Oligopoly with network effects: firm-specific versus single network

Abstract

We consider symmetric oligopolies with positive network effects where each firm has its own proprietary network, which is incompatible with that of its rivals. We provide minimal conditions for the existence of (non-trivial) symmetric equilibrium in a general setting. We analyze the viability of industries with firm-specific networks and show that the prospects for successful launch decrease with more firms in the market. This is a major reversal from the case of single-network industries. A central part of the paper compares the viability and market performance of industries with compatible and incompatible networks and shows that viability, output, (endogenous) demand, and social welfare are higher for the former. However, the comparison of industry price, profit and consumer surplus requires respective qualifications, of a general nature for the former two but not for the latter. Overall, these results provide theoretical grounding in a general but not universal sense for the conventional view that compatibility leads to superior performance, which was hitherto based on case studies and stylized facts.

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Notes

  1. 1.

    Veblen (1899) pioneered the study of interdependent demands in the form of fashion, fads and bandwagon effects and their novel implications of a macroeconomic nature (see also Leibenstein 1950).

  2. 2.

    A third, hybrid case often prevails in practice: An industry may have two incompatible standards, each being adopted by a subset of the firms, in such a way that firms with the same standard have mutually compatible products. The Betamax–VHS standard war went through a phase of such coalitional rivalry.

  3. 3.

    Among industries with firm-specific networks, some well-known cases that made headlines due to a standards war include: high-definition optical discs for high-definition video and audio (with Blu-ray overtaking HD DVD); videocassette recorders (Sony’s Beta format lost out to JVC’s VHS format after a long battle); video games consoles (Nintendo secured a near-monopoly after driving Atari out of the market); personal computers (early on, IBM and Macintosh computers were not compatible); digital music systems such as digital compact cassette and mini disc; operating systems (Microsoft’s DOS was chosen over Apple’s); and bank ATMs. For more details, see Church and Gandal (1992), Cusumano et al. (1992), Katz and Shapiro (1986) and Rohlfs (2003).

  4. 4.

    To be formally defined in the next section, the present notion of viability is meant to capture the idea that the industry will not unravel to the trivial equilibrium with zero output, due to the role of expectations in demand.

  5. 5.

    Among others, see Grilo et al. (2001), Crémer et al. (2000), Resende (2008), Markovich and Moenius (2009), Laussel and Resende (2014), Garcia and Vergari (2015), Laussel et al. (2015) and Guimaraes and Pereira (2017).

  6. 6.

    This emerging industry has indirect network effects defined as those that operate via the availability of complementary products, in this case charging stations and other specialized services.

  7. 7.

    The rich literature on two-sided markets is also related, due to the presence of cross-network effects (e.g., Rochet and Tirole 2006; Armstrong 2006; Jullien 2011, among others). A recent strand of the policy-focused literature argues that some of the industries viewed as two-sided markets (such as search engines) in fact reflect only one-sided network effects (e.g., Luchetta 2014; Filistrucchi et al. 2014). As such, these markets might thus better fit the present setting. Indeed, the main point put forward in the latter pair of studies is that, while advertisers care about the number of users, the opposite is often not true. This would add a potentially long list of examples that partially fit the present setting, including search engines such as Google and social media such as Facebook.

  8. 8.

    The assumptions on a standard Cournot model that make it a game of strategic substitutes are well known and quite general: see Novshek (1985) and Amir (1996a). These are widely adopted in various studies in industrial organization that are based on general functional forms (see, e.g., Vives 1999).

  9. 9.

    Recall that in their seminal work, Katz and Shapiro (1985) restrict consideration to network industries that are always viable and thus do not address the viability question. The same is true of the follow-up literature.

  10. 10.

    As in AL (2011), these notions are defined via the convergence of the said adjustment process to a nonzero equilibrium from any or from a sufficiently high, initial belief about the network size.

  11. 11.

    Gans et al. (2005) provide evidence on the speed of adoption of communication technologies with network effects, depending on compatibility (see also Liebowitz and Margolis 1994).

