Imperfect Competition, Information, and Privacy
∗
Rodrigo Montes†, Wilfried Sand-Zantman‡, and Tommaso Valletti§
March 20, 2015
Abstract
This paper studies how customer information and privacy affect the discriminatory behavior of a firm facing imperfect competition. We investigate
the effects of price discrimination on prices, profits and consumer surplus,
when one or more firms can use consumers’ private information to discriminate and consumers can pay a privacy cost to avoid it. The model predicts
that prices are larger than Hotelling prices, and they decrease with higher
privacy costs. This also implies that firms’ individual profits are decreasing
in the privacy cost and that consumer surplus is increasing. Finally, we show
that the owner of consumer data chooses to sell it exclusively to one firm
and that privacy can actually increase the price of information that can be
extracted by selling consumer information.
VERY PRELIMINARY
1
Introduction
This paper studies how customer information and privacy affect the discriminatory
behavior of a firm facing imperfect competition. In particular, we investigate the
effects of price discrimination on prices, profits and consumer surplus, when firms
∗
We thank Paul Belleflamme, Bruno Jullien, Marc Lebourges, Yassine Lefouili, as well as
participants at the 2015 Liege Workshop on Digital Economy. We gratefully acknowledge Orange
for financial support under the IDEI/Orange convention.
†
Toulouse School of Economics, [email protected]
‡
Toulouse School of Economics and ESSEC Business School, [email protected]
§
Imperial College London and University of Rome II, [email protected]
1
can use consumers’ private information to discriminate and consumers can potentially avoid being discriminated. This analysis allows us to draw some conclusions
on the value of customers’ information.
In the past few years, given the increasing amount of personal information collected by firms, price discrimination on the Internet has been documented many
times.1 The most (in)famous case was in 2000, when a customer complained that,
after erasing his cookies, he observed a lower price for a DVD on Amazon.com.2
More recently, the Wall Street Journal (2012a) published that the travel agency
Orbitz Worldwide was showing more expensive hotel offers to Mac users than to
PC users. A similar practice has been conducted by Staples.com: The Wall Street
Journal (2012b) uncovered the fact that this site was displaying different prices
once the location of the potential buyer had been identified. Other firms employing
similar methods were identified by this newspaper: Discover Financial Services,
Rosetta Stone and Home Depot. Another firm, Office Depot, even admitted to using “customers’ browsing history and geolocation to vary the offers and products it
displays to a visitor to its site” (Wall Street Journal 2012b).
Information need not be created by the firms that engage in price discrimination.
For example, behavioral targeting allows an advertising company (like Google’s
DoubleClick ) to plant a cookie in the user’s computer that can track the user’s
movements through any other website that uses DoubleClick. Then, a profile of the
user is created so that ads can be tailored specifically for her. Under this technology,
it is reasonable to expect that, due to network effects, big online advertising firms
are responsible for collecting data. In turn, this data can be sold to an online store
or given directly to another branch of the advertising company.
To study the role of information and privacy in markets, we use a model where
some firms can acquire information on consumers’ taste and then propose a set of
different personalized prices. We are interested in the situation where competition
(if any) occurs in two different markets. In the first market, individual information
cannot be extracted as in the standard Hotelling model. In the second market,
however, information is potentially available, i.e. this information can be bought
1
See Mikians et al. (2012, 2013) for a systematic data collection and Shaw and Vulkan (2012)
for an experimental approach.
2
https://en.wikipedia.org/wiki/Amazon.com_controversies#Differential_pricing
(Retrieved on 14/05/2014). See also Wall Street Journal (2012b)
2
from a data seller. One possible interpretation for this setting is that in the first
market, consumers are not very active online and do not leave any traces. On the
contrary, in the other market, consumers are very active and leave many traces of
their activity and need to engage ex-post in costly actions to erase those traces.
This means that if a firm does not know a consumer’s type after having bought
some information from the data seller, the firm cannot tell whether this consumer
is a less active consumer or a very active one who erased his browsing information.
The way very active consumers can protect their privacy is by paying a cost
to disappear from the firm’s database. We interpret this cost as the difficulty to
conceal one’s actions on-line. For example, as seen above, this cost could represent
how hard is to erase the cookies after shopping on the Internet or visiting a website.
In turn, regulators could make the cost smaller by imposing a full disclosure policy
on the use of cookies.3 This cost could be made larger if the firms were allowed to
trade customers’ data, so that they would need to request potentially many websites
to erase their data.
As a benchmark, we consider a case where privacy is not allowed. If the firm is
a monopolist, it sets price such that it fully supplies both markets. Under duopoly,
the uninformed firm is able to sustain a small operation in the Old market and
equilibrium prices will be lower than Hotelling prices. We show that monopoly
profits are larger than the informed firm’s profits.
Next we study the case where consumers can pay for privacy to avoid being in
the database. The model shows that monopoly prices are decreasing in the privacy
cost. Likewise, the amount of consumers who pay for privacy decreases with the
cost. However, the relationship between consumer surplus and the privacy cost is
ambiguous. Finally, profits always increase with the privacy cost.
In the duopoly case, prices are larger than Hotelling prices, and they decrease
with the privacy cost. Overall, these results come from the fact that the privacy cost
effectively reduces the size of the market, i.e. those consumers for whom there is no
information. Therefore, expensive privacy increases competition. This is consistent
with the fact that both firms’ individual profits are decreasing in the privacy cost
and that consumer surplus is increasing.
We prove that the owner of data chooses to sell it exclusively to one firm. This
3
In Europe, see for example Directive 2002/58/EC.
3
is true when privacy is in place and when it is not. The price of information will
be determined as in an auction with externalities: the difference in the profits of
owning the data when the competitor does not, minus the profits of not owning the
data when the competitor does. We show that when privacy is allowed, the price
of information is convex in the privacy cost.
Finally, we find that allocation of the data depends on the timing on the game.
In the main model, we will assume that information is sold at the beginning of the
game. Then, we prove that an interim sale of information (i.e. right before tailored
offers are made) never leads to exclusivity. However, the seller is always better off
with ex-ante sales.
This paper is related to two strands of the literature. The first one is behavior
based price discrimination. In particular, Fudenberg and Tirole (2000) study the
effects of poaching under duopoly in a two period model, where firms have to get
customer information from their own market shares in period one and can poach
from their competitor in period two. They found that poaching can lead to prices
lower than Hotelling. Our model is different in that we do not assume a period
in which there is no information. By doing so we abstract from the process of
creating information and focus on the privacy actions undertaken by consumers.
Esteves (2010) also focuses on the firms behavior towards selling in the first period
and acquire information. However, as opposed to our paper, consumers are naive
in that they ignore they could be discriminated in period two. Fudenberg and
Villas-Boas (2012) consider a dynamic model as well, but consumers are not myopic.
Nevertheless, the privacy and the consumption decision are tied together: consumers
can anticipate price discrimination and decide not to consumer in the first period. In
our model consumption is not strategic, given that consumers can pay to erase any
information that already existed on them, and this happens prior to consumption.
The second strand of the literature is related to consumers’ behavior towards
privacy and price discrimination. Taylor and Wagman (2014) survey an array of
models in which privacy can be incorporated. They show that competition drives
prices below Hotelling if both firms have perfect information about every consumer.
Other works - such as Conitzer et al. (2010) and Acquisti and Varian (2005) study situations when consumers can opt out from a monopolist’s database on past
purchases. Then, our main contribution to this literature is that we emphasize the
4
decision taken by the consumers to protect their privacy and study the effects of
this decision on competition.
