Quality competition and entry: a media market case

Abstract

The present paper provides an analysis of quality competition and entry in the context of two-sided media platforms. We provide a full characterization of a duopoly equilibrium in terms of advertising levels, subscription fees, and endogenous quality provision. Furthermore, we investigate the role of competition by considering potential entry. We show how the threat of entry shakes the equilibrium configuration by inducing lower quality differentiation. Finally, we describe the conditions under which a deterrence strategy in the form of a “limit quality” is feasible and profitable for the incumbent platform.

1 Introduction

Quality is a crucial feature of the media market. In fact, television and press markets exhibit a very limited degree of price competition as they compete primarily on product attributes, and in particular, quality. However, as the OECD points out: “Access to premium content is a serious bottleneck and a source of market power. In particular, premium sport events (e.g. Olympic Games or football matches) and new releases of movies, which have no substitutes, are essential to the successful functioning of pay TV providers. Barriers to accessing content can arise [...] Premium content may also have an impact on competition in other non-TV markets [...]” (OECD 2013).

Given that premium content such as important sporting events and television series are examples of quality in broadcasting (see, D’Annunzio 2017), the nature of competition among media platforms in terms of the quality of content, as well as the possibility of preventing entry by means of investing in quality, deserves closer attention in the media market debate. First, it is worth elucidating what quality is in media markets. At first glance, it appears to be associated with two different dimensions, namely technology and content. Indeed, technological innovations have greatly affected broadcasting quality, for example through higher-definition images and interactive services. Meanwhile, the quality of content is associated with providing not only entertainment but also education, learning, and cultural excellence, without ignoring niche interests (Costera Meijer 2005; Collins 2007). There is clear evidence in broadcasting that lowering production values to reduce costs tends to greatly reduce the appeal of a program to viewers (Barwise and Ehrenberg 1988; Jankowski and Fuchs 1996). Analogously, in the press market newspapers of higher quality enjoy higher circulation (Lacy and Fico 1991; Lacy and Martin 2004). However, providing high-quality broadcast programs and a high-quality bundle of press content is extremely expensive (Liu et al. 2004).¹ Second, according to the above assessment of media quality, there is the puzzle of determining which economic model is better suited to the analysis of media quality. In fact, there is no clear distinction between horizontal and vertical dimensions in media content; furthermore, what is conventionally defined as quality is not always assigned a positive value by all individuals. For example, were quality to be measured in terms of the extent to which a program informs and educates people (e.g., BBC programs²), it would be controversial to assume that all individuals considered this to be a net benefit.³ In this case, it would be more accurate to refer to a preference for variety and to use a horizontal differentiation framework.⁴ In contrast, clear examples of a vertical dimension to quality are the accuracy and immediacy of information, the presence of star journalists, and the ability to screen live performance events (including sports, music, and dance).

Finally, taking the crucial role of quality as a given, it is important to note that markets for news and entertainment are idiosyncratic examples of two-sided markets with media platforms competing on both sides. Platforms, as broadcasters or newspapers, compete for both audience⁵ and advertisers in order to maximize their profits; that is, they need to attract individuals as well as advertisers. On the one hand, advertising is typically considered to be a nuisance for the audience, or in other words, a negative externality; on the other hand, the audience exerts a positive externality on advertisers.

The present paper aims to analyze the strategic competition between platforms in a two-sided market characterized by vertical differentiation. We present a duopoly setup in which media platforms competing simultaneously to attract the audience set the advertising levels, subscription fees, and endogenous quality provision. Then, by considering a sequential duopoly setting in which the platforms move sequentially, we investigate how the possibility of potential entry affects not only the dominant position of the incumbent but also the quality provision of each platform and the degree of product differentiation. Finally, we focus on the incumbent platform’s selection of quality as an entry-deterring strategic device. In this respect, we explore the incumbent platform’s strategy of “limit quality” to maintain its dominant position in the presence of network externalities.⁶ More precisely, we illustrate the feasibility and profitability of the limit-quality deterrence strategy in a two-sided market. Thus, we shed new light on the outcome of entry deterrence based on an endogenous quality choice, elaborating on the features peculiar to a two-sided structure and the strategies that platforms use to defend their incumbency advantage.

Moreover, we assume single-homing readers/viewers but multi-homing advertisers; thus, platforms have monopoly power in providing their multi-homing side access to their single-homing customers. In this respect, the platforms act as “bottlenecks” between advertisers and consumers, by offering exclusive access to their respective sets of consumers.⁷ This assumption is crucial in explaining the prevailing competition on the consumer side. Furthermore, it is the driving force behind the “profit neutrality” outcome in a duopoly, where profit neutrality implies that advertising does not affect equilibrium profit. In fact, revenues from commercials are counterbalanced by a decrease in subscription fees. We also model advertisers as non-strategic: their payoffs do not depend on other advertisers’ actions but derive from an advertising benefit related to market demand. This behavior is appropriate in the case of informative advertising.

In this framework, the effects of endogenous quality provision on different market structures deserve closer attention. In fact, in this setup two forces are at stake. Higher quality induces consumers to pay higher subscription fees to join a platform. In turn, the platform can extract a surplus on the advertisers’ side and “invest” this in a reduction in subscription fees, implying that advertisers cross-subsidize single-homing consumers. Therefore, given profit neutrality a sort of substitution between quality and advertising arises. We show that the threat of entry may induce lower differentiation in terms of quality, whereby the degree of product differentiation is no longer maximal. Finally, we describe the conditions under which quality represents a bottleneck and a source of market power as an entry deterrence strategy for the incumbent platform.

1.1 Related literature

A media market represents an idiosyncratic example of a two-sided market (as seminal references, see Caillaud and Jullien 2001, 2003; Anderson and Coate 2005; Armstrong 2006; Hagiu 2006; Rochet and Tirole 2006). In this stream of the literature, the issue of quality has recently received noticeable attention. For example, Armstrong and Weeds (2007) and Weeds (2013) considered a model of endogenous quality provision in the two-sided context of digital broadcasters. By comparing competition in two different regimes, i.e. free-to-air and pay TV, they showed that program quality was higher for pay TV and was also optimal from a social point of view . However, they did not consider a sequential game or the entry issue. Anderson (2007) examined a similar setting with endogenous quality, but focused mainly on the effect of an advertising cap on both the quality provision of a monopoly broadcaster and welfare. He found that advertising time restrictions can improve welfare but may also reduce program quality. More recently, Lin (2011) extended his analysis to direct competition between two platforms: one operating as a free-to-air broadcaster, the second a pay TV broadcaster. In this framework, he found that the platforms vertically differentiated their programs according to the degree of viewers’ dislike for advertising. Again, the author did not extend the analysis to the sequential game or the possible strategic use of quality as a barrier to entry.⁸ Conversely, González-Maestre and Martínez-Sánchez (2015) developed a model using a publicly owned platform and a private one competing in a free-to-air broadcasting market. The platforms were differentiated on two dimensions: content (horizontal differentiation) and quality (vertical differentiation). Assuming that each platform chooses its advertising and quality levels, they calculated the subgame perfect equilibrium in terms of quality, advertising, and welfare under private and mixed duopoly competition. Our analysis departs from the above study by, on the one hand, considering a more general business model that allows for free-to-air competition as well as pay TV. Thus, we can endogenously determine not only advertising and quality but also subscription fees. On the other hand, González-Maestre and Martínez-Sánchez (2015) focused on welfare aspects and policy implications, while our main aim is to model competition, entry, and quality. Finally, the paper by Gabszewicz and Wauthy (2011) should be mentioned. These authors showed that in a platform competition with cross-network externalities, equilibrium outcomes resemble those obtained in standard models of vertical differentiation. In their setup, quality in one of the two markets is determined by outcomes in the other: the agents’ participation on each side determines the perceived quality of the other side. Therefore, it is the size of the network that endogenously determines willingness to join the platform and, in turn, the quality of the platform. In this way, the authors’ model diverges from the standard quality definition we use in our framework. Nevertheless, their results emphasize the importance of a vertical differentiation framework when dealing with competition between platforms. Note that all the above contributions except that of Anderson (2007)⁹ focused on the case of a duopoly, neglecting monopoly behavior. In contrast, we discuss the case of a monopoly in the Appendix 7.1. Furthermore, none of the above authors considered entry or pre-emption.

Ribeiro et al. (2016) and Roger (2017) extended the original one-sided analysis described by Gabszewicz and Wauthy (2011, 2014) to two-sided markets, nesting horizontal and vertical differentiation. Roger (2017) fully characterized a duopoly equilibrium in a pure strategy (and a mixed one) in a platform competition, with no specific reference to the media market. The same author also reported the existence of a quality distortion in the presence of cross-market externalities. In a similar setting and using nested horizontal and vertical differentiation, Ribeiro et al. (2016) showed that a negligible shock on the consumer side can be disruptive for market equilibrium when platforms compete on two sides. The authors also tackled the issue of the entry of a third platform, and found that inter-group externalities facilitated the deterrence of an inferior-quality entrant and the capture of the whole market by a superior-quality entrant. Apart from the different approach to modeling vertical differentiation, the present paper departs from these contributions by focusing on the incumbent platform’s decision to prevent entry by investing in higher quality.

In terms of competition between broadcasters, we refer in particular to Gabszewicz et al. (2001), Crampes et al. (2009) and Peitz and Valletti (2008). The first paper analyzed competition between two newspaper editors in a framework of horizontal differentiation of political opinion, in which readers were single-homing and advertisers multi-homing. The authors reported a tendency towards minimal differentiation in political messages, the so-called “La Pensée Unique,” in order to sell a larger audience to advertisers. We diverge from this paper in two aspects: on the one hand, we use a setting of vertical differentiation in which the platforms endogenously choose different levels of quality; and on the other, we consider different timings whereby subscription and advertising fees are set during the same stage. In this way, advertising tariffs do not strictly determine subscription fees or in turn, quality levels. These two aspects are behind our different outcome in terms of maximal quality differentiation. The second paper mentioned above, that by Crampes et al. (2009), examined the relationship between prices, advertising, and entry in a framework of horizontal differentiation, while we consider competition and entry with endogenous quality provision. Finally, Peitz and Valletti (2008) compared advertising intensity and content programming in a market with a duopoly of broadcasters choosing the degree of horizontal differentiation (i.e. the platforms chose the degree of program “diversity” in the horizontal space rather than vertical program quality). From this perspective, our model could be interpreted as a translation of Peitz and Valletti’s (2008) work into the vertical differentiation context, while also extending to the analysis of entry competition.

Finally, our paper relates to an older stream of the literature on industrial organizations in relation to vertical differentiation (see, e.g. Gabszewicz and Thisse 1979). In particular, we are indebted to the well-known study by Shaked and Sutton (1982, 1983), which demonstrated market equilibrium when firms compete in a vertically differentiated framework and are ranked according to their quality levels. More recently, Donnenfeld and Weber (1992, 1995) extended Shaked and Sutton’s simultaneous model to the case of sequential entry. In their 1992 paper, the authors produced a model that generated maximal differentiation between the equilibrium qualities of the first two firms to enter. Meanwhile, in both papers they showed how an incumbent firm can use limit quality to deter entry, as a result of which, according to a large fixed cost of entry, a high level of product differentiation is to be expected. With the same non-simultaneous entry, we extend their conditions to our two-sided framework to explain the role of entry in platform competition and the level of quality differentiation.

