Platform competition for advertisers and users in media markets

International Journal of Industrial Organization 30 (2012) 243–252

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International Journal of Industrial Organization journal homepage: www.elsevier.com/locate/ijio

Platform competition for advertisers and users in media markets☆ Markus Reisinger ⁎ Department of Economics, WHU-Otto Beisheim School of Management, Burgplatz 2, 56179 Vallendar, Germany

a r t i c l e

i n f o

Article history: Received 3 September 2010 Received in revised form 19 October 2011 Accepted 21 October 2011 Available online 29 October 2011 JEL classifications: D43 L13 L82

a b s t r a c t This paper analyzes a two-sided market model in which platforms compete for advertisers and users. Platforms are differentiated from the users' perspective but are homogenous for advertisers. I show that, although there is Bertrand competition for advertisers, platforms obtain positive margins in the advertising market. In addition, platforms' profits can increase in the users' nuisance costs of advertising. As a general insight, I obtain that factors affecting competition in the user market in a well-known direction without externalities now have opposing effects due to competition in the advertiser market. The model can also explain why private TV platforms benefit if their public rivals are regulated to advertise less—a result at odds with models in which there is no competition for advertisers. © 2011 Elsevier B.V. All rights reserved.

Keywords: Platform competition Two-sided markets Advertising Indirect externalities

1. Introduction Many markets are characterized by platforms that receive a large share of their revenues from advertising. 1 But in order to attract advertisers, these platforms at the same time need to attract users of their content who are potential consumers of advertisers' products. Users, however, may dislike advertising. For example, web portals like AOL or GMX receive revenues from advertisers who place banners on the portals' web pages.2 These banners are often a nuisance to users. But advertisers are willing to pay a higher price for banners the more time users spend on a platform. A similar structure can be observed for TV or radio stations. In these industries producers also pay high prices for commercials aired by stations. 3 However, they would not do so if the stations were not attractive to viewers or listeners, who in turn usually do not like interruptions by commercials.

☆ An earlier version of this paper has been circulated under the title “Two-Sided Markets with Negative Externalities”. I am very grateful to Yossi Spiegel (the editor) and two anonymous referees for very helpful comments and suggestions that greatly improved the paper. I also thank Simon Anderson, Tommy Gabrielsen, Martin Peitz, Sven Rady, Ray Rees, Klaus Schmidt and participants at various conferences for very helpful comments and discussions. ⁎ Corresponding author. Tel: + 49 261 6509 290; fax: + 49 261 6509 289. E-mail address: [email protected]. 1 For example, Strömberg (2004) reports that in the U.S. advertising revenues comprise of 60–80% for newspapers and even more for TV broadcasters. 2 A survey conducted by PricewaterhouseCoopers reports that in 2009 internet advertising revenues from banner ads amounted to around $5.0 bn. See http://www. iab.net/media/file/IAB-Ad-Revenue-Full-Year-2009.pdf. 3 For example, in the first quarter of 2010 TV advertising revenues in the U.S. amounted to $11.7 bn. See http://www.tvb.org/rcentral/adrevenuetrack/revenue/ad_figures_1.asp. 0167-7187/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ijindorg.2011.10.002

In the economics literature this two-sided market structure in which users exert a positive externality on advertisers while advertisers exert a negative one on users has been extensively considered in recent years. Starting with the seminal paper by Anderson and Coate (2005) on the TV market, many insights on the social welfare properties and the pricing structure in such markets have been gained by that and several follow-up papers. An important assumption in these papers is that the two-sided market structure takes the form of a competitive bottleneck, that is, users choose only one platform, i.e., they single-home, while advertisers may place ads on several platforms, i.e., they multi-home. 4 This implies that platforms compete for users but not for advertisers. 5 However, platforms often need to compete for advertisers as well. This is the case because e.g., marketing departments in many firms have prespecified budgets that they can use for advertising expenditures.6 As an example, consider the TV industry. In this industry, channels at any moment in time have a different viewer group, and advertisers may

4 For an in-depth analysis of competitive bottlenecks and its difference to two-sided single-homing, see Armstrong (2006) and Armstrong and Wright (2007). 5 Earlier literature on the television industry, e.g., Spence and Owen (1977), also recognized the two-sided structure of this industry. However, it neglects the negative effect of advertisers on viewers and assumes that advertising prices and levels are exogenously fixed. Wildman and Owen (1985) take the negative effects of advertising into account and consider endogenous advertising levels but they do not provide an equilibrium analysis. 6 As documented by Piercy (1987), setting the marketing budget is often a multistage process in which first the total marketing expenditure is set by the management, with only very little detail on how it is to be spent. (As Lilien and Little (1976) report, the budget is often around 5% of the historical sales revenue.) Afterwards, different groups in the marketing department consider in detail on how to allocate the money.

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wish to reach several groups with their messages. However, if the marketing department of a producer has only a limited advertising budget, it cannot place as many commercials at each channel as it wishes to, but either needs to select only one platform for its marketing campaign or divide its budget among several platforms. In both cases, platforms need to compete for shares of the producer's advertising budget. The aim of this paper is to provide a framework that, first, accounts for competition on the advertiser market and, second, incorporates other prevalent features of media markets. For example, previous literature takes usage time as fixed and therefore independent of the advertising level. However, users spend less time on a platform if it contains a higher level of advertising. 7 The framework presented in this paper accounts for these aspects. In particular, I present a model in which two platforms compete for user time and advertisers. I suppose that platforms are undifferentiated for advertisers and therefore compete à la Bertrand on this side. Arguably, this assumption is simplistic since advertisers may prefer one or the other platform dependent on the characteristics of their respective users. 8 But there are also markets in which platforms are very similar since their user composition is very alike. Examples are web portals that offer similar web services or TV stations that broadcast similar content. I also note that the assumption of undifferentiated platforms is in line with existing models considering multi-homing of advertisers, e.g., Anderson and Coate (2005) or Armstrong (2006). Interestingly, I obtain that, despite this assumption, platforms set the advertising price above marginal cost, implying that they earn positive profits. Therefore, relaxing the assumption and allowing for differentiation on the advertiser side would not alter the qualitative findings but just scale up platform profits. In accordance with most of the literature, I suppose that platforms compete for users in a Hotelling fashion. A user's utility and the time he spends on a platform are endogenous and fall at an increasing rate in the level of advertising of the platform. Advertisers wish to gain users' attention to tempt them to buy their products. Thus, advertisers' profits are increasing in the time users spend on a platform. I restrict attention to the case in which users obtain the services of a platform free of charge. In this setting I obtain a number of results. First, as mentioned earlier, I show that platforms' profits are positive although competition for advertisers is à la Bertrand. The reason is that a platform cannot attract all advertisers by undercutting its rival's price because an additional advertiser causes a negative externality on other advertisers, which implies that some advertisers prefer to stay on the platform with the higher price or abstain from advertising. This intuition holds generally for markets in which a group of buyers exerts negative externalities on another group. In that case a firm loses buyers from the second group when reducing the price for the first group—an effect that dampens competition. Second, even if competition in the user market is very intense, the equilibrium advertising volume can still be sizable. This is the case because via the market mechanism the amount of advertising will always be the same on both platforms. Therefore, the tendency to reduce advertising via increasing the price is dampened leading to a non-negligible advertising volume. Third, I show that platforms' profits may increase if the users' nuisance costs of advertising rise. This occurs because a platform's incentive to reduce the advertising price to attract more advertisers is 7 For example, Wilbur (2008) demonstrates that a highly rated television network in the U.S. can increase its audience size by 25% when reducing its advertising level by 10%. For the Internet, Cho and Cheon (2004) also demonstrate that users react aversely to advertisements on web pages. 8 Gal-Or et al. (2010) note that the composition of users of different platforms is important for advertisers since advertisers wish to target users who are very receptive to their ads. Taking this into account Gal-Or et al. (2010) investigate how heterogeneity among advertisers and differentiation in the advertising receptiveness of readers shape advertising behavior and influence the media bias of newspapers.