The most related paper is Koh et al. (2013) since they specifically introduce
a privacy decision for consumers that is independent and prior to consumption.
Consumers can accept being profiled by the firm or not. By doing so, consumers
partially alleviate the search cost they have to incur to find their most preferred
variety. In return, the firm gets a signal of the consumer’s valuation. In this model,
the privacy cost is the cost of staying hidden and is qualitatively equivalent to the
privacy cost in our model (only the private customers pay it). Nevertheless, while
Koh et al. (2013) consider only the monopoly case,4 we consider a duopoly situation
that endogenizes both the consumers’ privacy choice and the firms’ purchase of
information.
The paper is organized as follows. The next section presents the model. Section
3 solves the case where consumers are not allowed to have privacy. Section 4
introduces privacy. Sections 3 and 4 are further divided to study a monopoly and a
duopoly. Section 5 solves for extensions to the model. The last section concludes.
2
The model
In this section, we set a model of imperfect competition with heterogeneous consumers. There is a continuum of costumers, each of them with a unit demand. They
receive a utility v > 0 if they buy the product and nothing otherwise. Consumers
have horizontal preferences θ which are uniformly distributed on [0, 1]. Let pi be
the price a firm i charges for its product. Each firm’s marginal cost is normalized
to zero. Additionally, consumers have to pay a linear transportation cost t > 0 to
buy the good. Therefore, if firm i is located at point x ∈ [0, 1], a consumer θ makes
a net utility v − pi − |x − θ|t. When two firms are competing on the market, consumers buy from the firm that gives them the highest possible utility. Throughout
the paper we assume that v ≥ 2t. By doing so we ensure that the gross utility is
high enough so that all consumers would consume if the price was set at zero.5
4
They do acknowledge that an outside option (as a proxy for competition) can increase consumer surplus.
5
This assumption simplifies the analysis by reducing the number of case distinctions that have
to be studied without loss of economic insight.
5
Let us have two markets like the one just described, which differ in terms of
the information that can be obtained by firms on consumer preferences. We will
refer to the first one as the New market. In this market, there is no way to obtain
any detailed information about the taste of the consumers, so firms can only offer
basic or uniform prices that do not depend on θ. The second market will be called
the Old market. In this market there is some information that can potentially be
extracted by the firms. Therefore, firms could make tailored offers than depend on
θ in a way that we specify below. Both markets have a total mass of one.6
As discussed in the introduction, in the Old market consumers can pay a cost to
avoid detection by the firms. We call this the privacy cost c ≥ 0. This cost represents the actions consumers take to prevent any firm or third party from collecting
personal data about them. Throughout the paper we set c ≤ t. If c > t we can
show that no consumer would ever pay the privacy cost. We assume that c affects
the utility directly: if a consumer θ buys the good at pi from a firm located at x
and also paid the privacy cost, his utility is v − pi − |x − θ|t − c. For the rest of the
paper, unless explicitly mentioned, c will not depend on the consumer location. In
this way we can explore the effects of policies that facilitate or impede privacy on
the Internet in general.
Firm i can price differently for the two groups. A basic price will be offered to
all consumers it does not recognize. This group is comprised by every consumer
in the New market plus some consumers from the Old market, those who paid the
privacy cost. The basic price will be based on the firm’s belief about the average
willingness to pay in this group.
For the consumers the firm does recognize, it will make tailored offers, based on
each consumer’s location θ. This group is a subset of the Old market, comprised
by consumers who did not pay the privacy cost.
To interpret the two markets, consider an on-line retailer that supplies two
distinct sets of consumers. On the one hand, there are newcomers: those for whom
there is not enough data yet. These consumers do not necessarily need to be young,
only new to the Internet. Alternatively, these consumers could have preferences very
different from what the retailer has to offer, and therefore, even if they are active
6
In an extension, we discuss the impact of changing the relative size of the two markets (see
section 5.2).
6
on-line, that information will not help the firm. If, say, the retailer in question sells
sporting goods, even the most detailed data about consumers who never visited
any sports websites or never shopped anything sport related will not be useful for
the firm. On the other hand, there are old consumers, i.e., people who have been
actively using the Internet in the past, visiting websites, shopping, leaving reviews,
and so forth. These data are very likely to be informative about the consumer
preferences for the goods or services the retailer sells.
An Internet search engine or a social media site can gather data about the old
consumers. These data can be thought as personal taste, willingness to pay, brand
fidelity, etc. However, these consumers can also take actions to avoid being in the
database. There are many reasons to do so. First, privacy is a good on itself, and
for some people it is not right for a company to compile their personal information.
Second, some criminal or shameful activities could be actively concealed by some
people. Third, consumers might be aware of the possibility that a firm might
set different prices for different people, based on personal data. As seen in the
introduction, this have been documented many times and it is not usually in favor
of the consumers. It is this last particular aspect that our model captures.7
Lastly, competing firms can potentially buy this information from the search
engine. For those consumers who appear in the dataset, the firm will offer a take
it or leave it price, tailored according to the information the firm possesses. If the
firm has not information about a particular consumer, it cannot tell whether he is
a new or an old consumer who concealed his information. In these cases, a basic or
uniform price will be proposed.
3
No privacy
As a benchmark we first study the case where consumers cannot pay for privacy.
This is equivalent to say that the cost c is very large. First we analyze the monopoly
case, followed by a duopoly situation.
7
We emphasize that the only reason for paying the privacy cost in our model is anticipating
a tailored price higher than the basic price for a particular consumer θ. We do not include any
intrinsic benefits of privacy. Therefore, any positive effect of privacy in this model should be seen
as conservative. In turn, we also do not consider any negative external effects that privacy might
generate.
7
3.1
Monopoly
Consider a monopolist located at θ = 0 in the New and Old markets.8 The timing
of the model goes as follows:9
1. The firm can buy information at a price T about every consumer in the Old
market.
2. The firm chooses the basic price p.
3. The firm can offer tailored prices.
4. Consumers buy and consume.
Let us assume that the monopolist bought the information in period 1. The
monopolist’s market share in the New market would be all the consumers θ ∈ [0, θ1 ]
such that v − p − θ1 t = 0. This means that θ1 represents the indifferent consumer
between buying the good and getting the outside option of zero:10
θ1 = min{
v−p
, 1}
t
In the Old market, the firm offers tailored prices p(θ). Given that the outside
option is equal to zero, the tailored prices can be as high as to capture all the
surplus. Hence:
p(θ) = v − tθ,
where v ≥ 2t > t guarantees that that p(θ) > 0. Therefore, the monopolist solves
Z
max
p≥0
θ1
Z
pdθ +
0
1
p(θ)dθ
0
s.t. 0 ≤ θ1 ≤ 1.
With the assumption that v ≥ 2t, we have that p∗ = v − t and θ1 = 1. The price p∗
is the equilibrium basic price and it is only offered to consumers in the New market.
8
The results of this section would be qualitatively similar if the firm was located in the middle
of both markets.
9
Merging stages 2 and 3 would make no difference in this case, but we keep this distinction for
consistency with the case with competition that we analyze in Section 3.2.
10
Note that θ1 cannot be larger than one by assumption.
8
In the Old market, consumers are offered the tailored price p(θ) and are left with
no surplus. These results imply that both markets are fully covered.
