̃∗1 ),∗πA=α2∗ [(1 − δ)(1 − x̃∗2 ) + δ(1 − x̄∗1 )] + (1 − δ)2Z x̃∗2x̄∗
2p∗2 (x)dx − b(1 − δ)x̃∗2 .

To make such an equilibrium valid, we need 0 < x̃∗1 < 1, 0 < x̃∗2 < 1, 0 < x̄∗1 < 1, 0 < x̄∗2 < 1.
t−ωThese constraints are equivalent to t−2b+ω
2(t−b) < δ < 2(t−b) , which holds for any δ ∈ [0, 1] whenω ≤ 2b − t. In the following analysis, we assume ω ≤ 2b − t holds. In addition, we need to makesure no firm has the incentive to deviate from α1∗ and α2∗ , which requires
max{δ̂2 (b, ω), δ̃2 (b, ω), 0} ≤ δ ≤ min{δ̂1 (b, ω), δ̃1 (b, ω), 1}.

(13)

The expressions of δ̂i (b, ω) and δ̃i (b, ω) are the same as δ̂i (ε, ω) and δ̃i (ε, ω) in the proof of Section
5.1 after replacing ε with b. Condition (13) holds for any δ ∈ [0, 1] if and only if b̂(ω, t) ≤ b ≤ t.
In market B, the expected surpluses of firm Bi ’s targeted consumers are
Z x̃∗1
E[CSB1 ]=0=
E[CSB2 ]=(vA − α1∗ − tx)dx +Z x̄∗1x̃∗
1(vA − p∗1 (x) − tx + b)dx +

Z 1
x̄∗
1

(vA − α2∗ − t(1 − x) + b)dx

(1 − δ)2 ε2,vA − (1 + δ − δ 2 )t + (1 − δ)(2δ − 1)ε +t
Z x̄∗2Z x̃∗2Z 1∗∗(vA − α1 − tx + b)dx +(vA − p2 (x) − t(1 − x) + b)dx +(vA − α2∗ − t(1 − x))dx
x̄∗
2

0

= vA − (1 + δ − δ 2 )t − δ(2δ − 1)ε +x̃∗2

δ 2 ε2.
t

The indifferent consumer δ is determined by vB −tδ −β1 +gE[CSB1 ] = vB −t(1−δ)−β2 +gE[CSB2 ],2

1 +2β2 +gω+2t)
which leads to δ = 2b g−bg(ω+2t)+t(−2β. Two firms’ profits in market B are πB1 = β1 δ
4b2 g−2bg(ω+2t)+2t(gω+2t)

and πB2 = β2 (1 − δ). Firm 1 and firm 2 decide their uniform prices to maximize total profit∗∗
Π1 = π A+ πB1 and Π2 = πA
+ πB2 . The equilibrium uniform prices in market B are12β1∗ = β2∗ =16b2 g − b2 − 8bgω − 16bgt + 18bt + 8gωt + ω 2 − 10ωt + 8t2
.
16t

∗
∗
The indifferent consumer in market B is δ ∗ = 12 . We have πB= πB= β1∗ /2.
1
2

The equilibrium in market A is straightforward to get by replacing δ with 1/2. The profits of
2

2

∗
∗
firm A1 and firm A2 are πA= πA= 12t −5b −12tb+ω(6b+3ω−4t). The equilibrium profits of firm 1
32t
1
2
2

2

2

+10t
and firm 2 are Π∗1 = Π∗2 = b (8g−3)+b(3−4g)ω+b(3−8g)t+(4g−7)ωt+2ω
.
16t

53

Consumer surplus in market A is
1
∗
CSA
=
2

"Z ∗
x̃1

1
+
2

0

(vA − tx − α1∗ )dx +

"Z

x̄∗
2

0
2

=vA −

Z x̄∗1x̃∗
1(vA + ω − tx − p∗1 (x) + b)dx +(vA − tx − α1∗ + b)dx +Z x̃∗2
x̄∗
2

Z 1
x̄∗
1

#
(vA − t(1 − x) − α2∗ + b)dx

(vA + ω − t(1 − x) − p∗2 (x) + b)dx +

Z 1
x̃∗
2

#
(vA − t(1 − x) − α2∗ )dx

25t − b − 4tb ω
+ .
4t
2

Consumer surplus in market B is∗
CSB
(g = 0) =

Z 1/2
0

∗
CSB
(g = 1) =Z 1/2
0(vB − tx − β1∗ )dx +(vB − tx − β1∗ )dx +

Z 1
1/2

Z 1
1/2

(vB − t(1 − x) − β2∗ )dx = vB −

ω 2 − 10tω
12t2 + 18tb − b2
−
,
16t
16t

(vB − t(1 − x) − β2∗ )dx = vB −15b2 + 2bt + 12t2
ω 2 − ω(8b + 2t)
−
.
16t
16t

∗
∗
Under our assumption of ω ≤ 2b − t and b < t, CSB(g = 1) > CSB(g = 0) holds. Social welfare in22

market A is SWA∗ = vA − 8t −4tb+b+ ω(3ω+4t+6b). Social welfare in market B always equals vB − 4t .
16t
16t
When b = ε, the comparison to Section 5.1 is straightforward and is omitted here. When222b(ω+5t)+ 2ω −3ωt+10t. Since b ≥ b̂(ω, t) ≥ b̂(ω = 0, t) ≈ 0.67t andg = 1, we have Π∗1 = Π∗2 = 5b
16t −
16t
16t
∗
0.67t > ω+5t
10 , the optimal b = t. The following comparisons are straightforward. When g = 0, we222b(3ω+3t)
have Π∗1 = Π∗2 = − 3b
+ 2ω −7ωt+10t, implying the optimal b∗ = t+ω
16t +
16t
16t
2 . In this case, social

welfare in market A reduces if and only if
i
i
p
p
1 h
1 h
3(4ε − 3t) − 2 113ε2 + t2 − 98tε < ω <3(4ε − 3t) + 2 113ε2 + t2 − 98tε .
11
11