dampened if nuisance costs are higher. This implies that if platforms can coordinate on a common level of the nuisance costs, they would choose too high a level. Therefore, in this case there is scope for regulatory intervention to set an upper bound on the nuisance cost level. Fourth, despite the fact that only advertisers can be charged, the structure of platforms' profits strongly depend on the degree of differentiation on the user side. For example, I obtain that a lower level of competition on the users' side can increase competition on the advertisers' side, thereby reducing platforms' profits. As a general theme, the results show that if there is competition on both sides, several factors that influence competition in a one-sided market in a certain and well-known direction, do now have opposing effects due to the competitive pressure on the other side. I can also contrast these findings with those obtained in models with multi-homing of advertisers. In these models, due to the lack of competition for advertisers, the effects are more standard, i.e., the degree of differentiation affects platforms' profits positively while the nuisance costs affect them negatively. Finally, the framework with competition for advertisers is also able to resolve one of the puzzles that come out of the competitive bottleneck model of media. This model predicts that private television broadcasters benefit from legislations allowing public broadcasters to air commercials. However, usually private platforms strongly oppose such advertising increases of their public rivals. An example is the German television market: Under the current regulation public broadcasters are not allowed to air commercials after 8:00 p.m. In a recent discussion on whether this law should be abolished to raise the revenues of public broadcasters, private stations argued strongly in favor of keeping it. 9 In the competitive bottleneck framework such behavior cannot be explained since more commercials on public channels reduce the attractiveness of these channels, thereby inducing some viewers to switch to private channels, which in turn raises the profits of private channels. However, this argument neglects competition for advertisers. Once this competition is taken into account, as is done in the present framework, the behavior of private channels can be explained. Allowing public channels to air commercials after 8:00 p.m. forces private broadcasters to compete with public ones for advertisers during these time slots while they could set monopoly prices beforehand. Thus, their revenues fall. The remainder of the paper is organized as follows. The next section provides an overview of the related literature. Section 3 sets out the model. Section 4 presents the equilibrium and some comparative static analyses. In Section 5 I compare the results with those arising from a model without competition for advertisers. Section 6 presents two extensions and Section 7 concludes. 2. Related literature The paper relates to the literature on competition between media platforms—see Anderson and Gabszewicz (2006) for a survey. In the seminal paper Anderson and Coate (2005) consider competition between TV stations that need to attract viewers and advertisers. Viewers single-home and are differentiated à la Hotelling with respect to their preferences for content while advertisers multi-home and have heterogeneous advertising values.10 In this framework Anderson and Coate (2005) show, among many other things, that the equilibrium advertising level can either be higher or lower than the socially optimal one dependent on the monopoly power of stations over advertisers, the degree of competition between stations and the nuisance costs of viewers. They also demonstrate that a viewer charge can lead to a reduction in welfare. 9 Press release ‘Verband Privater Rundfunk und Telekommunikation (VPRT), 26.9.2003’. 10 Anderson and Coate (2005) also consider an extension with two periods in which some viewers switch channels between periods, i.e., some viewers multi-home as well.

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The model by Anderson and Coate (2005) has been extended in many directions. For example, Peitz and Valletti (2008) analyze endogenous content provision, Gabszewicz et al. (2004) allow viewers to mix between channels, and Choi (2006), Crampes et al. (2009) and Reisinger et al. (2009) consider different aspects of free entry by platforms. Dukes and Gal-Or (2003) also consider a duopoly model but allow for price negotiations between platforms and advertisers and explicitly consider competition between advertisers. 11 They focus on exclusivity contracts between platforms and advertisers. Since in their framework platforms individually decide to offer these exclusivity contracts or not, neither platforms nor advertiser compete in, respectively for, these contracts. Therefore, in contrast to my framework, they do not consider platform competition for advertisers. A different approach to modeling competition in media markets is used by Kind et al. (2007 and 2009). They consider a representative consumer instead of a Hotelling model on the user side, which implies that users multi-home. In this framework, Kind et al. (2007) analyze the case without user charge and determine conditions for over- or underprovision of advertising, while Kind et al. (2009) consider the case of user payments and analyze whether the platforms' main revenue sources are users or advertisers.12 Since in their framework users and advertisers multi-home, the present model can be seen as the opposite case with single-homing on both sides. In sum, contrary to the previous literature that focuses only on multi-homing of advertisers, the present paper considers competition for advertisers. A paper that also analyzes competition for advertisers is Ferrando et al. (2008). However, they use a different concept with respect to expectation building of advertisers and users than the previous literature and are mainly interested in asymmetric equilibria in which at least one platform does not use the advertising market as a revenue source. In addition, the aforementioned papers keep the overall usage time constant and thus independent of the advertising level while my paper allows for this dependence. 13 This dependence is a wellobserved feature, since, when a platform increases its advertising level, not only some users switch to the rival platform but also those who remain may reduce their usage time. Thus, there is not only an extensive margin but also an intensive margin. As I explain later, this assumption reinforces the effect that platforms earn positive profits despite competing à la Bertrand. Within a wider literature on two-sided markets the paper is closest to Armstrong (2006) who, in Section 4 of his paper, considers the case of two-sided single-homing and subscription charges on both sides.14 I deviate from Armstrong's (2006) analysis by (i) allowing for endogenous usage time and non-linear externalities, (ii) considering homogenous agents on one side, and (iii) analyzing pure advertiser charges. Armstrong and Wright (2007) extend the model by Armstrong (2006) and consider the case in which platforms are differentiated only for agents of one side but not for those of the other side. However, they allow agents of the latter side to multi-home.15

11 Gal-Or and Dukes (2006) analyze a similar framework but with more than two firms and show how mergers between platforms change the bargaining power and the distribution of profits. 12 For a model that considers not only a representative user but also a representative advertiser, see Godes et al. (2009). 13 Cunningham and Alexander (2004) also allow for endogenous usage time. However, they focus on a different aspect of the media market, industry concentration, and also use a different modeling framework. For example, in their model platforms are homogeneous from the user's perspective implying that all active platforms must have the same advertising level. They also abstract from competition for advertisers. 14 See Rochet and Tirole (2003) for a paper that focuses on per-transaction charges, and Rochet and Tirole (2006) for a model with both charges but a monopoly platform. 15 For models in which agents on both sides view platforms as homogeneous, see Caillaud and Jullien (2001, 2003). In contrast to the present paper, they suppose that agents on both sides exert positive externalities on each other.

