## Abstract

This article investigates a second-degree discrimination scheme where an online platform sells a two-version service to consumers involved in a random network. In particular, consumers choose between purchasing a premium or a free version of the service. The premium version is sold at a price and enables higher network externalities than the free version. The free version includes advertising about some product—unrelated to the service. Under the assumptions that (a) advertising rotates clockwise the inverse demand of the advertised product and (b) the platform receives a fixed portion of the revenue from the sales of the advertised product, I explore (1) how the random network, and the market conditions for the advertised product, relate to the optimal pricing of the service, and (2) the welfare implications for the platform and the consumers. Hazard rate functions are crucial for optimal pricing, and first-order stochastic dominance of the degree distribution characterizes the welfare implications. The model provides foundations for empirical analysis on degree distributions and hazard rate functions underlying complex social networks.

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## Notes

- 1.
- 2.
First, it requires the platform to have full information about the entire network, which may be unfeasible in complex (and, sometimes, rapid evolving) networks. Secondly, setting a particular price to each consumer depending on her position in the network may result too costly for the platform even if it is able to identify completely the neighbors of each consumer.

- 3.
A (non-exhaustive) list of companies that have an online platform to sell their services over social networks using technologies that allow them to attach ads includes Google and Yahoo (web search engine), Facebook, Twitter, Instagram, and Snapchat (social interaction), Whatsapp, Skype, and Line (communication) AirBnB (accommodation search), Amazon (retail, product search, and market matching), Waze (traffic and route forecasting), BBC, CNN, The New York Times (news), The Weather Channel (weather forecasting), Yelp and Foursquare (review and rating), YouTube, Vimeo, Apple—through the Apple Music application—and Spotify (entertaining and information), Strava, MapmyRun, Nike Training Club, and Azumio (exercise and health tracking), Box (second-hand trade), or Tinder (dating). The platforms owned by these companies provide either only the free version, only the premium version, or both versions. Over the recent years, many of these companies have switched between these policies, without a clear recognizable pattern.

- 4.
Though not in all cases, premium versions usually allow consumers to have (exclusive) access to features that exploit the network externalities to a further extent relative to the corresponding free versions. Examples include real-time, current-location weather information (The FollowMyWeather feature at The Weather Channel), access to network-based training programs (MapmyRun and Nike Training Club), access to post own’s information and to receive information from other users (Premium versions of MapmyRun and Strava), access to a wider set of matching options for trade (Prime premium option at Amazon), access to a broader set of network-based items (music files in Spotify and video files in the Red premium version of YouTube), more detailed network-based health analyses (premium versions of Azumio’s health services), access to sharing options (sharing music or listening to music shared by others in the premium version of Apple Music), or access to post news (in the premium version of the CNN application for tablets and smartphones). Leaving aside the benefits from removing the ads, the rationale for platforms to build or enable higher network externalities is to make the premium version far more attractive than the free one so as to compensate premium version adopters for the price charged.

- 5.
See, e.g., Jackson (2008) where empirical regularities of social networks such as

*small worlds*,*clustering*, or*assortativity*are discussed at length. - 6.
Intuitively, advertising makes “more disperse” or “more heterogeneous” the consumers’ valuations of the product.

- 7.
The online appendix presents an application to signaling advertising of the assumed demand rotation effects.

- 8.
Unlike the classical benchmark of second-degree discrimination proposed by Maskin and Riley (1984), wherein the two types of consumers “self-select” themselves based only on their exogenous valuations of some commodity, the model proposed here considers that consumers self-select themselves (to adopt one or another version of the service) depending on the price of the service, on their connections in the network, and on the difference in the network externalities across both versions of the service. Once such self-selection has taken place, advertising affects (endogenously) the valuations of the product (only) for those consumers that choose the free version. This, in turn, affects the platform’s choice on the price of the service.

- 9.
Google and Amazon are prominent examples of such advertising practices, which in practice may allow them to implement very complex compensation contracts on the advertised products.