Total profits π are given by the sum of the profits in the Old market π O and the
profits in the New market π N . The equilibrium values are
O
Z
1
(v − tθ)dθ = v −
π =
0
t
v − p∗ ∗
and π N =
p = v − t.
2
t
Hence total profits are π = 2v − 3t2 . Note that without consumers’ information, the
monopolist would make 2(v − t), i.e., twice the profit in the new market. Therefore,
the maximum price the firm is willing to pay to acquire the dataset is
t
T ∗ = π − 2(v − t) = .
2
Consumer surplus in the Old market CS O and in the New market CS N are given
by
O
Z
1
(v − (v − tθ) − tθ)dθ = 0 and CS
CS =
N
Z
=
0
0
1
t
(v − (v − t) − tθ)dθ = .
2
The equations above imply that total consumers surplus is simply CS = CS N = 2t .
3.2
Competition
We now study the case where 2 firms, A and B, compete.
We assume that in
both markets the locations are fixed, with firm A located at θ = 0 and firm B at
θ = 1. We look at different cases depending which firms have information, but we
are particularly interested in the case where only one firm has information, as the
data owner may sell information exclusively to one firm. We are interested in the
competitive effects of information when access to it is potentially asymmetric in the
market. In such a case we will assume, without loss of generality, that only A has
information about the consumers.
As in the previous section, information is only available in the Old market and
no information can be obtained about the consumers in the New market. The
timing for the competitive case is as follows:11
11
In this section, sequentiality in the choice of basic and tailored prices is necessary to have
an equilibrium in pure strategies. The problem with simultaneous moves is that, for example,
9
1. Firms A and B can buy information at a price T .
2. Firms A and B choose basic prices pA and pB .
3. Firms A and B can offer tailored prices.
4. Consumers buy and consume.
In stage 1, firms can buy information from an upstream data supplier. We
assume that only then information can be sold.12
The duopoly setting can be seen as one where a firm with customers’ information
is able to offer a rebate to the consumers on the competitor’s turf.13 For example,
this could be understood as the ability of the most informed firm to send targeted
coupons to some consumers.
As a benchmark, consider first a situation where neither firm has information.
This is equivalent to solving a Hotelling model in two equivalent markets. In each
of them, prices pA and pB are identical and equal to the transportation cost t, which
implies that the market is evenly split between firms.14 Then, profits for each firm
in the Old and New markets are
t
2
while total profits for each firm are πA = πB = t.
Finally, consumer surpluses in both markets are given by
O
CS = CS
N
Z
1/2
Z
3.2.1
(v − t − t(1 − θ))dθ = v −
1/2
0
Hence, CS = 2v −
1
(v − t − tθ)dθ +
=
5t
.
4
5t
.
2
Both firms have information
The case where both firms have information has been studied by Taylor and Wagman (2014). The difference lies here in that we consider two markets and two types
of prices. Proposition 1 shows the result.
A tailors a price that is optimally a function of B’s basic price, hence B can always reduce its
price and gain a large portion of the market. Sequentiality helps us ensure that a pure-strategy
equilibrium will exist, because the optimal tailored price will always exist.
12
We will relax this assumption in an extension where we allow for interim sales, that is, firms
can buy information after basic prices are set (see section 5.1).
13
See Fudenberg and Tirole (2000). Note that, in contrast, in the monopoly case studied before
there was no poaching and price discrimination simply allowed the firm to fully extract rent from
consumers.
14
See Belleflamme and Peitz (2010).
10
Proposition 1: Assume both firms A and B buy the information and consumers
cannot pay for privacy. Basic prices are:
pA = t and pB = t,
while tailored prices are:
pA (θ) = max{t(1 − 2θ), 0} and pB (θ) = max{t(2θ − 1), 0}.
Profits are:
πA = πB =
3t
,
4
and consumer surplus is:
CS = 2(v − t).
Proof. See Taylor and Wagman (2014).
Taylor and Wagman (2014) showed that if both firms had information and competed simultaneously only in the Old market, competition would intensify to the
point that both firms would make lower profits than they would by charging basic
prices only.
Notice that, compared to the case without information, profits decrease while
consumer surplus increases. This is due to the fact that competition reduces the
potential for rent extraction when both firms hold information about consumers.
3.2.2
Only firm A buys information
In this section we focus on the case where information is only sold to A. Later we
will address the conditions such that information is acquired. Note that now the
outside option for the consumers is not zero,15 but the utility obtained from buying
from B instead. Buying from A leads to a utility level given by
v − tθ − pA ,
15
We have guaranteed this by assuming v ≥ 2t, so that even consumers closer to A would make
strictly positive surplus if they were to buy from B at the equilibrium price.
11
while buying from B leads to
v − t(1 − θ) − pB ,
where pA and pB are A and B’s prices respectively. In the New market, market
shares [0, θ1 ] and [θ1 , 1] are determined by the indifferent consumer between buying
from A or B:
θ1 (pA , pB ) =
1 pB − pA
+
.
2
2t
In the Old market firm A offers in stage 3 a tailored price pA (θ) that makes
consumers indifferent between accepting and buying B’s product.16 This implies:
pA (θ) = pB + (1 − 2θ)t.
We observe that pA (θ) cannot be negative; otherwise, A would be better off by
offering a tailored price of zero. Let us call θ20 the consumer for whom pA (θ) = 0.
Hence:
pB + t
.
2t
θ20 (pB ) =
We now solve at stage 2. Firm A’s profits are given by:
Z
θ1
πA =
min{θ20 ,1}
Z
pA dθ +
0
pA (θ)dθ,
0
and B’s profits are:
Z
1
πB =
Z
1
pB dθ +
pB dθ.
min{θ20 ,1}
θ1
There are two possible cases: one when θ20 < 1 and another when θ20 ≥ 1. In the
former, B operates in the Old market, which means that prices are such that some
consumers would buy from B in the Old market. In the latter, it does not operate
in the Old market. Because pA (θ) does not depend on pA , A’s reaction function is
always the same, regardless of the case. This implies that it is B that decides the
16
Under indifference, consumers buy from the informed firm. In this case it is firm A.
12
outcome of the game based on its own profits. B can choose either to give away
every costumer in the Old market by setting a high price, or to lower the price to
keep some consumers on this market.
In Proposition 2 we show that firm B finds it optimal to operate in the Old
market, and therefore lowers its price compared to the case where it does not participate. In this sense, we say that competition is more aggressive.
Proposition 2: Assume only firm A buys the information and consumers cannot pay for privacy. Then the equilibrium prices on the new market are:
pA = t −
2t
t
and pB = t − ,
7
7
profits are:
πA =
54t
25t
and πB =
,
49
49
and consumer surplus is given by:
CS = 2v −
110t
.
49
Proof. We verify that B finds it profitable to set a price such that it operates in the
Old market (θ20 < 1). Remark that A’s reaction function is the same in every case:
arg max θ1 (pA , pB )pA
pA
If B operates in the Old market its reaction function is
arg max
pB
which leads to πB =
25t
.
49
1 − θ20 (pB ) + (1 − θ1 (pA , pB )) pB
On the contrary, if it does not operate in the Old market
its reaction function is
arg max [1 − θ1 (pA , pB )] pB
pB
which leads to πB = 2t . This implies that B operates in the Old market.