  12. 12.

    Although our approach can easily handle a more general cost function, we abstract away from cost curvature effects, since we wish to stress that the departures from standard oligopoly results that we are about to establish are all due to demand-side network effects. This is in contrast to the results of Amir and Lambson (2000) where counterintuitive findings are due to increasing returns to scale in production.

  13. 13.

    A well-known parallel is the fact that the price-taking assumption of perfect competition is more plausible in markets with many producers and thus more diffuse competition.

  14. 14.

    It is possible to think of RECE as a fully game-theoretic concept, but in the context of a two-stage game, as follows. In the first stage, a market maker (or a regulator) announces an expected network size s per firm. In the second stage, firms compete in Cournot fashion facing inverse demand P(zs). If the market maker’s objective function is to minimize the gap between the announced per-firm network and its Cournot equilibrium output, then to any subgame perfect equilibrium of this game corresponds a RECE of the Cournot market with network externalities, and vice-versa. This simple interpretation of the RECE solution also provides one natural approach for arriving at a RECE with the participation of a market maker, and in case of multiple equilibria, also for selecting a particular RECE. This construction is reminiscent of a Walrasian auctioneer in general equilibrium.

  15. 15.

    It is also closely related to the more familiar condition of marginal revenue decreasing in rival’s output, or \(P_{1}(z,s)+zP_{11}(z,s)\le 0\), used by Novshek (1985) and others.

  16. 16.

    This class of games has been studied extensively and applied to oligopoly theory early on; see Milgrom and Roberts (1990), Topkis (1998), Vives (1990, 1999), Amir (1996a) and Amir and Lambson (2000). More recent applications include Prokopovych and Yannelis (2017), Barthel and Sabarwal (2018), Barthel and Hoffmann (2019), Boyarchenko (2019), Cornand and Dos Santos Ferreira (2019), Jiménez-Martínez (2019), Gama and Rietzke (2019) and Laussel and Resende (2019).

  17. 17.

    Asymmetric equilibria are quite different in natur, and appear more suitable for a dynamic analysis. Thus, this is set aside for further research. Katz and Shapiro (1985) do consider asymmetric equilibria.

  18. 18.

    This process is well known in the literature and may be viewed as an adaptation of classical Cournot best-reply dynamics to a model with rational expectations of market size. In fact, one can note that (1) is to RECE what Cournot best-reply dynamics is to Cournot equilibrium.

  19. 19.

    Thus, the critical mass is the minimal initial network size that ensures convergence to a strictly positive equilibrium. Indeed, iterating the adjustment process starting from any \(s_{0}<\overline{s}\) will converge to the trivial equilibrium, interpreted as the extinction of the industry.

  20. 20.

    He also discusses the importance of technological progress in the production of network goods over time, but the role of this factor is more intuitive, and in line with regular (non-network) industries. The role of exogenous technological progress is captured by Proposition 2.

  21. 21.

    Gans et al. (2005) consider a long list of communication industries with network effects and offer a more nuanced picture with a case-by-case discussion of several issues of interest.

  22. 22.

    Interconnection may fail to be implemented because of a variety of reasons, including historical factors, inability of firms to compromise on adopting a rival’s standard, high costs sunk into firms’ current standards, perceived threat to firms’ prestige in case of abandonment of one’s standard, etc. A frequent reason is perception by an industry leader that competition with firm-specific networks will end up driving its rivals out of the market.

  23. 23.

    Here, \({z}_{n}^{I}\) solves \(q_{n}(s)=\frac{a+s}{n+1}=s\), while \({z}_{n}^{C}\) solves \(nq_{n}(s)=\frac{n(a+s)}{n+1}=s.\)

  24. 24.

    While this demand function leads to a Cournot game that is not of strategic substitutes, other examples may be constructed to show that Proposition 4(c) may fail when Condition (4) alone is violated. A similar remark applies to Condition (5).

  25. 25.