A.7

Proof for Section 6.1

We now prove firm Bi has no incentive to offer the choice of data erasure to consumers voluntarily. The game proceeds as in the main model, except that the two firms in market B
simultaneously decide whether to offer the choice before competition in market A begins. Let
us first characterize the market equilibrium when only firm B1 offers privacy management
to consumers. Firm A2 ’s optimal α2∗ = 2t , leading to x̄∗1 = 43 . Given x̃1 , firm A1 ’s optimalx̃1
α1 is α1 = t(1−δ)+2tδ
. Since consumers have rational expectations, based on (4), we have2(1−δ)∗
α1∗ = t(2+δ)−2δε
and x̃∗1 = (1−δ)(t−ε)
2(2−δ)t(2−δ) . Using α1 , firm A1 could successfully poach A2 ’s targeted54consumers on [0, x̄∗2 ], in which x̄∗2 = 2δε−3δt+2t
8t−4δt . Two firms’ personalized prices arep∗1 (x) =α∗ + t(1 − 2x)

when


0

when x ≥ 3/4

2

x̃∗1 < x < 3/4p∗2 (x) =0α1∗ + t(2x − 1)

when x ≤ x̄∗2
when x̄∗2 < x ≤ 1.

Two firms’ equilibrium profits in market A are∗πA= α1∗ [δ x̃∗1 + (1 − δ)x̄∗2 ] + δ
1

Z x̄∗1x̃∗
1p∗1 (x)dx,∗πA= δα2∗ /4 + (1 − δ)
2

Z 1
x̄∗
2

p∗2 (x)dx.

To make such an equilibrium valid, we need 0 < x̃∗1 < 1 and x̄∗2 > 0, which always hold under
our Assumption 1.
In market B, firm 1 and firm 2 decide their uniform prices to maximize total profits Π1 =∗∗, respectively. We can solve the optimal β1 and β2 and determineand Π2 = β2 (1−δ)+πA
β1 δ +πA2
1

all firms’ profits by numerical methods. Firm 1 and firm 2 simultaneously determine whether
to offer privacy management to consumers. The analysis of Nash equilibrium is straightforward
and is omitted here.

55

Online Appendix for “The Effects of Personal DataManagement on Competition and Welfare”

March 4, 2023

This online appendix provides more extensions to show the robustness of our insights inthe main model. It also contains omitted proofs of the analysis in the main file. Section
1 presents an extension where some consumers purchase nothing in market B. Section 2
extends the main model by assuming that consumers’ preferences in market A do not follow
the uniform distribution. In Section 3, we discuss two other possible settings for consumers’
privacy management, including ex ante and sequential privacy management. Section 4 andSection 5 show more detailed discussions on equilibria of production personalization (Section
5.1 of the main file) and data portability (Section 5.2 of the main file).

1

Some consumers do not purchase in market B

In this extension, we introduce some consumers who buy nothing in the data collection market. Specifically, there are three types of consumers in market B: purchasing from firm B1 ,purchasing from firm B2 , and consuming nothing in market B (call them free consumers).
Without loss of generality, we assume that the sizes of consumers who buy from B1 and B2 areδ and 1 − δ, respectively, and the size of free consumers is f ∈ [0, 1]. In addition, we assume
that consumers in market B do not anticipate the expected outcome in market A (i.e., g = 0) to
simplify the analysis. The equilibrium of g = 1 can be analyzed similarly.

1.1

Benchmark: no privacy management

Firm A1 can successfully poach the rival’s targeted consumers on [0, x̄2 ], in which x̄2 = (t −α1 )/(2t). The indifferent consumer in the segment of free consumers is x̂ = (t + α2 − α1 )/(2t).
Firm A1 ’s profit from its uniform price is α1 [(1 − δ)x̄2 + f x̂]. Firm A2 can successfully poach therival’s targeted consumers on [x̄1 , 1], in which x̄1 = (t + α2 )/(2t). Similarly, firm A2 ’s profit from1

its uniform price is α2 [δ(1 − x̄1 ) + f (1 − x̂)]. The equilibrium uniform prices are
α1n =t(f + δ)(2(1 − δ) + 3f )t(f + 1 − δ)(2δ + 3f ), α2n =.
4δ(1 − δ) + f (4 + 3f )4δ(1 − δ) + f (4 + 3f )In the equilibrium of market A,x̂n =4δ(1 − δ) + (5 − 2δ)f + 3f 2
6δ(1 − δ) + (6f + 7 − δ)f2δ(1 − δ) + (2 − δ)f, x̄n1 =, x̄n2 =.
2(4δ(1 − δ) + f (4 + 3f ))2(4δ(1 − δ) + f (4 + 3f ))2(4δ(1 − δ) + f (4 + 3f ))The profits of firms in market A are
Z x̄n1nπA 1 = δ(α2n + t(1 − 2x))dx + α1n [(1 − δ)x̄n2 + f x̂n ],0nπA= (1 − δ)
2

Z 1
x̄n
2

(α1n + t(2x − 1))dx + α2n [δ(1 − x̄n1 ) + f (1 − x̂n )].

The indifference consumer in market B is δ = (t + β2 − β1 )/(2t). Firms maximize their totalnnprofits from their uniform prices, β1 and β2 : Π1 = β1 δ + πA
, Π2 = β2 (1 − δ) + πA
. The equilibrium
12

prices in market B is
β1n = β2n =
8f 3 + 48f 2 + 45f + 9 t.
16(f + 1)2 (3f + 1)

(8f 3 +48f 2 +45f +9)t
n
n
=
=
π
The indifferent consumer in market B is δ n = 1/2. We have πBB
32(f +1)2 (3f +1)
2
1
3
2
16f
+40f
+36f
+11
t
(
)
n
n
. The equilibrium profits of firms 1 and 2 are Πn1 = Πn2 =and πA= πA=
32(f +1)2
1
2
4
3
2(24f +72f +98f +57f +10)t.
16(f +1)2 (3f +1)

1.2

Equilibrium under privacy management

Same with the main model, consumers on [0, x̃1 ] in firm A1 ’s turf and those on [x̃2 , 1] in firm A2 ’s
turf delete their data. Firm A1 ’s profit from its uniform price is α1 [δ x̃1 + (1 − δ)x̄2 + f x̂]. Firm
A2 ’s profit from its uniform price is α2 [(1 − δ)(1 − x̃2 ) + δ(1 − x̄1 ) + f (1 − x̂)]. The equilibrium
uniform prices areα1∗ =δε(f + 2) + (f + 1)t(−2δ + 3f + 3) − 2ε(f + 1)−δε(f + 2) + 2δ(f + 1)t − εf + 3f 2 t + 4f t + t, α2∗ =.
2
3f + 5f + 2
3f 2 + 5f + 2