We find that prices are lower than in the no-information case. As mentioned
above, this is because B stays in the Old market and competition is tougher. Finally,
13
note that consumer surplus is at its highest when both firms have information and
at its lowest when no firm has information. Concerning profits, the informed firm
makes the highest profits while the competitor makes the lowest. Furthermore, the
informed firm in this case makes higher profits than in any one of the previous
cases. This is something that can be exploited by the data owner and that we
analyze next.
3.2.3
The price of information
Suppose there is a firm F that owns the information and can sell it only in period
1. We call Ti the price paid when a number i of firms buys the information. This
price is a transfer made by the buyer to the seller also in period 1. The allocation
and the prices are set and payments realized in period 1. We further assume that
this is common knowledge. For example, F has to charge to the same price T2 to
both firms if it decides to sell it to both. Therefore, F makes profits
πF = max{T1 , 2T2 }.
This setting is similar to an auction with externalities as in Jehiel and Moldovanu
(2000). Indeed, when F sells to only one firm, the maximum price T1 that F can set
is the difference between winning and losing the auction, that is, the profits made
by the firm with information minus the profits the firm would make when its rival
has it. We obtain:
T1 =
29t
54t 25t
−
=
.
49
49
49
Analogously, if F sells to both firms:
T2 =
3t 25t
47t
−
=
4
49
196
Therefore, F chooses to sell the information only to one firm because T1 > 2T2 .
The next proposition presents the result of this section:
Proposition 3: Information is sold to only one firm at T =
for both firms are πA = πB =
25t
.
49
Proof. See text.
14
29t
49
and net profits
Proposition 3 shows that it is optimal for the owner of information to give
exclusive rights. This is because competition is too tough when both firms have
information and the potential for rent extraction is minimized. On the other hand,
an informed firm can extract more rents from the consumers if its competitor can
only use a single basic price.
4
Privacy
We now turn to the case where consumers can pay for privacy. As noted in the
description of the model, consumers can incur a cost c to be left out of the database
that firms can potentially buy.
4.1
Monopoly
There is a single firm, located a θ = 0 in the New and Old markets. Compared to
the timing used before, we add an initial stage when consumers can protect their
privacy.17 The timing of the model goes as follows:
1. Consumers make their privacy choice, i.e., decide whether or not to pay the
cost c.
2. The firm can buy information at a price T .
3. The f irm chooses the basic price p.
4. The firm can offer tailored prices.
5. Consumers buy and consume.
As in the case with no privacy, any consumer θ for whom the firm has information
will be charged the tailored price p(θ) = v − tθ > 0 and make zero utility. This
17
It is appropriate to put the consumers’ choice before the information is sold because information is created using consumers’ on-line records, which depend on the privacy behavior. In
terms of the game, this requires that, in equilibrium, consumers correctly anticipate whether the
firm buys the information or not. If the order of stages 1 and 2 were reversed, consumers would
not have to anticipate if the firm buys the information. This would change the interpretation of
privacy in the model. In the alternative timing, consumers would be “concealing” information
given that they know the firm knows their valuation. In terms of the outcome of the game, however, nothing would change. Now consumers react to stage 1 actions, which optimally take into
account consumers’ actions at stage 2. Then the equilibrium outcomes will be identical. The same
reasoning applies in the competition case below.
15
allows us to define the set of consumers who would pay the privacy cost as all
θ ∈ [0, θ2 ]. Since the choice of privacy is made in period 1 (i.e., before any price
has been set) we have to define an anticipated price pa ≥ 0. This price is the basic
price that consumers in the Old market expect to pay if they are not in the firm’s
database. Effectively, the basic price in the Old market is now pa + c. As this
price increases, clearly fewer consumers would avoid detection. If pa + c was small,
however, it would be very attractive for consumers to enforce their privacy in period
1. Hence, θ2 is such that:
v − tθ2 − pa − c = 0 ⇐⇒ θ2 =
v − pa − c
.
t
To solve the model, assume the firm bought the information in period 2 and this
purchase is anticipated by the consumers. In period 3, the monopolist maximizes
profits π(p) given the anticipated pa :
Z
θ1
Z
min{θ1 ,θ2 }
0
1
pdθ +
pdθ +
π(p) =
Z
p(θ)dθ.
(1)
θ2
0
The first term on the right hand side of Equation (1) represents the profits from
the New market. The second term covers the profits from the consumers in the Old
market who are not in the firm’s database, i.e., consumers who paid the privacy
cost. Note that the firm could set a large price p so that the actual market share
θ1 is smaller than the amount of people who pay the privacy cost.18 This is why
the upper bound in the second integral is min{θ1 , θ2 }. The last term represents the
rest of the Old market. These consumers are subject to price discrimination and
pay the tailored price p(θ).
The firm then maximizes the profits expressed in Equation (1) for p ≥ 0 subject
to the following constraints:
1 ≥ θ1 ≥ 0
(2)
1 ≥ θ2 ≥ 0
(3)
18
We call the actual market share the amount of people who buy in the Old market. This is
why this value will be also θ1 as in the New market.
16
Condition (2) requires that the market share in the New market stays in [0, 1].
Condition (3) states that the proportion of people who pay the privacy cost stays
bounded. Proposition 4 shows the solution.
Proposition 4: Assume the firm buys the information.
If c ≤ 3t − v, then
p∗ =
2v − c
3
and θ2∗ =
v − 2c
,
3t
and profits and consumer surplus are:
π=
2c2 + v 2 1
5c2 − 2cv + 2v 2
+ (2v − t) and CS =
.
6t
2
18t
If t ≥ c > 3t − v, then
c
and θ2∗ = 1 − ,
t
p∗ = v − t
and profits and consumer surplus are:
π=
c2
c2
+ 2(v − t) and CS = t + − c.
2t
2t
Proof. Note that θ2 > θ1 cannot be an equilibrium because consumers in period 1
must correctly anticipate the prices chosen by the monopolist in period 3: pa = p.
Then it has to be that in any equilibrium θ2 ≤ θ1 . Under this condition the firm
takes the quantity of consumers who pay the privacy cost θ2 as given. Ignore
positivity constraints for the moment. Assume that (2) and (3) do not bind. The
first order condition π 0 (p) = 0 implies:
p=
−pa − c + 2v
2
Finally, we apply the condition pa = p on the FOC. Note that under pa = p, θ1 ≤ 1
implies θ2 ≤ 1. Then this solution requires only θ1 ≤ 1 to be interior. The second
solution is the corner when θ1 = 1.
The basic price p∗ is now obtained based on the weighted average between the
17
willingness to pay in the New Market and the willingness to pay among consumers
who paid the cost c in the Old market. Also note that the average willingness to
pay from those who paid the privacy cost is larger than the average willingness to
pay in the New market, given that consumers who pay the cost are closer to the
firm. These facts imply that there are two effects at play to understand the effects
of an increase of c on the price.
The first one is the size effect. When c increases, the total number of consumers
who pay the basic price in the Old market goes down. This is from θ2∗ . As a result,
the overall weighted average willingness to pay is reduced. This effect leads to a
price decrease. The second one is the valuation effect. As c increases, consumers
who do not value the good as much will not pay the basic price any more and the
average valuation in the market will increase. This effect leads to a price increase.
that the size effect dominates, as the price goes down with c .
Note from p∗ = 2v−c
3
Corollary 1: Assume the firm buys the information. Profits are always increasing in c. Additionally, if c ≤ 3t − v, consumer surplus is convex and minimized at
v
.
5
If c > 3t − v, consumer surplus is always decreasing in c.
Proof. Direct from Proposition 4.