The paper is organized as follows: Sect. 2 introduces the general model; Sect. 3 provides a full characterization of the equilibrium in a duopoly. Section 5 deals with competition and entry; and Sect. 5.3 investigates the strategy of entry deterrence in particular. Some concluding remarks (Sect. 6) close the paper.

2 Duopoly: the set up

We consider a duopoly with two platforms. Without loss of generality we assume that $i=L$ is the low-quality platform, while $i=H$ is the high-quality one.

2.1 Individuals

There is a continuum of individuals of mass N. They constitute the buyer side in the market. If the individuals join a platform, they are exposed to media contents¹⁰ and some informative advertising about market products. All individuals value quality of information in the sense of vertical differentiation: the quality of the platform’s content is denoted by the parameter $q\in Q=[{\underline{q}} ,{\overline{q}}]$ with ${\overline{q}}>{\underline{q}}>0$. Individuals make a private valuation $\beta$ of the quality of information, which can be interpreted as their willingness to pay for it; their taste for quality $\beta$ is distributed uniformly on an interval $[{\underline{\beta }} ,{\overline{\beta }}]$ with ${\overline{\beta }}>{\underline{\beta }}>0$. Moreover, individuals are assumed to dislike advertising. In the presence of ads, their utility loss is $\delta a$, where a denotes the advertising level and $\delta$ the disutility parameter for being exposed to it. Unlike $\beta$, the parameter $\delta$ is assumed to be invariant across individuals. Individuals can access one platform at most (single-homing).¹¹

The utility of an individual from joining platform i of quality q is:

$$\begin{aligned} u_{i}=V-\delta a_{i}+\beta q_{i}-s_{i} \end{aligned}$$

(1)

where V is the utility of accessing the platform independently of its quality, $q_{i}$ denotes platform i’s quality and $a_{i}$ the level of advertising. Finally, $s_{i}$ stands for the subscription fee or the price to access the platform i. Each individual has a reservation utility $u_{0}=0$.

We can characterize the individual indifferent to either accessing the low-quality platform L, or not accessing at all:

$$\begin{aligned} \beta _{0L}=\frac{\delta a_{L}-V}{q_{L}}+\frac{s_{L}}{q_{L}} \end{aligned}$$

(2)

while the individual indifferent to either a high-quality platform or a low-quality one is described by the following equation:

$$\begin{aligned} \beta _{LH}=\frac{\delta \left( a_{H}-a_{L}\right) }{\left( q_{H} -q_{L}\right) }+\frac{\left( s_{H}-s_{L}\right) }{\left( q_{H} -q_{L}\right) } \end{aligned}$$

(3)

The expression of $\beta _{0L}$ and $\beta _{LK}$ define $B_{i}$, namely the share of individuals willing to join the platform i.

2.2 Advertisers

The supply side is made up of producers, who access the platform to advertise their products. They sell products of quality $\alpha$ produced at constant marginal costs, set equal to zero. Product quality $\alpha$ is distributed on an interval $\left[ 0,{\overline{\alpha }}\right]$ according to a distribution function $F(\alpha )$. Individuals are willing to pay $\alpha$ for a good of quality $\alpha$. Each producer has monopoly power and can therefore extract the full surplus from individuals by selling their product at price equal to $\alpha$. As is standard in this class of models, we assume advertising to be informative and that only those individuals who watch the advertisement buy the good. Hence, we refer to producers as advertisers. Advertisers are allowed to ‘multi-home’ and can advertise across none, one, or more platforms. Advertisers have to pay platform i an advertising charge $r_{i}$. Therefore, advertisers’ profits on platform i are:

$$\begin{aligned} \Pi _{a}=N\alpha _{i}B_{i}-r_{i} \end{aligned}$$

(4)

The advertising charge $r_{i}$ is endogenously determined by each platform. Due to our assumption of single homing on the buyer side, each media platform behaves as a monopoly in conveying its audience to advertiser. Therefore, the advertising charge $r_{i}$ is set in order to leave the marginal advertiser with zero profit, $\Pi _{a}=N\alpha _{i}B_{i}-r_{i}=0$, which implicitly defines:

$$\begin{aligned} \alpha _{i}=\frac{r_{i}}{NB_{i}} \end{aligned}$$

(5)

Thus, the amount of advertising for each platform is the share of advertisers with $\alpha >\alpha _{i}$:¹²

$$\begin{aligned} a_{i}=1-F\left( \frac{r_{i}}{NB_{i}}\right) \end{aligned}$$

(6)

2.3 Platforms

Media markets are characterized by a broad range of business models, under both private and public ownership:¹³ free-to-air TV under which broadcast platforms are financed through advertising revenues only; pay TV, under which they are financed through subscription revenues; and a mixed regime, under which they are financed through both subscription fees and advertising. Therefore, to encompass all these cases we consider a very general framework in which platforms are financed by both advertising and subscription fees.

Platforms set the advertising space, the subscription prices, which might be positive or negative (subsidies) and the quality. We assume neither constraints on advertising space (caps)¹⁴ nor the costs of running ads. Quality is, however, costly to provide. We assume that this quality cost is independent of the number of units and is fixed at K (see, e.g. Mussa and Rosen 1978; Hung and Schmitt 1988). This cost assumption can be justified under the theory of innovation and the idea that product innovations endowed with quality depend upon fixed investment in R&D. This assumption fits the structure of ICT and media markets, where fixed costs rather than marginal ones play a prominent role (see, e.g. Shapiro and Varian 1998; Anderson and Coate 2005; Choi 2006; Areeda and Hovenkamp 2014) very well. In other words, once the cost is incurred the higher-quality outlet can be distributed to individuals at no additional charge.¹⁵

Hence, a media platform collects revenue from both individuals and advertisers. For any platform i the objective function takes the form:

$$\begin{aligned} \Pi _{i}\left( s_{i},a_{i},r_{i},q_{i}\right) =NB_{i}s_{i}+a_{i} r_{i}-K \end{aligned}$$

(7)

2.4 Timing

We assume a three-stage game. In the first stage, platforms choose the quality levels of their contents. Then, in the second stage, subscription fees and advertising spaces are set. Finally, in the third stage individuals and advertisers simultaneously decide whether or not to join a platform. Individuals can join at most one platform (single-homing) while advertisers may join more than one (multi-homing). The game is solved backward in a duopoly configuration.

3 Duopoly: platform subscription fees and advertising levels

In this framework, we consider a market structure in which both firms are active (meaning that individuals’ demands for platform H and L are positive) and look for equilibrium in the covered market. Hence, we first rule out the trivial case in which the low-quality platform always faces zero demand in the price game. As is standard in the vertical differentiation literature (see Tirole 1988), the heterogeneity of the individuals must be sufficiently high:

$$\begin{aligned} {\overline{\beta }}>2{\underline{\beta }} \end{aligned}$$

(8)

Second, for the market to be covered, we introduce the following condition:¹⁶

$$\begin{aligned} {\underline{\beta }}q_{L}\ge \frac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }{3}-\left( \rho \left( a_{L}^{*}\right) -\delta a_{L}^{*}\right) \end{aligned}$$

(9)

which states that in equilibrium, the individual with a preference for the lowest quality also gets some positive utility from joining the low-quality platform with sufficiently small values of K. Notice that in this condition (9) compared with the condition ensuring market coverage in a single-side framework there is an additional part relating to the presence of externalities. If we choose the quality space $Q=[{\underline{q}},{\overline{q}}]$ such that condition (9) holds, we obtain ex-ante market coverage for every quality belonging to the technological range.¹⁷

Therefore, we define the demand function for the high-quality $NB_{H}$ and for the low-quality $NB_{L}$, respectively:

$$\begin{aligned} NB_{H}= & {} N\left( \dfrac{{\overline{\beta }}-\beta _{LH}}{\overline{\beta }-{\underline{\beta }}}\right) \nonumber \\= & {} N\left( \dfrac{{\overline{\beta }}}{{\overline{\beta }}-{\underline{\beta }}} -\dfrac{\delta (a_{H}-a_{L})}{\left( q_{H}-q_{L}\right) \left( {\overline{\beta }}-{\underline{\beta }}\right) }-\dfrac{s_{H}-s_{L}}{\left( q_{H}-q_{L}\right) \left( {\overline{\beta }}-{\underline{\beta }}\right) }\right) \end{aligned}$$

(10)

$$\begin{aligned} NB_{L}= & {} N\left( \dfrac{\beta _{LH}-{\underline{\beta }}}{\overline{\beta }-{\underline{\beta }}}\right) \nonumber \\= & {} N\left( \dfrac{\delta (a_{H}-a_{L})}{\left( q_{H}-q_{L}\right) \left( {\overline{\beta }}-{\underline{\beta }}\right) }+\dfrac{s_{H}-s_{L}}{\left( q_{H}-q_{L}\right) \left( {\overline{\beta }}-{\underline{\beta }}\right) }-\dfrac{{\underline{\beta }}}{{\overline{\beta }}-{\underline{\beta }}}\right) \end{aligned}$$

(11)

The amount of advertising for each platform becomes: $a_{i}=1-F\left( \frac{r_{i}}{NB_{i}}\right)$ with $i=H,L$.

The profit function (7) would be rewritten as follows, for the high-quality and low-quality platforms respectively:¹⁸

$$\begin{aligned} \Pi _{H}\left( s_{H},s_{L},a_{H},a_{L},r_{H},r_{L},q_{H},q_{L}\right)= & {} NB_{H}s_{H}+a_{H}r_{H}-K \end{aligned}$$

(12)

$$\begin{aligned} \Pi _{L}\left( s_{H},s_{L},a_{H},a_{L},r_{H},r_{L},q_{H},q_{L}\right)= & {} NB_{L}s_{L}+a_{L}r_{L}-K \end{aligned}$$

(13)

We can now solve the game backwards, from stage three. Duopoly platforms maximize profit subject to a positivity constraint on advertising level:

$$\begin{aligned} \left\{ \begin{array}{c} \underset{a_{H},s_{H}}{\max }\Pi _{M}=NB_{i}s_{i}+a_{i}r_{i}-K\\ s.t.\text { }a_{i}\ge 0 \end{array} \right. \end{aligned}$$

(14)

with $i=H,L$. The first order conditions with respect to the advertising spaces $a_{i}$ and subscription fees $s_{i}$ with $i=H,L$ are:

$$\begin{aligned} Ns_{i}\frac{\partial B_{i}}{\partial a_{i}}+r_{i}+a_{i}\frac{\partial r_{i} }{\partial a_{i}}\le & {} 0 \end{aligned}$$

(15)

$$\begin{aligned} NB_{i}+Ns_{i}\frac{\partial B_{i}}{\partial s_{i}}+a_{i}\frac{\partial r_{i} }{\partial s_{i}}= & {} 0 \end{aligned}$$

(16)

Then, in line with the literature, we define advertising revenues per individual as $\rho (a_{i})$

$$\begin{aligned} \rho (a_{i})=\frac{a_{i}r_{i}}{NB_{i}}=\frac{a_{i}F^{-1}(1-a_{i})NB_{i}}{NB_{i}}=a_{i}F^{-1}(1-a_{i}) \end{aligned}$$

(17)

We assume $\rho (a_{i})$ to be concave in the interval $a\in \left[ 0,1\right]$. Given that $\rho (a_{i})=0$ for $a_{i}=0$ and $a_{i}=1$, the function is single-peaked.¹⁹ Using the definition (17) for the duopoly platforms we can rewrite the optimality conditions, proving the following proposition.