In the equilibrium of market A,(2δ − 1)ε(f + 2) − (f + 1)t(4δ − 3f − 4),
2 (3f 2 + 5f + 2) t
δεf + 2δε − 2δf t − 2δt − 2εf − 2ε + 6f 2 t + 11f t + 5t
x̄∗1 =,
2(f + 1)(3f + 2)t−δεf − 2δε + 2δf t + 2δt − εf − f t − t,x̄∗2 = −2(f + 1)(3f + 2)t
ε 2δ(f + 2) − 3f 2 − 6f − 4 − (f + 1)t(4δ − 3f − 4)x̃∗1 =,
2 (3f 2 + 5f + 2) t2δε(f + 2) − 4δ(f + 1)t + (3f + 4)(εf + f t + t)x̃∗2 =.
2 (3f 2 + 5f + 2) t

x̂∗ =

2

The profits of firms in market A are
∗
πA
=δ
1

Z x̄∗1x̃∗
1(α2∗ + t(1 − 2x))dx + α1∗ [δ x̃∗1 + (1 − δ)x̄∗2 + f x̂∗ ],∗πA= (1 − δ)2Z x̃∗2x̄∗
2(α1∗ + t(2x − 1))dx + α2∗ [(1 − δ)(1 − x̃∗2 ) + δ(1 − x̄∗1 ) + f (1 − x̂∗ )].

The indifference consumer in market B is δ = (β2 − β1 + t)/(2t). The firms maximize the∗∗following profits from their uniform prices, β1 and β2 : Π1 = β1 δ + πA1, Π2 = β2 (1 − δ) + πA2. The

equilibrium prices and profits are:

12f 3 + 24f 2 + 15f + 2 + 4ε −2f 3 − 2f 2 + f + 1 t + 4(f + 1)2 (7f + 4)t2,
16(f + 1)2 (3f + 2)t
ε2 4f 2 + 8f + 3 − 4ε 2f 2 + 3f + 1 t + 4(f + 1)2 (4f + 3)t2
Π∗A1 = Π∗A2 =,32(f + 1)2 t
−ε2 12f 3 + 24f 2 + 15f + 2 + 4ε −2f 3 − 2f 2 + f + 1 t + 4(f + 1)2 (7f + 4)t2∗∗,
ΠB1 = ΠB2 =
32(f + 1)2 (3f + 2)t
ε2 4f 2 + 5f + 2 − 2ε 8f 3 + 15f 2 + 8f + 1 t + 4(f + 1)2 6f 2 + 12f + 5 t2Π∗1 = Π∗2 =.
16(f + 1)2 (3f + 2)t−εβ1∗ = β2∗ =2In the equilibrium, other endogenous variables arex̃∗1 =

1.3

t−ε
t−ε
ε
ε
ε
, x̃∗2 = 1 −, x̄∗1 = 1 −, x̄∗2 =, α1∗ = α2∗ = t −.
2t
2t
4f t + 4t
4f t + 4t
2(f + 1)Equilibrium comparison

It is straightforward to conclude that αi∗ > αin and βi∗ < βin hold, implying price competition
mitigates in market A and intensifies in market B under privacy management. We can also
check that all firms get lower profits under privacy management.
Consumer surplus in market A without privacy management is
"Z
#
Z 1
3/4
1
n
CSA
=
(vA − tx − pn1 (x))dx +(vA − t(1 − x) − α2n )dx
2 0
3/4
"Z
#
Z 1
1/4
1
n
n
+
(vA − tx − α1 )dx +(vA − t(1 − x) − p2 (x))dx
2 0
1/4
"Z
#
Z 1
1/2
n
n
+f
(vA − tx − α1 )dx +(vA − t(1 − x) − α2 )dx
0

1/2



5f
=(1 + f )vA − 1 +t.
43Consumer surplus in market A under privacy management is#"Z ∗Z x̄∗1
Z 1
x̃1
1
∗
∗
∗
∗
(vA − t(1 − x) − α2 )dx
CSA =(vA − tx − α1 − ε)dx +(vA − tx − p1 (x))dx +
2 0
x̄∗
x̃∗
1
1
"Z ∗
#
Z x̃∗2
Z 1
x̄2
1
∗
∗
∗
+
(vA − tx − α1 )dx +(vA − t(1 − x) − p2 (x))dx +(vA − t(1 − x) − α2 − ε)dx
2 0
x̄∗
x̃∗
2
2
"Z
#
Z 1
1/2
+f
(vA − tx − α1∗ )dx +(vA − t(1 − x) − α2∗ )dx
0

1/2
2

=(1 + f )vA −25(1 + f )t − ε
.
4t

∗
n
It is straightforward to check that CSA< CSA
holds.

Consumer surplus in market B without and with privacy management are
n
CSB
=

Z 1/2

(vB − tx − β1n )dx +

0
∗
CSB
=

Z 1/2
0

Z 1

(vB − t(1 − x) − β2n )dx = vB −1/2

(vB − tx − β1∗ )dx +Z 1


20f 3 + 76f 2 + 65f + 13 t,
16(f + 1)2 (3f + 1)

(vB − t(1 − x) − β2∗ )dx

1/2



ε2 12f 3 + 24f 2 + 15f + 2 + 4ε 2f 3 + 2f 2 − f − 1 t − 8(f + 1)2 (5f + 3)t2.
= vB +
16(f + 1)2 (3f + 2)t
n
n
∗
∗
n
∗
holds if and
+ CSB
< CSA
+ CSB
always holds. Moreover, CSA> CSB
We can check that CSB

only if t̂ ≤ ε ≤ t, in whichs2352f 7 t2 + 10096f 6 t2 + 18200f 5 t2 + 17824f 4 t2 + 10295f 3 t2 + 3567f 2 t2 + 708f t2 + 64t2 2 2f 3 t + 2f 2 t − f t − t
t̂ =
−
.
2
24f 3 + 56f 2 + 43f + 10
(3f + 1) (24f 3 + 56f 2 + 43f + 10)
Social welfare in market A without privacy management is SWAn = (f +1)vA −(4f 3 +16f 2 +16f +5)t16(f +1)2.