Corollary 1 states that profits are increasing in the privacy cost. Profits increase
in c because, in spite of having a lower basic price, the share of consumers who pay
the privacy cost decreases. Then, higher profits in the Old market from charging a
personalized price compensate the losses in the New market.
Consider the interior case c ≤ 3t − v, where consumer surplus is convex in c.
This is because consumers in the New market always gain with a larger c, while
consumers in the Old market always lose. In the former market, a larger cost is
associated with a larger market share and a lower price. In the latter market, it is
the opposite. This can be seen by noting that the effective basic price in the Old
market is p∗ + c. What the Corollary shows is that, for c ≤ v5 , the losses in the
Old market associated to c dominate, while for c >
v
5
the gains in the New market
dominate.
Finally, when c > 3t − v, c has no effect in the New market because full coverage
is achieved and changing the price has no impact on the profits in that market.
Then, when c increases consumer surplus decreases in the Old market as fewer
18
consumers pay for privacy and have to pay their full valuation.19
As in the case without privacy, the data owner can extract any profit the monopolist makes in excess of 2(v − t), which are the profits it could earn without
having any information at at all.
4.2
Competition
Now we consider the duopoly case where consumers can pay for privacy. This
section is similar to 3.2 but it includes an initial stage in which consumers can
choose to pay for privacy. The timing in this section changes to accommodate for
the consumers’ actions.
1. Consumers make their privacy choice, i.e. decide whether or not to pay the
cost c.
2. Firms A and B can buy information at a price T .
3. Firms A and B choose basic prices pA and pB .
4. Firms A and B can offer tailored prices.
5. Consumers buy and consume.
Some consumers might pay the privacy cost to avoid the tailored price in the
Old market. This share is given by those who prefer paying the privacy cost c - and
buying from A at the basic price - than buying from A at the tailored price:
v − paA − θt − c ≥ v − paA (θ) − θt = v − t(1 − θ) − paB .
As in the previous section, this share must be based on anticipated prices, given
that consumers make their privacy choices in period 1 and prices are set in periods
3 and 4. In other words, by the time prices are set, firm A treats the proportion of
consumers who paid the privacy cost as fixed. Then, the indifferent consumer is
θ2 =
19
1 paB − paA − c
+
.
2
2t
When c = t Proposition 4 leads to the No Privacy result, with θ2∗ = 0.
19
(4)
Finally, if neither firm has information, the fact that consumers can pay for
privacy does not change the benchmark result in section 3.2. That is, if consumers
anticipate that no firm will have information, no consumer will want to pay the cost
c.
4.2.1
Both firms have information
Consider the case where both firms buy information in period 2. The following
Lemma describes the consumers’ behavior anticipating firms buy. In short, tough
competition leads to tailor prices low enough so that it is not profitable for any
consumer to pay for privacy.
Lemma 1: If all consumers anticipate that both firms will buy the information,
no consumer pays the privacy cost.
Proof. Take any subset of consumers in the Old market who did not pay the privacy
cost. Competition in this subset is for every consumer, therefore, tailored prices are
equivalent to those in Proposition 1. This implies that any consumer who hides does
it to be allowed to pay the basic price of the nearest firm, because the alternative
(the tailored price) is already matching the competitor’s offer. The rest of the proof
relies on two facts:
1. If a consumer θ̂ ∈ [0, 21 ] pays the privacy cost, every consumer θ ∈ [0, θ̂] pays
the privacy cost. By symmetry the same is true for B in [ 21 , 1]. Note that if
consumer θ̂ pays, it implies that:
t − c − pA
≥θ >θ−
2t
2. If a strictly positive mass of consumers pays the privacy cost then pA , pB > t
Say consumers θ ∈ [0, θˆA ] pay the privacy cost to hide from B. Then A solves:
Z
max
pA
θ̂
Z
pA dθ +
p −p
1
+ B 2t A
2
pA dθ
0
0
B solves and analogous problem. Then prices are:
pA = t +
2t(2θA + 1 − θB )
2t(θA + 2(1 − θB ))
and pA = t +
3
3
20
Fact 1 implies that consumers to the left of
1
2
would pay the privacy cost if and
only if pA + c < t(1 − 2θ). By Fact 2 that implies c < −2θ, which cannot hold for
c > 0. Therefore, no consumer would pay the privacy cost.
In the Old market, competition will be for every consumer and tailored prices
will be very low. Therefore, to observe consumers paying the privacy cost, basic
prices should be even lower than tailored prices, after taking the cost c into account,
and such basic prices are not profitable.
To summarize, when consumers can pay for privacy and information is symmetric among firms, in equilibrium, consumers do not pay for privacy. This means
that the fact that consumers are entitled to privacy has no effect over the outcome
of the model when both firms have information. Therefore, Proposition 1 holds:
Hotelling prices are set in the New market and lower, targeted prices are set in the
Old market. To have a positive amount of consumers who do pay for privacy, only
one firm should have bought information.
4.2.2
Only firm A buys information
As before, assume, without loss of generality, that firm A buys the information
and B does not. The equilibrium is given in the following result.
Proposition 5: Assume only firm A buys the information and consumers can
pay for privacy. There exists a threshold c < t such that if c ≤ c the equilibrium is:
pA = t +
t−c
t−c
3(t − c)
, pB = t +
and θ2 =
,
2
4
8t
and if c > c, the equilibrium is the one in Proposition 2.
Proof. First we verify that B chooses a price such that it does not operate in the
Old market (θ20 ≥ 1). Note that A’s reaction function is always given by:
arg max [θ1 (pA , pB ) + θ2 (paA , paB )] pA
pA
If B does not operate in the Old market, its reaction function is:
arg max (1 − θ1 (pA , pB )) pB
pB
21
These two equations lead to pA (θ2 ) and pB (θ2 ). Finally, because in equilibrium
pA = paA and pB = paB (which implies θ2 < θ1 ), we can solve for θ2 =
lead to πB =
(5t−c)2
32t
3(t−c)
.
8t
Prices
. On the contrary, if B operates in the Old market, its reaction
function is
arg max
pB
1 − θ20 (pB ) + (1 − θ1 (pA , pB )) pB
Note that θ20 is a direct consequence of pA (θ) ≥ 0. Given that pA (θ) = pB + (1 − 2θ)t
is always chosen by A in period 4, θ20 is a function of pB and not of paB .
Following the same procedure as above, we obtain θ2 =
Then to keep θ2 ≥ 0 we must have c ≤
6t
.
7
6t−7c
20t
and πB =
(8t−c)2
.
100t
For a larger c no consumer pays for
privacy and prices are given in Proposition 2. Therefore, it is enough to establish
√
2
(8t−c)2
20
c
=
5t
−
2t ≈ 0.96t such
≥
.
Then,
we
can
find
a
for c ≤ 6t7 that (5t−c)
32t
100t
7
that for c ≤ c,
(5t−c)2
32t
≥
25t
.
49
We find that prices are larger than Hotelling prices for c ≤ c̄. Recall that
prices in the two Hotelling lines are based on the weighted average willingness to
pay among both markets. From A’s perspective, the average willingness to pay in
the segment [0, θ2 ] is higher than the average willingness to pay in the New market,
because consumers are closer to A on average. Then it follows that the weighted
average willingness to pay in the New market plus the [0, θ2 ] segment is larger than
the average willingness to pay in the New market alone.20 This leads to pA ≥ t.