- 10.
Examples of platforms that have some level of integration with the company that sells the advertised product are The Weather Channel with IBM, Nike Training Club with Nike, MapmyRun with Under Armour, Apple Music with Beats, or Amazon with Whole Foods Market.

- 11.
In their seminal study of the World Wide Network, Barabási and Albert (1999) documented that the degree distribution of the nodes on the Internet adjusts to a power law distribution. More recently, Clauset et al. (2009) found empirical evidence that both the nodes on the Internet (at the level of autonomous systems) and the number of links in websites adjust to a power law distribution. Also, Stephen and Toubia (2009) argue that the emerging social commerce network through the Internet follows a typical power law distribution. In perhaps the largest structural analysis conducted up to date, Ugander et al. (2011) argue that the Facebook social network features decreasing hazard rates as well, though not necessarily fitting a power law distribution. Also, decreasing hazard rates are always generated by two ubiquitous models of the theoretical literature on random networks: the

*preferential attachment model*proposed by Barabási and Albert (1999) and the*network-based search model*suggested by Jackson and Rogers (2007). - 12.
- 13.
See also Dessein et al. (2016) within the organizational literature.

- 14.
- 15.
For instance, Galeotti and Goyal (2009) relate optimal marketing strategies, both under

*word-of-mouth*communication and adoption externalities, to the characteristics of the random network. Under the assumption that consumers only inform their neighbors if they themselves purchase the product, Campbell (2013) proposes a dynamic model of optimal pricing and advertising in random networks where information diffusion is endogenously generated. More recently, Fainmesser and Galeotti (2016), building upon the insights into perfect discrimination by Candogan et al. (2012), and Bloch and Quérou (2013), explore pricing and welfare implications when consumers are heterogenous with respect to their influence abilities. Using random networks, Leduc et al. (2017) have recently proposed a model of sequential product adoption where the monopolist offers referrals payments. Interestingly, their results on optimal pricing and optimal profits can be related to the degree distribution of the underlying random network as well. - 16.
Formally, versioning corresponds to an interior optimal choice whereas serving only either the premium version or the free version is associated with corner optimal choices.

- 17.
Parameter \(\alpha \) can be interpreted as the level of integration that exists between the platform that sells the two-version service and the monopolist that sells the advertised product. The value \(\alpha =1\) reflects that both companies are totally integrated. This happens to be the case in some real-world instances, such as IBM and the Weather Channel, Under Armour and MapmyRun, Amazon and Whole Foods Market, Amazon and Zappos, or Apple Music and Beats. While Facebook is reported to have acquired 65 different businesses since 2005, Apple participates in 91 different companies. Over the last 16 years, Google is reported to have acquired (either fully or partially) over 200 companies from 16 different countries. The products offered by the companies integrated with the companies that own the platform are often advertised by the corresponding platform.

- 18.
As indicated in Introduction, the plausible complex contractual relationships that in practice regulate compensations from advertising in the explored environment are not the subject matter of this paper, which is why the analysis abstracts from them by imposing a reasonably parsimonious and tractable assumption.

- 19.
The upper limits \(\overline{\omega }\) and \(\overline{\theta }\) are allowed to tend to infinity.

- 20.
The social network includes the links that are provided by the platform as part of its service but a consumer’s neighbors can include connections derived from other online platforms, or from informal relations such as family, friendship, or working relations. In this sense, the platform allows the consumers to form links, but it does not have the power to design itself the entire network. In most situations, platforms can be assumed as not being able to break or create connections of family, friendship, co-working, or social links facilitated by other platforms or channels.

- 21.
In practice, such externalities often take the form of informational gains (e.g., weather forecast, traffic monitoring, news services, or review and rating services), collaborative gains (e.g., joint entrepreneurial projects or online gaming), gains from being able to interact with a higher number of people (e.g., communication, exercise tracking, or dating services), or gains from facilitating transactions (e.g., retail matching or second-hand trade).

- 22.
In some applications, one might consider \(\underline{n}=0\). Also, the set \([\underline{n},\overline{n}]\) could be unbounded as well and, in particular, \(\overline{n}\) could tend to infinity in some applications.