Social welfare in market A under privacy management is
ε2 8f 2 + 16f + 7 − 4ε 2f 2 + 3f + 1 t − 4(f + 1)2 (f + 2)t2∗
SWA = (1 + f )vA +.
16(f + 1)2 t
We have SWA∗ < SWAn . Social welfare in market B always equals vB − 4t .

2

Non-uniform distribution of consumers in market A

In this section, a consumer’s location x in market A follows the probability density function:x k(1−k) , x ∈ [0, k]1h(x, k) =x ∈ [k, 1 − k]
1−k ,



 1−x , x ∈ [1 − k, 1],k(1−k)In which k ∈ [0, 1/2] is a parameter that uniquely characterizes the distribution. Figure 1 shows
the shape of h(x, k), a general symmetric trapezoid. Two extreme cases are k = 1/2 (i.e., h(x, k)4

is a triangle) and k = 0 (i.e., h(x, k) is a rectangle corresponding to uniform distribution). Whenk increases, consumers are more concentrated around the center. Consumers in market B
follow a uniform distribution, as in the main model.

k = 120 < k < 12
k=0

1
2

0

1

Figure 1: Shape of h(x, k)

2.1

Benchmark: no privacy management

Suppose that firm B1 has won consumers on [0, δ] and firm B2 has won consumers on [δ, 1] inmarket B. Let us show how to solve firm Ai ’s optimal uniform price. Firm A2 can poach therival’s consumers on [0, x̄1 ] with uniform price, in which x̄1 = α2t2 + 21 > 12 . Its profit from theuniform price is




α2

Z 1

1−x
dx +
k(1
− k)

Z 1−k
1

1−x


dx δ,
α2
x̄1 k(1 − k)

Z 1−k
x̄1

!
1
dx δ,
1−k

x̄1 ∈ [k, 1 − k],x̄1 ∈ [1 − k, 1].

It turns out when k ≤ 1/3, the optimal α2n = t(1 − k)/2, implying x̄1 = (3 − k)/4 and firm A2 ’s
profit from uniform price is tδ(1 − k)/8; when k ≥ 1/3, the optimal α2n = t/3, implying x̄1 = 2/3
and firm A2 ’s profit from uniform price is tδ/(54k(1 − k)).
Firm A1 ’s optimal uniform price is symmetric. When k ≤ 1/3, the optimal α1n = t(1 − k)/2,
implying x̄2 = (1+k)/4 and firm A1 ’s profit from uniform price is t(1−δ)(1−k)/8; when k ≥ 1/3, theoptimal α1n = t/3, implying x̄2 = 1/3 and firm A2 ’s profit from uniform price is t(1−δ)/(54k(1−k)).
Therefore, when k ≤ 1/3, firm A2 ’s profit is
!
Z 1
1
1−x
n
dx +
(α1 + t(2x − 1))dx (1 − δ)
1+k
1−kk(1 − k)
1−k
4
!
Z 1
Z 1−k
1−x
1
n
+ α2
dx +
dx δ
3−k
1−k
1−k k(1 − k)4t(1 − δ) 31k 2 − 54k + 27tδ(1 − k)=+.
48(1 − k)
8

n
πA
=
2

Z 1−k

(α1n + t(2x − 1))5When k ≥ 1/3, firm A2 ’s profit is
Z k

n
=
πA
2

1
3

+ α2n
=

(α1n + t(2x − 1))
Z 1
2
3

x
dx +
k(1 − k)!

Z 1−k(α1n + t(2x − 1))

k

1
dx +
1−k

Z 1

(α1n + t(2x − 1))

1−k

!
1−x
dx (1 − δ)k(1 − k)1−xdx δk(1 − k)t(1 − δ) −27k 2 + 27k + 1tδ
+.
81(1 − k)k
54k(1 − k)Similarly, firm A1 ’s profit istδ 31k 2 − 54k + 27t(1 − δ)(1 − k)


+
,

48(1
−
k)
8
n

πA
=
2
1
tδ −27k + 27k + 1t(1 − δ)+,81(1 − k)k
54k(1 − k)when k ≤ 13 ,when k ≥ 31 .

It is straightforward to check that E[CSB1 ] = E[CSB2 ] holds for any k because the equilibrium
in market A is symmetric. The indifferent consumer in market B is δ = (t + β2 − β1 )/(2t) as innthe standard Hotelling model. Firm 1 chooses β1 to maximize Πn1 = β1 δ +πA, and firm 2 chooses
1nβ2 to maximize Πn2 = β2 (1 − δ) + πA.
2t(−25k2 −6k+27), implying δ n = 12 . The equilibrium
When k ≤ 1/3, the equilibrium β1n = β2n =48(1−k)t(−25k2 −6k+27)t(37k2 −66k+33)nnnn
profits are: πB
= πB
=
, πA
= πA
=
, and Πn1 = Πn2 = t(5−k). When96(1−k)96(1−k)
8
1
2
1
2
2
t
−108k
+108k+1()
k ≥ 1/3, the equilibrium β1n = β2n =, implying δ n = 21 . The equilibrium profits
162(1−k)k222t(−108k +108k+1)t(−54k +54k+5)+27k+1)
n
n
n
n
=
= πB
=
are: πB= πA, πA, and Πn1 = Πn2 = t(−27k.
324(1−k)k324(1−k)k54(1−k)k
2
1
2
1

2.2

Privacy management

Suppose that firm B1 has won consumers on [0, δ] and firm B2 has won consumers on [δ, 1] inmarket B. We assume the range of δ satisfies l ≤ δ ≤ 1 − l, in which 0 < l < 1/2. The parameterl indicates firm Bi ’s segment of loyal consumers.1 When privacy management is available, themarginal consumers in market A who choose to erase their data are the same as in the mainmodel:x̃1 ≡ε1 αa − α1aε
1 α2a − α1a+−and x̃2 ≡ + 2
+ .
2
2t
2t
2
2t
2t

We now analyze the equilibrium uniform prices in market A after consumers engage in privacy management. Suppose the two firms’ uniform prices are α1 and α2 . Firm A1 ’s opt-out−α1consumers purchase from firm A1 if and only if x < x̂ ≡ 12 + α22t
. Consumers’ price antici-

pations should be correct in equilibrium due to rational expectations, implying that inequalityx̃1 < x̂ < x̃2 should hold in equilibrium.
1 We assume that firm B

i cannot charge a monopoly price for these consumers because they are still considering

the other firm’s poaching offer.