Noteworthy, the fact that pA ≥ pB is a consequence of the difference in the average
willingness to pay for each product, as no consumer in [0, θ2 ] buys from B.
The result that the price pB ≥ t comes from the fact that B does not find it
profitable to operate in the Old market. In other words, A supplies the whole Old
market. This means θ20 ≥ 1, which in turn implies pB ≥ t. The change of regime at
c̄ is given by B’s decision to participate in the Old market,21 then B’s equilibrium
profits are continuous at c̄. However, θ2 is discontinuous at c̄ and drops to zero, i.e.,
no consumer pays for privacy. Hence, we will focus on c ≤ c̄ to study the effects of
the privacy cost on the equilibrium.
Analogous to the monopoly case, a change in c has two effects: the size and
the valuation effect. Then, because prices decrease when c increases, the size effect
20
To see this, call the average willingness to pay in the New market v N and v O the average
N
+θ2 v O
willingness to pay in the segment [0, θ2 ] in the Old market. Since v O > v N , then v 1+θ
> vN .
2
21
The details can be found the in proof of Proposition 5.
22
dominates and A reduces its price as c goes up. In this equilibrium, B only operates
in the New market and its reaction function corresponds to the standard Hotelling
model, which implies that prices are strategic complements, then pB also decreases
with c. Furthermore, p0A (c) > p0B (c) given that c affects pA directly as fewer consumers pay for privacy but also trough pB . However, c only affects pB through the
competitor’s price.
Finally, an important implication from Proposition 5 is that price discrimination
cannot be avoided when enforcing privacy is free. The reason is that for some
consumers it is better to accept the tailored price than to pay pA . This fact is
not related to privacy, rather it is a property of the Hotelling model. Consumers
relatively closer to B will accept the tailored price, because such price is decreasing
in θ. When c = 0 there is no difference between the people who would pay for
privacy and the people who would accept pA in stage 2, and this share can never
be one. When c > 0, however, some consumers would accept pA had c been zero
but decide to pay the tailored price instead if the gains pA (θ) − pA do not offset the
cost c. The next result presents profits and consumer surplus.
Corollary 2: Assume only firm A buys the information and consumers can
pay for privacy. Then, the profits of both firms are decreasing in c while consumer
surplus is increasing in c.
Proof. If c ≤ c profits and consumer surplus are:
Z
pB −pA +t
2t
πA =
Z
pA dθ +
0
θ2
Z
0
πB =
pB −pA +t
2t
CS =
1
pB −pA +t
2t
Z
(v−pA −θt) dθ+
0
(pB + (1 − 2θ)t) dθ =
θ2
Z
Z
1
pA dθ +
pB dθ =
Z
+
Z
1
(v−pA −θt−c) dθ+
(v−(pB +(1−2θ)t)−θt) dθ
θ2
1
pB −pA +t
2t
(5t − c)2
32t
θ2
0
11c2 − 22ct + 107t2
64t
(v − pB − (1 − θ)t) dθ =
10ct + 5c2 − 103t2
+ 2v
32t
and if c > c, the equilibrium is the one in Proposition 2. We can show that
πA0 (c), πB0 (c) < 0 since c < t and CS 0 (c) > 0. In the proof of Proposition 5 we
showed that there is a c above which firm B poaches some consumers from the Old
23
market which leads to basic prices low enough so that consumers do not pay for
privacy and the equilibrium in Proposition 2 applies. Then profits and consumer
surplus are linear in c > c. However, the results in the Proposition continue to hold
for any c ≤ t.
Note that as c increases, the proportion of consumers who pay the privacy cost
decreases, this pulls down the average willingness to pay for A’s basic price. As a
response, A reduces its price and B will do the same to keep up with competition
in the New market. Because pB decreases, the tailored price will also decrease: A
now has to leave more surplus to the consumers to make them indifferent between
buying from it or from B. This brings down the profits of both firms.
In turn, consumers benefit from this competition. From Corollary 2, consumer
surplus is always increasing in c. This is because when c increases, all prices go
down and the amount of consumers who pay the privacy cost goes down as well.
These two effects are more than enough to compensate for the consumers who pay
a higher c. Furthermore, it can be checked that consumer surplus is discontinuous
at c̄ because firms compete more aggressively when B operates in the Old market.
4.2.3
The price of information
The price of information when consumers can pay for privacy is calculated in the
same way as in section 3.2.3. The seller of information F compares the price it
can get from selling to one firm T1 to the twice the price of selling to both firms
2T2 . Proposition 6 shows that, even when consumers can pay for privacy, F finds
it optimal to sell the information only to one of the firms downstream.
Proposition 6: If c ≤ c, then information is sold to one firm at T =
and net profits are πA = πB =
(5t−c)2
.
32t
9c2 −2ct+57t2
64t
If c > c, Proposition 3 holds.
Proof. The case where c > c̄ has been already solved. For c ≤ c̄ we use the proof of
Corollary 2 to check that T1 > 2T2 , where T1 = πA − πB and T2 =
πA − πB ≥ 2
and therefore T = T1 .
24
3t
− πB
4
3t
4
− πB . Indeed,
The result that the data owner prefers to sell information exclusively to one
downstream firm is therefore robust to introducing privacy. Remarkably, privacy
actually allows the information seller F to set a larger fee T , compared to Proposition 3, because the profits of firm A increase by a larger amount than the profits
of firm B. Intuitively, the informed firm (A) knows that, when privacy is allowed,
it can charge higher prices because it faces consumers with a larger average valuation (only consumers close to the informed firm will pay for privacy). Therefore,
competition is less aggressive and there are larger gains in acquiring information exclusively. because of this, privacy reinforces the result that information is allocated
to only one firm.
On the other hand, given that T in Proposition 6 is a convex function in c,
we are able to show that the corner c = c̄ yields the highest revenue for firm F .
Furthermore, since T is minimized at c =
t
,
9
the value of information is for the
most part increasing in c. The reason is that the outside option of not buying πB
decreases faster than the profits generated from buying the information πA when
c > 9t .
Note that pA decreases faster in c than pB , then, B’s market share in the New
market decreases with c. This means that B only loses when c increases. Firm A,
on the contrary, faces two effects in the New market as its market share increases
but its price decreases with c. In the Old market, a larger privacy cost leads to
fewer consumers paying for it, which increases the revenue from tailored prices, even
though p(θ) decreases. At the same time, more consumers pay for the basic price
in the Old market. This makes A’s profits less sensitive to c than B’s.
5
Extensions
5.1
Interim sales of information
So far we have assumed that the seller F sells the information ex-ante at a price T .
We now consider interim sales. To model such a situation, we change the timing
so that buying information occurs when it is used, i.e., after basic prices are set.
Furthermore, we avoid any change of regime by assuming c ≤ c̄. The timing is as
follows:
25
1. Consumers make their privacy choice, i.e., decide whether or not to pay the
cost c.
2. Firms A and B choose basic prices pA and pB .
3. Firms A and B can buy information at a price T .
4. Firms A and B can offer tailored prices.
5. Consumers buy and consume.
The main change in this section is that firms can deviate at stages 2 and 3.
At stage 3, each firm can deviate to buying or to not buying the information,
which depends on the type of equilibrium being evaluated. For example, in an
equilibrium where both firms buy, we need to make sure no firm deviates to not
buying the information, taking both prices as given from period 2. At stage 2,
firms can deviate setting a different price and change their buying behavior as well
as their competitor’s. For instance, in an equilibrium where only firm A buys,
B could deviate and set a price such that A decides not to buy while B buys.