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3. The model There are three types of agents in the model, platforms, users (consumers), and advertisers (producers). I assume that if platforms are web portals, radio stations, or television channels, users possess the hardware to gain access to these platforms. 3.1. Platforms There are two platforms i = 1, 2. Platforms do not charge users for access and make money by selling advertisements. 16 The profit function of platform i is given by Π i ¼ pi ni ; i ¼ 1; 2: Here, pi is the price that platform i demands for placing an advertisement, and ni denotes the number of advertisements. To simplify the exposition, I assume that the costs of both platforms are zero. Each advertiser can place one advertisement only, implying that ni is also the number of advertisers using platform i, and has to decide on which platform to advertise, if on any. Therefore, platforms compete for advertisers in prices. Competition for advertisers is captured here in an extreme way by imposing single-homing. However, as explained in Section 5, what is necessary for the results to hold is that the number of advertisers on platform j falls if platform i lowers its price due to competition for advertisers. 17 The assumption is also in line with the empirical analysis by Kaiser and Wright (2006) who use data from the German magazine market and find strong evidence for platform competition for advertisers. 3.2. Users There is a mass 1 of users. Users are uniformly distributed on a line of length one on which platform 1 is located at point 0 and platform 2 at point 1. Each user decides in favor of one platform only. A user also decides how much time to spend on his chosen platform. I normalize the amount of available leisure time of a user to 1. The utility that a user derives from spending time t ≥ 0 on platform i is denoted by v(t), where v(t) is an increasing and strictly concave function that fulfills the Inada conditions, i.e., v(0) = 0, v′(0) = ∞, v′(1) = 0, and v(t)″ b 0. Thus, 1 − t is the time a user spends on doing other things than using the platform's services. To simplify the exposition, the utility a user receives from these other things is normalized to 1 per unit of time. The maximization problem of a user who is located at x and uses platform i located at xi ∈ {0, 1} with advertising level ni can then be written as 18 maxt

U i ¼ 1−t þ vðt Þ−γtnλi −τ jx−xi j

ð1Þ

The parameter γ > 0 measures the nuisance costs of advertising per unit of time while τ is the transportation cost parameter and represents the degree of differentiation between platforms. Lastly, λ measures the curvature of the utility function in ni. I assume that λ ≥ 1, i.e., the utility function is weakly concave in ni. This assumption is in line with the one made by Wildman and Owen (1985) and Gabszewicz et al. 16 This reflects e.g., a technological constraint that prevents platforms to exclude users. 17 From this perspective the assumption can be seen as a short-cut for the fact that advertisers have only a certain budget for marketing expenditures and platforms compete for shares of this budget. If one platform then reduces its price and advertisements on platforms are substitutes, an advertiser will spend a smaller share of his budget on the rival platform. 18 The advantage of this formulation is that the decision of users how much time to spend on a platform is separated from the decision which platform to use. See e.g., Anderson et al. (1992).

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(2004). As Owen and Wildman (1992) note, a viewer's utility from watching TV falls slowly at first, but faster as the number of commercials increases.19 From Eq. (1) I obtain that the amount of time that a user who joins platform i spends on this platform is implicitly given by ′

λ

v ðt Þ ¼ 1 þ γni :

ð2Þ

From the Implicit Function Theorem t′(ni) = (γλniλ − 1)/v″(t) b 0, i.e., t falls with ni. The indirect utility of a user located at x who joins platform i can then be written as U(x, ni) = UG(ni) − |τx − xi|, where UG(ni) ≡ 1 − t(ni) + v(t(ni)) − γt(ni)niλ. Thus, UG(ni) is the gross utility of a user excluding transportation costs. Due to the Envelope Theorem UG(ni) = − γt(ni)λniλ − 1 b 0. The marginal user xm who is indifferent between which platform to use is given by xm ¼

1 1 þ ðU ðn Þ−U G ðn2 ÞÞ: 2 2τ G 1

Thus, a mass X1 = xm of users choose platform 1 and the remaining mass X2 = (1 − xm) choose platform 2. Via advertising a producer informs users, who are prospective consumers, about the existence of its product. Advertisers' products are assumed to be independent of each other. If a user spends more time on a platform, the likelihood that he becomes aware of an advertisement increases. Since t(ni) ∈ [0, 1], one can interpret t(ni) as the probability that a user of platform i becomes aware of a product's ad. 20 I assume that advertisers are homogeneous in the sense that all users have a valuation of K with probability β and of 0 with probability 1 − β for each advertiser's good. 21 This formulation simplifies the analysis because it implies that users do not receive a positive value from becoming aware of ads. This is the case because each producer will sell its product at a price of K, i.e., each advertiser's price is equivalent to the consumers' valuation. Therefore, a consumer's utility of becoming aware of a new good is zero, which implies that users do not obtain informational benefits from using a platform with a lot of ads. In the following, I define k ≡ βK. 3.3. Advertisers There is a mass N of advertisers. As mentioned, advertisers choose only one platform to advertise on. If an advertiser chooses platform i, its profit is Xit(ni)k − pi. In this expression, Xi is the mass of users who use platform i. With probability t(ni) a user becomes aware of the advertisement. In this case, the user has an expected value of k for each advertiser's product. An advertiser pays platform i a fixed price pi which is independent of the number of users. 22 Finally, I assume for simplicity that production costs for advertisements and products are zero. If a producer does not advertise, its profit is zero. 19 Also, for the magazine market, Ha and Litman (1997) provide evidence that a decrease in a magazine's circulation is particularly large if the advertising level exceeds the historic average, while for the Internet, Cho and Cheon (2004) find that users avoid web sites for which they perceive the advertising level to be excessive. 20 If the amount of disposable time were not normalized to unity but denoted by t ≠ 1 instead, a user's probability of becoming aware of an advertisement would be t(ni)/t. All results are then qualitatively unchanged but the probability needs to be replaced by t(ni)/t. 21 This formulation is a simplified version of the one developed by Anderson & Coate (2005). They use the same stochastic structure but allow advertisers to differ in the probability with which users put a positive valuation on the product. This results in a decreasing advertising demand function of a platform. As will turn out later, in the present model with competition for advertisers such a decreasing demand function will result although advertisers are homogeneous. 22 I follow the existing literature, e.g., Anderson and Coate (2005), Peitz and Valletti (2008) and Crampes et al. (2009), by assuming that platforms charge a fixed advertiser price. However, I note that assuming that platforms charge a per-user fee instead of a fixed price does not change the qualitative results.

3.4. Game structure I consider a two-stage game. In the first stage, the two platforms decide simultaneously about their prices p1 and p2. In the second stage, advertisers and viewers simultaneously take their decisions: advertisers decide on which platform they want to advertise, if on any, and users decide which platform to join and how much time to spend on the respective platform. Afterwards, profits and utilities are realized. Note that I consider a model in which platforms compete in prices, which implies that the amount of advertising adjusts to prices. I do so because in many media industries platforms are very flexible in adjusting their advertising levels. For example, web portals can easily adjust their design to the number of advertisers or, as Kind et al. (2007) report, TV channels or radio stations are usually flexible with respect to the amount of advertising that is aired. However, I also note that the main results of the analysis are unchanged if I assume competition in advertising levels, because the mechanism underlying the results is not driven by the mode of competition. Finally, I state three assumptions on the parameters to simplify the analysis: (i) The user market is covered. This implies that in equilibrium v(t(ni)) is large relative to γ and τ so that it is optimal for all users to join one or the other platform. This assumption is imposed merely to rule out that I have to deal with the case in which platforms do not compete for users and are therefore local monopolists on this side of the market. (ii) v″(t(N/2))t(N/2) + 2γλ(N/2) λ b 0, i.e., |v″(t)| is sufficiently large when exactly half of the advertisers join each platform. As will become evident later—see footnote 25—this assumption simplifies the exposition of the equilibrium but is not crucial to the results. (iii) v‴(t) is not too large, i.e., it is either negative or not very positive. This is purely a technical assumption to guarantee an interior solution of the platforms' optimization problems. 4. Equilibrium and comparative statics Suppose that all producers advertise. Since advertisers are homogeneous, in equilibrium each one of them must be indifferent between platforms 1 and 2. Hence,
 1 1 kt ðn1 Þ þ ðU G ðn1 Þ−U G ðN−n1 ÞÞ −p1 2 2τ
 1 1 ¼ kt ðN−n1 Þ þ ðU G ðN−n1 Þ−U G ðn1 ÞÞ −p2 : 2 2τ

