Taxation and Privacy Protection on Internet Platforms∗ Francis Bloch† Gabrielle Demange‡ October 4, 2015

Abstract This paper studies data collection by a monopolistic internet platform We show that the optimal strategy of the platform is either to cover the market or to choose the highest data exploitation level, excluding users with high privacy costs from the platform. We analyze the effect of different tax instruments on the level of data collection and show that user-based taxes lead to an increase in data collection and the exclusion of users. Taxation with different rates according to the source of revenues, with higher tax level for revenues generated by data exploitation can reduce data collection. We also analyze the effect of opting-out options, letting users access the platform with no data collection under different financial transactions (absence of transfers, subscription fee for the opt-out option, compensation for the opt-in option). Finally, we consider the effect of competition from a platform offering access without data collection. JEL classification numbers: Keywords: options

H23, L86, L50

Digital services, Privacy protection, Taxation, Opt-out and opt-in

∗ We are grateful to France Strategie for funding this research within the framework of a Research Project on the ”Evolution of the Value created by the Digital Economy and its Fiscal Consequences”. We have benefited from comments and discussion with M. Bacache, P.J. Benghozi, M. Bourreau, B. Caillaud,, J. Cremer, S. Gauthier, L. Gille, J. Hamelin, L. Janin and J.M. Lozachmeur. † Universit´e Paris 1 and Paris School of Economics, 106-112 Boulevard de l’Hopital, 75647 Paris CEDEX 13, France. Email [email protected]. ‡ Paris School of Economics-EHESS, 48 Boulevard Jourdan, 75014 Paris, France . Email [email protected].

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1

Introduction

The precipitous decline in the cost of data collection and storage linked to the development of information technologies has transformed business models in advertising and commerce. While records on customers and sales histories have always existed, the digital economy now enables firms to exploit data at a much larger scale, opening up new opportunities for profit as well as new concerns about privacy and exploitation of personal data. Large sales platforms can now use detailed records of past sales histories to target users and engage in discriminatory dynamic pricing. Other platforms, like search engines or online social networks, use data on immediate search to auction off advertising spaces to clients, or sell search histories to intermediaries who accumulate data to better target users with ads. In fact, business models of all giant internet platforms rely at different degrees on the collection and exploitation of personal data. The use of personal data is clearly one of the main specificities of the digital sector in modern industrial economies. The development of ”big data” and its potential exploitation raises two separate questions. First, data are a valuable input for internet platforms, but users voluntarily upload their data without any payment. One can argue that internet platforms are engaged in a barter agreement, where platforms deliver a valuable service (targeted proposals for products, targeted ads, outcomes of search, access to friends) in exchange for the uploading of data. But absent any price and financial transaction, it is difficult to assess whether this barter is ”fair” and if users receive a fair share of the surplus. The immense profits of (some not all) internet platforms suggests that it may not be the case and that platforms benefit from a ”free” input which is not paid at its true value. Furthermore, in the absence of financial transaction, governments cannot properly tax the benefit of personal data, creating a distortion with respect to other sectors, clouding the territoriality principle for the taxation of profits, and leading to extremely low levels of taxation of internet platforms through a clever use of transfer prices and the absence of records of financial transaction in countries where users reside. Second, users are rightly afraid that the collection of personal data infringes on their privacy. In addition, the resale of data to unknown intermediaries through opaque arrangements results in a loss of control on the dissemination of personal data to third parties. The exploitation of data, while it provides a valuable service to users by improving targeting, also necessarily involves a cost in privacy loss. Even though the two problems of the absence of fair payment of data and privacy loss seem unrelated at first glance, they are in fact closely connected. In this paper, we study how regulatory instruments, and in particular taxation, can be used to solve both problems at once. We first investigate the effect of different forms of taxation – corporate profit taxation, taxation based on users or data flows, specific tax paid by users, revenue taxation based on differentiated rates according to the origin of the platform revenues. We then study regulation which mandates platforms to offer a service without data collection (letting for example users decide whether they want to leave cookies or not), either with no financial transaction, or with two forms of financial transaction: one where the platform collects a subscription fee from users choosing the opt-out option and one where the platform compensates users who choose the opt-in option. We construct a model where users are differentiated along their privacy cost. The collection of data enables the platform to better target offers to users and users to products or advertisers, 2

resulting in an increase in the benefit to users as well as the value of users to advertisers or to the firm. Initially, we suppose, following current usage in the digital industry, that platforms do not charge users for their service and that their entire revenue comes from the other side of the market (advertisers or sales of future goods). A monopolistic platform chooses (and commits to) a degree of data exploitation balancing two effects: on the one hand, an increase in data collection increases revenues by increasing the value of the user to advertisers or for targeted pricing, on the other hand, an increase in data collection may deter users with high privacy cost to access the platform. In this model, we first compute the optimal level of data exploitation chosen by the platform and the users. We show that the platform either chooses to ”cover the market” making sure that all users access the platform, or chooses the maximal level of data exploitation, thereby excluding some users from the platform. The welfare of users is also maximized at one of three values: either at a low value, or at the level of market coverage or at the maximal degree of data exploitation. While the optimal levels of data collection from the point of view of the platform and users cannot be directly compared (they depend on the particular shape of the revenue and benefit functions), anecdotal evidence suggests that the share of revenues linked to data collection is higher than the benefits of users from better service due to data exploitation, so that the platform’s optimal level of data collection is likely to be excessive from the point of view of users. Given that the platform chooses an excessive degree of data exploitation, we then study how different forms of taxation affect data collection. We first observe that a tax on profits (or equivalently a tax on revenues because variable costs are negligible) does not affect the choice of the platform. A tax paid by the platform per user does not affect the marginal benefit of data exploitation, but it reduces the profit made on the marginal user accessing the platform, thereby reducing the cost of data collection. Hence, a tax per user (or per flow of data as users do not choose the level of data they upload), results in an increase in data exploitation. A specific tax paid by users (like a tax on internet service providers) produces ambiguous effects on the degree of data exploitation, but always increases the region where exclusion occurs. The only tax which allows to correct for excessive data collection is a tax on revenues which treats differentially platform’s revenues accruing from one-time use (like auction revenues based on current keywords) and revenues linked to data collection (like resale of data to intermediaries). If fiscal authorities charge a higher tax level on resale of data than on auction revenues, taxes deter the platform from exploiting the data, playing the classical role of a Pigovian tax correcting for externalities. We then explore the effect of the introduction of an option for the users to access the platform with no data collection. This allows the platform to collect access revenue from those users who choose to opt-out, segmenting the market into two groups with different revenue levels. Users will now all access the platform – some with data collection and others without. We show that this changes the level of data collection chosen by the platform, resulting in a decrease in the maximal level of data exploitation for which the market is covered, but in an increase in the region of parameters for which the platform chooses the maximal degree of data exploitation. Hence, both from the point of view of the platform and from the point of view of users, the

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introduction of an opt-out option has ambiguous effects. When the access value to users is small, both the platform and the users benefit from the introduction of the zero-option, but when the access value to users is large, users are indifferent and the platform prefers the uniform policy. We then allow for further discrimination between the different types of users by letting the platform use pricing instruments. We first consider the case where the platform charges a subscription fee from users selecting the opt-out option. We show that the fee will always be set at the maximal level, transferring all the surplus from users choosing the opt-out option to the platform. The platform may choose to switch its business model entirely, foregoing data collection completely and collecting subscription revenues. Alternatively, the platform may choose to implement market coverage or set the highest level of data collection. Because the platform collects revenues from users choosing the opt-out option, its incentive to exclude users from the opt-in option is higher, and we observe exclusion more often than in the benchmark case. The platform always gains by proposing an opt-out option with subscription fee, and the effect on user welfare is ambiguous. We also analyze the effect of competition with a platform proposing the opt-out option. If a competing platform proposes the opt-out option at no cost, the level of data exploitation under market coverage goes down and the region of parameters for which exclusion occurs increases, reflecting the competition created by the opt-out option. The platform is hurt by the presence of a competitor but users benefit when the degree of data exploitation is initially excessive. If the competing platform selects a subscription fee, the equilibrium of the game where one platform selects the degree of data exploitation and the other platform a subscription fee results in two possible configurations: one where a single platform operates on the market, at a low level of data exploitation which guarantees market coverage and one where one platform selects the highest level of data collection and the other platform collects revenues charging the optimal subscription fee. Our analysis of the effect of taxation and regulation on data collection relies on an original model, but is related to two strands of the literature. First, it is related to the literature on the economics of media, which considers a media (television, newspaper) as a platform in a twosided market connecting readers with advertisers. (See Gabsewicz, Laussel and Sonnac (2001) and (2004) for early contributions to the literature and the survey by Anderson and Gabszewicz (2006)). Advertisement in these models play the same role as data collection in ours. As in our model, users are assumed to suffer a linear cost from advertisements. In the initial papers in the literature, the platform only collects revenues from the advertising market. Later papers, like Peitz and Valletti (2008), Choi (2006), Crampes, Haritchabalet and Jullien (2009)) also allow for subscription prices charged to viewers, and compare regimes of ”free-to-air” with ”pay-for-view” televisions. Reisinger, Ressner and Schmidtke (2009) and Reisinger (2012) pay close attention to competition among advertisers on different platforms. Most of the literature (with the exception of Anderson and Coate (2005)) considers competition between platforms which are horizontally differentiated. Anderson and Coate (2005) analyze the behavior of a monopolistic platform when the market is not covered. In their model, users are differentiated by their intrinsic benefit from the good and not their aversion to advertising, resulting in very different demand functions and different conclusions – for example, they find that platforms typically choose too