Proposition 1

For each platform i, if the profit maximizing advertising level is strictly positive, then it is constant and determined by

$$\begin{aligned} \rho ^{\prime }(a_{i})=\delta \end{aligned}$$

Proof

Platforms maximize profits, (13) and (12), subject to $a_{i}\ge 0$ with $i=H,L$. Equations (15) and (16) define the first order conditions with respect to the advertising spaces $a_{i}$ and subscription fees $s_{i}$. Given (17) for platform H we have $r_{H}=\dfrac{NB_{H}\rho (a_{H})}{a_{H}}$ and:

$$\begin{aligned} \frac{\partial r_{H}}{\partial s_{H}}&=\frac{1}{a_{H}}N\rho (a_{H} )\frac{\partial B_{H}}{\partial s_{H}}\nonumber \\ \frac{\partial r_{H}}{\partial a_{H}}&=\frac{[NB_{H}\rho ^{\prime } +N\rho (a_{H})\frac{\partial B_{H}}{\partial a_{H}}]a_{H}-NB_{H}\rho (a_{H} )}{a_{H}^{2}} \end{aligned}$$

(18)

Therefore optimality condition (16) and (15) rewrite:

$$\begin{aligned} B_{H}+(s_{H}+\rho (a_{H}))\frac{\partial B_{H}}{\partial s_{H}}= & {} 0 \end{aligned}$$

(19)

$$\begin{aligned} B_{H}\rho ^{\prime }(a_{H})+\left( \rho (a_{H})+s_{H})\frac{\partial B_{H}}{\partial a_{H}}\right)\le & {} 0 \end{aligned}$$

(20)

Since:

$$\begin{aligned} \frac{\partial B_{H}}{\partial a_{H}}=\delta \frac{\partial B_{H}}{\partial s_{H}} \end{aligned}$$

(21)

(20) becomes:

$$\begin{aligned} \frac{\rho ^{\prime }(a_{H})}{\delta }B_{H}+(\rho (a_{H})+s_{H})\frac{\partial B_{H}}{\partial s_{H}})\le 0 \end{aligned}$$

(22)

Together with (19), we obtain the following conditions:

$$\begin{aligned} {\left\{ \begin{array}{ll} -B_{H}=(s_{H}+\rho (a_{H}))\frac{\partial B_{H}}{\partial s_{H}}\\ (\frac{\rho ^{\prime }(a_{H})}{\delta }-1)B_{H}\le 0 \end{array}\right. } \end{aligned}$$

(23)

If $a_{H}>0$ the above inequality is satisfied with equality. Therefore, given that $\rho (a_{H})$ is single-peaked, $a_{H}$ is uniquely determined by the following condition:

$$\begin{aligned} \rho ^{\prime }(a_{H}^{*})=\delta \end{aligned}$$

Analogously, for platform L, if $a_{L}>0$ we get:

$$\begin{aligned} \rho ^{\prime }(a_{L}^{*})=\delta \end{aligned}$$

$\square$

Proposition 1 states that, for both platforms, a fixed advertising space is the best reply. In particular, the equilibrium level of advertising depends on the advertising disutility of individuals, suggesting that both platforms only compete on individuals. In this respect, platforms act as “bottlenecks” between advertisers and individuals, by offering sole access to their respective set of individuals. Notice also that the platforms do not set the maximum level of advertising or the amount that maximize revenues per viewer, i.e. $\rho ^{\prime }(a_{i})=0$, unless consumers are neutral to ads ($\delta =0$). This result replicates that of Weeds (2013) in a context of vertical differentiation but with quadratic costs.²⁰ We also share the same insight: that what really matters for competition in two-sided markets is the single-homing part.

We now have a solution for the subscription fees, $s_{H}$ and $s_{L}$, which enables us to compute the equilibrium values of the advertising prices, $r_{H}$ and $r_{L}$.

Proposition 2

Platform H set a higher subscription fee and a higher advertising price, with respect to platform L: $s_{H}^{*}\left( q_{H},q_{L}\right) >s_{L}^{*}\left( q_{H},q_{L}\right)$ and $r_{H}^{*}\left( a,\rho \right) >r_{L}^{*}\left( a,\rho \right)$. Moreover, they share the market in a fixed proportion: $B_{H}^{*} >B_{L}^{*}$.

Proof

In the second stage of the game, with $\rho (a_{i})$ concave, we obtain the equilibrium prices $s_{H}^{*},s_{L}^{*}$ and $r_{H}^{*},r_{L}^{*}$ as function of qualities, revenues per viewer and advertising. From condition (19) for platform H and the analogous condition for platform L, we get:

$$\begin{aligned} \left\{ \begin{array}{c} s_{H}=\dfrac{s_{L}+{\overline{\beta }}\left( q_{H}-q_{L}\right) -\delta (a_{H}-a_{L})-\rho (a_{H})}{2}\\ s_{L}=\dfrac{s_{H}-{\underline{\beta }}\left( q_{H}-q_{L}\right) +\delta (a_{H}-a_{L})-\rho \left( a_{L}\right) }{2} \end{array} \right. \end{aligned}$$

(24)

Then, the solution of the above system becomes:

$$\begin{aligned}&s_{H}^{*}\left( q_{H},q_{L},\rho (a_{H}),\rho \left( a_{L}\right) \right) \nonumber \\&\quad =\frac{2}{3}{\overline{\beta }}\left( q_{H}-q_{L}\right) -\frac{1}{3}{\underline{\beta }}\left( q_{H}-q_{L}\right) -\frac{1}{3}\delta \left( a_{H}-a_{L}\right) -\frac{2}{3}\rho (a_{H})-\frac{1}{3}\rho \left( a_{L}\right) \qquad \quad \end{aligned}$$

(25)

$$\begin{aligned}&s_{L}^{*}\left( q_{H},q_{L},\rho (a_{H}),\rho _{L}\right) \nonumber \\&\quad =\frac{1}{3}{\overline{\beta }}\left( q_{H}-q_{L}\right) -\frac{2}{3}{\underline{\beta }}\left( q_{H}-q_{L}\right) +\frac{1}{3}\delta \left( a_{H}-a_{L}\right) -\frac{1}{3}\rho (a_{H})-\frac{2}{3}\rho \left( a_{L}\right) \qquad \quad \end{aligned}$$

(26)

If we plug $s_{H}^{*}$ and $s_{L}^{*}$ in the demand function obtained at stage three, (10) and (11), we get:

$$\begin{aligned}&B_{H}^{*}\left( q_{H},q_{L},\rho (a_{H}),\rho \left( a_{L}\right) \right) \nonumber \\&\quad =\tfrac{{\overline{\beta }}}{{\overline{\beta }}-{\underline{\beta }}} -\tfrac{\tfrac{\left( {\underline{\beta }}+{\overline{\beta }}\right) \left( q_{H}-q_{L}\right) +\delta \left( a_{H}-a_{L}\right) -\rho (a_{H})+\rho \left( a_{L}\right) }{3}}{\left( {\overline{\beta }} -{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) } \end{aligned}$$

(27)

$$\begin{aligned}&B_{L}^{*}\left( q_{H},q_{L},\rho (a_{H}),\rho \left( a_{L}\right) \right) \nonumber \\&\quad =\tfrac{\tfrac{\left( {\underline{\beta }}+{\overline{\beta }}\right) \left( q_{H}-q_{L}\right) +\delta \left( a_{H}-a_{L}\right) -\rho (a_{H})+\rho \left( a_{L}\right) }{3}}{\left( {\overline{\beta }} -{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }-\tfrac{{\underline{\beta }}}{{\overline{\beta }}-{\underline{\beta }}} \end{aligned}$$

(28)

Finally, considering:

$$\begin{aligned}&r_{L}\left( \left( s_{H},s_{L},a_{H},a_{L},q_{H},q_{L}\right) \right) \nonumber \\&\quad = F^{-1}(1-a_{L})NB_{L} \end{aligned}$$

(29)

$$\begin{aligned}&r_{H}\left( s_{H},s_{L},a_{H},a_{L},q_{H},q_{L}\right) \nonumber \\&\quad = F^{-1}(1-a_{H})NB_{H} \end{aligned}$$

(30)

we end with:

$$\begin{aligned}&r_{H}^{*}\left( q_{H},q_{L},\rho (a_{H}),\rho \left( a_{L}\right) \right) \nonumber \\&\quad = \dfrac{\rho \left( a_{H}\right) }{a_{H}}N\left( \dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) -\delta \left( a_{H}-a_{L}\right) +\rho (a_{H})-\rho \left( a_{L}\right) }{3\left( {\overline{\beta }}-{\underline{\beta }}\right) \left( q_{H} -q_{L}\right) }\right) \qquad \end{aligned}$$

(31)

$$\begin{aligned}&r_{L}^{*}\left( q_{H},q_{L},\rho (a_{H}),\rho \left( a_{L}\right) \right) \nonumber \\&\quad = \dfrac{\rho \left( a_{L}\right) }{a_{L}}N\left( \dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) +\delta \left( a_{H}-a_{L}\right) -\rho (a_{H})+\rho \left( a_{L}\right) }{3\left( {\overline{\beta }}-{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }\right) \qquad \end{aligned}$$

(32)

If $a_{L}=a_{H}=a^{*}$ then $\rho (a_{H})=\rho (a_{L})=\rho (a^{*})$, it will be straightforward to see:

$$\begin{aligned}&s_{H}^{*}\left( q_{H},q_{L},a^{*}\right) =\dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }{3}-\rho (a^{*})>\nonumber \\&\quad \dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) \left( q_{H} -q_{L}\right) }{3}-\rho (a^{*})=s_{L}^{*}\left( q_{H},q_{L},a^{*}\right) \end{aligned}$$

(33)

and

$$\begin{aligned}&r_{H}^{*}\left( a,\rho \right) =\dfrac{\rho \left( a^{*}\right) }{a^{*}}N\left( \dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }{3\left( {\overline{\beta }}-\underline{\beta }\right) \left( q_{H}-q_{L}\right) }\right) >\nonumber \\&\quad \dfrac{\rho \left( a^{*}\right) }{a^{*}}N\left( \dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }{3\left( {\overline{\beta }}-{\underline{\beta }}\right) \left( q_{H} -q_{L}\right) }\right) =r_{L}^{*}\left( a,\rho \right) \end{aligned}$$