ð3Þ

The left-hand side is the profit of an advertiser on platform 1 and the right-hand side is the profit of an advertiser on platform 2, given that the numbers of advertisers on the platforms are n1 and n2 = N − n1. Thus, ni is implicitly defined by Eq. (3) and depends on pi and pj. The maximization problem of platform i is given by maxpi

  Π i ¼ ni pi ; pj pi ;

i ¼ 1; 2:

The respective first-order condition is ∂ Πi/∂ pi = (∂ ni(pi, pj)/∂ pi)pi + ni(pi, pj) = 0. Using Eq.(3) I get ∂ ni(pi, pj)/∂ pi = 1/κ, with 
 1 U G ðni Þ−U G ðN−ni Þ U ðn Þ þ U G ðN−ni Þ þ þ kt ðni Þ G i 2 2τ 2τ

 1 U ð N−n Þ−U ð n Þ U ðN−ni Þ þ U G ðni Þ i G i þ kt ðN−ni Þ G : þkt ′ ðN−ni Þ þ G 2 2τ 2τ

κ ≡ kt ′ ðni Þ




ð4Þ

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In a symmetric equilibrium ni⋆ = N/2. Inserting this into the firstorder condition and using Eq. (4) I can determine the equilibrium price to get23 " ⋆

λ−1

pi ¼ kNγλðN=2Þ

# t ðN=2Þ2 1 − ″ > 0; τ 2v ðt ðN=2ÞÞ

ð5Þ

where I used t′(N/2)= γλ(N/2)λ − 1/v″(t(N/2)) and UG(N/2)= −γt(N/ 2)λ(N/2)λ − 1. From Eq.(5) it is evident that pi⋆ is positive. Thus, although platforms are undifferentiated from the advertisers' perspective and compete à la Bertrand, they nevertheless obtain positive profits. The reason is that a platform, which undercuts the price of its rival, attracts more advertisers but this results in a higher negative externality on all of them. This externality is two-fold: First, some users switch to the rival platform —the extensive margin—and, second, the remaining users lower their usage time—the intensive margin. Since both effects reduce advertisers' profits, some advertisers will stay on the other platform or abstain from advertising. A consequence of this is that price competition is dampened.24 The profit of each platform is given by ⋆ Πi

" # 2 t ðN=2Þ 1 − ″ ¼ kNγλðN=2Þ : τ 2v ðt ðN=2ÞÞ λ

ð6Þ

However, this can only be an equilibrium if each advertiser indeed receives non-negative profits. Inserting pi* given by Eq. (5) into the profit function of an advertiser who has chosen platform i, which is equal to kt(N/2)/2 − pi*, I obtain after rearranging that advertiser profits are positive only if τ≥

4v″ ðt ðN=2ÞÞðN=2Þλ γλt ðN=2Þ2 ≡ τ; v″ ðt ðN=2ÞÞt ðN=2Þ þ 2γλðN=2Þλ

with τ > 0 because of assumption (ii). 25 As shown in the Appendix, for τ b τ, there is no competition for advertisers. Here I obtain that for τ b Pτ , with τ≡

P

v″ ðt ðN=2ÞÞðN=2Þλ γλt ðN=2Þ2 b τ; v″ ðt ðN=2ÞÞt ðN=2Þ þ γλðN=2Þλ

ni⋆ in the symmetric equilibrium is implicitly defined by the unique solution to t ðni Þ þ

λ

2

λ

γλni γt ðni Þ λni ¼ 0; − τ v ðt ðni ÞÞ ″

ð7Þ

leading to platform profits of Πi⋆ = (kt(ni⋆)ni⋆)/2, while for Pτ ≤ τ b τ, the symmetric equilibrium is characterized by n1⋆ = n2⋆ = N/2 and platform profits of Πi⋆ = kNt(N/2)/4. It is of particular interest to look at the case in which platforms are almost undifferentiated. Consider the case in which τ = ε, with ε > 0, but very small. Since almost all users would then decide in favor of the same platform if platforms have a different advertising volume, one might expect that the advertising volume is also of magnitude ε. However, as can be seen from Eq. (7), this is not the case if λ > 1, i.e., ni⋆ is of magnitude ε 1/λ > ε, implying that although τ is very small, ni⋆ is not necessarily very small. The intuition is as follows: Suppose 23

In the Appendix I show that the second-order conditions are indeed satisfied. This result differs from the one obtained in models in which agents exert positive externalities on each other. If in such models the agents can coordinate on the platform that gives them the highest surplus, prices will be driven down to zero because of the standard Bertrand argument. For overviews of this literature, see Katz and Shapiro (1994) or Farrell and Klemperer (2007). 25 If assumption (ii) was not imposed, I additionally need to consider the case τ
≤0. However, if τ
≤0, it turns out that the equilibrium is given by Eq. (7). Therefore, imposing assumption (ii) just reduces the set of cases to consider and simplifies the exposition. 24

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both platforms have a non-negligible amount of advertising and one of them increases its price. This reduces its advertising volume and usually results in almost all users choosing this platform. However, advertising on this platform then becomes more valuable which leads to an increase in the advertising volume and to a reduction in users again. Therefore, since platforms are almost undifferentiated, via the market mechanism they will have the same number of advertisements, although they charge different prices. As a consequence, even if τ is very small a platform must weigh the price effect against the volume effect, and the second effect does not necessarily dominate. 26 So with competition for advertisers and users neither the advertising price nor the advertising volume is brought down to zero even if there is almost perfect competition on both sides. 27 It is evident from the earlier analysis that, although users cannot be charged, platforms' profits are determined to a large extent by τ, the degree of differentiation on the users' side. I now take a closer look at how a change in the degree of differentiation affects platforms' profits. Proposition 1. Platforms' profits are non-monotone in the degree of differentiation τ. In particular, they increase in the degree of differentiation as long as differentiation is relatively small (τ b Pτ ), stay constant for intermediate values ( Pτ ≤ τ b τ) and decrease for high values of differentiation (τ ≥ τ). Proof. See the Appendix If τ is small, platforms compete fiercely for users. Since platforms cannot charge users, they set a low level of advertising in equilibrium. Therefore, advertising prices are high and only few producers advertise. If τ increases, advertising prices fall and platforms' profits rise because platforms attract more advertisers. This is possible since in this region platforms do not compete for advertisers, i.e., n1⋆ + n2⋆ b N. If τ becomes larger than Pτ , all producers choose to advertise. However, in the region between Pτ and τ
, profits are independent of τ. This is the case because it does not pay off for one platform to lower its price and steal an advertiser from the rival platform because too many users would switch to the rival. But if τ is above τ, the degree of differentiation on the user side is high enough, so that each platform has an incentive to lower its advertising price because only few users will switch to the other platform. But this causes profits to fall since both platforms lower their prices while ni⋆ stays at N/2. I now turn to the effects of γ—the nuisance cost of advertising—on platforms' profits. An increase in γ has two effects on profits. First, since users spend less time on platforms, platforms' advertising revenues get smaller. Second, a change in γ also affects the competitive pressure in the advertising market and may dampens competition. The next result shows that this second effect can dominate the first one. Proposition 2. Platforms' profits fall in the nuisance costs of advertising if differentiation between platforms is relatively small, i.e.,τ b τ. However, for high levels of differentiation, i.e.,τ ≥ τ, platforms' profits can either rise or fall in the nuisance costs. In particular, they rise for low levels of the nuisance costs. Proof. See the Appendix The intuition for this result is the following. From the analysis earlier, we know that for τ b τ, platforms do not compete for advertisers. In this case, the first effect explained before Proposition 2 dominates. If the nuisance costs of advertising rise, each user spends less time on a platform. Thus, advertisers are willing to pay less, and so advertising prices and platforms' profits are lower. But if platforms compete for advertisers, profits may increase in γ. The reason for this can be grasped in the 26