6

We formulate the objectives of firms A1 and A2 to derive their optimal uniform prices. There
are many cases to discuss based on the relative locations between x̄1 and 1 − k, x̃1 and k, x̄2
and k, and x̃2 and 1 − k. In the following analysis, we focus on symmetric equilibrium and thescenario of x̄1 < 1 − k, x̄2 > k, x̃1 > k, and x̃2 < 1 − k to simplify the analysis. Firm A1 ’s profit
from its uniform price α1 is
!
!#
"
Z x̄2
Z k
Z x̃1
Z k
1
x
1
x
α1 (1 − δ)dx +dx + δ
dx +
dx
.
1−k
1−k
k
0 k(1 − k)k
0 k(1 − k)Similarly, firm A2 ’s profit from its uniform price α2 is
!
!#
" Z
Z 1
Z 1−k
Z 1
1−k
1−x
1
1−x
1
α2 δ
dx +
dx + (1 − δ)
dx +
dx
.
1−k
1−k
1−k k(1 − k)x̃2
1−k k(1 − k)x̄1
The equilibrium uniform prices areα1∗ =1(−2δ(ε + (k − 1)t) − kt + t),21α2∗ = (δ − 1)ε + (2δ − 3)(k − 1)t.
2

The equilibrium cutoffs are
ε
 ε k
ε
 k
x̃∗1 = δ+ k − 1 − − + 1, x̃∗2 = δ+ k − 1 − + 1,
t
t
2
t
2
2δ(ε + (k − 1)t) + (k + 1)t2(δ − 1)ε + t(2δ(k − 1) − 3k + 5)∗∗
, x̄2 =
.
x̄1 =
4t
4t
√
Under the assumption of ε ≥ 3t/2, the requirements x̄∗1 < 1 − k, x̄∗2 > k, x̃∗1 > k, and x̃∗2 < 1 − kholding for any δ ∈ [l, 1 − l] are equivalent to
0≤k≤

t−ε
kt
1
and
<l< .
2t
2(1 − k)t − 2ε2(1)

Firms’ equilibrium profits in market A are!
!#"
Z x̄∗2Z k
Z x̃∗1
Z k
x
x
1
1
∗
∗
dx +
dx + δ
dx +dx
πA1 =α1 (1 − δ)1−k1−k0 k(1 − k)
0 k(1 − k)kk
Z x̄∗11dx,+δ(α2∗ + t(1 − 2x))
∗
1
−
k
x̃
" 1Z
!
!#
Z 1
Z 1−k
Z 1
1−k
1
1
−
x
1
1
−
x
∗
πA
=α2∗ δ
dx +dx + (1 − δ)
dx +
dx
2
1−k
1−k
x̄∗
1−k k(1 − k)x̃∗
1−k k(1 − k)12
Z x̃∗2
1+ (1 − δ)(α1∗ + t(2x − 1))
dx.
1−k
x̄∗
2
We now analyze the equilibrium in market B. It is straightforward to check thatZ k
Z x̃∗1Z x̄∗1
1
1x
E[CSB1 ] =dx +(vA − tx − α1∗ − ε)dx +(vA − tx − α2∗ − t(1 − 2x))dx(vA − tx − α1∗ − ε)
∗
k(1
−
k)
1
−
k
1
−
k
k
x̃1
0
Z 1−k
Z 1
1
1−x
+
(vA − t(1 − x) − α2∗ )dx +(vA − t(1 − x) − α2∗ )dx,
1−kk(1 − k)
x̄∗
1−k
1
Z k
Z x̄∗2Z x̃∗21
1x∗∗(vA − tx − α1 )dx +(vA − tx − α1 )dx +(vA − t(1 − x) − α1∗ − t(2x − 1))dx
E[CSB2 ] =k(1 − k)
1−k
1−k
k
x̄∗
0
2
Z 1−k
Z 1
1
1−x
∗
+
(vA − t(1 − x) − α2 − ε)dx +(vA − t(1 − x) − α2∗ − ε)
dx.
∗
1−k
k(1 − k)x̃2
1−k7

We have E[CSB1 ] − E[CSB2 ] = (2δ−1)ε((1−k)t−ε). The indifferent consumer δ is determined by(1−k)tvB − β1 − tδ + gE[CSB1 ] = vB − β2 − t(1 − δ) + gE[CSB2 ], g ∈ {0, 1},
which leads toδ=(k − 1)t(β1 − β2 − t) + ε2 g + εg(k − 1)t.
2 (ε2 g + εg(k − 1)t − (k − 1)t2 )∗Firm 1 chooses β1 to maximize Π1 = β1 δ + πA
and firm 2 chooses β2 to maximize Π2 = β2 (1 −1∗δ) + πA. The equilibrium prices in market B are2

ε2 (1 − 16g) − 2ε(8g − 1)(k − 1)t + 8β1∗ = β2∗ =k 2 − 1 t2

16(k − 1)t,

implying δ ∗ = 1/2. It is straightforward to check that β1∗ (g = 1) < β1∗ (g = 0) holds.
In the equilibrium, we haveεα1∗ = α2∗ = (1 − k)t − ,2x̃∗1 =

1
ε
− ,
2 2t

x̃∗2 =

1
ε
+ ,
2 2t

x̄∗1 = 1 −



k
ε
+
2 4t



x̄∗2 =

,

k
ε
+ .
2 4t

The equilibrium profits are
ε2 (1 − 16g) − 2ε(8g − 1)(k − 1)t + 8
∗
∗
πB
= πB
=
1
2
32(k − 1)t

∗
∗
πA
= πA
=
1
2

1
32




k 2 − 1 t2

,


3ε2
− 4ε − 12(k − 1)t ,t − ktε2 (8g + 1) + ε(8g + 1)(k − 1)t + 2
Π∗1 = Π∗2 = −k 2 − 6k + 5 t2

16(k − 1)t.