(34)

Finally,

$$\begin{aligned} B_{H}^{*}=\frac{2{\overline{\beta }}-{\underline{\beta }}}{3\left( {\overline{\beta }}-{\underline{\beta }}\right) }>\dfrac{{\overline{\beta }}-2{\underline{\beta }}}{3\left( {\overline{\beta }}-{\underline{\beta }}\right) }=B_{L}^{*} \end{aligned}$$

$\square$

Looking at equilibrium subscription fees and market shares, $B_{H}^{*}$ and $B_{L}^{*}$ , it is easy to see a “profit neutrality” result: revenues from the advertising side are counterbalanced by a decrease in subscription fee. In fact, advertising does not directly affect market shares or equilibrium profits but has an impact only on subscription fees. Given that subscription fees positively depend on quality, a sort of substitutability between advertising and quality emerges.

$$\begin{aligned} s_{H}^{*}\left( q_{H},q_{L},a^{*},\delta \right)= & {} \frac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }{3}-\rho (a^{*}) \end{aligned}$$

(35)

$$\begin{aligned} s_{L}^{*}\left( q_{H},q_{L},a^{*},\delta \right)= & {} \frac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }{3}-\rho (a^{*}) \end{aligned}$$

(36)

4 Duopoly: platform quality

We can now solve the initial stage of the game, namely quality choice. To anticipate the results, profits increase with quality distance as is standard in vertical differentiation models with a single-side. As a matter of fact, given our assumption about costs, platforms have the incentive to differentiate themselves maximally.

Proposition 3

In equilibrium, the high-quality platform chooses the maximum quality level, $q_{H}^{*}={\overline{q}}$, the low-quality platform the minimum quality level, $q_{L}^{*}={\underline{q}}$.

Proof

Rewriting profit function for H and L respectively, (12) and (13) we have:

$$\begin{aligned} \Pi _{H}^{*}\left( q_{H},q_{L}\right)= & {} s_{H}^{*}NB_{H}^{*}+\rho NB_{H}^{*}-K=N\tfrac{\left( {\underline{\beta }}-2{\overline{\beta }}\right) ^{2}\left( q_{H}-q_{L}\right) }{9\left( {\overline{\beta }}-\underline{\beta }\right) }-K \end{aligned}$$

(37)

$$\begin{aligned} \Pi _{L}^{*}\left( q_{H},q_{L}\right)= & {} s_{L}^{*}NB_{L}^{*}+\rho NB_{L}^{*}-K=N\tfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}\left( q_{H}-q_{L}\right) }{9\left( {\overline{\beta }}-\underline{\beta }\right) }-K \end{aligned}$$

(38)

Computing the FOCs, under the assumption of non-negativity constraint of quality we obtain:

$$\begin{aligned} \frac{\partial \Pi _{H}^{*}}{\partial q_{H}}= & {} \dfrac{\left( \underline{\beta }-2{\overline{\beta }}\right) ^{2}}{9({\overline{\beta }}-{\underline{\beta }})}N>0 \end{aligned}$$

(39)

$$\begin{aligned} \frac{\partial \Pi _{L}^{*}}{\partial q_{L}}= & {} -\dfrac{\left( \overline{\beta }-2{\underline{\beta }}\right) ^{2}}{9({\overline{\beta }}-{\underline{\beta }})}N<0 \end{aligned}$$

(40)

Hence:

$$\begin{aligned} q_{H}^{*}={\overline{q}}\text {, }q_{L}^{*}={\underline{q}} \end{aligned}$$

$\square$

In the special case of a uniform distribution of advertisers, the following lemma holds:

Lemma 4

Under the assumption of the p.d.f. of advertisers F uniform on $\left[ 0,1\right]$ equilibrium values are:

$$\begin{aligned} a_{L}^{*}= & {} a_{H}^{*}=a^{*}=\frac{1-\delta }{2} \end{aligned}$$

(41)

$$\begin{aligned} s_{H}^{*}\left( q_{H},q_{L},\delta \right)= & {} \frac{\left( 2\overline{\beta }-{\underline{\beta }}\right) \left( {\overline{q}}-{\underline{q}}\right) }{3}-\frac{1-\delta }{2}\left( \frac{1+\delta }{2}\right) \end{aligned}$$

(42)

$$\begin{aligned} s_{L}^{*}\left( q_{H},q_{L},\delta \right)= & {} \frac{\left( \overline{\beta }-2{\underline{\beta }}\right) \left( {\overline{q}}-{\underline{q}}\right) }{3}-\frac{1-\delta }{2}\left( \frac{1+\delta }{2}\right) \end{aligned}$$

(43)

Notice that the advertising level decreases with the disutility parameter $\delta$. Conversely, both subscription fees $s_{L}^{*}$ and $s_{H}^{*}$ increase with $\delta$. As expected, profits are neutral with $\delta$. This result is in line with our findings for profit neutrality: a higher $\delta$ implies lower advertising revenues to be used in the reduction of fees. Finally, equilibrium market shares and quality are not affected by the assumption of p.d.f.

5 Entry

In this section we focus on the effects of competition on market structure and platform quality. We have already considered the comparison between a monopoly and a duopoly. However, a framework with simultaneous choices does not allow us to deal with potential competition or the issue of incumbency advantage. Therefore, to tackle this issue we analyze quality differentiation in a framework of sequential entry. We modify our timing slightly by considering an incumbent platform and an entrant platform and splitting the quality choice stage: the incumbent platform (I) sets quality first, followed by the entrant platform (E). The technology structure and profit function are the same but for the entry cost F on the side of the entrant, as is standard in the literature. Intuitively, this cost can be interpreted as an asymmetry between platforms concerning quality costs.²¹ In this framework we focus on the existence conditions of a duopoly equilibrium and check its robustness by looking at the entry deterrence strategy of the incumbent.

5.1 Sequential duopoly

As already mentioned, in order to deal with a sequential equilibriumequilibrium, we slightly modify the timing of the game. Nothing changes for stages 3 and 2, while at stage 1 we separate out the quality decision of the two platforms: first, the incumbent platform selects the quality; then the entrant platform sets its quality. After the choice of quality, the two platforms simultaneously set their prices for advertising and subscription fees, $r_{i}$ and $s_{i}$, as in the previous setting. Hence, the equilibrium solutions for stages 3 and 2 still hold (see Proposition 2). Remember that the equilibrium profit of the high-quality platform is higher than that of the low-quality one; in other words, higher quality commands higher profit. Therefore, the incumbent platform will exploit its advantage by behaving like a high-quality one and leaving room only for entry at the low-quality level. Equilibrium solutions for the simultaneous framework, with $E=L$ for the entrant and $I=H$ for the incumbent are as follows.

Equilibrium subscription fees:

$$\begin{aligned} s_{I}^{*}= & {} \dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) \left( q_{I}-q_{E}\right) }{3}-\rho (a^{*})\nonumber \\ s_{E}^{*}= & {} \dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) \left( q_{I}-q_{E}\right) }{3}-\rho (a^{*}) \end{aligned}$$

(44)

Equilibrium demands:

$$\begin{aligned} NB_{I}^{*}= & {} N\dfrac{2{\overline{\beta }}-{\underline{\beta }}}{3\left( {\overline{\beta }}-{\underline{\beta }}\right) }\nonumber \\ NB_{E}^{*}= & {} N\dfrac{{\overline{\beta }}-2{\underline{\beta }}}{3\left( {\overline{\beta }}-{\underline{\beta }}\right) } \end{aligned}$$

(45)

Equilibrium advertising prices:

$$\begin{aligned} r_{I}^{*}= & {} N\dfrac{\rho (a^{*})}{a^{*}}\dfrac{2\overline{\beta }-{\underline{\beta }}}{3\left( {\overline{\beta }}-{\underline{\beta }}\right) \left( q_{I}-q_{E}\right) }\nonumber \\ r_{E}^{*}= & {} N\dfrac{\rho (a^{*})}{a^{*}}\dfrac{\overline{\beta }-2{\underline{\beta }}}{3\left( {\overline{\beta }}-{\underline{\beta }}\right) \left( q_{I}-q_{E}\right) } \end{aligned}$$

(46)

Equilibrium profits:

$$\begin{aligned} \Pi _{I}^{*}= & {} N\dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }\left( q_{I} -q_{E}\right) -K\nonumber \\ \Pi _{E}^{*}= & {} N\dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }\left( q_{I} -q_{E}\right) -K-F \end{aligned}$$

(47)

Moving to the quality stage, the entrant platform fixes its quality in order to maximize profits given the quality choice of the incumbent platform.

$$\begin{aligned} \Pi _{E}^{*}= & {} N\dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }\left( q_{I} -q_{E}\right) -K-F \\ \frac{\partial \Pi _{E}^{*}}{\partial q_{E}}|_{q_{I}}= & {} -N\dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }<0 \end{aligned}$$

Given the negative sign of the derivative, platform E has the incentive to choose the minimum quality ${\underline{q}}$.

The final stage involves the quality choice of the incumbent platform:

$$\begin{aligned} \Pi _{I}^{*}= & {} N\dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }(q_{I} -{\underline{q}})-K \end{aligned}$$

(48)

$$\begin{aligned} \frac{\partial \Pi _{I}^{*}}{\partial q_{I}}= & {} N\dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }>0 \end{aligned}$$

(49)

Given the positive sign of the derivative, platform I has the incentive to choose the maximum quality. In equilibrium, the profits of the sequential duopoly are:

$$\begin{aligned} \Pi _{I}^{*}= & {} N\dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }\left( {\overline{q}}-{\underline{q}}\right) -K \end{aligned}$$

(50)

$$\begin{aligned} \Pi _{E}^{*}= & {} N\dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }\left( {\overline{q}}-{\underline{q}}\right) -K-F \end{aligned}$$

(51)

As in the simultaneous case, we obtain a result of maximal differentiation. Revenues are not changed for either platform. However, I has the incumbency advantage of being first on the market, behaving as the high-quality platform and saving entry costs.

5.2 Threat of entry

In this section we analyze the effect of potential competition by means of the entrance of new competitors. As mentioned above, the incumbent platform behaves like a high-quality one, leaving room for entry at the low-quality level only. Given this framework, we consider the impact of potential competition on the quality and degree of vertical differentiation of the platforms. On the one hand, it is worth noticing that with a fixed cost of entry a potential entrant cannot profitably leapfrog the high-quality incumbent. Quality is already at a maximum and therefore the only option available is to charge lower prices for the same quality. However, the existence of an entry cost prevents this strategy from being profitable. On the other hand, the existence of positive profits for a low-quality platform makes it convenient for a potential entrant to enter the market. In this case, by setting a slightly higher quality the entrant will capture all the low-quality demand. According to Shaked and Sutton (1982), a traditional model of vertical differentiation with simultaneous decisions will have at most two firms with positive market shares and covering the entire market with different levels of quality, given an appropriate heterogeneity of individuals.²² We show that this condition applies to a two-sided context too.²³

Lemma 5

Let $2{\underline{\beta }}<{\overline{\beta }}<4{\underline{\beta }}$ . Then, of any n platforms offering distinct levels of quality, exactly two will have positive market shares on the buyers’ side (audience) at equilibrium. Moreover, at equilibrium the market will be covered.