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little advertising whereas we observe that the level of data collected by the platform is likely to be excessive. Another difference stems from the shape of the utility that users obtain from ads. All models, with the exception of Crampes, Haritchabalet and Jullien (2009) assume that users suffer a linearly increasing utility loss for ads, whereas we assume that, in addition to a linearly decreasing loss, they obtain a concave utility gain due to improved service. (Crampes, Haritchabalet and Jullien (2009) allow users to have a positive value for low levels of ads, which is similar but not equivalent to our assumption.) The main difference between our analysis and the literature on media as two-sided platforms stems from the questions raised. The literature on media focuses on program differentiation and competition, while we are mostly interested in regulatory r´egimes and taxation to improve privacy protection. In that sense, our paper is more closely connected to recent work on ad-avoidance (Anderson and Gans (2011), Tag (2009) and Johnson (2013)), but differences in the models preclude a direct comparison between our results and theirs. Second, our paper is more distantly related to the literature on taxation on two-sided markets (see Kind et al. (2008), (2010a), (2010b), Kind et al. (2013) and Kotsogiannis and Serfes (2010)), but again the focus of the analyses are different, as we focus on the effect of taxes on privacy protection rather than revenues and distortion. Finally, we note that our paper is closely related to the two contributions by Cr´emer (2015) and Bourreau, Caillaud and De Nijs (2015) as all three papers consider the effect of specific taxation on the behavior of internet platforms, even though the three papers rely on different models of internet platforms and focus on different choice variables. The rest of the paper is organized as follows. We introduce our model in the next Section. Section 3 is devoted to the baseline model of a uniform data collection policy. Section 4 analyzes the effect of different tax instruments. Section 5 considers the introduction of a free option to access the platform with no data collection. Section 6 studies the binary model with financial transfers (either the platform paying users for data collection or users playing the platform for option zero). Section 7 allows for competition from a platform with no data collection. Section 8 concludes. All proofs not given in the texts are collected in Section 9.

2 2.1

The Model Platform and users

We consider an internet platform which provides services to users. The platform collects revenues either directly (as in the case of e-commerce) or from third parties like advertisers in the case of search engines or digital social networks. The platform also collects data from users, recording their history of activity on the platform. We distinguish between two sources of revenues: some of the revenues like sales revenues from e-commerce or instantaneous search-base advertising revenues are collected immediately, whether the platform records data on the customer or not. Other revenues, like the sale of personal data to aggregators or the expected revenue from future targeted advertising, only accrue if personal data are collected and stored. We suppose that the platform commits to the degree of exploitation of personal data, denoted x ∈ [0, 1]. The degree x can be interpreted along different dimensions. It can represent the duration of time during which personal histories are stored by the platform, the fraction of personal data which are sold 5

by the platform to third parties or kept for direct exploitation, or any specified limitation on the use of personal data. We denote by v0 the fixed value generated by a user independently of data collection and by v(x) the value generated by data exploitation, where v(·) is supposed to be increasing and concave. The total value of a user to the platform is V (x) = v0 + v(x). Users get a benefit from using the platform, which is also decomposed into a fixed component, u0 and a component which depends on the collection of data x. Data collection allows the platform to better match the user with products and hence results in an improvement in the service. We let u(x) denote the benefit from the improvement in match quality, so that the benefit of a user can be written as U (x) = u0 + u(x), where u(·) is increasing and concave. In addition, we suppose that users are aware that they may be harmed by the collection of storage and personal data, and suffer a privacy cost which is proportional to the degree of data exploitation. We suppose that users are heterogeneous in their sensitivity to privacy loss, and let θ denote the characteristic of the user with privacy cost θx. We normalize the payoff of a user who does not access the platform to zero. We let F (·) denote the distribution of privacy costs in the population of users. Given x, the marginal user accessing the platform, T (x) is given by the solution to the equation: u0 + u(x) − T (x)x = 0, or u0 + u(x) . x and, assuming that the marginal cost of providing digital services is negligible, the profit of the platform is given by T (x) =

u0 + u(x) ). x The total surplus of users accessing the platform with data collection x is given by Π(x) = [v0 + v(x)]F (

Z

T (x)

[u0 + u(x) − θx]dθ = [u0 + u(x)]F (

W (x) = 0

x u0 + u(x) 2 u0 + u(x) ) − F( ) x 2 x

(1)

(2)

Our model can be compared to models of interaction on media markets when users are averse to advertisement. In these models, x is interpreted as the level of advertising, v(x) the revenues from advertising and θx the cost of ads to users. Models of advertising in media markets typically assume that v0 = 0 (the media does not make any direct revenue from users) and that u(x) = 0 6

(users have no positive value for advertising).1 A more important difference stems from the dimension of heterogeneity of users. We assume that users have the same value for the service but different privacy costs, whereas models of media markets suppose that users have the same cost of advertising but differ in their value of the service or in their ideal point in models of horizontal differentiation across media. This difference between our model and other models of media with advertising avoidance results in non-trivial consequences. When users differ in their value for the service or ideal points, demand is linearly decreasing in x ; when users differ in their advertising or privacy cost, demand is no longer linear and in fact includes a part ux0 which is decreasing and convex in x. This simple observation will lead to a major difference in the characterization of optimal levels of data exploitation.

2.2

Regulatory instruments

The objective of the paper is to assess how different regulatory instruments affect the level of data collection chosen by the platform and the welfare of users. We distinguish between tax instruments and regulations imposing opt-out options. We investigate the incidence of four different tax instruments: • A tax τ levied on the revenue of the platform • A tax tP levied on the platform per user • A differentiated revenues tax system, with two different tax rates τ1 and τ2 applied to the revenues v0 and v(x) • A tax tU levied on each user for accessing the platform. As an alternative regulation, we consider a policy mandating that the platform offer a binary choice to the user with an opt-out option letting users access without data collection. We look at two possibilities • The opt-out option is offered at no charge • The platform charges a subscription price f for the opt-out option • The platform charges a subscription price f for the opt-out option and pays users p for the opt-in option • The platform faces competition from an other platform offering the opt-out option 1

One exception is Haritchabalet et al. (2009) who allow users to have a positive value for low levels of advertising as we do.

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3

The benchmark model

In the benchmark model, there is no public intervention and the platform freely chooses the degree of data exploitation x. For tractability, we suppose that both the platform’s value v(x) and the user’s benefit u(x) are iso-elastic in x and that the distribution of privacy costs is uniform on [0, 1]: Assumption 1 Suppose that v(x) = bxβ and u(x) = axα with a, b > 0 and α, β ∈ [0, 1]. In addition, let the distribution F (θ) be uniform over [0, 1]. Under Assumption 1, we compute the maximal level of data collection for which all users access the platform – the coverage level – as a function of the benefit of the platform u0 and the parameters a, α. The coverage level ξ(u0 ) is the solution to the equation: u0 + axα − x = 0.