It is easy to check that αi∗ > αin and βi∗ (g = 1) < βin always hold, and βi∗ (g = 0) < βin holds if and
q

√ 
√
√ 
√

3 − 2 + 12 3 28 − 16 3 and 3t/2 < ε < t − 1 + 2 3 3 kt.
only if 0 ≤ k < 32
Consumer surplus and social welfare in the two markets are(15 − 24k + 8k 2 )t2 − 3ε2,12(1 − k)t 2
1 ε (16g − 1)∗
CSB =vB ++ 2ε(8g − 1) − 4(2k + 3)t ,16(k − 1)t
k 2 + 6k − 6 t7ε2ε∗SWA =vA +− −,16t − 16kt 4
12(k − 1)t
SWB∗ =vB − .
4
∗
CSA
=vA −

8

When (1) holds, consumer surpluses and social welfare under no privacy management are
Z k
Z x̄n1x
1n(vA − tx − α2n − t(1 − 2x))
CSA
=
dx +
(vA − tx − α2n − t(1 − 2x))
dx
k(1
−
k)
1
−
k
0
k
Z 1−k
Z 1
1
1−x
n
(vA − t(1 − x) − α2 )(vA − t(1 − x) − α2n )
+
dx +
dx,
n
1
−
k
k(1
− k)
x̄1
1−k
2−k
=vA −
t,
2

Z 1/2
Z 1
25k 2 + 18k − 39 t
n
n
n
CSB =
(vB − tx − β1 )dx +(vB − t(1 − x) − β2 )dx = vB −,48(k − 1)0
1/2
15 − 6k − 13k 2 t
SWAn =vA −,48(1 − k)t
SWBn =vB − .
4
n
∗
n
∗
always hold, and(g = 1) > CSB
and CSB< CSA
It is straightforward to check that CSA
q
√
√
√


n
∗
holds if and only if 0 ≤ k < 23
(g = 0) > CSB
CSB
3 − 2 + 12 3 28 − 16 3 and 3t/2 < ε <

√ 
∗
∗
n
n
n
n
∗
∗
,
+ CSB
(g = 0) < CSA
t − 1 + 2 3 3 kt. Moreover, CSA
+ CSB
+ CSB
(g = 1) > CSA
+ CSB
, CSA

SWA∗ < SWAn and SWB∗ = SWBn always hold.

3

Privacy management settings

In this section, we compare four models of privacy management. Specifically, the four models
are:(i) No privacy management: consumers opt in for data collection and application by default
and have no rights to manage their data, as in the main model;(ii) Ex post privacy management: consumers opt in for data collection by default and canerase data before data application, as in the main model;(iii) Ex ante privacy management: consumers decide whether to opt in for data collection andapplication before data collection begins, and have no rights to manage their data after
that; if they opt in, firms can collect and apply their data; if they choose opt-out, firms
cannot collect or apply their data;(iv) Sequential privacy management: consumers decide whether to opt in for data collection
before data collection begins; if they choose opt-out, firms cannot collect or apply their
data; if they opt in, they can decide whether to erase data before data application.
We assume all consumers are forward looking. The models “no privacy management” and“ex post privacy management” have been analyzed in the main model. The following analysis
examines two other models.
9

3.1

Ex ante privacy management

We start with the following observation: firm Bi ’s consumers who opt in (opt out) for datainoutcollection get the same expected consumer surplus E[CSB] (E[CSB]) in market A because theiiconsumer’s location in market A, a random variable, follows the same distribution. Notice thatfirm Bi ’s consumers get utility vB − txi − βi (xi = 1 − xj ) in market B, independent of opt-in and
opt-out. Firm Bi ’s consumers choose opt-in ex ante if and only if
in
out
E[CSB
] > E[CSB
].
i
i

(2)

The expected surpluses of firm B1 ’s consumers in market A are:
out
E[CSB
]=
1
in
E[CSB
]=
1

Z x̂∗

(vA − tx − α1∗ )dx +0

Z x̄∗1Z 1x̂∗(vA − tx − p∗1 (x))dx +(vA − t(1 − x) − α2∗ )dx,

Z 1
x̄∗
1

0

(vA − t(1 − x) − α2∗ )dx =Z 1(vA − t(1 − x) − α2∗ )dx.

0

Then we have
out
in
E[CSB
] − E[CSB
]=
1
1

Z x̂∗

(α2∗ − α1∗ + t(1 − 2x))dx.

0

out
in
In the symmetric equilibrium, x̂∗ = 1/2 and α2∗ = α1∗ . Therefore, E[CSB] > E[CSB] always
1
1

holds, implying all of firm B1 ’s consumers choose to opt out. Similarly, all of firm B2 ’s consumers choose to opt out in the symmetric equilibrium. As a result, competitions in the twomarkets are classic Hotelling type. The equilibrium and welfare are the same as in our extension of “no cross-market data transfer” (Section 6.2 of the main file).

3.2

Sequential privacy management

Firm B1 ’s consumers who opt out of data collection get the same expected consumer surplus
out
E[CSB
] in market A, in which
1
out
E[CSB
]=
1

Z x̂∗

(vA − tx − α1∗ )dx +

Z 1

0

x̂∗

(vA − t(1 − x) − α2∗ )dx.

Firm B1 ’s consumers who opt in of data collection get the same expected consumer surplus
in
E[CSB] in market A, in which
1
in
E[CSB
]=
1

Z x̃∗1Z x̄∗1(vA − α1∗ − tx − ε)dx +

x̃∗
1

0

Z x̃∗1=Z 1(vA − α1∗ − tx − ε)dx +

x̃∗
1

0

(vA − p∗1 (x) − tx)dx +

Z 1
x̄∗
1

(vA − α2∗ − t(1 − x))dx(vA − α2∗ − t(1 − x))dx.