Proof

See Appendix 7.2. $\square$

Therefore by assuming $2{\underline{\beta }}<{\overline{\beta }}<4{\underline{\beta }}$ we know that in equilibrium the market is covered by the two platforms with the highest levels of quality. Hence, the low-quality platform can only survive if no other competitors enter the market. Therefore, in order to deter entry the low-quality platform should drive down its profits to zero. In this way, there is no incentive for any other platform to enter. Given that, we can examine how the quality levels of the incumbent (high quality) and entrant (low quality) are affected.

Proposition 6

Under the threat of entry the equilibrium quality of the incumbent platform $q_{I}^{*}$ lies in the interval $[\max ({\widetilde{q}} _{I},\tilde{\tilde{q_{I}}}),{\overline{q}}]$ while the quality choice of firm E is such that $q_{E}^{*}=q_{I}^{*}-\left( K+F\right) \frac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}$.

Proof

We start with platform E. Platform E should drive its profit to zero, in order to prevent the entrance of a new platform:

$$\begin{aligned} \Pi _{E}^{*}=N\dfrac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }\left( q_{I} -q_{E}\right) -K-F=0 \end{aligned}$$

(52)

then

$$\begin{aligned} q_{E}^{*}=q_{I}-\left( K+F\right) \frac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}} \end{aligned}$$

(53)

Given the choice of platform E the profit of the incumbent becomes:

$$\begin{aligned} \Pi _{E}^{*}\left( q_{E}^{*}\right)= & {} 0\nonumber \\ \Pi _{I}^{*}\left( q_{E}^{*}\right)= & {} \dfrac{3{\overline{\beta }} ^{2}-3{\underline{\beta }}^{2}}{\left( {\overline{\beta }}-2\underline{\beta }\right) ^{2}}K+\dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) ^{2}}{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}F \end{aligned}$$

(54)

incumbent’s profits are constant (i.e. independent of quality) and positive. However, we need to assess a range of quality for the platform I compatible with the duopoly equilibrium, such that a second platform can just survive as a low quality. We calculate two threshold values for the incumbent, ${\widetilde{q}}_{I}$ and $\tilde{\tilde{q_{I}}}$, such that the profits of the entrant are driven to zero if it enters with the lowest quality ${\underline{q}}$ or with the highest quality ${\overline{q}}$ respectively:

$$\begin{aligned} \Pi _{E}^{*}({\widetilde{q}}_{I},{\underline{q}})= & {} 0\nonumber \\ {\widetilde{q}}_{I}= & {} {\underline{q}}+\left( F+K\right) \dfrac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( \overline{\beta }-2{\underline{\beta }}\right) ^{2}} \end{aligned}$$

(55)

and

$$\begin{aligned} \Pi _{E}^{*}(\tilde{\tilde{q_{I}}},{\overline{q}})= & {} 0\nonumber \\ \tilde{\tilde{q_{I}}}= & {} {\overline{q}}-\left( K-F\right) \dfrac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( 2{\overline{\beta }}-{\underline{\beta }}\right) ^{2}} \end{aligned}$$

(56)

Indeed, if $q_{I}>{\widetilde{q}}_{I}$ then it is possible for platform E to enter at the low level with quality $q_{E}^{*}$. If, also, $q_{I} >\tilde{\tilde{q_{I}}}$ then platform E cannot leapfrog the high quality. Hence under the threat of entry a duopoly equilibrium exists for $q_{I}^{*}\in [\max ({\widetilde{q}}_{I},\tilde{\tilde{q_{I}}}),{\overline{q}}]$ and $q_{E}^{*}=q_{I}^{*}-\left( K+F\right) \frac{9\left( \overline{\beta }-{\underline{\beta }}\right) }{N\left( {\overline{\beta }}-2\underline{\beta }\right) ^{2}}$. $\square$

In equilibrium, under the threat of entry the quality differentiation may decrease: $\left( q_{I}^{*}-q_{E}^{*}\right) \le \left( \overline{q}-{\underline{q}}\right)$. This outcome follows on from Proposition 6, in that the incumbent platform does not necessarily reach the maximum quality while the entrant platform sets a quality above the minimum unless the costs F and K are sufficiently high. Notice that if we assume $K=0$ and consider the minimum ${\underline{q}}=\frac{\rho \left( a^{*}\right) -\delta a^{*}}{{\overline{\beta }}}$ as in the monopoly case, then $\left( q_{I}^{*}-q_{E}^{*}\right) <\left( {\overline{q}} -{\underline{q}}\right)$ certainly holds if $\ {\overline{q}}>F\frac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}+\frac{\rho \left( a^{*}\right) -\delta a^{*}}{{\overline{\beta }}}$ .

The threat of entry shakes the equilibrium configuration. The quality of platform I might decrease, while the quality of platform E might increase. Consequently, the quality differentiation may shrink. In this respect, there is no evidence that increasing competition positively affects the high quality of the incumbent. Conversely, potential competition, namely the threat of entry, can boost the quality of the entrant up from the minimum level. Potential competition in a two-sided setup brings about lower quality differentiation than would be the case in its absence. Our insights are along similar lines to Hung and Schmidt’s (1988) results for a traditional one-sided market with no cost of production. Conversely, our result for vertical differentiation differs from that of Donnenfeld and Weber’s (1992, 1995) one-sided model with sequential entry and maximal differentiation in equilibrium.

5.3 Entry deterrence

Following on from the previous analysis, in this section we examine whether investment in quality might be a successful deterrence strategy. More precisely, we state the conditions under which an incumbent prevents entry into the market by means of producing a limit quality. First, we show that under certain conditions deterrence could be a feasible strategy. Then, in order to ascertain whether a deterrence strategy is also profitable, we compare deterrence and accommodation profits. Finally, by means of numerical simulation we show that for appropriate values of the fixed cost (F), accommodation profits are lower than deterrence ones, making quality preemption a profitable strategy.

In this way, we endogenize the monopoly structure in a two-sided framework with a quality choice. The difference in equilibrium quality between the accommodation case (duopoly) and the deterrence case (threatened monopoly) is a measure of the effects of potential competition on the provision of premium content.

The sequence of moves is as follows: the incumbent selects the quality of its product; having observed this quality, the potential entrant decides to enter if it can guarantee itself positive profits. Therefore, we analyze the entrant platform’s decision to enter the market or to stay out of it when the incumbent platform is already in it.

To examine whether deterrence is a feasible strategy, we calculate the profits of platform I in the case of deterrence. If platform I decides to preempt the entry of a potential entrant, it behaves as a threatened monopolist. In this case, all the assumptions of the monopoly hold.²⁴ Having defined threshold values ${\widetilde{q}} _{I}$ and $\tilde{\tilde{q_{I}}}$ (see Eqs. (55) and (56)), as in Proposition 6, we prove the following statement.²⁵

Proposition 7

Given ${\widetilde{q}}_{I}$ and $\tilde{\tilde{q_{I} }}$ and K not too high, if:

• ${\widetilde{q}}_{I}<$$\tilde{\tilde{q_{I}}}$ the monopoly platform cannot prevent entry for $q\in ({\underline{q}},{\overline{q}})$, therefore deterrence is not a feasible strategy (a)

• ${\widetilde{q}}_{I}>$$\tilde{\tilde{q_{I}}}$ the monopoly platform can prevent entry for $q_{I}^{D}={\widetilde{q}}_{I}-\varepsilon$, with $\varepsilon$ close enough to zero, therefore deterrence is a feasible strategy (b)

Proof

• (a) According to Proposition 6, to prevent the entry of a high quality platform, the incumbent should set $q_{I}>$$\tilde{\tilde{q_{I}}}$, while it prevents entry on low quality level if $q_{I}<$${\widetilde{q}}_{I}$. Therefore it is straightforward to see that if ${\widetilde{q}}_{I}<$$\tilde{\tilde{q_{I}}}$ holds, it will not exist any $q_{I}$ such that entry is prevented at both high quality and low quality levels.

• (b) According to Proposition 6, we know that for ${\widetilde{q}} _{I}>$$\tilde{\tilde{q_{I}}}$ it exist a value of $q_{I}$ such that the platform I can prevent the entry on both high quality and low quality. In particular for $\forall q_{I}\in \left( \tilde{\tilde{q_{I}}},{\widetilde{q}} _{I}\right)$ entry can be deterred. Recalling that for a quality $q\ge {\underline{q}}=\dfrac{\left( 1-\delta \right) ^{2}}{4{\overline{\beta }}}$ the monopoly profits are increasing in quality. Hence, the incumbent optimal deterrence strategy is to set $q_{I}^{D}=$${\widetilde{q}}_{I}-\varepsilon$ close enough to ${\widetilde{q}}_{I}$.

$\square$

The above proposition states the conditions under which platform I is able to deter entry. In case (a) the only equilibrium strategy is accommodation, while in case (b), entry deterrence is feasible but it is not necessarily an equilibrium. To be an equilibrium, the monopoly’s profit from the deterrence quality $q_{I}^{D}$ must be higher than that of the duopoly (accommodation). Otherwise, platform I should opt to accommodate even if ${\widetilde{q}}_{I}>\tilde{\tilde{q_{I}}}$.

According to Proposition 7, if ${\widetilde{q}} _{I}>\tilde{\tilde{q_{I}}}$ platform I can prevent entry for $q_{I} ^{D}={\widetilde{q}}_{I}-\varepsilon$, with an amount of $\varepsilon$ close enough to zero. Now, we need to examine when an entry deterrence strategy is profitable compared with an accommodation strategy. We calculate deterrence profit in. $q_{I} ^{D}$:

$$\begin{aligned} \Pi _{M}(q_{I}^{D})=\frac{N\left( {\overline{\beta }}q_{I}^{D}+\left( \frac{1-\delta }{2}\right) ^{2}\right) ^{2}}{4q_{I}^{D}}-K \end{aligned}$$

(57)

Considering $q_{I}^{D}=$${\widetilde{q}}_{I}-\varepsilon$ and taking the limit of (57), we obtain:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Pi _{M}(q_{I}^{D})=\dfrac{N\left( {\overline{\beta }}\left( \dfrac{\left( 1-\delta \right) ^{2}}{4{\overline{\beta }}}+\left( F+K\right) \dfrac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}\right) +\left( \frac{1-\delta }{2}\right) ^{2}\right) ^{2}}{4\left( \dfrac{\left( 1-\delta \right) ^{2}}{4{\overline{\beta }}}+\left( F+K\right) \dfrac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}\right) }{\small -K}\nonumber \\ \end{aligned}$$

(58)

Deterrence profits (58) must be compared with duopoly profits (accommodation case) as previously calculated in Proposition 6:

$$\begin{aligned} \Pi _{I}=\dfrac{3{\overline{\beta }}^{2}-3{\underline{\beta }}^{2}}{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}K+\dfrac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) ^{2}}{\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}F \end{aligned}$$

(59)

Notice that, given the deterrence feasibility condition, ${\widetilde{q}} _{I}>\tilde{\tilde{q_{I}}}$, (see Proposition 7), the quality in the case in which the incumbent prevents entry is $q_{I} ^{D}={\widetilde{q}}_{I}-\varepsilon$. Therefore the deterrence quality is lower than the accommodation quality, $q_{I}^{*}\in [\max ({\widetilde{q}} _{I},\tilde{\tilde{q_{I}}}),{\overline{q}}]$.