(3)

If u0 + a > 1, all users access the platform even at the maximal degree of data exploitation x = 1. We focus attention on situations where the benefit of the platform to users is lower, so that the coverage level ξ is an interior value in [0, 1]. The following Lemma computes the effect of changes in the parameters on the coverage level ξ: Lemma 1 The coverage level ξ(u0 ) is increasing in u0 and a and decreasing in α. 1

Finally, notice that ξ(1) = 1 and ξ(1 − a) = a 1−α .

3.1

Platform profit

To compute the optimal level of data collection of the platform, we write down the platform’s profit as

Π(x) = v0 + v(x) if x ≤ ξ(u0 ), = [v0 + v(x)]T (x) if x ≥ ξ(u0 ). When the level of data exploitation is below the coverage level, all users access the platform and the profit is increasing in x. When the level of data exploitation is above the coverage level, the profit is no longer increasing in x everywhere. Instead, we compute the derivative of the profit for x ≥ ξ(u0 ) and derive the elasticity of profit with respect to data exploitation: ∂Π x ∂x Π

= =

xv 0 (x) T 0 (x)x + v0 + v(x) T (x) β α + u0 −α − 1. v0 −β +1 +1 a x b x 8

The elasticity of profit with respect to data exploitation is thus the sum of two convex functions of x. Figure 1 illustrates the shape of the profit of the platform as a function of x. Because of the fixed terms u0 and v0 , the elasticity of profit is strictly increasing in x. The profit of the platform is thus maximized at one of the two extreme values ξ(u0 ) or 1. There are two possible r´egimes depending on the values of the parameters: a r´egime of uniform coverage where the platform chooses x = ξ(u0 ) and serves all users and a r´egime of exclusion where the platform selects x = 1 and serves users with a privacy cost smaller than u0 + a. We summarize this finding in the next Proposition. < Insert Figure 1> Proposition 1 The platform optimally chooses a degree of data exploitation x ∈ {ξ(u0 ), 1}. It β 0) chooses ξ(u0 ) if and only if vb0 ≥ ν(u0 ) ≡ u0 +a−ξ(u 1−u0 −a . Proposition 1 establishes the existence of a threshold ν(u0 ) such that the platform chooses uniform coverage the market when the fraction vb0 is larger than ν(u0 ) and exclusion otherwise. This result is easily understood. The fraction vb0 measures the relative importance of the value due to one-time access v0 over the value due to data exploitation v(x). When the value due to access is large relative to the value due to data exploitation, the platform has an incentive to serve all users and hence will choose x low enough so that it covers the market. If the value of data exploitation is large relative to the value of access, the opposing intuition holds and the platform optimally chooses to restrict access and set a high degree of data exploitation. The next Lemma provides comparative statics results on the effect of changes in the parameters on the threshold ν(u0 ). Lemma 2 The threshold ν(u0 ) is increasing in α and β and decreasing in u0 . By Proposition 1, the platform is more likely to choose uniform coverage when ν(u0 ) is small. Lemma 2 thus shows that the platform chooses optimal coverage for low values of α and β (when values and benefits are not too concave) and for high values of u0 (when the benefit of users for access is high).

3.2

User welfare

We next turn to the characterization of the degree of data exploitation which maximizes the welfare of users, W . Under Assumption 1, we compute the sum of welfare of users as Z

T (x)

[u0 + u(x) − xθ]dθ

W (x) = 0

or x if x ≤ ξ(u0 ) 2 u0 + u(x) x u0 + u(x) 2 1 = [u0 + u(x)] − [ ] = [u0 + u(x)]2 if x ≥ ξ(u0 ). x 2 x 2x

W (x) = u0 + u(x) −

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When the level of data exploitation is below the coverage level, all users access the platform, 1 and welfare is strictly concave in x. The maximum is attained at x b = (2aα) 1−α , whenever x b ≤ ξ. Lemma 3 The value x b is independent of u0 , and increasing in a When the level of data exploitation is above the coverage level, the expression for welfare captures the fact that some users are excluded from the platform, and a computation of the elasticity of welfare with respect to the degree of data collection shows that ∂W x = ∂x W

2 − 1. +1

u0 axα

As u0 > 0, the elasticity of welfare is increasing in x, so that the maximum is either attained at ξ(u0 ) or at 1. Figure 2 illustrates the shape of the welfare of users as a function of x. The user welfare is either maximized at x b, ξ(u0 ) or 1. < Insert Figure2> Proposition 2 The welfare maximizing level of data collection is either x b, ξ(u0 ) or 1. If α ≤ 21 , 1 1 ≥ α ≥ 12 , the welfare maximizing level is x b. If α ≥ 2a , the welfare maximizing level is 1. If 2a the welfare maximizing level is either x b, ξ(u0 ) or 1. If 21 −a+ab xα − xb2 < 0, the welfare maximizing 1 level is 1. Otherwise, let u0 be the unique value of the access benefit for which W (b x) = W (1) and 2 1 2 u0 the unique value of the access benefit for which x b = ξ(u0 ). If u0 > u0 , welfare is maximized b for u > u10 . If u10 ≤ Uo2 , there exists a unique value u30 of the access at 1 for u0 < u10 and at x benefit for which W (ξ) = W (1) and welfare is maximized at 1 if u < u30 , at ξ(u0 ) if u30 < u0 < u20 and at x b if u > u20 . Proposition 2 details the configurations of parameters for which welfare is maximized at the low value of data collection x b, the market coverage level ξ(u0 ) or the highest level 1. While the conditions are complex, they reflect the following intuitions. When the benefit function is very concave (low values of α), user welfare is maximized at the lowest value of data collection, x b, which maximizes the welfare of the average user. When, on the other hand, the benefit function is not too concave (high values of α), welfare is maximized with the maximal level of data collection generating exclusion. In the intermediate range of α, the welfare maximizing level depends on the access benefit u0 . If the access benefit is large, user welfare is maximized by covering the market ; if it is low, user welfare is maximized by excluding users with high privacy cost. For one particular configuration of parameters, the three possible values x b, ξ(u0 ) and 1 emerge as welfare maximizing levels as the access benefit u0 decreases. Propositions 1 and 2 show that the optimal level of data collections of the platform and of users may differ and cannot be ranked uniformly for all parameters. The platform is more likely to choose exclusion when the access revenue is small and the revenue function not too concave; similarly, users are more likely to prefer exclusion when the access benefit is small and the benefit function is not too concave. The comparison between the optimal choice of the platform and of users thus depends on a comparison between the parameters u0 and v0 and α and β. Based on simple anecdotal evidence, we argue that users benefit more from access and the platform cares 10

more about data collection, so that the parameters satisfy u0 >> v0 and v(x) >> u(x). We thus expect the optimal level of data collection of the platform to exceed the welfare maximizing level of data collection of users.

4

Taxation

In this Section, we analyze the effect of the imposition of a tax on the optimal data exploitation strategy of the platform. The imposition of different taxes can be interpreted as changes in the parameters of the model. A tax τ on the revenues only scales down the profit and has no effect on the strategy of the platform. A tax per user paid by the platform reduces the access revenue from v0 to v0 − τP . A differentiated revenues tax reduces the access revenue from v0 to v0 − τ1 and the revenues generated from data collection from v(x) to v(x)(1 − τ2 ). A tax per user paid by users reduces the access benefit from u0 to u0 − τU . The effect of taxes on the optimal choice of the platform can thus be computed by using the comparative statics results of the previous Section. They are summarized in the following Proposition. Proposition 3 The imposition of a tax has the following effects on the degree of data exploitation by the platform: • An ad valorem tax τ on the profits or revenues of the platform has no effect on the degree of data exploitation • A tax paid by the platform per user τP results in an increase in the degree of data exploitation • In a differentiated revenues tax system, a tax τ1 on the access revenues results in an increase in the degree of data exploitation, whereas a tax τ2 on the revenues generated by data exploitation results in a decrease in the degree of data exploitation • A tax paid by users for accessing the platform τU has ambiguous effects on the degree of data collection but increases the probability that exclusion occurs. Proposition 3 shows that most taxes are likely to induce the platform to increase its degree of data exploitation. The only exception is a differentiated tax on revenues, which targets profits due to the use of personal data and, as any Pigovian tax, leads to a reduction in data exploitation. With the exception of the tax on users which generates ambiguous results, all other tax effects are robust and hold for general benefit and payoff functions. Clearly, a uniform tax on profits, or revenues are marginal cost are negligible, is neutral, and does not affect the platform’s choice of data exploitation. A tax per user paid by the platform has no effect on the marginal benefit of data exploitation on the users accessing the platform, but it lowers the loss of not serving the marginal user who chooses not to access on the platform. Hence a tax per user results in a higher degree of data exploitation. Similarly, a tax τ1 on the access revenue of users leaves the marginal benefit of data exploitation unchanged but reduces the loss of not serving the marginal user. A tax targeted at the revenues linked to data exploitation necessarily 11