I thank an anonymous referee for pointing this out. If τ → 0, i.e., competition on the user market becomes perfect, as can be seen from Eq.(7), ni⋆ also approaches zero, but at a smaller rate than τ. 27

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easiest way by looking at the case at which γ is close to zero. In this case there is almost perfect competition for advertisers and profits are close to zero. If the nuisance costs of advertising increase, competition is dampened since a platform loses more users when attracting new advertisers and the remaining users spend less time on the platform. This reduced-competition effect dominates the effect that advertising is less valuable because of reduced usage time if γ is low and profits are close to zero. Hence, equilibrium prices rise if γ increases. If γ is relatively large, either of the two effects may dominate, and so the result is ambiguous. The result does not imply that a platform individually has an incentive to make its advertisements more annoying. In fact, it can be easily shown that if platforms have different nuisance parameters and can independently choose their levels, each one has an incentive to reduce this parameter. However, if platforms can coordinate on a common nuisance parameter, they would choose a positive one in case of competition for advertisers. This leads to lower welfare compared to a nuisance parameter of zero. Therefore, in this case regulators have an incentive to intervene in the market and impose an upper bound for the nuisance level.28 In summary, the two propositions show that in a two-sided market with competition on both sides, the direct effect of a change in an exogenous variable in one market can be dominated by the indirect effect that this change has via the adjacent market. For example, a lower degree of competition in the user market can increase the competitive pressure in the advertising market and overall lead to lower profits. Therefore, variables can have opposing effects compared to those expected from well-known results of markets with only one group of buyers or several groups that do not exert externalities on each other. 5. Multi-homing of advertisers In this section I consider the same model assumptions as stated in Section 3 but allow advertisers to multi-home. This enables me to show that, first, the economics behind Propositions 1 and 2lies inherently in the fact that there is competition for advertisers and, second, that the framework can explain why platforms prefer that their rivals are forced to advertise less while a model with advertiser multihoming predicts the opposite result. Since there is no advertiser competition, each platform i charges an advertiser price equal to the expected profit from advertising, that is,
  1 1  pi ¼ kt ðni Þ þ U G ðni Þ−U G nj ; j≠i; i ¼ 1; 2: 2 2τ

ð8Þ

From Eq. (8) I can determine ∂ ni/∂ pi and insert the result in the first-order condition given by     ∂Π i ∂ni pi ; pj ¼ pi þ ni pi ; pj ¼ 0 ∂pi ∂pi

to obtain the equilibrium number of advertisers. Doing so yields that ni⋆ is implicitly given by 29  2  λ   γλn⋆ λ γt n⋆i λ n⋆i ⋆ ¼ 0; t n i þ ″   i ⋆  − τ v t ni 28

ð9Þ

This could be done for example by regulating the points of time in movies or shows during which airing of commercials is allowed. 29 In the same way as in the last section it is easy to check that the second-order conditions are satisfied for v‴(t) negative or not too positive and that Eq.(9) exhibits a unique solution.

which gives a platform profit of ⋆

Πi ¼

 ⋆ kt ni ⋆ ni : 2

Similar to the case of two-sided single-homing, this equilibrium is only valid as long as ni⋆ ≤ N. Inserting ni⋆ = N in Eq.(9), this is satisfied if τ≤

v′′ ðt ðNÞÞNλ γλt ðNÞ2 ≡ τ^ : v′′ ðt ðNÞÞt ðN Þ þ γλNλ

For τ > τ^ , ni⋆ = N, i.e., competition for users is mild so that all advertisers join both platforms. Performing a comparative static analysis with respect to τ and γ gives the following result: Proposition 3. If advertisers can multi-home, platforms' profits always (weakly) increase in the degree of differentiation and always (strictly) decrease in the nuisance costs of advertising. These results can be shown in the same way as those of Propositions 1 and 2. Proposition 3 shows that the comparative static results on the degree of differentiation and the nuisance costs differ starkly between the two models. The intuition for the results in the multi-homing framework is that, since there is no competition for advertisers, an increase in the nuisance costs of users just lowers usage time and intensifies competition for users which drives advertising levels down. Similarly, if platforms are less differentiated, they reduce their advertising levels, which lowers their profits. Since there are no further effects of these two variables on the advertising market due to the lack of competition there, the comparative statics are similar to those in a one-sided market. I can now explore in more detail the effects of a platform's price change on the equilibrium advertising levels in the two frameworks. I start with the case of two-sided single-homing. If there is competition for advertisers, in a symmetric equilibrium ni⋆ = N/2, and the change of ni in pi and pj can be determined from Eq. (3). This yields dni τ b0 ¼  dpi k t ′ ðN=2Þτ þ 2t ðN=2ÞU ′G ðN=2Þ and dni dn ¼ − i > 0: dpj dpi Therefore, platform i benefits from an increase in the rival's advertising price since more advertisers decide in favor of platform i, which rises its profits. In case of multi-homing of advertisers, it is evident from Eq. (8), that pj does not affect ni directly but only indirectly via nj and its effect on the user demand. Now differentiating Eq. (8) for i and j, and solving for dni/dpi and dni/dpj yields that in equilibrium h      i 2 t ′ n⋆i τ þ t n⋆i U ′G n⋆i dni     b 0 ¼     dpi kt ′ n⋆i t ′ n⋆i τ þ 2t n⋆i U ′G n⋆i and     2t n⋆ U ′ n⋆ dni ¼ ′  ⋆  ′  ⋆  i G i ⋆  ′  ⋆  b 0: dpj kt ni t ni τ þ 2t ni U G ni In contrast to the case with competition for advertisers, platform i now suffers from an increase in the rival's advertising price. This is because when platform j has a lower advertising volume due to the higher price, platform j's demand on the user side rises, leading to a profit decrease for platform i. Proposition 4 sums up this analysis.

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Proposition 4. If τ
≤τ≤τ^ , then a price increase of platform j raises the profit of platform i in the model with competition for advertisers while it lowers the profit of platform i in the model with multi-homing of advertisers. The result in the case of advertiser multi-homing is at odds with the observed fact that private TV stations prefer their public rivals to be forced to broadcast few or no commercials, which corresponds to a very high or infinitely large advertising price. However, the example from the German TV market given in the Introduction suggests that changes in the advertising levels of rival platforms have an effect not only through the user market but that there is also a direct effect through the advertising market. The present framework, by including competition for advertisers, presents a way to show how this mechanism can work. 6. Extensions In this section I consider two extensions of the baseline model. First, I compare the equilibrium advertising volume with the socially efficient one and explain how competition for advertisers brings in a new effect in this comparison. Second, I show that the main insights of the baseline model also hold if users can multi-home. 6.1. Social welfare I start with a comparison of the socially optimal advertising volume with the equilibrium one. Here I obtain the following result: Proposition 5. The equilibrium advertising volume tends to be below the socially optimal one if the degree of differentiation is relatively low and above the socially optimal one if the degree of differentiation is relatively high. Proof. See the Appendix There are two effects that drive a wedge between the socially optimal and the equilibrium advertising volume. First, a social planner considers the users' utility loss from advertising directly while platforms do so only indirectly since they receive lower advertising revenues if users spend less time on the platforms. This effect leads to over-provision of advertising. Second, platforms compete for users by reducing the level of advertising. The extent of this competition depends on the degree of differentiation, τ. The smaller is τ, the fiercer is competition for users and the lower is the equilibrium advertising volume. This effect leads to under-provision of advertising. As a consequence, I obtain that if competition for users is fierce, there tends to be too little advertising, while if competition for users is mild, there tends to be too much advertising. The result that there are two countervailing effects when comparing the efficient with the equilibrium level of advertising is also present in Anderson and Coate (2005). 30 However, there is an additional effect in the present model that arises only due to competition for advertisers. Suppose that platform i reduces its advertising price, thereby expanding its advertising level. If advertisers multi-home, this price reduction has no direct effect on the advertising level of platform j. However, with competition for advertisers some advertisers now switch from platform j to platform i. Since