Since both $\Pi _{M}(q_{I}^{D})$ and $\Pi _{I}$ depend on entry costs (F) and a set of parameters ($\delta ,{\overline{\beta }},{\underline{\beta }}$), we perform this comparison using numerical simulation (see Appendix 7.3). According to this simulation, we can finally determine the condition under which deterrence is profitable compared with accommodation, in terms of fixed costs. Considering the difference between deterrence $\Pi _{M}(q_{I}^{D})$ and accommodation $\Pi _{I}$ profits, (57) and (59), we can define a threshold value for the fixed cost of entry ${\widehat{F}}\left( \delta ,{\overline{\beta }},{\underline{\beta }}\right)$ that makes accommodation and deterrence profits equal (see Fig. 1).

Table 1 in Appendix 7.3 shows the simulation of ${\widehat{F}}\left( \delta ,{\overline{\beta }},{\underline{\beta }}\right)$ where $\delta =0.5$. For sufficiently high levels of ${\overline{\beta }}$ deterrence profits are always larger than accommodation ones for every value of F. Therefore in these cases, without calculating the threshold value ${\widehat{F}}$, we can state that a deterrence strategy dominates accommodation strategy. In the remaining cases, we have values of ${\widehat{F}}$ such that deterrence and accommodation profits are equal. Therefore, for $F>{\widehat{F}}$ deterrence is admissible and profitable.

Therefore, for $F>$${\widehat{F}}\left( \delta ,{\overline{\beta }} ,{\underline{\beta }}\right)$ accommodation profits are lower than deterrence ones, making preemption a profitable strategy. Moreover, this difference increases with F. With a high cost F the incumbent prefers to prevent entry, by choosing the deterrence quality $q_{I}^{D}$. Conversely, for $F<{\widehat{F}}\left( \delta ,{\overline{\beta }},{\underline{\beta }}\right)$ the incumbent prefers to accommodate entry and behaves like a duopoly’s high-quality platform.²⁶

Fig. 1

${\widehat{F}}\left( 0.5 ,{\overline{\beta }},{\underline{\beta }}\right)$

6 Conclusions

This paper provides an analysis of entry and vertical differentiation of two-sided platforms with competition prevailing on one side of the market, namely the audience. Our major aim was to determine whether the serious concerns of policy makers about the existence of market dominance because of premium content in the broadcasting market and consequent market power is theoretically grounded.

To this end, we provided a full characterization of the equilibrium relating to advertising, subscription fees, market share, and quality in the case of a duopoly. Three results emerged. First, for each platform, when the optimal advertising level was positive it was constant and solely determined by the disutility parameter $\delta$. Furthermore, we showed that regardless of market structure, the strategic advertising choice was the same. Second, in a duopoly there was a full profit neutrality effect: there was a pass-through of advertising revenues into lower pay-per-view prices. This result is strongly related to the issue of competitive bottlenecks and prevailing competition on the individuals’ side. Therefore, in this specific setting competition is beneficial because it enhances overall investment in advertising without further harming single-homing consumers. Third and finally, duopoly platforms chose to engage in maximal differentiation.

Given the equilibrium configuration, the core of our paper focused on the role of competition by considering potential entry into a two-sided market and the behavior of the incumbent platform. In the case of a sequential duopoly threatened by entry, we extended Shaked and Sutton’s (1982) findings to a two-sided structure: under weak conditions of individuals’ heterogeneity, we showed that for any n platforms offering distinct quality, precisely two will have positive market shares on the buyers’ side (audience) in equilibrium, covering the entire market. Thus, the threat of entry shakes the equilibrium configuration of a sequential duopoly. Indeed, the incumbent platform’s quality might decrease while the quality of the entrant platform might increase, thereby reducing quality differentiation. In this respect, the model predicts that competition, or the threat of entry, does not necessarily result in higher quality. Furthermore, we showed that the two-sidedness of the market for news and information eliminates the standard result of maximal differentiation.

In addition, we showed that, for appropriate values of ${\underline{F}} (\delta ,{\overline{\beta }},{\underline{\beta }})$, entry deterrence is a feasible and profitable strategy for the incumbent platform. Therefore, our results suggest that investment in quality might be a useful strategy for an incumbent platform in order to maintain its dominant position, restricting competition. In this respect, our model provides a theoretical basis for the concerns of the OECD.

Furthermore, our prediction concerning quality as an entry-deterrence strategy is consistent with real-world business practices in media markets. For instance, the UK Office of Communication (Ofcom) reports that the content quality of traditional TV broadcasters plays an important role in competition with pay TV and video on demand (VOD) services. In fact, most pay TV and VOD service subscribers claim that the quantity of the channels offered is not matched by quality, making the choice of subscribing not worthwhile (Ofcom 2017). Conversely, the emphasis on high-quality drama has intensified. High-quality broadcast TV programmes attract episode audiences of over eight million (e.g. “Call the Midwife” and Broadchurch”) (Ofcom 2018). Similarly, Bajon and Villaret (2004) and Menezes and de Quadros Carvalho (2009) have shown empirically that in selected countries high-quality content commands higher profits and higher market concentration. Finally, a recent study by the Italian Antitrust Authority (AGCOM 2015) showed that premium content (such as sports events, movies, and popular TV series) is the crucial strategic variable for both traditional broadcasters (on DTT and DTH) and new video services providers.

According to the above results, a crucial question is whether competition is beneficial in terms of media market performance. As expected, competition (a duopoly) performs better than a high-quality monopoly in terms of prices and market coverage. However, the effects are ambiguous when we introduce asymmetry and potential competition. In this situation, two effects arise. On the one hand, an incumbent with a persistent monopoly results in a homogeneous provision of quality (no quality differentiation) and, in some circumstances, an uncovered market. On the other hand, the presence of potential entrants forces the incumbent platform to reduce the quality of its content, inducing lower subscription fees and better market coverage compared with the unthreatened monopoly.

Furthermore, our numerical simulation suggests that increasing barriers to entry (F) induces greater asymmetry among platforms and could in turn generate anti-competitive outcomes. For an appropriate range of parameter values, a sufficiently high value of entry cost would strengthen the incentive to defend a dominant position.

Appendix

1.1 The monopoly case

1.1.1 Monopoly: individuals and advertisers demand

By considering the individual to be indifferent to either accessing the monopoly platform or not accessing at it all, we obtain the demand function of individuals.

From (2), by assuming $V=0$:

$$\begin{aligned} NB_{M}=N({\overline{\beta }}-\beta _{0M})=N\left( \dfrac{{\overline{\beta }} q_{M}-s_{M}-\delta a_{M}}{q_{M}}\right) \end{aligned}$$

(60)

Note that the demand is positive if:

$$\begin{aligned} {\overline{\beta }}q_{M}\ge \delta a_{M}+s_{M} \end{aligned}$$

(61)

From (6), the share of advertisers willing to join the platform becomes:

$$\begin{aligned} a_{M}=1-F\left( \frac{r_{M}}{NB_{M}}\right) \end{aligned}$$

(62)

Having defined the demand function of individuals and advertisers for given prices $r_{M}$ and $s_{M}$, by simultaneously solving Eqs. (60) and (62) we get:

$$\begin{aligned} r_{M}\left( s_{M},a_{M},q_{M}\right) =F^{-1}(1-a_{M})N(\dfrac{{\overline{\beta }}q_{M}-s_{M}-\delta a_{M}}{q_{M}}) \end{aligned}$$

(63)

This equation describes how advertising charges react to changes in subscription price, advertising and quality.

1.1.2 Monopoly: platform subscription fees and advertising level

According to the above assumptions, the monopoly platform maximizes profit subject to a positivity constraint on advertising level:

$$\begin{aligned} \left\{ \begin{array}{c} \underset{a_{H},s_{H}}{\max }\Pi _{M}=NB_{M}s_{M}+a_{M}r_{M}-K\\ s.t.a_{M}\ge 0 \end{array} \right. \end{aligned}$$

(64)

First order conditions are:

$$\begin{aligned} \frac{\partial \Pi _{M}}{\partial a_{M}}=N\frac{\partial B_{M}}{\partial a_{M} }s_{M}+r_{M}+a_{M}\frac{\partial r_{M}}{\partial a_{M}}\le 0 \end{aligned}$$

(65)

and

$$\begin{aligned} \frac{\partial \Pi _{M}}{\partial s_{M}}=NB_{M}+N\frac{\partial B_{M}}{\partial s_{M}}s_{M}+a_{M}\frac{\partial r_{M}}{\partial s_{M}}=0 \end{aligned}$$

(66)

Then, using the definition (17) for the monopoly platform we can rewrite the optimality conditions, proving that Proposition 1 also holds in the case of a monopoly. This means that for a monopoly platform, the best option is to set a fixed advertising space dependent solely on the disutility of the individuals as measured by parameter $\delta$. However, the platform does not set the maximum level of advertising or the amount that maximize revenues per viewer, i.e. $\rho ^{\prime }(a_{i})=0$, unless consumers are neutral towards ads ($\delta =0$). Note that our result is in contrast with Peitz and Valletti’s (2008) suggestion that the market is covered and the monopoly advertising space is at its maximum at $\rho ^{\prime }\left( a_{M}\right) =0$.

Moreover, we compare advertising intensity in a monopoly with the results for the duopoly case.

Remark 8

The strategic advertising choice is the same, regardless of market structure:

$$\begin{aligned} \rho ^{\prime }(a_{i}^{*})=\delta \text { for }i=H,L,M \end{aligned}$$

However, in a duopoly structure the total amount of advertising is double that in a monopoly. In particular, in the uniform case,

$$\begin{aligned} a_{L}^{*}+a_{H}^{*}=1-\delta =2a_{M}^{*} \end{aligned}$$

The above Remark specifies that platforms’ strategic advertising choice is neutral with respect to competitive market structure.²⁷

Solving the optimality conditions for $s_{M}$ and $B_{M}$, we obtain the following equilibrium values: 

$$\begin{aligned} s_{M}^{*}= & {} \frac{{\overline{\beta }}q_{M}-\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*}}{2} \end{aligned}$$

(67)

$$\begin{aligned} B_{M}^{*}= & {} \frac{{\overline{\beta }}q_{M}+\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*}}{2q_{M}} \end{aligned}$$

(68)

Expressions (67) and (68) confirm the results of profit neutrality of Proposition 3 for the monopoly setting. Revenues from the advertising side are counterbalanced by a decrease in the subscription fee. However, only half of the revenues from advertising are involved in this pass-through effect (see Eq. (67)). Moreover, given that subscription fees positively depend on quality, a sort of substitutability between advertising and quality emerges.