0

(x) leads to a reduction in the ratio xV V (x) while leaving the effect on demand unchanged. It thus reduces the platform’s incentive to collect and exploit data and results in a lower optimal value of data exploitation. The imposition of a tax per user τP may induce another negative effect when users are differentiated along an additional dimension: a value η ∈ [0, η] parametrizing the revenue of the user to the platform (measured for example in frequency of access or demographic characteristics). If the legal system allows the platform to discriminate among users, it will optimally choose to exclude users with low value, namely those users for whom

ηi V (x) ≤ Any user with low value who accesses the platform will thus switch from a positive benefit U (x) − θ(x) to a benefit of 0: exclusion following the imposition of a tax per user will further reduce the welfare of users.

5

Binary policy with no payment

In this section, we suppose that the platform is mandated to propose an opt-out policy, where users are guaranteed that data will not be collected but suffer from the resulting loss in service quality. For example, users may refuse to be geo-localized or refuse to let cookies monitor their activity, but will in consequence obtain a lower quality of service on the platform. Users now freely choose between two options: one with no data collection and one with a single positive level x of data collection.2 As each user prefers the free service with no data collection to the outside option (u0 > 0) all users access the platform. We examine the optimal level of data collection with an opt-out option, compare the profit of the platform and the welfare of users with and without the opt-out option. A θ-user chooses the positive level of data collection x instead of the opt-out option if and only if u(x) + u0 − θx ≥ u(0), or θ <

u(x) x .

With the opt-out option, the coverage level is the solution to axα − x = 0, 1

or ξ(0) = a 1−α , the coverage level defined in the benchmark model when u0 = 0. The profit of the platform is

Π(x) = v0 + v(x) if x ≤ ξ(0), u(x) Π(x) = v0 + v(x) if x ≥ ξ(0). x 2

For reasons of tractability and technical implementability, we do not consider more complex policies where the platform discriminates among users by proposing several positive degrees of data collection.

12

Notice that the platform now collects the access revenue v0 on all users, and that the demand for data collection is independent of the access benefit u0 . Inspection of the profit shows that Π(x) is increasing in x for x ≤ ξ(0) and either increasing or decreasing in x for x ≥ ξ(0) depending on whether α + β exceeds 1 or not. Hence we characterize the profit-maximizing degree of data collection as follows. Proposition 4 With a binary policy without payment, the platform optimally chooses a degree of data exploitation x ∈ {ξ(0), 1}. It chooses ξ(0) if and only if α + β ≤ 1. Proposition 4 shows that, as in the benchmark case, the platform either chooses to cover the market or sets the highest possible degree of data exploitation. It covers the market when the revenue and benefit functions are highly concave, α + β ≤ 1. It chooses the maximal degree of data exploitation whenever the revenue and benefit functions are less concave. The comparison between the level of data collection of the platform with and without the opt-out option produces ambiguous results. On the one hand the coverage level ξ(0) is always lower than the coverage level in the benchmark case, ξ(u0 ). The existence of the opt-out option forces the platform to reduce x. On the other hand, the region of parameters for which the platform chooses exclusion may increase. The platform now collects the access revenue v0 on all users and does not have an incentive to lower x to attract users when the access revenue is high as in the benchmark case. This effect may lead the platform to choose exclusion at the highest degree of data exploitation when the opt-out option is present, whereas it chooses to cover the market in the benchmark case. We next compare the profit of the platform in the benchmark case and with the opt-out option. Proposition 5 The profit of the platform is higher in the benchmark case than under the binary policy without payment whenever α + β < 1. If α + β ≥ 1, there exists a threshold u ˜0 ∈ (0, 1 − a) of the access benefit such that the profit of the platform is higher in the benchmark case than in the binary policy without payment if and only if u0 > u ˜0 . Proposition 5 shows that the only situation where the platform willingly proposes a binary policy without payment is when the access benefit of the users is low and the revenue and profit functions are not very concave. In all other circumstances, the platform is hurt by the introduction of the opt-out policy. The intuition underlying Proposition 5 is easy to grasp. When α + β < 1, the platform covers the market at level ξ(0) < ξ(u0 ). Given that, when the market is covered, profit is increasing in x, this policy is always dominated by a coverage policy at ξ(u0 ) which can be chosen by the platform in the benchmark case. When α + β ≥ 1, the argument is based on the observation that the profit of the platform in the binary policy is independent of u0 . Instead, in the benchmark case, the profit of the platform is strictly increasing in u0 , as an increase in u0 uniformly increases demand to the platform. At u0 = 0, the choice between coverage and exclusion is identical under the uniform and binary policies (both choose to exclude when α + β ≥ 1), and the profit of the platform is strictly higher under the binary policy as it collects access revenue v0 on all users. On the other hand at u0 = 1 − a, the profit of the platform is higher under the uniform policy, as it attracts all users by setting the maximal level 13

of data collection. Hence there exists a threshold u ˜o ∈ (0, 1 − a) under which profit is higher under the binary policy and over which profit is higher in the benchmark case. We now ask whether users benefit from the switch from a uniform to a binary policy. The answer to this question is not easy. On the one hand, if the uniform policy involves exclusion, excluded users benefit from the introduction of the opt-out option. On the other hand, the introduction of the opt-out option may lead to a deviation of the level of data collection away from the optimum, hurting users who access the platform. The following Proposition details situations under which the uniform and binary options are preferred by users. Proposition 6 If α+β ≤ 1, user welfare is larger under the binary policy than in the benchmark case if and only if (i) the platform chooses market coverage under the uniform policy and x b< 2

1 1−α

ξ(u0 ) or (ii) the platform chooses exclusion under the uniform policy and (u0 +a) < u0 + a 2 . If 2 α + β ≥ 1, user welfare is larger under the binary policy than in the benchmark case if and only 2 if (i) the platform chooses market coverage under the uniform policy and a2 > aξ(u0 )α − ξ(u20 ) or (ii) the platform chooses exclusion under the uniform policy Proposition 6 characterizes situations under which users favor the introduction of the optout option. When both policies result in market coverage, users prefer the binary policy which results in lower levels of data collection if and only if the welfare maximizing level of data collection is x b < ξ(u0 ). When both policies result in exclusion, users prefer the binary policy which provides access to excluded users the access benefit u0 . When the two policies result in different levels of market coverage, the comparison is unclear and depends on the values of the parameters.

6

Binary policy with payments

6.1

Subscription fee for the opt-out option

We analyze in this Subsection the situation where the platform charges a subscription fee f for the opt-out option. The policy is thus characterized by two variables: the degree of data exploitation x and the subscription fee f . Charging a subscription fee results in two positive effects for the platform: it provides an additional source of revenue and makes the opt-out option less attractive, increasing the user’s incentive to choose the option with data collection. Notice that the platform can also choose to concentrate on the subscription fee, by committing not to exploit any data. We denote this option by ∅. Given the policy (x, f ) a θ-user chooses the positive level of data collection x rather than the opt-out option if u(x) + u0 − θx ≥ max(u0 − f, 0). It is never optimal for the platform to charge a subscription price f larger than u0 .3 Thus we assume f ≤ u0 . Hence a θ-user picks the opt-out option if and only if θ ≥ u(x)+f . The x 3

For f > u0 , agents who do not choose x do not access the platform at all. By charging the price f = u0 −  for a positive and small , the platform could generate a positive profit equal to u0 + v0 −  on each of these users.