30 In addition, there is a third effect in the model of Anderson and Coate (2005) which occurs because advertisers are heterogeneous with respect to the expected benefit they receive in case a user becomes aware of the respective ad. This heterogeneity creates a decreasing demand curve on the advertisers' side. Since platforms have monopoly power over their advertisers, a platform optimally charges a high price to advertisers, thereby keeping the level of advertising inefficiently low. Since advertisers are homogeneous in my model, this effect is not present there.

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this reduces the advertising volume on platform j, this platform becomes more attractive to users, thereby diminishing the positive effect of the price reduction for platform i. As a consequence, in equilibrium platforms set a relatively high price which tends to lead to an inefficiently low level of advertising. 6.2. Multi-homing of users In several purely advertising-financed markets users have the possibility to multi-home. For example, some users browse several internet newspapers or do e-mailing on two or more platforms. In this subsection I show that the main insights of the baseline model carry over to this case. Consider the model introduced in Section 3, but now allow each user to use both platforms if he wishes to do so. This implies that a user chooses platform i located at x ∈ {0, 1} if the utility he receives from using platform i is positive. So the marginal user xm of platform i is described by UG(ni) − τ|xm − x| = 0. Determining the demand for platform i I obtain that it is given by UG(ni)/τ. For the sake of brevity I focus on the case in which all producers advertise, i.e., there is competition for advertisers. Since a producer's expected revenue from advertising on platform i is kt(ni)UG(ni)/τ, advertisers are indifferent between the two platforms if kt ðni ÞU G ðni Þ kt ðN−ni ÞU G ðN−ni Þ −pi ¼ −pj : τ τ

ð10Þ

As before, the first-order condition of platform i is given by (∂ ni(pi, pj)/∂ pi)pi + ni(pi, pj) = 0. Using Eq. (10) to determine ∂ ni/∂ pi I obtain that at a symmetric equilibrium ∂ni τ : ¼  ∂pi 2k t ′ ðN=2ÞU G ðN=2Þ þ t ðN=2ÞU ′G ðN=2Þ Inserting ∂ ni/∂ pi into the first-order condition, solving for pi and using t′(N/2) = γλ(N/2)λ − 1/v″(t(N/2)) and UG(N/2) = − γt(N/2)λ(N/ 2)λ − 1 yields ⋆ pi

" # λ kN 1−t ðN=2Þ þ vðt ðN=2ÞÞ−γt ðN=2ÞðN=2Þ λ−1 2 ¼ t ðN=2Þ − γλðN=2Þ τ v′′ ðt ðN=2ÞÞ

which gives a platform profit of ⋆

Πi ¼

" # λ kN 1−t ðN=2Þ þ vðt ðN=2ÞÞ−γt ðN=2ÞðN=2Þ λ 2 γλðN=2Þ t ðN=2Þ − : τ v′′ ðt ðN=2ÞÞ

ð11Þ It is evident that Πi⋆ is positive if γ is positive. The intuition for the result is the same as in case of single-homing of users. Although platforms are homogenous for advertisers, prices and profits are not driven down to zero if γ is positive, since with negative indirect externalities a platform cannot attract all users. I can also perform a comparative static analysis with respect to γ to get

⋆



 ∂Π i U ðN=2Þ ∂ U ðN=2Þ 2 2 t ðN=2Þ − ′′ G ¼ sign t ðN=2Þ − ′′ G þγ : sign ∂γ ∂γ v ðt ðN=2ÞÞ v ðt ðN=2ÞÞ

The first two terms are positive while the third one is ambiguous. Since the third term is multiplied by γ, ∂ Πi/∂ γ > 0 for γ close to zero, because in this case the first two terms dominate. I therefore obtain the following result which is in line with the one of Section 4: Proposition 6. If users can multi-home and there is competition for advertisers, platforms' profits increase in the nuisance costs of advertising, γ, for small values of γ.

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Finally, as can be seen from Eq. (11), Πi⋆ falls in τ. This result is standard in models of Hotelling differentiation in which the two firms are not in competition with each other. In that case, τ represents by how much the utility of a consumer who is located further away from a platform's product falls with distance. If τ is larger, fewer users join a platform, given the level of advertising, resulting in a decrease in platforms' profits. 7. Conclusion This paper analyzed a media market model in which platforms compete for advertisers and users. Platforms are differentiated from the users' perspective but are homogenous for advertisers. I show that platforms receive positive profits although competition is à la Bertrand. The reason is that users dislike advertising, and so a platform cannot attract all advertisers by undercutting its competitor's price. Platforms profits can even increase if this negative externality rises although usage time falls. I also demonstrate that, despite the fact that users cannot be charged, the advertising price and the shape of platforms' profits strongly depend on the degree of differentiation on the user side. As a central theme, I obtain that because of competition on both sides factors that influence competition for users in a certain direction now have opposing effects due to the indirect effect on competition for advertisers. I also show that this cannot happen in a model with multi-homing of advertisers. In addition, the framework can explain why private broadcasting platforms benefit if their public rivals are regulated to advertise less, an observation at odds with the advertiser multi-homing model but in line with casual evidence. In the analysis I kept the locations of the platforms exogenous and assumed maximal differentiation. It is of interest to explore if platforms indeed have an incentive to maximally differentiate themselves from each other, in particular, since the results of previous papers are mixed. As shown by Peitz and Valletti (2008), the incentive to differentiate depends on the users' disutility of advertising and increases in this disutility. By contrast, Gabszewicz et al. (2004) obtain that maximal differentiation occurs only if the disutility is small. It might therefore be worthwhile to analyze if competition for advertisers brings in additional effects that affect platforms' differentiation incentives. Since competition is then on both sides, it seems plausible that platforms' incentives to differentiate are relatively high. A different direction for further research is to consider a model with more than two firms and free entry. This could be pursued by using a circular model à la Salop, as is done by e.g., Choi (2006), Crampes et al. (2009) or Reisinger et al. (2009). These papers generally find that both excessive and insufficient entry may occur in equilibrium. Since the model presented in the present paper involves competition on both sides, new effects can arise when analyzing the socially optimal number of firms. First, since competition is on both sides, fewer firms may enter since overall profits are lower. On the other hand, the business stealing effect may be larger due to competition on both sides which leads to excessive entry. It is an interesting direction for future research to analyze which of these effects tends to dominate. Appendix A. Derivation of the equilibrium of Section 4 In the main text I determined pi⋆ for the case in which all advertisers are active, i.e., ni⋆ = N/2. I need to check that pi⋆ indeed constitutes a maximum, i.e., that the profit function is concave. The second derivative with respect to pi is given by