- 23.
Formally, under the assumptions of the

*configuration model*, the expected consumption of some version of the service of a consumer*i*’s neighbor can be computed as$$\begin{aligned} \frac{\int ^{\overline{n}}_{\underline{n}} m h_{s}(m) \left[ \mathbb {P}(z_{i}=1 \, | \, n_{i}=m)+\mathbb {P}(z_{i}=0 \, | \, n_{i}=m) \right] \text {d}m}{\int _{\underline{n}}^{\overline{n}}n h_{s}(n)\text {d}n}=1 \end{aligned}$$because, given the assumption \(\underline{\theta }>0\) (so that each consumer adopts at least the free version of the service), we trivially have \(\mathbb {P}(z_{i}=1 \, | \, n_{i}=m)+\mathbb {P}(z_{i}=0 \, | \, n_{i}=m)=1\) for each degree \(m \in [\overline{n},\underline{n}]\).

- 24.
The

*degree independence assumption*states that the nodes of the network regard their shared links as independently chosen from the random network. This is a very common assumption in the literature on random networks and has been used, among others, by Jackson and Yariv (2007), Galeotti et al. (2010), Fainmesser and Galeotti (2016), and Shin (2016). - 25.
Formally, for the continuous distribution case, \(r_{s}(n)\) is the probability that the number of neighbors of a randomly selected consumer lies in the interval \((n-\varepsilon ,n+\varepsilon )\), for \(\varepsilon >0\) sufficiently small.

- 26.
Specifically, advertising is persuasive when it raises the consumer’s propensity to pay for the advertised product and it is complementary when the consumption of the product is complementary to that of the ads.

- 27.
While the general effects of the three types of advertising are rather similar, in the sense that all of them ultimately affect the consumers’ tastes for the product, they are conceptually different and might lead to different implications. In particular, persuasive and complementary advertising always raise individual demands, which leads to an upwards shift of the inverse demand function, whereas informative ads need not do so because consumers might learn that their tastes are indeed not well suited to the product’s characteristics.

- 28.
The conditions described by Assumption 1 are totally analogous to those stated in Definition 1 of Johnson and Myatt (2006). Johnson and Myatt (2006), though, consider instead that the inverse demand of the product is parameterized by a continuous set of parameters and allow the crossing points to vary with the parameter. The conditions stated in Assumption 1 give us the analog for the case with two parameters where there is a fixed crossing point.

- 29.
When both distributions \(F_{0}(\omega )\) and \(F_{1}(\omega )\) have the same mean, the relation described by Assumption 1 implies that \(F_{1}(\omega )\) second-order stochastically dominates \(F_{0}(\omega )\).

- 30.
Importantly, given the structure of revenues described by Eq. (4), considering only two possible degrees of advertising is without loss of generality in the current benchmark. Johnson and Myatt (2006) consider that demand rotates clockwise for the case where the parameter

*a*is drawn from a bounded interval and, under (the appropriate version of) Assumption 1, show that (the appropriate version of) the profit function in (4) is quasi-convex in*a*. It follows from their key insight (Proposition 1) that, if the monopolist were allowed in the current setting to choose an advertisement level \(a \in [0,1]\), then it would optimally choose either \(a=0\) or \(a=1\). - 31.
Otherwise, the problem would be trivial as the monopolist would prefer to provide no advertising and, thus, the platform would offer only the premium version of the service.

- 32.
Because of its second-mover advantage, by solving its decision problem, the platform determines the optimal profits of the monopolist.

- 33.
Since each density \(h_{s}(n)\) is strictly positive on the support \([\underline{n},\overline{n}]\) and \(\overline{\theta } \ge \beta \overline{n} \equiv \overline{q}\), the derived form for the demand of the premium version the service entails that such interior choices are necessarily associated with prices \(\underline{\theta }<q^{*}_{s} <\overline{\theta }\). In other words, the platform does in fact implement discrimination through versioning when we restrict attention to interior optimal prices.