Since x̃∗1 < x̂∗1 holds in market A, we have
out
in
E[CSB
] − E[CSB]=11

Z x̃∗1

Z x̂∗
εdx +
x̃∗
1

0

10

(α2∗ − α1∗ + t(1 − 2x))dx.

outin
In the symmetric equilibrium, x̂∗ = 1/2 and α2∗ = α1∗ . Therefore, E[CSB] > E[CSB] always
1
1

holds, implying all of firm B1 ’s consumers choose to opt out. Similarly, all of firm B2 ’s consumers choose to opt out in the symmetric equilibrium. As a result, competitions in the twomarkets are classic Hotelling type. The equilibrium and welfare are the same as in our extension of “no cross-market data transfer” (Section 6.2 of the main file).
In summary, ex ante and sequential privacy management lead to the same equilibriumoutcome as Section 6.2 of the main file. The reason is that if consumers can choose whether
to opt in for data collection, they all choose opt-out because opt-in brings a lower expectedsurplus in the data application market. Therefore, compared to no privacy management andex post privacy management, Proposition 10 of the paper holds.

4

More equilibrium analysis for Section 5.1

In this section, we determine the condition under which firm A1 and firm A2 do not deviate
from α1∗ = t(1/2 + δ) − δε − ω2 and α2∗ = t(3/2 − δ) − (1 − δ)ε − ω2 given any δ.
Firm A1 has two possible deviations. The first one is increasing α1 above t − ω and does notpoach the rival’s targeted consumers. The second is reducing α1 to attract firm A2 ’s consumers
who have erased their data (i.e., consumers on [x̃∗2 , 1]).
Firm A1 ’s deviation one: α1d ≥ t − ω.

Notice that vA − t − α2∗ > 0 holds if and only if

ω > 5t − 2ε − 2vA − 2δ(t − ε), which holds based on our assumption of ω > 5t − 2ε − 2vA . So firm
A1 cannot become a local monopoly for its consumers on [0, x̃∗1 ]. Given its deviation price α1d ,the indifferent consumer x̂d = 12 +

d
α∗
2 −α1
2t .

Firm A1 maximizes its deviation profit from uniformd= δα1d x̂d subject to x̂d ≤ x̃∗1 , which is equivalent to α1d ≥ ε − δε + tδ + t−ω
price πA
2 . The optimal
1

deviation price is α1d =

t+α∗
t+α∗
t−ω
d
2
when 2 2 > ε − δε + tδ + t−ω2
2 and is α1 = ε − δε + tδ + 2 otherwise.

d∗
We find that firm A1 ’s deviation profit πA
is less than its equilibrium profit πA
if and only if
11

δ ≤ min{δ̂1 (ε, ω), 1}. In the shaded region of Figure 2, firm A1 ’s deviation is not profitable.
Firm A1 ’s deviation two: x̂d ≥ x̃∗2 . Firm A1 ’s deviation price should satisfy α1d ≤ tδ − ε − δε +t−ωd∗2 to make x̂ ≥ x̃2 hold. Firm A1 ’s deviation profit from the uniform price isdπA= δα1d x̃∗1 + (1 − δ)α1d (x̄d2 + x̂d − x̃∗2 ) = δα1d x̃∗1 + (1 − δ)α1d1
1 α1d + ω 1 α2∗ − α1d∗−+ +− x̃2 .
2
2t
2
2t

d
Firm A1 maximizes πAsubject to x̂d ≥ x̃∗2 . Firm A1 ’s optimal deviation price is α1d = tδ − ε − δε +
1
t−ω
t−ω
2 if and only if tδ − ε − δε + 2 ≤ (6tδ + 3t − 3ω − 6δε − 2ε)/8. Otherwise, its optimal deviationdprice is α1d = (6tδ + 3t − 3ω − 6δε − 2ε)/8. We find that firm A1 ’s deviation profit πA
is less than1∗
its equilibrium profit πA
if and only if δ ≤ min{δ̃1 (ε, ω), 1}. In the shaded region of Figure 3, firm
1

A1 ’s deviation is not profitable.
11

1.0

1.0

0.8

0.8

0.6

0.6

δ


δ1

δ
0.4

0.4


δ1

0.2

0.2

0.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

ϵ

0.6

0.8

1.0

0.8

1.0

ϵ

(a) t = 1 and ω = 13(b) t = 1 and ω = − 31

Figure 2: The region of A1 ’s unprofitable deviation when x̂d ≤ x̃∗1
1.0

1.0

0.8

0.8


δ1

0.6


δ1

0.6

δ

δ
0.4

0.4

0.2

0.2

0.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

ϵ

0.2

0.4

0.6
ϵ

(a) t = 1 and ω = 13(b) t = 1 and ω = − 31

Figure 3: The region of A1 ’s unprofitable deviation when x̂d ≥ x̃∗2
Therefore, firm A1 does not deviate if and only if δ ≤ min{δ̂1 (ε, ω), δ̃1 (ε, ω), 1}.
Similarly, Firm A2 has two possible deviations. The first one is increasing α2 above t − ω anddoes not poach the rival’s targeted consumers. The second is reducing α2 to attract firm A1 ’s

12consumers who have erased their data (i.e., consumers on [0, x̃∗1 ]).
Firm A2 ’s deviation one: α2d ≥ t − ω.

Notice that vA − t − α1∗ > 0 holds if and only if

ω > 3t − 2vA + 2δ(t − ε), which holds based on our assumption of ω > 5t − 2ε − 2vA . So firm
A2 cannot become a local monopoly for its consumers on [x̃∗2 , 1]. Given its deviation price α2d ,
αd −α∗
the indifferent consumer x̂d = 21 + 2 2t 1 and x̂d ≥ x̃∗2 is equivalent to α2d ≥ δε + 23 − δ t − 12 ω.
dFirm A2 maximizes its deviation profit from uniform price πA= (1 − δ)α2d (1 − x̂d ) subject to2d∗is less than its equilibrium profit πA
α2d ≥ δε + 23 − δ t − 12 ω. Firm A2 ’s deviation profit πA22

if and only if δ ≥ max{δ̂2 (ε, ω), 0}. In the shaded region of Figure 4, firm A2 ’s deviation is notprofitable.
1.0

1.0

0.8

0.8


δ2

0.6

0.6

δ

δ
0.4

0.4


δ2
0.2

0.2

0.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.8

1.0

Firm A2 ’s deviation two: x̂d ≤ x̃∗1 . Firm A2 ’s deviation price should satisfy α2d ≤