2

∂ Πi ¼2 ∂ðpi Þ2

  ∂ni pi ; pj ∂pi

þ pi

  ∂2 ni pi ; pj ∂ðpi Þ2

:

It is easy to check that at the equilibrium ∂ 2ni(pi, pj)/∂ (pi) 2 = 0 which implies that   ∂ni pi ; pj ∂2 Π i 2τ b 0: ¼2 ¼ ′ ∂pi τt ðN=2Þ þ 2t ðN=2ÞU G ðN=2Þ ∂ðpi Þ2 Thus, the profit function is locally concave at the equilibrium. Tedious but otherwise routine calculations reveal that the profit function is also globally concave if v‴(t) is not too positive which I assumed at the outset. As a consequence, pi⋆ is the unique solution to the game in which all advertisers are active. Now I turn to the case in which there is no competition for advertisers. The number of advertisers on platform i is then implicitly given by the zero-profit condition for advertisers:
  1 1  kt ðni Þ þ U G ðni Þ−U G nj −pi ¼ 0: 2 2τ From the last equation I obtain " !#   t ′ ðn ÞU ðn Þ −1 ∂ni ðpi Þ 1 1  ′ i G i þ U G ðni Þ−U G nj þ ¼ k t ðni Þ : 2τ 2 2τ ∂pi ð12Þ Inserting Eq. (12) into the first-order condition for profit maximization of firm i, which is given by ni(pi) + pi(∂ ni(pi)/∂ pi) = 0 and using that pi = kt(ni)[1/2 + (UG(ni) − UG(nj))/(2τ)] yields that in a symmetric equilibrium ni is implicitly defined by the solution to t ðni Þ þ

γλnλi γt ðni Þ2 λnλi ¼ 0; − τ v″ ðt ðni ÞÞ

ð13Þ

where I used t′(ni) = γλ(ni) λ − 1/v″(t(ni)) and UG(ni) = − γt(ni)λ(ni) λ − 1. Before showing that a solution to Eq.(13) exists and is unique, I need to check that the second-order condition is satisfied. The second-order condition is given by 2(∂ ni(pi, pj)/∂ pi) + pi (∂ 2ni(pi, pj)/∂ (pi) 2) b 0. Using Eq. (12) to determine ∂ 2ni(pi, pj)/∂ (pi) 2, I obtain that, at a symmetric equilibrium, the second-order condition is ′

″

2t ðni Þ þ t ðni Þ þ

U G ðni Þt ðni Þni þ U G ðni Þt ′ ðni Þni þ U G ðni Þt ðni Þ b0: τ

ð14Þ

To evaluate the second-order condition I need to determine t″(ni) and UG(ni). Doing so yields ″

t ðni Þ ¼

i γλðni Þλ−2 h ″ 2 λ ‴ ðλ−1Þv ðt ðni ÞÞ −γλðni Þ v ðt ðni ÞÞ ″ 3 v ðt ðni ÞÞ

and " λ−1

U G ðni Þ ¼ −γλðni Þ

# γλnλi t ðni Þðλ−1Þ þ : ni v″ ðt ðni ÞÞ

Inserting these expressions into Eq.(14) I obtain γλðni Þ

h i 2 ″ 3 v ðt ðni ÞÞ2 τ ð1 þ λÞ þ 2γλt ðni Þnλi −λt ðni Þ2 v″ ðt ðni ÞÞ3 −γλτnλi v‴ ðt ðni ÞÞ 5b0: τv″ ðt ðni ÞÞ3

λ−1 4

ð15Þ The denominator of the left-hand side of Eq. (15) is negative while the numerator is positive if v‴(t(ni)) is not very positive, which holds true by assumption. Thus, the inequality (15) is fulfilled, and so the profit function is locally concave. Again, routine calculations show

M. Reisinger / Int. J. Ind. Organ. 30 (2012) 243–252

that the profit function is also globally concave if v‴(t) is not too positive, which holds because of assumption (iii). Now I show that a solution to Eq.(13) exists and is unique. First, note that at ni = 0 the left-hand side of Eq. (13) is positive because t(0) > 0. Second, I know that the profit function is only valid if there is no competition for advertisers, i.e., at ni⋆ b N/2. Inserting ni = N/2 into Eq.(13) and solving for τ yields τ ¼ Pτ . From Eq. (15), I know that the left-hand side of Eq. (7) is strictly decreasing in ni. Since it is strictly increasing in τ, it follows that the profit function is only relevant if τb Pτ . Thus, I have that at ni = N/2 the lefthand side of Eq. (13) is negative for all τb Pτ . As a consequence, the left-hand side of Eq.(13) is positive at ni = 0 and negative at ni = N/2 which implies that a solution with ni⋆ ∈ (0, N/2) exists. In addition, since I know that the left-hand side of Eq. (13) is strictly decreasing in ni, the solution is unique. Prices and profits of the platforms at the unique solution are   kt n⋆i ⋆ pi ¼ 2

and

  kt n⋆i ⋆ ⋆ Πi ¼ ni : 2

ð16Þ

Finally, the solution is only valid for ni⋆ ≤ N/2. Inserting ni = N/2 into Eq. (7) and solving for τ yields τ¼

v″ ðt ðN=2ÞÞðN=2Þλ γλt ðN=2Þ2 ≡ τ; v″ ðt ðN=2ÞÞt ðN=2Þ þ γλðN=2Þλ P

with 0b Pτ bτ. Therefore, the solution given by Eq. (7) is only valid for τb Pτ . By contrast, in the region τ≤τbτ, platforms are just at the brink of competing for advertisers. Thus, in a symmetric equilibrium n1⋆ =n2⋆ = N/2, but the marginal advertiser receives zero profits, that is, pi⋆ = kt(N/2)/2. Equilibrium profits are then Πi⋆ = kNt(N/ 2)/4. Proof of Proposition 1. If τb Pτ , a platform's profit is given by Eq.(16) and the optimal number of advertisers is implicitly defined by Eq.(7). Using the Implicit Function Theorem, I get from Eq. (7)

⋆     ∂ni 1 ⋆ ⋆ ⋆ > 0: sign ¼ sign − 2 kni t ni U G ni ∂τ 2τ Differentiating Eq. (16) with respect to τ yields ⋆ ∂Π i

∂τ

¼

   ∂n k   ⋆ ′ ⋆ ⋆ i t ni þ t ni ni : 2 ∂τ ⋆

Since ∂ ni⋆/∂ τ > 0, I obtain

⋆ n    o ∂Π i ⋆ ′ ⋆ ⋆ ¼ sign t ni þ t ni ni : sign ∂τ But from Eq. (7) I know that  2  λ     t n⋆i γλ n⋆i ⋆ ′ ⋆ ⋆ t ni þ t ni ni − ¼ 0: τ Since the last term of the left-hand side on the last equation is negative, t(ni⋆) + t′(ni⋆)ni⋆ > 0. It follows that ∂ Πi⋆/∂ τ > 0. If Pτ ≤τbτ, platforms' profits are given by Πi⋆ = kNt(N/2)/4 which is independent of τ. If τ≥τ, platforms' profits are given by Eq.(6). Differentiating Eq. (6) with respect to τ gives " # 2 ∂Π ⋆i λ t ðN=2Þ ¼ −kNγλðN=2Þ b0: ∂τ τ2 ■

251

Proof of Proposition 2. I start with the case τb Pτ . Differentiating the equilibrium profit given by Eq. (16) with respect to γ I get " #    ∂t n⋆  ∂Π ⋆i k ∂n⋆i   ⋆  ′ ⋆ ⋆ ⋆ i ¼ t ni þ t ni ni þ ni : 2 ∂γ ∂γ ∂γ