- 34.
- 35.
- 36.
The

*configuration model*was originally developed by Bender and Canfield (1978) and, since then, it has been extensively used by a number models, such as Bollobás (2001), Newman et al. (2001), Jackson and Yariv (2007), Fainmesser and Galeotti (2016), and Shin (2016), among many others. A nice discussion of the configuration model, and of its relation with other random graph models, is provided by Jackson (2008). The idea behind the configuration model is that each consumer*i*with degree \(n_{i}\) gets randomly linked to a set of size \(n_{i}\) of other consumers according to a weighted uniform distribution where the weights are determined by the corresponding degrees \(n_{j}\) of the consumers in the remaining sample. As already mentioned, the preference specification in (1) encompasses the assumptions of the configuration model because, under the assumption that \(\underline{\theta }>0\), all consumers purchase some version of the service. In other words, the expected proportion of consumers that purchase some version of the service equals one, as captured by (1), when the configuration model is formally used in our setting. - 37.
For instance, if we let \({\varOmega }_{s}(n) \equiv [{1-H_{s}(n)}]/{r_{s}(n)}\), then it follows that \(r^{\prime }_{s}(n)>0\) ensures \({\varOmega }^{\prime }_{s}(n)<0\). The right-hand side of the inequality provided by Proposition 3 can then be written as \(\beta {\varOmega }\big (n(q^{\mathrm {pm}}_{s}) \big )\), where the cutoff degree

*n*(*q*) increases with*q*. For the special case of degree distributions with increasing hazard rate functions, we observe that the right-hand side of the inequality in the sufficient condition provided by Proposition 3 decreases as the optimal service price—corresponding to the case where only the premium service being served—increases. This incentivizes the platform to raise the price of the service. If such incentives are maintained as the price increase—which algebraically translates into that the platform’s profits are always increasing in \(q_{s}\)—then the platform ultimately offers only the free version of the service. - 38.
Recently, an increasing number of platforms (such as Twitter, YouTube, Facebook, or Google, just to mention a few of the most prominent ones) had adhered to this strategy. Anecdotally, such platforms have often reported on the media to regard potential expansions of the social networks where they operate as beneficial for their businesses because they allow them to increase considerably its audience for ads.

- 39.
As mentioned earlier—footnote 3—some of the most prominent platforms have reportedly changed their policies regarding to versioning over the recent years, without a clearly recognizable pattern. Just to mention a few examples, YouTube and Amazon have recently transitioned from policies with only the free version of their services to new ones where they pursue versioning by providing premium versions as well (Red YouTube and Prime Amazon) that enhance the network externalities enjoyed by the consumers. On the other hand, platforms such as AirBnB or Kayak have switched from policies with the two versions of their services to others where only the free version is provided.

- 40.
While the model proposed in this paper is fairly general, obtaining a more detailed specification of the platform’s optimal pricing strategy requires to be more specific about the distribution of consumers’ valuations. The proposed framework is general with respect both to the class of demand functions considered and the stochastic laws that govern the social network. In a companion paper, Gonzalez-Guerra and Jimenez-Martinez (2017) restrict attention to the signaling role of advertising and use a uniform distribution of the consumers’ valuations to obtain sharper predictions of how the platform discriminate through versioning.

- 41.
In a recent study on the Flickr social network, Scholz (2015) provides empirical evidence that sparsely connected networks adjust reasonably well to power law distributions.

- 42.
The density measure of a network specifies the proportion of possible links that are actually present in the network. Algebraically, it is given by the coefficients \(l/n(n-1)\), for directed networks, and \(2l/n(n-1)\), for undirected networks, where

*l*denotes the number of links present in the network and*n*indicates the number of nodes in the network. - 43.
Casual observation suggests that this is the case for platforms such as Nike Training Club, Strava, or MapmyRun. Such local networks are certainly much smaller than the networks where Google or Twitter operate and, conceivably, each user can access to the information posted by most of their local users.