The next Lemma allows us to rule out equilibria with an asymmetric allocation of
information.
Lemma 2: There is no T that supports an equilibrium where only one firm buys
information.
Proof. Consider an equilibrium where only firm A buys information. In period 3,
A could deviate and not buy the information. It does so if and only if:
Z
pB −pA +t
2t
pA dθ =
2
0
9t2 − c2
11c2 − 22ct + 107t2
>
−T
8t
64t
The first term are the deviation profits, which come from the basic prices from
Proposition 5 (pA = t +
t−c
2
and pB = t +
t−c
)
4
applied to both markets and the
second term are the equilibrium profits.
Let us turn attention to the uninformed firm, B. Analogous to A, we study a
deviation in period 3. In this period, pA and pB are as in Proposition 5 and then B
could deviate and purchase the information. According to Proposition 1, tailored
prices set in period 4 would be such that B will capture the segment [ 12 , 1] in the
26
Old market22 , as opposed to zero, as it is implied by Proposition 5. It can be shown
that these new profits are equal to 4t . Therefore, B deviates if and only if T < 4t .
The above implies that, if there is an equilibrium where only one firm buys, it has
h
i
2
2
to be that T ∈ 4t , 35t −22ct+19c
≡ C. It can be verified that the interior always
64t
exists.
Now we look for a deviation in which B sets a price pB such that A does not buy
while B buys. We will show that such deviation always exists in C. Define a
deviation price as pD
B . Such a deviation is profitable for B and for A if and only if:
NT
∗
D
T
∗
D
∗
∗
πBT (pD
B , pA ) − T > πB and πA (pA , pB ) > πA (pA , pB ) − T
where πiT and πiN T are the profits from buying and not buying the information,
respectively (and the competitor behaves as stated in the deviation). Then reD
∗
mark that consumers’ actions are fixed so θ2 is fixed and denote θ1 (pD
B , pA ) = θ1 .
Assuming θ2 ≤ θ1D (which will be verified ex-post), the inequalities above imply:
Z
1
θ1D
Z
θ2
p∗A dθ
Z
+
0
pD
B dθ
1
Z
p∗A + t(2θ − 1)dθ − T >
+
θ2
θ1D
p∗A dθ
Z
>
0
Then a deviation exists for
1
2
Z
t(1 − 2θ)dθ +
θ2
pD
B
=
p∗B
θ2
0
if T ∈ D ≡
(5t − c)2
32t
p∗A dθ
Z
+
θ1D
p∗A dθ − T
0
9c2 +6ct+t2 −21c2 +10ct+75t2
,
64t
64t
. There-
∗
fore, it exists for pD
B in the neighborhood of pB . Finally, it can be verified that
C ⊂ D.
Intuitively, firm B (the uninformed firm) can set a slightly higher basic price at
stage 2 and force firm A not to buy at stage 3 while B buys itself. This strategy
works because consumers’ privacy choice is fixed at stage 2 and only those close to
A paid for privacy. Therefore, if B buys the information, A’s benefits from buying
are reduced. The next results characterizes equilibria.
Proposition 7: Suppose there is a price of information T . For c ≤ c,
1. If T ≤ 4t , both firms buy the information and consumers do not pay for privacy.
2. If T ≥ 2t , neither firm buys the information and consumers do not pay for
22
Note that θ2 <
1
2
in this case.
27
privacy.
Proof. Part 1: When both firms buy information, consumers do not pay for privacy,
and the equilibrium outcome is the one given in Proposition 1. In period 2 firm B
(wlog) can deviate and change its price such that (a) A does not buy and B buys
and such that (b) A does not buy and B does not buy. Similar to the proof of
Lemma 2, we can show that (a) is profitable when T >
t
4
and (b) is profitable when
T > 2t . Now consider the deviation in period 3 only where B does not buy the
information given pA = pB = t. It can be verified that if B does not buy it cannot
operate in the Old market (θ20 = 1) and gets half of the New market. This implies
that B deviates if:
3t
t
t
−T <
⇐⇒ T >
4
2
4
(5)
Hence, Equation (5) is enough to pin down an equilibrium where both firms buy
the information.
Part 2: Consider an equilibrium where no firm buys information. If firm B deviated
in period 3 alone it would make profits πBD = t + 2t . Hence, B deviates if:
t<t+
t
t
− T ⇐⇒ T <
2
2
It can be verified that there are no deviations that involve periods 2 and 3 together.
A direct consequence of Proposition 7 is that the seller F would set T =
t
4
and
both firms will buy. Then, by considering deviations at the buying stage we are able
to break exclusivity as an allocation rule. In this setting, the seller of information
makes less revenue, as compared to T in Proposition 6. Furthermore, consumers do
not have to pay for privacy because the new allocation strengthens competition, as
seen in Lemma 1. Indeed, it can be checked that consumer surplus is smaller under
exclusivity.
5.2
Markets of different sizes
We now allow the New market to be of size m > 0 while the size of the Old market is
still normalized to 1. We are interested in the effect of changing the relative market
size on the incentives to acquire information in the presence of privacy. This is why
28
we assume c ≤
t(3m+3)
3m+4
< t so that at least some consumers pay for privacy in every
case (θ2 ≥ 0).
The main effect of m on the equilibrium of the game is that it affects the incentives of B to operate in the Old market.23 We recall that, for B to operate in the
Old market, it needs to set a low price (smaller than t) because it faces disadvantageous competition from A, who holds costumer information. Furthermore, B can
only set one price for both markets. Therefore, if m is large it is more profitable for
B to keep a higher price and compete only in the New market. Conversely, if m is
small, forsaking the Old market will not be profitable. The next Lemma posits the
equilibrium when B does not operate in the Old market and when it does.
Lemma 3: Assume only firm A buys the information and consumers can pay
for privacy. There are two types of equilibria:
1. If B does not operate in the Old market:
pA = t +
t−c
3m(t − c)
2(t − c)
, pB = t +
and θ2 =
1 + 3m
1 + 3m
2t(1 + 3m)
2. If B operates in the Old market:
pA = t+
t − 2c
cm + (1 + m)t
3m(m + 1)(t − c) − cm
, pB = t−
and θ2 =
2 + 3m
(1 + m)(2 + 3m)
2(m + 1)(3m + 2)t
Proof. In equilibrium we must have pA = paA and pB = paB , then θ2 < θ1 . However,
reaction functions ought to be calculated taking the consumers’ actions as given.
In case 1, B’s reaction function is:
arg max m (1 − θ1 (pA , pB )) pB
pB
while A’s is:
arg max [mθ1 (pA , pB ) + θ2 (paA , paB )] pA
pA
These two equations lead to pA (θ2 ) and pB (θ2 ). Finally, because in equilibrium
pA = paA and pB = paB , we can plug these reaction functions into Equation (4).
Doing so, leads to the expression for θ2 .
In case 2 we solve in the same manner but we change B’s reaction function to
23
Note that B’s choice is always independent of A’s actions.
29
account for its operation in the Old market:
arg max
pB
1 − θ20 (pB ) + m (1 − θ1 (pA , pB )) pB
Remark that θ20 is a direct consequence of pA (θ) ≥ 0. Given that pA (θ) = pB +
(1 − 2θ)t is always chosen by A in period 4, θ20 is a function of pB and not of paB .