ð17Þ

So the sign of ∂ Πi⋆/∂ γ depends on the sign of the squared bracket on the right-hand side of Eq.(17). From the proof of the last proposition I know that t(ni⋆) + t′(ni⋆) ⋆ ni > 0. Using the Implicit Function Theorem, I can determine the sign of ∂ ni⋆/∂ γ from Eq. (7) to get, after rearranging, (   λ     2 )

⋆ τð1 þ λÞ−γλt n⋆i n⋆i −v″ t n⋆i t n⋆i λ ∂ni    sign : ð18Þ ¼ sign ∂γ τv″ t n⋆i Multiplying Eq. (7) with τ/t(ni⋆) yields     λ τ ′ ⋆ ⋆ ⋆ ⋆ τ þ t ni ni  ⋆  −γλt ni ni ¼ 0: t ni Since t′(ni⋆) b 0, it follows that τ − γλt(ni⋆)(ni⋆) λ > 0 which implies that τ(1 + λ) − t(ni⋆)γλ(ni⋆) λ > 0. Since v″(t(ni⋆)) b 0, the numerator of the right-hand side of Eq.(18) is positive. By the same reason the denominator is negative. As a consequence, ∂ ni⋆/∂ γ b 0. Therefore, the first term in the squared bracket of the right-hand side of Eq. (17) is negative. It remains to determine ∂ t(ni⋆)/∂ γ. Applying the Implicit Function Theorem to Eq. (2) yields ∂ t(ni⋆)/∂ γ = (ni⋆) λ/ (v″(t(ni⋆)) b 0. Thus, the second term is negative as well. It follows that ∂ Πi⋆/∂ γ b 0. If Pτ ≤ τ b τ, a platform's profit is given by Πi⋆ = kNt(N/2)/4. Differentiating Πi⋆ with respect to γ yields ∂Π ⋆i βKN ∂t ðN=2Þ ¼ b0 4 ∂γ ∂γ since ∂ t(N/2)/∂ γ = (N/2) λ/(v″(t(N/2))) b 0. If τ≥τ, a platform's profit is given by Eq. (6). Differentiating Eq. (6) with respect to γ yields (

⋆ ∂Π i ðt ðN=2ÞÞ2 1 2γt ðN=2ÞðN=2Þλ − ″ sign þ ¼ sign τ ∂γ τv″ ðt ðN=2ÞÞ v ðt ðN=2ÞÞ ) γðN=2Þλ v‴ ðt ðN=2ÞÞ þ :   2 v″ ðt ðN=2ÞÞ 3 The first two terms are positive, the third term is negative and the fourth term is positive or only slightly negative due to the assumption that v‴(t(ni)) is not too positive. Thus, the overall sign is ambiguous. However, for γ close to zero, the third and fourth terms are negligible, and the first two terms are the dominating ones. This implies that in this case ∂ Πi⋆/∂ γ > 0. ■ Proof of Proposition 5. I start by deriving the socially optimal number of advertisements on each platform. Total welfare can be written as
 1 1 ðU G ðn1 Þ−U G ðn2 ÞÞ WF ¼ ½kn1 t ðn1 Þ þ U G ðn1 Þ þ 2
2τ  1 1 ðU G ðn2 Þ−U G ðn1 ÞÞ þ½kn2 t ðn2 Þ þ U G ðn2 Þ þ 2 2τ
2
2 τ 1 1 τ 1 1 þ þ − ðU G ðn1 Þ−U G ðn2 ÞÞ − ðU G ðn2 Þ−U G ðn1 ÞÞ : 2 2 2τ 2 2 2τ ð19Þ The first term represents the gains from trading advertisers' products and the utility of users gross of transportation costs that accrue on platform 1. The second term represents the same for platform 2 while the last two terms are the transportation

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costs. Differentiating Eq.(19) with respect to ni, i = 1, 2, yields first-order conditions of
 h   i ∂WF 1 1  ′ þ U G ðni Þ−U G nj k t ðni Þ þ ni t ðni Þ þ U G ðni Þ ¼ 2 2τ ∂ni h     i τ þ U G ðni Þ k t ðni Þ−t nj þ U G ðni Þ−U G nj ð20Þ 2  1 0 U G ðni Þ−U G nj A ¼ 0; i≠j; i; j ¼ 1; 2: −τU G ðni Þ@ τ Due to assumption (iii) the maximization problem is strictly quasi-concave. Since the first-order conditions for n1 and n2 are symmetric, n1W = n2W. After simplifying Eq. (20) I obtain that the socially efficient number of advertisements can be written as follows: If

N ′ k t ðN=2Þ þ t ðN=2Þ þ U G ðN=2Þ > 0; 2

ð21Þ

niW = N/2 is efficient. If Eq.(21) does not hold, the efficient number of advertisers niW, i = 1, 2, is implicitly defined by 31       W W ′ W W þ U G ni ¼ 0: k t ni þ ni t ni

ð22Þ

First, I look at the last case in which niW is smaller than N/2 and is implicitly given by Eq. (22). I know that ni⋆ = N/2 if τ≥ Pτ . Thus, it follows that ni⋆ > niW if τ≥ Pτ . If τb Pτ , ni⋆ is implicitly given by t ðni Þ þ

λ

2

λ

γλðni Þ t ðni Þ γλni ¼ 0: − τ v″ ðt ðni ÞÞ

ð23Þ

Inserting niW given by Eq. (22) into Eq. (23) and simplifying yields   λ−1 nW γλt nW i i k

−

 2  λ γλt nW nW i i τ

0

or   W W τkni t ni : Thus, if τ ≥ kniWt(niW), the left-hand side of Eq. (23) is zero at ni = ni⋆ but it is larger than zero at ni = niW. Hence, ni⋆ ≥ niW. Conversely, ⋆ W if τ b kniWt(niW), I have It therefore follows that ni⋆ ≥ niW if  ni b n⋆i . W W W τ≥min Pτ ; knW t n b n if τ b min Pτ ; knW , and n . i i i i i t ni W Finally, I turn to the case ni = N/2, i.e., k(t(N/2) + N/2t′(N/2)) + UG(N/2) > 0. Here, I know that in equilibrium ni⋆ = N/2 if τ ≥ Pτ and ni⋆ b N/2 if τ b Pτ . Therefore, for τ ≥ Pτ the equilibrium advertising level is efficient, while for τ b Pτ there is underprovision of advertising. Summing up, I obtain that for τ≥ Pτ there is either overprovision of advertising or, if niW = N/2, the equilibrium advertising level is efficient, while for τ b Pτ , there is underprovision of advertising with the excep W tion if Eq. (22) and knW ■ b Pτ hold. i t ni References Anderson, S.P., Coate, S., 2005. Market provision of broadcasting: a welfare analysis. Review of Economic Studies 72, 947–972. Anderson, S.P., de Palma, A., Thisse, J.-F., 1992. Discrete Choice Theory of Product Differentiation. MIT Press, Cambridge, Massachusetts. Anderson, S.P., Gabszewicz, J.J., 2006. The media and advertising: a tale of two-sided markets. In: Ginsburgh, V.A., Throsby, D. (Eds.), Handbook of the Economics of Art and Culture, Volume 1. Elsevier B.V, North-Holland, pp. 567–614.

31 Note that niW = 0 can never be efficient since it would imply that kt(0) + UG(0) b 0. However, since kt(0) > 0 but UG(0) = 0 due to the fact that UG(ni) = − γt(ni)λ(ni)λ − 1, it follows that kt(0) + UG(0) > 0, contradicting that niW = 0 is efficient.

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