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## Acknowledgements

I am grateful to the Associate Editor (Rabah Amir) for his advice and encouragement, and to Oscar González, for earlier joint work in this project. For very useful feedback and valuable advice, I thank two anonymous referees, Francis Bloch, Marc Escrihuela, Benjamin Golub, Navin Kartik, Luciana Moscoso, Christopher Sánchez, Joel Sobel, Joel Watson, Jorge Zatarain, and seminar participants at UC San Diego and at the Third Conference on Network Science and Economics at Washington University, St. Louis. Part of this project was conducted while visiting the Department of Economics at UC San Diego. I am deeply grateful to this institution for its generous hospitality and support. Any remaining errors are my own.

## Author information

### Affiliations

### Corresponding author

## Additional information

This is a substantial revision of a paper by Oscar González and Antonio Jiménez-Martínez formerly circulated under the title “Second-Degree Discrimination with Advertising in Random Networks”. In addition to the previous version, the current paper supersedes portions of Jimenez-Martinez (2017)’s “‘Versioning’ with Advertising in Social Networks”. I acknowledge financial support from CONACYT, Grant 41826.

## Electronic supplementary material

Below is the link to the electronic supplementary material.

## Appendix: Omitted Proofs

### Appendix: Omitted Proofs

**Proof of Lemma** 1

### Proof

Consider a given degree distribution \(H_{s}(n)\) and an optimal price for the consumption product \(p^{*}_{s} \in (\omega ^{R},\overline{\omega })\). Using the profit specification in (5), we know that the platform’s goal is to choose a price for the service \(q^{*}_{s} \in (\underline{\theta },\overline{\theta })\) so as to maximize its profits

The first-order condition for interior optimal prices to this problem is

Since the density \(h_{s}(n)\) is strictly positive in the support \([\underline{n},\overline{n}]\) and \(\overline{\theta } \ge \beta \overline{n} \equiv \overline{q}\), interior optimal prices \(q^{*}_{s} \in \left( \underline{\theta },\overline{\theta } \right) \) ensure that \(x_{s}(q^{*}_{s}) \in (0,1)\). Therefore, by dividing the first-order condition above over \(x_{s}(q^{*}_{s})\), we obtain that if \(q^{*}_{s}\) is an interior optimal price then, necessarily,

From the preference specification in (1) and the assumptions on the degree distribution, it follows that \(x_{s}(q^{*}_{s})=1-H_{s}\left( n(q^{*}_{s})\right) \) and \(x^{\prime }_{s}(q^{*}_{s})=-\left( \frac{1}{\beta }\right) h_{s}\left( n(q^{*}_{s})\right) \). Furthermore, from the definition of hazard rate function of the degree distribution in (3), we obtain

Then, an interior optimal price for the service \(q^{*}_{s}\) must satisfy the first-order condition

As for the second-order condition for interior optimal prices to the platform’s problem, differentiation of the obtained first-order condition yields

From the equalities \(x_{s}(q^{*}_{s})=1-H_{s}\left( n(q^{*}_{s})\right) \) and \(x^{\prime }_{s}(q^{*}_{s})= -\left( \frac{1}{\beta }\right) h_{s}\left( n(q^{*}_{s})\right) \), we observe that the first-order condition for interior optimal prices derived earlier can be rewritten as

Using this expression for the first-order condition, together with \(x^{\prime \prime }_{s}(q^{*}_{s})=-\left( \frac{1}{\beta }\right) ^{2}h^{\prime }_{s} \big (n(q^{*}_{s})\big )\), the required second-order condition can then be expressed as

Then, by rearranging terms in the inequality above, the required second-order condition can be written as

Finally, to ensure that the platform optimally chooses an interior optimal price \(q^{*}_{s}\) rather than serve only the free version of the service, we need to impose the condition that the optimal profits to the platform from serving only the free version, \(\alpha \pi _{1}(\omega ^{*}_{1})\), do not exceed the optimal profits from such an interior optimal price \(q^{*}_{s}\). By combining the first-order condition for interior optimal prices with the expression for the platform’s profits derived in (5), it follows that the optimal profits for an interior optimal price are given by