1.1.3 Monopoly: platform quality

In order to solve the quality stage, we maximize monopoly profits $\Pi _{M}\left( s_{M}^{*},a_{M}^{*},r_{M}^{*},q_{M}\right)$ with respect to quality $q_{M}$. We obtain the following FOC, subject to $q_{M} \ge 0$ :

$$\begin{aligned} \dfrac{\partial \Pi _{M}}{\partial q_{M}}=N\left( \dfrac{{\overline{\beta }} ^{2}q_{M}^{2}-(\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*})^{2} }{4q_{M}^{2}}\right) =0 \end{aligned}$$

(69)

By computing the second order condition, we show the convexity of the profit function:

$$\begin{aligned} \dfrac{\partial ^{2}\Pi _{M}}{\partial q_{M}^{2}}=\dfrac{N(\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*})^{2}}{2q_{M}^{3}}>0 \end{aligned}$$

(70)

Unfortunately, in this general framework we cannot analytically determine the equilibrium solution $q_{M}^{*}$. However, we restrict ourselves on the increasing slope of the profit function (64). Therefore, we restrict the technical range of quality (Q) to a narrower set ($Q_{R}$) :

$$\begin{aligned} Q_{R}=\left[ {\underline{q}},{\overline{q}}\right] \text { with } {\underline{q}}=\frac{\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*} }{\beta } \end{aligned}$$

Therefore, we can prove the following result,

Proposition 9

In equilibrium, under $Q_{R}$, the monopoly platform chooses the maximum quality.

Proof

By comparing monopoly profit functions in ${\underline{q}}$ and ${\overline{q}}$, given $Q_{R}=\left[ {\underline{q}},{\overline{q}}\right]$ with ${\underline{q}}=\frac{\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*} }{\beta }$, respectively:

$$\begin{aligned} \Pi _{M}^{*}\left( {\underline{q}}\right)= & {} \dfrac{N({\overline{\beta }}{\underline{q}}+\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*})^{2}}{4{\underline{q}}}-K \end{aligned}$$

(71)

$$\begin{aligned} \Pi _{M}^{*}\left( {\overline{q}}\right)= & {} \dfrac{N(\overline{\beta }{\overline{q}}+\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*} )^{2}}{4{\overline{q}}}-K \end{aligned}$$

(72)

we get:

$$\begin{aligned} \Pi _{M}^{*}\left( {\overline{q}}\right) -\Pi _{M}^{*}\left( {\underline{q}}\right) =\frac{1}{4q}N\left( \delta a_{M}^{*}-\rho \left( a_{M}^{*}\right) +{\overline{\beta }}{\overline{q}}\right) ^{2}>0 \end{aligned}$$

For $q\in Q_{R}$ profit are convex and increasing in quality. Therefore to maximize profit the monopoly platform set $q_{M}^{*}={\overline{q}}$. $\square$

Given our result on quality, we obtain equilibrium values for subscription fee and individuals’ share:

$$\begin{aligned} s_{M}^{*}= & {} \frac{{\overline{\beta }}{\overline{q}}-\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*}}{2} \end{aligned}$$

(73)

$$\begin{aligned} B_{M}^{*}= & {} \frac{{\overline{\beta }}{\overline{q}}_{M}+\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*}}{2{\overline{q}}_{M}} \end{aligned}$$

(74)

Equilibrium profits are:

$$\begin{aligned} \Pi _{M}^{*}\left( {\overline{q}}\right) =\dfrac{N({\overline{\beta }} {\overline{q}}+\rho \left( a_{M}^{*}\right) -\delta a_{M}^{*})^{2}}{4{\bar{\theta }}}-K \end{aligned}$$

(75)

Notice that all the equilibrium values depend on ${\overline{\beta }}$, on the technological constraint, namely the upper bound ${\overline{q}}$, and the disutility of advertising $\delta$.

We also provide equilibrium results considering a uniform distribution of advertisers ( p.d.f. of advertisers F is uniform on $\left[ 0,1\right]$), as a special case.

$$\begin{aligned} a_{M}^{*}= & {} \frac{1-\delta }{2} \end{aligned}$$

(76)

$$\begin{aligned} s_{M}^{*}= & {} \frac{4{\overline{\beta }}{\overline{q}}-\left( 1-\delta \right) \left( 1+3\delta \right) }{8} \end{aligned}$$

(77)

$$\begin{aligned} B_{M}^{*}= & {} \frac{{\overline{\beta }}{\overline{q}}+\left( \frac{1-\delta }{2}\right) ^{2}}{2{\overline{q}}} \end{aligned}$$

(78)

Note that under the assumption of fixed costs, the monopoly profit function is convex in relation to quality. One might expect that this shape strictly depends on the assumption of K, that is, the fixed cost of quality. In fact, in a single-sided framework, the standard model of vertical differentiation is solved using the quadratic costs of quality, inducing concavity in the profit function. However, in a two-sided setting, the issue of concavity in the profit function is more complex. As expected, a linear cost of quality does not solve the problem of convexity in the profit function. However, more surprisingly, even increasing the costs of quality does not guarantee a well-shaped monopoly profit function (for further details, see Battaggion and Drufuca 2015). For instance, the quadratic costs of quality (see Weeds 2013) do not make the monopoly profit function concave with regard to quality without ad hoc assumptions concerning the derivatives. One possible option would be to have implicit quality cost functions (see Anderson 2007), but that would render us unable to provide a close solution to the model. Therefore, we choose to introduce the simplest cost function and a technological range bounding the levels of quality, allowing us to characterize the equilibrium configuration.

1.2 Proof of Lemma 5

Proof

We have already stated that for $2{\underline{\beta }}<{\overline{\beta }}$ the low-quality platform has a positive audience (see Sect. 3).

For ${\overline{\beta }}<4{\underline{\beta }}$ we follow Shaked and Sutton (1982) with appropriate transformations to fit our two-sided structure.

From Sect. 3, we know that in equilibrium subscription fees are:

$$\begin{aligned} s_{H}^{*}&=\frac{\left( 2{\overline{\beta }}-{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }{3}-\rho (a^{*}) \end{aligned}$$

(79)

$$\begin{aligned} s_{L}^{*}&=\frac{\left( {\overline{\beta }}-2{\underline{\beta }}\right) \left( q_{H}-q_{L}\right) }{3}-\rho (a^{*}) \end{aligned}$$

(80)

Looking at equilibrium subscription fees, it is straightforward to see that the “profit neutrality” result still holds. Advertising revenues per viewers $\rho (a^{*})$ are entirely spent in reducing subscription fees $s_{i} ^{*}$. Due to this neutrality result, we can apply the following transformation to equilibrium demands in order to have a single price, which is always positive:

$$\begin{aligned} p_{i}=s_{i}+\rho (a_{i})>0 \end{aligned}$$

(81)

In this way we are able to obtain a framework similar to the one of Shaked and Sutton (1982). We consider a situation of n platforms ordered by their quality $q_{1}<q_{2}<\cdots <q_{n}$ competing for an uniform audience (same assumptions as in previous sections) covering the entire market.²⁸

Given the equilibrium of stage 2 ($a_{1}=a_{2}=\cdots =a_{n}=a^{*}$), indifferent viewers are defined as follows:

$$\begin{aligned} \beta _{2}&=\dfrac{p_{2}-p_{1}}{\left( q_{2}-q_{1}\right) }\nonumber \\ \beta _{3}&=\dfrac{p_{3}-p_{2}}{\left( q_{3}-q_{2}\right) }\nonumber \\&\vdots \nonumber \\ \beta _{n}&=\dfrac{p_{n}-p_{n-1}}{\left( q_{n}-q_{n-1}\right) } \end{aligned}$$

(82)

Demands become:

$$\begin{aligned} NB_{1}&=N\left( \dfrac{p_{2}-p_{1}}{\left( q_{2}-q_{1}\right) }-{\underline{\beta }}\right) \nonumber \\ NB_{2}&=N\left( \dfrac{p_{3}-p_{2}}{\left( q_{3}-q_{2}\right) } -\dfrac{p_{2}-p_{1}}{\left( q_{2}-q_{1}\right) }\right) \nonumber \\&\vdots \nonumber \\ NB_{n}&=N\left( {\overline{\beta }}-\dfrac{p_{n}-p_{n-1}}{\left( q_{n}-q_{n-1}\right) }\right) \end{aligned}$$

(83)

Platforms’ Revenues are:

$$\begin{aligned} R_{1}&=p_{1}NB_{1}=p_{1}N(\beta _{2}-{\underline{\beta }})\nonumber \\ R_{2}&=p_{2}NB_{2}=p_{2}N(\beta _{3}-\beta _{2})\nonumber \\&\vdots \nonumber \\ R_{n}&=p_{n}NB_{n}=p_{n}N({\overline{\beta }}-\beta _{n}) \end{aligned}$$

(84)

Profit maximizations w.r.t. qualities give the following optimality conditions:

$$\begin{aligned} (\beta _{2}-{\underline{\beta }})+p_{1}(\dfrac{-1}{\left( q_{2}-q_{1}\right) })&=0\nonumber \\ (\beta _{3}-\beta _{2})+p_{2}(-\dfrac{1}{\left( q_{3}-q_{2}\right) }-\dfrac{1}{\left( q_{2}-q_{1}\right) })&=0\nonumber \\&\vdots \nonumber \\ \left( {\overline{\beta }}-\beta _{n}\right) +p_{n}\left( -\dfrac{1}{\left( q_{n}-q_{n-1}\right) }\right)&=0 \end{aligned}$$

(85)

Recall from indifference conditions:

$$\begin{aligned} \beta _{n-1}=\dfrac{p_{n-1}-p_{n-2}}{\left( q_{n-1}-q_{n-1}\right) } =p_{n-1}\frac{1}{\left( q_{n-1}-q_{n-1}\right) }-p_{n-2}\frac{1}{\left( q_{n-1}-q_{n-1}\right) } \end{aligned}$$

(86)

which can be written as:

$$\begin{aligned} p_{n-1}\frac{1}{\left( q_{n-1}-q_{n-1}\right) }=\beta _{n-1}+p_{n-2}\frac{1}{\left( q_{n-1}-q_{n-1}\right) } \end{aligned}$$

(87)