    We remind the reader that the issue of industry viability had not received any theoretical treatment in the extensive literature on network effects before AL (2011) for single-network industries, despite its prominence in the policy debates on network industries (Rohlfs 2003; Shapiro and Varian 1998; Shy 2001).

  26. 26.

    Gans et al. (2005) offer a less unambiguous view on the problem of interconnection and of the role of government in some industries (see also Shapiro and Varian 1998).

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Correspondence to Rabah Amir.

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It is a great pleasure for the authors to dedicate this paper to their colleague and friend Nicholas Yannelis, in recognition of his many contributions to economic theory. The authors are grateful to Jean Gabszewicz, Filomena Garcia, Gautam Gowrisankaran, Natalia Lazzati, Ana Elisa Pereira, Joana Resende, Stanley Reynolds, and John Wooders, as well as to three anonymous referees, for helpful feedback and suggestions on the subject and the contents of this paper.

Appendix

Appendix

This appendix contains the proofs of all the results in this paper.

Proof of Theorem 1

  1. (a)

    Let the expected size of the network be s for each firm. From Amir and Lambson (2000), we know that, due to (A3), a unique and symmetric Cournot equilibrium exists for all n and s. Let the corresponding per-firm equilibrium output be \(q_{n} (s)\). By the upper hemi-continuity of the equilibrium correspondence for strategic games (e.g., Fudenberg and Tirole 1991), \(q_{n} (s)\) is upper hemi-continuous as a correspondence. Since it is also single-valued, it must be a continuous function. Finally, by the (smooth) Implicit Function Theorem and the fact that P is \(C^{2}\), \(q_{n} (s)\) is \(C^{1}\) in s.

  2. (b)

    Since the set of RECE coincides with the set of fixed points of \(q_{n} (s),\) the existence of a RECE follows directly from Brouwer’s fixed- point theorem applied to the function \(q_{n} (s)\), which is defined from [0, K] to itself. \(\square \)

Proof of Lemma 1

By definition, an individual (and hence, industry) output of 0 is a symmetric RECE if \(0\in x(0,0)\). This holds if and only if \(\pi (0,0,0)\ge \pi (x,0,0)\,\forall x\in [0,K]\), i.e., \(0\ge xP(x,0)-cx\) for all \(x\in [0,K]\). Moreover, \( q_{n}(0)=0\) if and only if \(\pi (0,0,0)\ge \pi (x,0,0)\) for all \(x\in [0,K]\) if and only if \(q_{n+1}(0)=0\). \(\square \)

The following lemmas will be useful to prove Theorem 2.

Lemma 3

The function \(q_{n}(s)\) is differentiable in s, and, if \(q_{n}(s)\in (0,K)\) for \(s>0\), one has

$$\begin{aligned} \frac{\partial q_{n}(s)}{\partial s}=-\frac{ P_{2}(nq_{n}(s),s)+q_{n}P_{12}(nq_{n}(s),s)}{ (n+1)P_{1}(nq_{n}(s),s)+nq_{n}P_{11}(nq_{n}(s),s)}. \end{aligned}$$
(7)

In particular, if \(q_{n}(0)=0,\)

$$\begin{aligned} \frac{\partial q_{n}(0)}{\partial s}=-\frac{P_{2}(0,0)}{(n+1)P_{1}(0,0)}. \end{aligned}$$
(8)

Proof of Lemma 3

If \(q_{n}(s)\) is interior, it satisfies the first-order condition

$$\begin{aligned} P(nq_{n}(s),s)+q_{n}(s)P_{1}(nq_{n}(s),s)-c=0. \end{aligned}$$
(9)

Since \(q_{n}(s)\) is \(C^{1}\), one can differentiate both sides of Eq. (9) with respect to s. Reordering terms yields (7). Evaluating at \(s=0\) and \(q_{n}(0)=0\) yields (8). \(\square \)

Let \(\varPi (z,s)\triangleq \frac{n-1}{n}\left[ \int _{0}^{z}P(t,s)\mathrm{d}t-cz\right] + \frac{1}{n}[zP(z,s)-cz]\), a weighted average of welfare and industry profits when s is the same for all firms. Similar to AL (2011), we have the following result relating, for given s, argmax’s of \(\varPi (z,s)\) and symmetric Cournot equilibria.