Therefore, to guarantee that the corner choice where the platform serves only the free version does not yield higher profits than the interior choice \(q^{*}_{s}\), we need to impose the condition

as stated. \(\square \)

**Proof of Proposition** 2

### Proof

Consider the function defined as

so that \({\varGamma }\big (p^{*}_{s}(q^{*}_{s}), q^{*}_{s} \big )=0\) gives us the first-order condition (7) that was obtained in Lemma 1. It follows that

and

Then, by applying the *implicit function theorem*, we obtain

Moreover, for an interior optimal price for the service \(q^{*}_{s} \in \left( \underline{\theta }, \overline{\theta } \right) \), the first-order condition to the problem \(\max _{p \in (\omega ^{R},\overline{\omega })} \pi (p,q^{*}_{s})\), where \(\pi (p,q)\) are the profits from the product’s sales specified in (4), leads to

Since the density \(h_{s}(n)\) is strictly positive in the support \([\underline{n},\overline{n}]\) and \(\overline{\theta } \ge \beta \overline{n} \equiv \overline{q}\), interior optimal prices \(q^{*}_{s} \in \left( \underline{\theta },\overline{\theta } \right) \) imply that \(x_{s}(q^{*}_{s}) \in (0,1)\). Therefore, by using the condition in (14), which must be satisfied by the fraction of premium version adopters, we obtain that \(x_{s}(q^{*}_{s})<1 \Rightarrow \pi ^{\prime }_{0}(p^{*}_{s})<0\). Then, the requirement \(x_{s}(q^{*}_{s})>0\) leads to either (a) \(\pi ^{\prime }_{1}(p^{*}_{s})>0\) or (b) \(\pi ^{\prime }_{1}(p^{*}_{s})<\pi ^{\prime }_{0}(p^{*}_{s})\). \(\square \)

**Proof of Proposition** 3

### Proof

The stated sufficient condition follows by considering a price for the product, not necessarily optimal, which induces an upper bound on the platform’s profits when it pursues discrimination through versioning. As the platform finds optimal to offer the free version only if \(\pi _{1}(p) \ge \pi _{0}(p)\), then, by picking \(p=\omega ^{R}\), it follows from the expression in (5) for the platform’s profits that

gives us the supremum of \({\varPi }(p,q)\) when only the price of the product is allowed to vary, under the restriction that the platform benefits from providing the free version of the service. Then, the argument proceeds by comparing the optimal profits to the platform when it provides only the free version of the service, \(\alpha \pi _{1}(\omega ^{*}_{1})\), with the above derived upper bound on the profits from pursuing versioning. By applying the first-order condition for interior optima to the problem \(\max _{q \in (\underline{\theta },\overline{\theta })} \nu (q)\), it follows that

where the price \(q^{\mathrm {pm}}_{s}\) corresponds to the optimal choice of the platform when it offers only the premium version of the service. The sufficient condition provided in the statement of the proposition follows by requiring that the optimal profits from offering only the free version of the service are no less than the upper bound obtained above. \(\square \)

**Proof of Lemma** 2

### Proof

Using the preference specification in (1), consumer surplus at prices (*p*, *q*) can be written as

The proof proceeds by decomposing each of the two terms obtained above. First, notice that

Secondly, notice that

Finally, given the two formulas derived above, the expression stated in (12) follows from the result that \(\mathbb {P}\big (n \ge n(q) \, \big | \, s \big )=1-x_{s}(q)\). \(\square \)