14

subscription price f formally plays a similar role than the access benefit u0 in the benchmark case. The coverage level is thus ξ(f ) and the profit of the platform is equal to Π(x, f ) = v0 + v(x) if 0 < x < ξ(f ), u(x) + f Π(x, f ) = [v(x) − f ]( ) + v0 + f if x > ξ(f ) x When the option ∅ is chosen, the platform collects f + v0 from each user.4 The platform charges the maximal fee: f = u0 and obtains a profit Π(∅, u0 ) = u0 + v0 . This profit associated to policy (x, f ) is smaller than the profit achieved at ∅ if v(x) < u0 , which necessarily happens if b < u0 . In that case, the platform has an incentive to focus on the subscription fee and forego any revenue due to data exploitation. Otherwise, as in the benchmark case, the platform chooses a positive level of data exploitation which is either equal to the coverage level or to the maximal level. Proposition 7 Under a binary policy where the platform charges for the opting-out option, the platform always chooses the maximal fee f = u0 . If b < u0 , the platform commits not to collect data: the optimal policy is ∅ and the revenues are generated by the subscription fee collected on all users. If b > u0 , the platform collects data and optimally chooses x ∈ {ξ(u0 ), 1}. It chooses the lower level ξ(u0 ) when bν(u0 ) + u0 ≤ 0. Proposition 7 shows that the optimal policy is to focus on the subscription fee and serve all users when the value of data is low relative to the access benefit, b < u0 . In that case, the maximal revenue that the platform can obtain from data collection is v(x) ≤ b < u0 = f and the platform has no incentive to collect and exploit data. If instead, the value of data is high relative to access benefit, b > u0 , the platform optimally proposes an option with positive data collection. The maximal level x = 1 can only improve the profit over the option ∅ because it leads to the extraction of value b (rather than u0 ) from users who choose to provide data. We then show, as in the benchmark case, that the profit reaches a maximum either at the coverage level ξ(u0 ) or at the maximal level 1, and characterize the condition under which profit is reached as a function of the threshold ν(u0 ) defined in the benchmark case. We now compare the optimal policy with the uniform optimal policy described in Proposition 1. In contrast to the uniform policy, the optimal choice does not depend on the access revenue v0 because this revenue is collected from each user independently of the data collection policy. Furthermore, the maximal degree of exploitation is chosen more often than under the 0 uniform policy. Covering the market with the binary policy is only profitable if ν(u0 ) < −u b v0 whereas market coverage is the optimal policy in the benchmark case when ν(u0 ) < b . Hence there is an interval of threshold values [ ub0 , vb0 ] for which the platform chooses exclusion under the 4

The policy (∅, f ) is not equivalent to (x = 0, f ): x = 0 allows the user to access the service without a fee charge. This explains why the profit Π(0, f ) differs from Π(∅, f ).

15

binary policy and market coverage under the uniform policy. This can be explained intuitively as follows. Under market coverage, the platform does not collect any fee, and the revenues are the same under the two policies. Under exclusion at x = 1, the users who choose the costly option at f = u0 are exactly those who do not access the platform under the uniform policy. As the platform collects revenues from the users opting out, it receives a higher profit under the binary policy with payment than under the uniform policy. This argument also shows that the profit of the platform is always higher under the binary policy than in the benchmark case. User welfare is lower under the binary policy when the platform commits not to collect data, as all users receive a zero utility. When b > u0 , users’ welfare is identical under the two policies when they result in the same market coverage. If 0 ν(u0 ) < −u b , all users access the platform at the level of data collection ξ(u0 ) for both policies. When ν(u0 ) > vb0 , under both policies exclusion occurs at the maximal level of data collection, and users who do not access the platform under the uniform policy or select the opt-out option v0 0 under the binary policy receive a zero utility. When −u b < ν(u0 ) < b , the comparison between user welfare under the two policies depends on the parameters. User welfare is higher under the binary policy if and only if ξ(u0 ) < (u0 + a)2 , a condition which, as we saw in the proof of Proposition 2, is satisfied whenever α > 21 and ξ(u0 ) < x b.

6.2

Subscription price for the opt-out option and payment for data collection

In this Subsection, we suppose that the platform can not only charge a fee for the opt-out option but also pay users for their data.5 The platform thus chooses three instruments: the degree of data exploitation x, the fee f paid by users for the opt-out option and the price p paid to users who choose the opt-in option. As in the previous subsection, we can focus attention on subscription fees f ≤ u0 and assume that all users access the platform. As in the previous Subsection, the platform can choose to commit not to collect data (∅) and charge the subscription fee u0 from all users, resulting in the profit Π(∅) = u0 + v0 . As in the previous Subsection, this policy is optimal if and only if b ≤ u0 . We now consider the case b > u0 . A θ-user chooses the positive level of data collection x rather than f the opt-out option if and only if u(x) + u0 − θx + p ≥ u0 − f. +p Hence a θ-user picks the opt-out option if and only if θ ≥ u(x)+f . The coverage level is equal x to ξ(f + p) and the situation is similar to the benchmark case when u0 is replaced by p + f . Observe that the subscription fee and the data price have the same impact on the user’s choice between the opt-out and opt-in options. The profit of the platform is

Π(x, f, p) = v(x) − p + v0 + f if x ≤ ξ(f + p), u(x) + p + f Π(x, f, p) = (v(x) − p − f )( ) + v0 + f if x ≥ ξ(f + p). x Notice that in order to induce users to choose the opt-in option, the platform can either use the data price or the subscription fee. The fee results in additional revenues whereas the price results 5

We assume that the price paid to users is always positive. We do not consider the case where the platform asks for a subscription fee both from users choosing the opt-out option and users choosing to opt in.

16

in additional costs. Hence, the platform always has an incentive to increase the subscription fee f rather than the price p. And a positive price only emerges when the platform cannot raise the subscription fee anymore, i.e. when the subscription fee is equal to u0 . In order to analyze situations where the platform chooses a positive price for the data, we may thus restrict attention to the situation where f = u0 . In addition, we observe that it cannot be optimal for the platform to choose a degree of data exploitation x < ξ(p + u0 ) so that we must have x ≥ ξ(p + u0 ). For any fixed x, we now derive the optimal price data price p chosen by the platform, by computing the derivative of profit with respect to p: ∂Π v(x) − u(x) − 2(p + u0 ) = if x > ξ(u0 + p) ∂p x At x = ξ(u0 + p) the profit has a kink with a right derivative equal to −1. The optimal price satisfies p = 0 if v(x) − u(x) − 2u0 ≤ 0 p = x − u(x) − u0 if 0 < x − u(x) − u0 ≤ p =

v(x) − u(x) − u0 2

v(x) − u(x) v(x) − u(x) − u0 if 0 < − u0 ≤ x − u(x) − u0 2 2

≥ x − u(x) is equivalent to x ≤ ˚ x where ˚ x is the unique positive The condition v(x)−u(x) 2 6 value that satisfies v(˚ x) + u(˚ x) − 2˚ x = 0. To summarize, if v(x) − u(x) − 2u0 ≤ 0 the price is null, and if v(x) − u(x) − 2u0 ≥ 0 the price is given by p = x − u(x) − u0 if x ≤ ˚ x v(x) − u(x) p = − u0 if x ≥ ˚ x 2 For x ≤ ˚ x, the price is chosen to cover the market and the profit is given by v(x) + u(x) − x + u0 + v0 . The platform optimally selects the level of data exploitation x e that maximizes v(x) + u(x) − x over [ξ(0), ˚ x], assuming ξ(0) < ˚ x. 7 In that case, offering a positive data price allows the platform to increase the degree of data exploitation at which the market is covered. Notice that, since v(x) is the value of data x to the platform and u(x) − x the benefit of a user with maximal privacy cost, the degree of data exploitation x e maximizes the sum of the surplus of the platform and of the user with highest privacy cost. For x > ˚ x, the optimal price leads to exclusion. As in the benchmark case, the elasticity of profit is increasing in x so that the platform optimally chooses the highest degree of data exploitation. We summarize our findings in the following Proposition. 6 The existence of a unique value ˚ x satisfying this condition follows from the fact that the condition is equivalent v(x)+u(x) to v(x)+u(x) ≥ 2 and that is decreasing, larger than 2 for x = 0 and smaller than 2 for x = ∞ Notice x x that ˚ x does not depend on u0 . 7 Again, notice that x e does not depend on u0 .