Lemma 4

If \(z^{*}\in \arg \max \{\varPi (z,s),0\le z\le nK\}\), then, \(x^{*}\equiv \frac{z^{*}}{n}\in q_{n}(s)\), for all \(s\in [0,K]\).

Proof of Lemma 4

Since the cost function is linear, it is convex. By AL (2011), Lemma 14, for any \(n\in N\) and \(s\in [0,K]\), if \(z^{*}\in \arg \max \{\varPi (z,s),0\le z\le nK\}\), then, \(z^{*}\in Q_{n}(s)\), where \(Q_{n}(s)\) is the total output equilibrium correspondence for a given s. Then, by symmetry, \(z^{*}\in Q_{n}(s)\) implies that \(x^{*}\equiv \frac{z^{*}}{n}\in q_{n}(s)\). \(\square \)

Proof of Theorem 2

  1. (a)

    If the trivial outcome is not part of the equilibrium set, Theorem 1 guarantees there is a symmetric RECE with strictly positive individual output.

  2. (b)

    Parts (b) and (c) use the following argument. By the proof of Theorem 1, \(q_{n}(s)\) is \(C^{1}\) and maps [0,K] into itself. In addition, suppose that there exists \(s^{\prime }\in (0,K)\) such that \( q_{n}(s^{\prime })>s^{\prime }\), then, by Brouwer’s fixed-point theorem, it exists at least one fixed point, say \(s^{\prime \prime }\), such that \( s^{\prime \prime }>s^{\prime }\) and hence \(s^{\prime \prime }>0\), i.e., there exists a non-trivial symmetric RECE, \(s^{\prime \prime }\). Therefore, we only need to show that such \(s^{\prime }\) exists; to this end, it suffices to have \(q_{n}^{\prime }(0)>1\), since it implies that there is a small \(\epsilon >0\) for which \(q_{n}(\epsilon )>\epsilon \). By hypothesis, \( q_{n}(0)=0\) and by Lemma 3, \(q_{n}^{\prime }(0)>1\) if \( (n+1)P_{1}(0,0)+P_{2}(0,0)>0\), which proves our result.

  3. (c)

    The third condition in this part implies that \(\varPi _{1}(z,s)\ge 0\) for some \(s\in (0,K]\) and for all \(z\le ns\), i.e., there exist \(s\in (0,K]\) and \(z^{\prime }\ge ns\) such that \(\varPi (z^{\prime },s)\ge \varPi (z,s)\) for all \( z\le ns\). Hence, the largest argmax of \(\varPi (z,s)\), say \(z^{*}\), must be greater than or equal to ns, i.e., \(z^{*}\ge ns\) and \(z^{*}/n\ge s\). By Lemma 4, \(z^{*}/n\in q_{n}(s)\) so there is an \( s\in (0,K]\) such that an element of \(q_{n}(s)\) is greater or equal than s. By the argument in part (b), it follows that a non-trivial symmetric equilibrium exists for n firms.

\(\square \)

Proof of Proposition 1

From Amir and Lambson (2000, Theorem 2.3), \(q_{n}(s)\) is decreasing in n for each fixed s when P is log-concave. (This is just the condition for the regular Cournot game to be of strategic substitutes, for each fixed s.) Consequently, the viability of the industry decreases in n (since the critical mass increases). \(\square \)

Proof of Lemma 2

By Lemma 1 in AL (2011), and (A1)–(A4), every selection of the best-response correspondence z(ys) strictly increases in both y and s. (This follows from a strengthening of Topkis’s Theorem, see Amir 1996b or Topkis 1998, p. 79.) Then, the correspondence

$$\begin{aligned}&B_{s}{:}\,[0,(n-1)K]\rightarrow 2^{[0,(n-1)K]},\\&y\rightarrow \frac{n-1}{n}z(y,s) \end{aligned}$$

has a unique fixed point, which corresponds to the symmetric Cournot equilibrium (Amir and Lambson 2000). By Milgrom and Roberts (1990), such fixed point, say \(y_{n}(s)\), increases in s. Hence, by symmetry and z(ys) increasing in s, the function (by Proof of Theorem 1) \(q_{n}{:}\,[0,K]\longrightarrow [0,K]\), \(q_{n}(s)=y_{n}(s)/(n-1)\), is increasing in s. \(\square \)