Note that the assumption c ≤
t(3m+3)
3m+4
is necessary to ensure that the proportion of
consumers who pay the privacy cost is positive.
Prices are decreasing in m when B does not operate in the Old market. In
fact, as m tends to infinity, prices converge to Hotelling prices. Intuitively, as the
New market becomes relatively more important, firms seek optimality only in that
market. This is not always true when B operates in the Old market. The next
result proves that a threshold m∗ (c) above which B does not operate in the Old
market exists.
Lemma 4: Assume only firm A buys the information and consumers can pay
for privacy. There exists m∗ (c) > 0, such that for m ≥ m∗ (c), B does not operate
in the Old market.
Proof. Compute the difference in B’s profits when B does not operate in the Old
market (case 1 in Lemma 3) and when B does (case 2 in Lemma 3):
2
m(c − (3m + 2)t)2 (−cm + 3m2 t + 4mt + t)
∆π =
−
2(3m + 1)2 t
2(m + 1)(3m + 2)2 t
The first term on the right hand side are the profits B makes when it does not
operate in the Old market. The second term are the profits when it does operate
there. It can be verified that ∆π goes to
t
6
as m increases to infinity. On the other
hand, as m goes to zero, ∆π tends to − 8t . Furthermore, it can be proved that ∆π
is always increasing. Therefore m∗ > 0 exists.
Finally, we study the information seller’s behavior for any m > 0. Clearly,
Lemma 4 gives us two cases to analyze based on whether m is greater of smaller
than m∗ (c). By Lemma 1 (which can be generalized to any m > 0), if both firms
buy information, consumers do not pay for privacy. Then, following Proposition 1,
firms make equal profits in both markets:
30
mt
2
in the New and
t
4
in the Old. The
next proposition establishes the main result of the section.
Proposition 8: For any m > 0, it is always more profitable for F to sell the
information only to one firm.
Proof. For m ≥ m∗ (c), B does not operate in the Old market. Then, as in the proof
t(2m+1)
− πB , where
of Proposition 6, it suffices to show πA − πB ≥ 2
4
Z
pB −pA +t
2t
πA = m
Z
θ2
Z
1
(pB + (1 − 2θ)t) dθ
pA dθ +
pA dθ +
0
0
Z
πB = m
θ2
1
pB −pA +t
2t
pB dθ
and pA , pB and θ2 are given by Lemma 3, 1.
For m < m∗ (c), B operates in the Old market. Again, as in the proof of Proposition
−
π
6, it suffices to show πA − πB ≥ 2 t(2m+1)
B , where
4
Z
pB −pA +t
2t
πA = m
Z
θ2
pA dθ +
0
Z
pA dθ +
0
Z
πB = m
pB +t
2t
1
pB −pA +t
2t
(pB + (1 − 2θ)t) dθ
θ2
Z
pB dθ +
1
pB +t
2t
pB dθ
and pA , pB and θ2 are given by Lemma 3, 2.
Proposition 8 implies that exclusivity in the allocation of information is robust
to any relative market size, provided that information is sold ex-ante.
6
Conclusions
In this paper we studied the effects of consumers’ privacy choice and price discrimination on prices, profits and consumer surplus. We highlighted the role of
consumers by endogenizing the privacy choice, which in turn depended on expected
prices and on privacy costs. Additionally, the model allowed us to relate the privacy
cost to the value of information that can be acquired from data providers.
The model predicts that privacy reduces competition. We find that a rise in
the privacy cost strengthens competition due to a reduction in the valuation of the
informed firm’s product. This leads to lower profits and higher consumer surplus.
31
We also show that the owner of consumer data chooses to sell it exclusively to one
firm, and that privacy strengthens this result as it increases the price of information.
Although the exclusivity result is robust to different relative market sizes, it crucially
depends on the ability of the data owner to commit to selling information prior to
customer prices being set. If instead information is sold on a spot market, after
basic uniform prices are chosen, exclusivity does not arise anymore. Thus the timing
of the sale of information is an important factor that affects both the positive and
the normative analysis of privacy and competition.
In terms of policy, consumers are better off when both firms buy consumer data.
This can be achieved endogenously in the model by allowing the sales to occur at
any time. In this situation, consumers would not pay for privacy. Furthermore,
both firms having information is efficient in the sense that it minimizes the total
transportation costs in both markets, even if the seller of information is better off
when it deals only with one firm.
References
[1] Acquisti, A. and H. Varian (2005). “Conditioning Prices on Purchases History”,
Marketing Science, Vol. 14(3), 367-381.
[2] Belleflamme, P and M. Peitz (2010). Industrial Organization, Markets and
Strategies, Cambridge University Press .
[3] Conitzer, V., C. Taylor and L. Wagman (2010). “Hide and Seek: Costly Consumer Privacy in a Market with Repeat Purchases”, Marketing Science, Vol.
31(2): 277-292.
[4] Directive 2002/58/EC of the European Parliament and of the Council of 12 July
2002 concerning the processing of personal data and the protection of privacy
in the electronic communications sector, 2002 O.J. L 201/37 .
[5] Esteves, R. (2010). “Pricing with customer recognition”, International Journal
of Industrial Organization, Vol. 28, 669-681.
[6] Fudenberg, D. and J. Tirole (2000). “Customer Poaching and Brand Switching”,
RAND Journal of Economics, Vol. 31(4), 634-657.
32
[7] Fudenberg, D. and J. Villas-Boas (2012). “Price Discrimination in the Digital
Economy”, In: M. Peitz and J. Waldgofel (Eds.), The Oxford Handbook of the
Digital Economy.
[8] Jehiel, P. and B. Moldovanu (2000). “Auctions with Downstream Interaction
among Buyers”, RAND Journal of Economics, Vol. 31(4), 768-791.
[9] Koh, B., S. Raghunathan and B. Nault (2013). “Is voluntary profiling welfare
enhancing?”, Working Paper.
[10] Mikians, J., L. Gyarmati, V. Erramilli and N. Laoutaris (2012). “Detecting
price and search discrimination on the internet”, Hotnets’12, October 29-30,
2012, Seattle, WA, USA., 79-84.
[11] Mikians, J., L. Gyarmati, V. Erramilli and N. Laoutaris (2013). “Crowdassisted Search for Price Discrimination in E-Commerce: First results”, Mimeo.
[12] Shaw, I. and N. Vulkan (2012). “Competitive Personalized Pricing: An Experimental Investigation”, Working Paper.
[13] Taylor, C. and L. Wagman (2014). “Customer privacy in oligopolistic markets:
winners, losers, and welfare”, International Journal of Industrial Organization,
Vol. 34, 80-84.
[14] Wall Street Journal (2012a). “On Orbitz, Mac Users Steered to Pricier
Hotels”,
23/08/2012,
retrieved
on
17/05/2014,
http://online.wsj.
com/news/articles/SB10001424052702304458604577488822667325882?
mg=reno64-wsj\&url=http\%3A\%2F\%2Fonline.wsj.com\%2Farticle\
%2FSB10001424052702304458604577488822667325882.html
[15] Wall Street Journal (2012b). “Websites Vary Prices, Deals Based on Users’
Information”, 24/12/2012, retrieved on 17/05/2014, http://online.wsj.
com/news/articles/SB10001424127887323777204578189391813881534?
mg=reno64-wsj&url=http\%3A\%2F\%2Fonline.wsj.com\%2Farticle\
%2FSB10001424127887323777204578189391813881534.html
33