17

Proposition 8 Let b ≥ u0 . The platform always chooses the maximal fee f = u0 . If v(x) − u(x) < 2u0 for any x, the platform never chooses to pay for data, and the choice is described in proposition 7. If v(x) − u(x) ≥ 2u0 , the platform 1. either chooses to cover the market, with a level of data exploitation is ξ(u0 ) or x e if ξ(u0 ) < x e < ˚ x. In the first case, the price is null, in the second case, it is positive given by x e − u(e x) − u0 ; 2. or to exclude and pick x = 1. The price is null if otherwise.

b−a 2

< u0 , or positive given by p =

b−a 2 −u0

Proposition 8 shows that two situations can happen when the platform chooses to pay users for their data. In the first case, when x e is chosen, the platform covers the market and uses the payment to increase the degree of data exploitation. In the second case, the platform selects exclusion at the highest degree of data exploitation and uses payment to increase the number of users choosing the opt-in option. The platform’s choice between the two strategies depends on the parameters in a complex way. The platform’s degree of data exploitation can be higher or lower than the optimal degree when the platform does not pay users fro the data. We illustrate the optimal choice of the platform in the simple case where u0 = 0. In that case, with no payment for data, by Proposition 4, the optimal degree of data exploitation is 1 ξ(0) = a 1−α if α + β ≤ 1 and 1 if α + β ≥ 1. Note that that ˚ x is the unique solution to the α β equation ax = bx = 2x and x e the unique solution to the equation αaxα−1 + βbxβ−1 = 1. 1−β Assume first that ξ(0) ≥ ˚ x (a 1−α ≥ b) or x e < ξ(0) (a + βb ≤ 1) or ˚ x a. We observe that if α + β < 1, the platform may switch from market coverage in the r´egime without payment to exclusion in the r´egime with payment with a positive price for the data.8 Next consider the case where ξ(0) < x e<˚ x. In that case, positive prices may emerge both when the market is covered and under exclusion. We simplify the analysis by assuming α = β. 1 1 1−α and x We then have ˚ x = ( a+b e = (α(a + b)) 1−α and ξ(0) < x e if a < α(a + b), x e<˚ x if α < 12 . 2 ) The optimal policy when the market is covered is to choose x e with a profit equal to (a + α b)e x −x e = (1/α − 1)e x (up to v0 ). The optimal policy when the market is not covered is to 2 choose x = 1, offering the positive price (b − a)/2 with a profit equal to ( a+b 2 ) (up to v0 ). 2 Thus, the optimal policy is to cover the market if (1/α − 1)e x ≥ ( a+b 2 ) . When α = 1/2, a+b 2 x e = ( 2 ) and the two profit expressions are equal. This profit is obtained either by covering b−a market with x e and a price equal to x e − u(e x) = ( a+b 2 )( 2 ), or by proposing x = 1 with the larger price ( b−a x decreases with α we conclude that the exclusion r´egime is chosen for 2 ). As (1/α − 1)e α < 1/2, as in the case without payment for the data. 8

1−β

1−β

For example if a 1−α ≥ b, this switch may happen if a 1−α ≥ b > a, a condition which requires β > α and b > a.

18

7

Competition with a platform without data collection

In this Section, we analyze the effect of competition from another platform offering access without data collection.9 Competition between advertising-based and fee-based platforms has been analyzed in the economics of media (see Choi (2006), Peitz and Valletti (2008), Crampes, Haritchabalet and Jullien (2009)). In the context of our model, this competition can easily be analyzed, both when the competing platform only collects access revenues v0 and when can it charge a subscription fee f .

7.1

Competition with a platform without subscription fee

If the platform faces competition from another platform which does not charge a subscription fee, a θ-user accesses the platform if and only if u0 + u(x) − θx ≥ u0 . Hence the coverage level is the same as when the platform proposes the opt-out option for free 1 and given by ξ(0) = a 1−α ≤ ξ. As in the benchmark case, we observe that Π(x) is either maximized at ξ(0) or at 1 and obtain the following characterization: Proposition 9 If the platform faces competition from another platform offering the opt-out option for free, it optimally selects x ∈ {ξ(0), 1}. It selects the coverage level ξ(0) if and only if v0 b ≥ ν(0). Proposition 9 shows that the presence of a competing platform reduces the degree of data collection when the market is covered from ξ to ξ(0). In addition, observe that, by Lemma 2, ν is decreasing in u0 so that ν(0) ≥ ν(u0 ). This last observation shows that competition with a platform proposing an opt-out option results in an increase in the region of parameters for which exclusion occurs. Hence, as in the case of a platform proposing a binary option without payment, competition results in a decrease in the level of data collection but an increase in the region of parameters for which exclusion occurs. The platform is clearly hurt by competition, because its profit goes down both under market coverage (from v0 + bξ(u0 )β to v0 + bξ(0)β ) and under exclusion (from (v0 + b)(u0 + a) to (v0 + b)a). Under market coverage, users benefit if the level of data collection ξ(u0 ) is excessive ; under exclusion, users benefit as the excluded users now have access to the service proposed by the competing platform.

7.2

Competition with a platform with subscription fee

We now consider a situation where the competitor chooses a subscription fee f for his service. The two platforms are involved in a noncooperative game where one platform chooses the level of data exploitation x and the other one the subscription fee f , and we characterize the Nash 9

A recent example of of a platform proposing access without advertising nor data collection is the digital social network Ello which was launched in March 2014 and claims above a million users.

19

equilibrium of the game played by the platforms. A θ- user accesses the platform with data collection if and only if u0 + u(x) − θx ≥ u0 − f, and observe that the market coverage is ξ(f ) where ξ(f ) is known, by Lemma 1 to be increasing in f . The profit of platform 1, the platform with data collection is Π1 (x) = v0 + v(x) if x ≤ ξ(f ), u(x) + f ] if x ≥ ξ(f ). Π1 (x) = [v0 + v(x)][ x while the profit of platform 2, the platform with the opt-out option, is Π2 (f ) = 0 if x ≤ ξ(f ), u(x) + f Π2 (f ) = (v0 + f )(1 − ) if x ≥ ξ(f ). x We check that the best response of platform 1 to f is to select x = ξ(f ) whenever v0 +bξ(f )β ≥ (v0 + b)(f + a) and x = 1 otherwise. The best response of platform 2 to x is to select any value f if ξ(0) ≥ x, to choose f = 0 if x ≥ ξ(0) and v0 + u(x) > x and the interior value f ∗ = x−v02−u(x) if v0 + u(x) < x. We will suppose that a + v0 < 1. Based on the derivation of the best response functions, we characterize the Nash equilibria of the game of platform competition. Proposition 10 If vb0 ≥ ν(0), the game of platform competition has a pure strategy equilibrium where platform 1 chooses market coverage at ξ(0) and platform 2 chooses a subscription fee f = 0. If vb0 ≤ ν(f ∗ ) , the game of platform competition has a pure strategy equilibrium where platform 1 chooses exclusion at the maximal level of data exploitation and platform 2 chooses subscription fee f ∗ . If ν(f ∗ ) < vb0 < ν(0), the game of platform competition does not have an equilibrium in pure strategies. Proposition 10 shows that two situations can arise: (i) one where platform 1 covers the market and platform 2 cannot attract any user even at a subscription fee equal to zero and (ii) one where platform 1 chooses exclusion and platform 2 selects the optimal subscription fee f ∗ to maximize its profit. For some region of the parameters, a pure strategy equilibrium does not exist and the mixed strategy equilibrium involves platform 1 randomizing between market coverage and exclusion and platform 2 randomizing between the optimal fee f ∗ and a zero subscription fee.

8

Conclusion

This paper studies data collection by a monopolistic internet platform We show that the optimal strategy of the platform is either to cover the market or to choose the highest data exploitation 20

level, excluding users with high privacy costs from the platform. We analyze the effect of different tax instruments on the level of data collection and show that user-based taxes lead to an increase in data collection and the exclusion of users. Taxation with different rates according to the source of revenues, with higher tax level for revenues generated by data exploitation can reduce data collection. We also analyze the effect of opting-out options, letting users access the platform with no data collection under different financial transactions (absence of transfers, subscription fee for the opt-out option, compensation for the opt-in option). Finally, we consider the effect of competition from a platform offering access without data collection. Our analysis provides a first step in understanding how different regulatory instruments affect the optimal degree of data collection of an internet platform ( or equivalently the optimal level of advertising of a media platform). Our analysis is based on a simple model of behavior of platforms and users and three directions seem particularly fruitful to follow in future research. First, we wish to study how another dimension of heterogeneity among users (for example demographic characteristics and the frequency of access) affects our results. Second, we would like to open the black box of the value of data to the platform, and construct a full-blown model of advertising or dynamic pricing to derive the revenues generated by the data. Finally, we plan to expand the analysis by allowing the platform to choose a larger menu of policies with different degrees of data exploitation in order to discriminate more finely among users.