**Proof of Proposition** 6

### Proof

Consider two degree distributions \(H_{s}(n)\) and that \(H_{s^{\prime }}(n)\) that induce, respectively, two optimal prices for the product \(p^{*}_{s}\) and \(p^{*}_{s^{\prime }}\) such that \(p^{*}_{s}<\omega ^{R}\) and \(p^{*}_{s^{\prime }}>\omega ^{R}\). It follows from Proposition 1 that price \(p^{*}_{s}\) induces the platform to optimally serve only the premium version. Suppose that price \(p^{*}_{s^{\prime }}\) induces the platform to optimally serve both versions of the service. Let \(q^{*}_{s^{\prime }}\) be the optimal price of the service induced by the optimal price of the product \(p^{*}_{s^{\prime }}\). Notice first that, as \(p^{*}_{s}<\omega ^{R}\), Assumption 1 implies that \(z_{0}(p^{*}_{s})>z_{1}(p^{*}_{s})\). Secondly, as \(z_{1}(p)=1-F_{1}(p)\) decreases in *p*, then we have \(z_{1}(p^{*}_{s})>z_{1}(p^{*}_{s^{\prime }})\). Using the expression obtained in (12) for the consumer surplus, it then follows from the relation \(z_{0}(p^{*}_{s})>z_{1}(p^{*}_{s^{\prime }})\) established above that

because \(\big [1+z_{0}(p^{*}_{s^{\prime }})-z_{1}(p^{*}_{s^{\prime }}) \big ]<1\). \(\square \)

**Proof of Proposition** 7

### Proof

Consider two degree distributions \(H_{s}(n)\) and that \(H_{s^{\prime }}(n)\) that induce, respectively, two optimal prices for the product \(p^{*}_{s}>\omega ^{R}\) and \(p^{*}_{s^{\prime }}>\omega ^{R}\). Suppose that price \(p^{*}_{s}\) induces the platform to optimally serve only the free version and that price \(p^{*}_{s^{\prime }}\) induces the platform to optimally serve both versions of the service. Let \(q^{*}_{s^{\prime }}\) be the optimal price of the service induced by the optimal price of the product \(p^{*}_{s^{\prime }}\). Using the expression obtained in (12) for the consumer surplus, it then follows that

As \(z_{1}(p)=1-F_{1}(p)\) decreases in *p*, if (a) \(p^{*}_{s^{\prime }}<p^{*}_{s}\), then it follows directly that \(\text {CS}(p^{*}_{s^{\prime }},q^{*}_{s^{\prime }})>\text {CS}(p^{*}_{s})\). On the other hand, if (b) \(p^{*}_{s^{\prime }} > p^{*}_{s}\), we observe that \(\text {CS}(p^{*}_{s^{\prime }},q^{*}_{s^{\prime }})>\text {CS}(p^{*}_{s})\) if and only if

as stated. \(\square \)

**Proof of Proposition** 8

### Proof

By plugging the first-order condition to the monopolist’s problem stated in (4), the expression in (13), which gives us the change in consumer surplus induced by a local change in the optimal price of the service, can be rewritten as

Therefore, \({\partial \text {CS}(s) }/{ \partial q^{*}_{s}}>0\) if and only if \({\partial p^{*}_{s}}/{ \partial q^{*}_{s}}<0\) with

Recall that the definition of the hazard rate function of the degree distribution in (3), together with the results obtained earlier in (2) on the optimal fraction of premium version adopters, leads to that

Then, using the expression provided by Eq. (10) of Proposition 2 for the change \({\partial p^{*}_{s}}/{ \partial q^{*}_{s}}\), we obtain that the inequality in (15) above is satisfied if and only if

where

Finally, we know that \(z_{0}(p^{*}_{s})+ [(1/x_{s}(q^{*}_{s}))-1] z_{1}(p^{*}_{s})>0\) and \(\big [(p^{*}_{s}-c) +\pi _{0}(p^{*}_{s})-\pi _{1}(p^{*}_{s}) \big ]= (p^{*}_{s}-c)\big [1 +z_{0}(p^{*}_{s})-z_{1}(p^{*}_{s}) \big ]>0\) so that \(\xi (p^{*}_{s},q^{*}_{s})>0\) for each pair of interior optimal prices. \(\square \)

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Jiménez-Martínez, A. Discrimination through versioning with advertising in social networks.
*Econ Theory* **67, **525–564 (2019). https://doi.org/10.1007/s00199-018-1107-y

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### Keywords

- Social networks
- Second-degree discrimination
- Advertising
- Demand rotation
- Degree distributions
- Hazard rate

### JEL Classification

- D83
- D85
- L1
- M3