Hence we re-write optimality condition for $(n-1)$th platform :

$$\begin{aligned} (\beta _{n}-\beta _{n-1})-p_{n-1}\dfrac{1}{\left( q_{n}-q_{n-1}\right) } -\beta _{n-1}-p_{n-2}\frac{1}{\left( q_{n-1}-q_{n-1}\right) }&=0\nonumber \\ \beta _{n}-2\beta _{n-1}-p_{n-1}\dfrac{1}{\left( q_{n}-q_{n-2}\right) } -p_{1}\dfrac{1}{\left( q_{n-1}-q_{n-2}\right) }&=0 \end{aligned}$$

(88)

This condition implies that

$$\begin{aligned} \beta _{n}>2\beta _{n-1} \end{aligned}$$

(89)

We do the same for the optimality condition of nth platform , obtaining:

$$\begin{aligned} {\overline{\beta }}-2\beta _{n}-p_{n-1}\dfrac{1}{\left( q_{n}-q_{n-1}\right) }=0 \end{aligned}$$

(90)

which implies

$$\begin{aligned} {\overline{\beta }}>2\beta _{n} \end{aligned}$$

(91)

Taking conditions (89) and (91) together we get:

$$\begin{aligned} {\overline{\beta }}\ge 2\beta _{n}>4\beta _{n-1} \end{aligned}$$

(92)

which implies:

$$\begin{aligned} {\overline{\beta }}>4\beta _{n-1} \end{aligned}$$

(93)

Having assumed ${\overline{\beta }}<4{\underline{\beta }}$ we end up with:

$$\begin{aligned} 4\beta _{n-1}<{\overline{\beta }}<4{\underline{\beta }} \end{aligned}$$

(94)

This inequality implies:

$$\begin{aligned} \beta _{n-1}<{\underline{\beta }} \end{aligned}$$

(95)

The inequality (95) implies that market is completely covered by the $(n-1)$th and the nth platform, namely those with the highest qualities. This means that all other platforms face a zero market share on viewers’ side.

Notice that we can also show that in a triopoly case, given ${\overline{\beta }}<4{\underline{\beta }}$, only the two platform with highest qualities survive and cover the market.

We consider the same framework as in the duopoly case but with three platforms ranked by quality $q_{1}<q_{2}<q_{3}$. Under market coverage, indifferent consumers are identified by:

$$\begin{aligned} \beta _{12}= & {} \frac{\delta \left( a_{2}-a_{1}\right) +\left( s_{2} -s_{1}\right) }{q_{2}-q_{1}} \end{aligned}$$

(96)

$$\begin{aligned} \beta _{23}= & {} \frac{\delta \left( a_{3}-a_{2}\right) +\left( s_{3} -s_{2}\right) }{q_{3}-q_{2}} \end{aligned}$$

(97)

Demands from the consumers’ side are respectively:

$$\begin{aligned} NB_{1}= & {} N\dfrac{1}{{\overline{\beta }}-{\underline{\beta }}}\left( \beta _{12}-{\underline{\beta }}\right) \nonumber \\= & {} N\dfrac{1}{{\overline{\beta }}-{\underline{\beta }}}\left( \dfrac{\delta \left( a_{2}-a_{1}\right) +\left( s_{2}-s_{1}\right) -{\underline{\beta }}\left( q_{2}-q_{1}\right) }{q_{2}-q_{1}}\right) \end{aligned}$$

(98)

$$\begin{aligned} N\, B_{2}= & {} N\dfrac{1}{{\overline{\beta }}-{\underline{\beta }}}\left( \beta _{23}-\beta _{12}\right) \nonumber \\= & {} N\dfrac{1}{{\overline{\beta }}-{\underline{\beta }}}\left( \dfrac{\delta \left( a_{3}-a_{2}\right) +\left( s_{3}-s_{2}\right) }{q_{3}-q_{2}}-\dfrac{\delta \left( a_{2}-a_{1}\right) +\left( s_{2}-s_{1}\right) }{q_{2}-q_{1} }\right) \end{aligned}$$

(99)

$$\begin{aligned} NB_{3}= & {} N\dfrac{1}{{\overline{\beta }}-{\underline{\beta }}}\left( \overline{\beta }-\beta _{23}\right) \nonumber \\= & {} N\dfrac{1}{{\overline{\beta }}-{\underline{\beta }}}\left( \dfrac{\overline{\beta }\left( q_{3}-q_{2}\right) -\delta \left( a_{3}-a_{2}\right) -\left( s_{3}-s_{2}\right) }{q_{3}-q_{2}}\right) \end{aligned}$$

(100)

Resolution for stages 3 and 2 is standard. From optimality conditions we obtain:

$$\begin{aligned} s_{i}=\frac{-B_{i}}{\partial B_{i}/\partial s_{i}}-\rho \end{aligned}$$

(101)

where $\rho =\rho (a_{i})$. Since in equilibrium $a_{1}=a_{2}=a_{3}=a^{*}$ and $\rho =\rho (a^{*})=\rho ^{*}$, we get the following system:

$$\begin{aligned} s_{1}&=\frac{s_{2}-{\underline{\beta }}\left( q_{2}-q_{1}\right) }{2} -\frac{\rho ^{*}}{2}\nonumber \\ s_{2}&=\frac{s_{3}\left( q_{2}-q_{1}\right) +s_{1}\left( q_{3} -q_{2}\right) }{2\left( q_{3}-q_{1}\right) }-\frac{\rho ^{*}}{2}\nonumber \\ s_{3}&=\frac{s_{2}+{\overline{\beta }}\left( q_{3}-q_{2}\right) }{2} -\frac{\rho ^{*}}{2} \end{aligned}$$

(102)

Equilibrium access prices are:

$$\begin{aligned} s_{1}^{*}&=\frac{1}{6\left( q_{3}-q_{1}\right) }\left( \left( {\overline{\beta }}-{\underline{\beta }}\right) \left( q_{2}-q_{1}\right) \left( q_{3}-q_{2}\right) -3{\underline{\beta }}\left( q_{3}-q_{1}\right) \left( q_{2}-q_{1}\right) \right) -\rho ^{*}\\ s_{2}^{*}&=\frac{1}{3\left( q_{3}-q_{1}\right) }\left( \overline{\beta }-{\underline{\beta }}\right) \left( q_{3}-q_{2}\right) \left( q_{2}-q_{1}\right) -\rho ^{*}\\ s_{3}^{*}&=\frac{1}{6\left( q_{3}-q_{1}\right) }\left( \left( {\overline{\beta }}-{\underline{\beta }}\right) \left( q_{3}-q_{2}\right) \left( q_{2}-q_{1}\right) +3{\overline{\beta }}\left( q_{3}-q_{2}\right) \left( q_{3}-q_{1}\right) \right) -\rho ^{*} \end{aligned}$$

Given $s_{1}^{*}$ , $s_{2}^{*}$ and $s_{3}^{*}$, we check whether or not $\beta _{12}$ > ${\underline{\beta }}$ under the condition of $\,4{\underline{\beta }}>{\overline{\beta }}$. If this is the case, platform 1 faces zero demand and platforms 2 and 3 cover the whole consumer market, confirming the result of Shaked and Sutton (1982).

$$\begin{aligned} \beta _{12}&=\frac{\left( s_{2}^{*}-s_{1}^{*}\right) }{q_{2}-q_{1} }\nonumber \\&=\frac{1}{6\left( q_{3}-q_{1}\right) }\left( \left( \overline{\beta }-{\underline{\beta }}\right) \left( q_{3}-q_{2}\right) +3\underline{\beta }\left( q_{3}-q_{1}\right) \right) \end{aligned}$$

(103)

$$\begin{aligned} \beta _{12}-{\underline{\beta }}&=\frac{1}{6\left( q_{3}-q_{1}\right) }\left( \left( {\overline{\beta }}-{\underline{\beta }}\right) \left( q_{3}-q_{2}\right) -3{\underline{\beta }}\left( q_{3}-q_{1}\right) \right) \end{aligned}$$

(104)

Which is negative since $\left( {\overline{\beta }}-{\underline{\beta }}\right) <3{\underline{\beta }}$ and $\left( q_{3}-q_{2}\right) <\left( q_{3} -q_{1}\right)$.

Hence $\beta _{12}<{\underline{\beta }}$: the two platforms with highest qualities cover the market, leaving no room for the low-quality platform. $\square$

1.3 Numerical simulation

Under the assumptions $K=0$, $N=1$ and $2{\underline{\beta }}<{\overline{\beta }}<4{\underline{\beta }}$ we computed (57) and (59) for different values of the parameters.

We started by restricting the set of values of ${\overline{\beta }},{\underline{\beta }}$ according to Lemma 5. Doing that we set the value of ${\overline{\beta }}$ as a function of different values of ${\underline{\beta }}$. Then, we computed deterrence (58) and accommodation (59) profits for every combination of ${\underline{\beta }}$ and ${\overline{\beta }}\left( {\underline{\beta }}\right)$.²⁹ This simulation was repeated for three different values of $\delta \in [0,1)$, namely 0.01, 0.5 and 0.9. The numerical simulation made no noticeable difference according to the change in $\delta$, therefore, for the sake of simplicity, we only considered the case: $\delta =0.5$.

Note that both deterrence and accommodation profits, $\Pi _{M}(q_{1}^{D})$ and $\Pi _{1}$, decrease with preference for quality ${\overline{\beta }}\left( {\underline{\beta }}\right)$. To better understand this result, we focused on the relationship between quality and ${\overline{\beta }}$ in the entry deterrence case. From Proposition (7) we have $q_{1}^{D}={\widetilde{q}}_{1}-\varepsilon$. By taking the limit of $q_{1}^{D}$ for $\varepsilon$ close to zero, we get:

$$\begin{aligned} q_{1}^{D}={\underline{q}}+\left( F+K\right) \tfrac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2}}=\tfrac{\left( 1-\delta \right) ^{2}}{4{\overline{\beta }} }+\left( F+K\right) \tfrac{9\left( {\overline{\beta }}-{\underline{\beta }}\right) }{N\left( {\overline{\beta }}-2{\underline{\beta }}\right) ^{2} } \end{aligned}$$

(105)

By simple calculation from Eq. (105) we have:

$$\begin{aligned} \frac{\partial q_{1}^{D}}{\partial {\overline{\beta }}}<0 \end{aligned}$$

(106)

Table 1 Values of $\ {\widehat{F}}\left( \delta ,\overline{\beta },{\underline{\beta }}\right)$ for $\delta =0.5$

Table 2 Values of ${\underline{ F}}$$\left( \delta ,{\underline{\beta }}({\overline{\beta }})\right)$ for $\delta =0.5$

Given that monopoly profits increase with quality, it is straightforward to see that deterrence profits $\Pi _{M}(q_{1}^{D})$ decrease with ${\overline{\beta }}$. The intuition is similar to the accommodation case. Given that the entrant’s profits increase with ${\overline{\beta }}$, the incumbent platform can prevent entry with a lower level of quality. Therefore, for platform 1 the cost of deterrence increases with ${\overline{\beta }}$. The same happens in the case of a decrease in ${\underline{\beta }}\left( {\overline{\beta }}\right)$ (Tables 1, 2).