9

Proofs

Proof of Lemma 1: By implicit differentiation of equation 3 ∂ξ ∂u0 ∂ξ ∂a ∂ξ ∂α

= = =

1 > 0, 1 − αaξ α−1 ξα > 0, 1 − αaξ α−1 aξ α log ξ < 0. 1 − αaξ α−1

Proof of Proposition 1: We compare Π(ξ) and Π(1). We have Π(ξ) = c0 + bxβ0 and Π(1) = (v0 + b)(u0 + a) This immediately gives that ξ is chosen when

v0 b

≤ ν.

Proof of Lemma 2: We compute the signs of ∂ν ∂α ∂ν ∂β

= −

∂ξ > 0, ∂α

= − log xxβ > 0. 21

(4)

To compute the effect of changes in u0 , we compute the sign of ∂ξ ∂ν = 1 − βξ β−1 (1 − u0 − a) − ξ β . ∂u0 ∂u0 ∂ν ∂u0

When u0 converges to 1 − a, with respect to u0 : ∂2ν ∂u20

= βξ β−1

converges to zero. We also compute the second derivative of ν

∂ξ ∂ξ 2 − β(β − 1)ξ β−2 [ ] (1 − u0 − a) ∂u0 ∂u0

∂2ξ ∂ξ (1 − u(0) − a) − βξ β−1 2 ∂u0 ∂u0 ∂2ξ ∂ξ 2 ] (1 − u0 − a) − βξ β−1 2 (1 − u(0) − a) = −β(β − 1)ξ β−2 [ ∂u0 ∂u0

− βξ β−1

Next note that aα(α − 1)ξ α−2 ∂ξ ∂2ξ = < 0, (1 − aαxα−1 )2 ∂u0 ∂u20 so that finally we conclude guarantees that

∂ν ∂u0

∂2ν ∂u20

> 0 which together with the fact that

∂ν ∂u0

= 0 at u0 = 1 − a

< 0.

Proof of Lemma 3: By immediate differentiation 1 α ∂b x 1 = (2α) 1−α a 1−α > 0. ∂a 1−α

1

1

Proof of Proposition 2: Suppose that α ≤ 21 . At u0 = 0, ξ = a 1−α ≥ (2aα) 1−α so ξ ≥ x b. Because ξ is increasing in u0 and x b independent of u0 by Lemmas 1 and 3, ξ ≥ x b for all u0 . Next compute the difference between welfare at ξ and welfare at 1:

∆ ≡ =

u0 + aξ α (u0 + a)2 − . 2 2 ξ (u0 + a)2 − . 2 2

To prove that ∆ > 0, it is sufficient to prove that the market is covered at x = (u0 + a)2 . To this end, compute the utility of an agent with privacy cost 1 as U

= u0 + a[(u0 + a)]2α − (u0 + a)2 , ≥ u0 + a[u0 + a] − (u0 + a)2 , = u0 [1 − a − u0 ] ≥ 0. 22

1

1 Next suppose that α ≥ 2a . At u0 = 1 − a, ξ = 1 < x b = (2aα) 1−a . Because ξ is increasing in u0 and x b independent of u0 by Lemmas 1 and 3, ξ < x b for all u0 . Now compute the derivative of ∆ with respect to u0 :

∂∆ 1 = − 2(u0 + a). ∂u0 1 − αaξ α−1 1 Because ξ < x b, αaξ α−1 > 21 so that 1−αaξ α−1 > 2 ≥ 2(u0 + a) and hence the difference ∆ is increasing in u0 . At u0 = 1 − a, ξ = 1 and ∆ = 0, so that ∆ < 0 for all u0 < 1 − a, establishing that user welfare is maximized at 1. 1 Finally suppose that 12 ≤ α ≤ 2a . At u0 = 0, ξ ≤ x b and at u0 = 1 − a, ξ ≥ x b. Because ξ is increasing in u0 and x b independent of u0 by Lemmas 1 and 3, there exists a unique threshold u20 such that ξ = x b if and only if u0 = u20 , ξ > x b if u0 > u20 and ξ < x b if u0 < u20 . Compute also the difference between welfare at x b and welfare at 1:

x b (u0 + a)2 − , 2 2 u2 a2 x b = − 0 − au0 + u0 − + ab xα − . 2 2 2

∆0 ≡ u0 + ab xα −

Because 1 − u0 − a > 0, ∆0 is increasing in u0 . At u0 = 0, 2∆0 = −a2 + 2ab xα − x b, 1

≤ −a2 + 2ab x2 − x b, 1

= −(a − x 2 )2 < 0. At u0 = 1 − a, 1 x b − a + ab xα − . 2 2 If ∆0 < 0 at u0 = 1 − a, then ∆0 < 0 for all u0 , and welfare is maximized at 1, because x) if W (1) > W (b x) ≥ W (ξ). If ∆0 > 0, there exists a unique threshold u10 such that W (1) = W (b 1 u = u0 , W (1) > W (b x) if u < u10 and W (1) < W (b x) if u > u10 . To conclude the proof, suppose that u0 > max{u10 , u20 }, then ξ ≥ x b and W (b x) > W (1) so welfare is maximized at x b. If u0 < u10 , then W (1) > W (b x) ≥ W (ξ) so welfare is maximized at 1. Otherwise, if u20 > u0 > u10 , we remark that, as ξ < x b, ∆ is increasing in u0 . At u0 = u20 , W (ξ) = W (b x) > W (1) and ∆ > 0. At u = u10 , W (1) = W (b x) > W (ξ) and ∆ < 0. Hence there exists a unique value u30 such that W (ξ) = W (1) if u = u30 , W (ξ) < W (1) if u10 < u0 < u30 and W (ξ) > W (1) if u30 < u0 < u20 . ∆0 =

Proof of Proposition 3: From Proposition 1, we observe that a reduction in v0 increases the region of parameters for which exclusion is chosen while not affecting ξ. A reduction of v(x), which can be interpreted as a reduction in b, leaves ξ unaffected and reduces the region 23

of parameters for which exclusion is chosen. A reduction in u0 increases the bound ν, thereby increasing the region of parameters for which exclusion occurs. However, a reduction in u0 also reduces the coverage level ξ, thereby reducing the degree of data exploitation when the platform chooses to cover the market. Proof of Proposition 6: If α + β ≤ 1, the platform chooses xB = ξ0 < ξ under the binary policy. If the platform chooses market coverage under the uniform policy, the users are better off under the binary policy of and only if x b < ξ. If the platform chooses to exclude under the uniform policy, the expected welfare under the uniform policy is W = (u0 + a)2 whereas the expected welfare of a user under the binary policy is 1 1 W 0 = u0 + a 1−α 2 The conclusion follows. If α + β ≥ 1, the platform chooses the maximal level of data collection under the binary policy. If it also selects x = 1 under the uniform policy, excluded users are clearly better off under the binary regime whereas users accessing the platform have the same utility. Suppose next that the platform chooses market coverage under the uniform policy. The expected welfare under the uniform policy is ξ W = u0 + aξ α − , 2 whereas the expected welfare of a user under the binary policy is W 0 = u0 +

a2 , 2

The conclusion follows. Proof of Proposition 7: The case b ≤ u0 has been considered in the text. Assume b > u0 . We know that ∅ is sub-optimal. Consider a policy (x, f ) with x 6= ∅. The profit is π(x, f ) = (v(x) − f )T (x, f ) + v0 + f with T (x, f ) = min(

u(x) + f , 1). x

Assume first u(x)+f < 1. Then T (x, f ) = u(x)+f is decreasing in x – fewer users choose x x the x option when x increases– and increasing in f – more users choose the x-option when f increases. Hence, for x, f such that x < 1 and f < u0 , we can simultaneously increase x and f while leaving T (x, f ), the number of users choosing option x, unchanged. Now, for a fixed value of T (x, f ), the platform’s profit is increasing in x and non-decreasing in f (it is increasing in f if the value of T (x, f ) is strictly less than 1 and constant if T (x, f ) is equal to 1). This implies that x = 1 and/or f = u0 at an optimal policy. We show that f < u0 is never optimal. Assume by contradiction that f < u0 at an optimal policy. We must have x = 1. The optimal value of f is min( b−a+1 2 , u0 ). Thus, f < u0 can be 24

optimal only if b−a+1 < u0 , which is equivalent to b < 2u0 +a−1. By assumption u0 +a−1 < 0, 2 thus surely b < u0 . But for b < u0 we know that (∅, u0 ) is the optimal policy. Assume now f = u0 . Using the same argument as before, the optimal choice of the platform is either x = ξ, which is equal to the value x that covers the market in the uniform case, or x = 1. We compare the profit at the two values x0 = ξ(u0 ) and x = 1 for which we have respectively T (x0 , u0 ) = 1 and T (1, u0 ) = u0 + a: π(x0 , u0 ) = bxβ0 + v0 and π(1, u0 ) = (b − u0 )(u0 + a) + u0 + v0 .