3
2 −δ

ϵ

0.6
ϵ

(a) t = 1 and ω = 13(b) t = 1 and ω = − 31

Figure 4: The region of A2 ’s unprofitable deviation when x̂d ≥ x̃∗2t−(2 − δ)ε − ω2 to make x̂d ≤ x̃∗1 hold. Firm A2 ’s deviation profit from the uniform price is
α2d + ω
d
d
∗
d
d
∗
d
d
∗
d 1
πA
=
(1
−
δ)α
(1
−
x̃
)
+
δα
(1
−
x̄
+
x̃
−
x̂
)
=
(1
−
δ)α
(1
−
x̃
)
+
δα
(
−
+ x̃∗1 − x̂d ).
2
2
2
1
1
2
2
2
2
2
2t

d
Firm A2 maximizes πAsubject to α2d ≤ 23 − δ t − (2 − δ)ε − ω2 . Firm A2 ’s optimal deviation price2is α2d = 23 − δ t − (2 − δ)ε − ω2 if and only if 32 − δ t − (2 − δ)ε − ω2 ≤ (9t − 6tδ + 6δε − 3ω − 8ε)/8.
Otherwise, it optimal deviation price is α2d = (9t − 6tδ + 6δε − 3ω − 8ε)/8. We find that firm A2 ’sd∗deviation profit πA
is less than its equilibrium profit πA
if and only if δ ≥ max{δ̃2 (ε, ω), 0}. In22

the shaded region of Figure 5, firm A2 ’s deviation is not profitable.
Therefore, firm A2 does not deviate if and only if δ ≥ max{δ̂2 (ε, ω), δ̃2 (ε, ω), 0} holds.
13

1.0

1.0

0.8

0.8

0.6

0.6


δ2

δ


δ2

δ

0.4

0.4

0.2

0.2

0.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

ϵ

0.6

0.8

1.0

0.8

1.0

ϵ

(a) t = 1 and ω = 13(b) t = 1 and ω = − 31

Figure 5: The region of A2 ’s unprofitable deviation when x̂d ≤ x̃∗1
1.0

1.0

0.8

0.8


δ2


δ2

0.6

0.6

0.4

0.4


δ1


δ1
0.2

0.2

0.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

(a) t = 1 and ω = 13

0.2

0.4

0.6

(b) t = 1 and ω = − 31

Figure 6: The region that two firms do not deviate

In summary, no firm has incentive to deviate if and only if
max{δ̂2 (ε, ω), δ̃2 (ε, ω), 0} ≤ δ ≤ min{δ̂1 (ε, ω), δ̃1 (ε, ω), 1}.

(3)

We find that this inequality holds for any δ ∈ [0, 1] when ε > t̂(ω, t). The cutoff t̂(ω, t) increases
14with ω and satisfies 0.35t < t̂(ω, t) < t. In the dark blue region of Figure 6, firm A1 and firm A2
do not deviate.

5

More analysis for data portability

In this section, we establish the condition under which firms in market A do not deviate fromt−ε+δε∗given δ and ε ∈ (0, t/2).
α1∗ = t−δε
2−δ and α2 = 1+δ

Firm A1 has two possible deviations. The first one is increasing α1 and does not poach therival’s targeted consumers. The second is reducing α1 to attract firm A2 ’s consumers who haveerased their data (i.e., consumers on [x̃∗2 , 1]). We discuss them one by one.
Firm A1 ’s deviation one Notice that vA − t − α2∗ > 0 always holds, implying firm A1 cannot
become a local monopoly for its consumers on [0, x̃∗1 ]. Given its deviation price α1d , the indifferent
consumer x̂d = 21 +

d
α∗
d
d d
2 −α1
2t . Firm A1 maximizes its deviation profit from uniform price πA1 = δα1 x̂2ε, implying x̂d = x̃∗1 , and thesubject to x̂d ≤ x̃∗1 . The optimal deviation price is α1d = 3t+3δε−3δ
2+δ−δ 2

deviation profit from the uniform price is
d
πA
=
1

3δ(1 − δ)(t − 2ε)(t + δε − δ 2 ε).
2t(1 + δ)(2 − δ)22∗∗dIts equilibrium profit from the uniform price is πA= (1−δ)(t−δε)2t(2−δ)2 . Then we find that πA1 > πA11noholds if and only if δ < min 1, δ1 (t, ε) . We find that 1 < δ1 (t, ε) holds if and only if ε ≥ 0.37t.

Firm A1 ’s deviation two To make x̂d ≥ x̃∗2 , firm A1 ’s deviation price should satisfy α1d ≤ −ε,
which cannot be profitable.
Firm A2 has two possible deviations. The first one is increasing α2 and does not poach therival’s targeted consumers. The second is reducing α2 to attract firm A1 ’s consumers who haveerased their data (i.e., consumers on [0, x̃∗1 ]). We discuss them one by one.
Firm A2 ’s deviation one

Notice that vA − t − α1∗ > 0 always holds, implying firm A2 cannot

become a local monopoly for its consumers on [x̃∗2 , 1]. Given its deviation price α2d , the indifferent
consumer x̂d = 21 +

∗
αd
2 −α1
2t

2

ε
and x̂d ≥ x̃∗2 is equivalent to α2d ≥ 3t+3δε−3δ. Firm A2 maximizes its
2+δ−δ 22εddeviation profit from uniform price πA= (1−δ)α2d (1− x̂d ) subject to α2d ≥ 3t+3δε−3δ. The optimal2+δ−δ 222εdeviation price is α2d = 3t+3δε−3δ, implying x̂d = x̃∗2 , and the deviation profit from the uniform2+δ−δ 2

price isdπA=
2

3δ(1 − δ)(t − 2ε)(t + δε − δ 2 ε).
2t(1 + δ)2 (2 − δ)2∗∗dIts equilibrium profit from the uniform price is πA= δ[t−(1−δ)ε]2t(1+δ)2 . Then πA2 > πA2 holds if and2noonly if δ > max 0, δ2 (t, ε) . We find that δ2 (t, ε) < 0 if and only if ε ≥ 0.37t.

Firm A2 ’s deviation two To make x̂d ≤ x̃∗1 , firm A2 ’s deviation price should satisfy α2d ≤ −ε,
which is unprofitable.
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In summary, firm A1 and firm A2 have incentive to deviate if and only if()()
max 0, δ2 (t, ε)≤ δ ≤ min 1, δ1 (t, ε) .

This condition holds for any δ ∈ [0, 1] when ε ≥ 0.37t.

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