(5)

The difference in profits write π(x0 , u0 ) − π(1, u0 ) = b(xβ0 − u0 + a) − u0 (1 − u0 − a) or, using the definition of ν, −(bν + u0 )(1 − u0 − a). immediately gives that x0 is chosen when bν + u0 ≤ 0. In particular, ν must be non-positive, a case where x0 is surely chosen under the uniform policy. Proof of Proposition 8: Let b > u0 . We know that x ≥ ξ(u0 + p) at an optimal policy. Fix p and consider the profit in x for x > ξ(u0 + p), i.e., for levels under which the market is not covered. Arguing as in the case of a uniform policy, Section 3, the profit is convex in x. Thus the profit is maximal at one of the boundary values for x: either at x = ξ(u0 + p) and the market is covered or at the maximal level x = 1. Given x, the profit is concave in p. The derivative of the profit with respect to p is strictly ∂π lower than that with respect to f : ∂π ∂p = ∂f − 1. It immediately follows that p can be positive at the optimal policy only if f is equal to the maximal level u0 . Since when p = 0 the fee is optimally set equal to u0 , we necessarily have f = u0 at the optimum. We determine the optimal price given x and f = u0 : p = x − u(x) − u0 if x ≤ ˚ x v(x) − u(x) p = − u0 if x ≥ ˚ x 2 Lemma 4 1. If ξ(u0 ) > ˚ x, then the market is not covered at any x for which the optimal price is positive. 2. Let ξ(u0 ) ≤ ˚ x. For x in ]ξ(u0 ), ˚ x], the optimal price is positive, given by x − u(x) − u0 , and all users select the opt-in option (covered market). In particular, if ˚ x ≥ 1, i.e., if b + a ≥ 2, the market is surely covered. Proof of Lemma 4. The first point follows from the definition of ˚ x and the fact that x ≥ ξ(u0 ). For the second point, consider the condition for the positivity of p. Write v(x) − u(x) − 2u0 = [v(x) + u(x) − 2x] + 2[x − u(x) − u0 ]. The first term inside the square bracket is positive for x<˚ x, the second term is positive for any x larger than ξ(u0 ). The result follows. Let us consider the two possible regimes. Covered market From Lemma 4, this case can happen only if ξ(u0 ) ≤ ˚ x and x belongs to (ξ(u0 ), ˚ x]. The profit is ΠC (x) = v(x) + u(x) − x + u0 + v0 . ΠC which is concave in x. Recall 25

that x e is defined as the maximizer of ΠC over x ∈ [ξ(0), ˚ x]. The optimal level of data collection is thus • ξ(u0 ) if x e ≤ ξ(u0 ); in that case, the price is null • x e if x e ∈]ξ(u0 ), ˚ x]; in that case, the price is positive. In particular, using 2 of Lemma 4, if ˚ x > 1, i.e., if b + a ≥ 2, the optimal policy is the maximum of ξ(u0 ) or x e, and the price is positive in the latter case. If furthermore αa + βb > 1, x e = 1: the market is covered, data exploitation is maximal and the price is positive. Finally, if ξ(u0 ) < x e=˚ x, we show that x e cannot be the global optimum: the derivatives (v(x)+u(x))2 of v(x) + u(x) − x and are equal at ˚ x. Since the derivative is positive, raising x is 4x profitable. Thus x = 1. Exclusion We know that x = 1 if exclusion is chosen. The price is null if b − a − 2u0 < 0 with a profit given by the right hand side of (5) or equal to b−a 2 − u0 otherwise with a profit equal to a+b 2 ( 2 ) + u0 + v0 . Proof of Proposition 10: The computation of the best response of platform 1 follows the same steps as the computation of optimal level of data exploitation in previous sections. For platform 2, if x ≤ ξ(0), the profit is zero for any level of the fee f . If x > ξ(0), profit is a concave function of f and is positive at f = 0. If u(x) + v0 > x, profit is decreasing in f for all f > 0 and the optimal choice is f = 0 ; if u(x) + v0 < x, profit is increasing in f at f = 0 and attains its 0 −x maximum at f ∗ = u(x)+v . If at f = 0, platform 1 prefers market coverage, then equilibrium 2 is attained when f = 0, x = ξ(0). If at f ∗ , platform 1 prefers exclusion, equilibrium is attained when x = 1 and f = f ∗ . Otherwise, there cannot be a pure strategy equilibrium. There is no pure strategy equilibrium where x 6= ξ(f ), 1 as platform 1 will only choose ξ(f ) or 1. If x = ξ(f ) and f > 0, platform 2 has an incentive to lower its fee and get a positive profit. If x = 1 and f 6= f ∗ , platform 2 has an incentive to change its fee to f ∗ .

10

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Choi, J.P. (2006) “Broadcast competition and advertising with free entry: Subscription vs. free-to-air,” Information economics and Policy, 18, 181-196. Crampes, C., C. Haritchabalet and B. Jullien (2009) “Advertising, competition and entry in media industries ,” Journal of Industrial Economics , 57, 7-31. ´mer, J. (2015) “Taxing network externalities ,” mimeo., PSE Toulouse. Cre Gabszewicz, J.J., D. Laussel and N. Sonnac (2001) “Press advertising and the ascent of the pensee unique ,” European Economic Review , 45, 645-651. Gabszewicz, J.J., D. Laussel and N. Sonnac (2004) “Programming and advertising competition in the broadcasting industry ,” Journal of Economics and Management Strategy, 13, 657-669. Johnson, J. (2013) “Targeted advertising and advertising avoidance ,” RAND Journal of Economics, 44, 128-144. ¨ hler, F. (2013) “Newspaper Differentiation and Kind, H.J. and Schjelderup, G. and Sta Investments in Journalism: The Role of Tax Policy,” Economica, 80, 131-148. Kind, H.J. and Koethenbuerger, M. and Schjelderup, G. (2010a) “Tax responses in platform industries,” Oxford Economic Papers, 62, 764-783. Kind, H.J. and Koethenbuerger, M. and Schjelderup, G. (2010b) “On revenue and welfare dominance of ad valorem taxes in two-sided markets,” Economics Letters, 104, 86-88. Kind, H.J. and Koethenbuerger, M. and Schjelderup, G. (2008) “Efficiency enhancing taxation in two-sided markets,” Journal of Public Economics, 92, 1531-1539. Kotsogiannis, C. and Serfes, K. (2010) “Public goods and tax competition in a two-sided market,” Journal of Public Economic Theory, 12, 281-321. Peitz, M. and T. Valletti (2008) “Content and advertising in the media: Pay-tv vs freeto-air” International Journal of Industrial Organization, 26, 949-965. Reisinger, M., L. Ressner and R. Schmidtke (2009) “Two-sided markets with pecuniary and participation externatlities” Journal of Industrial Economics, 57, 32-57. Reisinger, M. (2012)) “Platform competition for advertisers and users in media markets” International Journal of Industrial Organization, 30, 243-252. Tag, J. (2009) “Paying to remove advertisements,” Information Economics and Policy, 21, 245-252.

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11

Figures

p(x)

0

x0

Figure 1: Profit of the platform Π(x)

28

1

x

W(x)

0

x

x0

1

Figure 2: Welfare of users W (x)

29

x