Proof of Proposition 3

By Proposition 1, if an industry with n incompatible networks is viable, the industry with one incompatible network is too. In other words, \({q}_{1}(s)\) has a fixed point different than zero which is also a non-trivial RECE for the monopolist with complete compatibility. Hence, the viability of the n-oligopoly with incompatible networks implies the viability of a monopolist in the single-network model. By AL (2011, Theorem 7), the viability of the industry with complete compatibility increases in n, which completes our proof. \(\square \)

Proof of Proposition 4

  1. (a)

    The (largest) RECE of the oligopoly with incompatible networks is the (largest) fixed point of \({q}_{n}(s)\), say \({q}_{n}(s^{\prime })=s^{\prime }\) . Hence, \(\overline{x}_{n}^{I}=s^{\prime }.\)

On the other hand, the (largest) RECE of the oligopoly with a single-network may be seen as the (largest) intersection point of \({q}_{n} (s)\) with the line (through the origin) s / n, say \({q}_{n} (s^{ \prime \prime }) =s^{ \prime \prime }/n\). Hence, \(\overline{x}_{n}^{C} ={q}_{n} (s^{ \prime \prime }).\)

Since \({q}_{n} (s)\) is increasing in s, it is easy to see that \(\overline{x }_{n}^{C} ={q}_{n} (s^{ \prime \prime }) =s^{ \prime \prime }/n \ge \overline{x}_{n}^{ I} ={q}_{n} (s^{ \prime }) =s^{ \prime }\).

It follows that industry output goes the same way, i.e., \(\overline{z} _{n}^{C}=n\overline{x}_{n}^{C}\ge n\overline{x}_{n}^{I}=\overline{z}_{n}^{I} .\)

  1. (b)

    The endogenous inverse demand functions in Cases C and I are, respectively, \(P(\cdot ,n\bar{x}_{n}^{C})\) and \(P(\cdot ,\bar{x}_{n}^{I})\) . The conclusion follows from part (a).

  2. (c)

    For this proof and the proof of Proposition 5(a), we introduce an auxiliary parametrized Cournot oligopoly with an exogenous demand shifter \(\alpha \). Thus, we consider the profit function

    $$\begin{aligned} \varPi (x,\alpha )=x[P(x+y,\alpha )-c] \end{aligned}$$
    (10)

    and re-frame the questions as ones of comparative statics with respect to \( \alpha \) in this Cournot game.

Now, to show that \(P_{n}^{C}=P(n\bar{x}_{n}^{C},n\bar{x}_{n}^{C})\ge P_{n}^{I}=P(n\bar{x}_{n}^{I},\bar{x}_{n}^{I})\) we show that the Cournot equilibrium price in (10), call it \(p(\alpha ),\) is increasing in \( \alpha \), under condition (4). This clearly implies our desired conclusion here, i.e., \(P_{n}^{C}=p(n\bar{x}_{n}^{C})\ge p(\bar{x} _{n}^{I})\ge P_{n}^{I},\) in light of part (a).

The first-order condition for a Cournot equilibrium in (10) may be rewritten as (with \(z(\alpha )\) as equilibrium total output)

$$\begin{aligned} nP(z(\alpha ),\alpha )+z(\alpha )P_{1}(z(\alpha ),\alpha )-nc=0. \end{aligned}$$
(11)

Since there is a unique Cournot equilibrium under our assumptions, we can differentiate with respect to \(\alpha \) throughout (11) and collect terms to obtain

$$\begin{aligned} z^{\prime }(\alpha )=-\frac{nP_{2}(z(\alpha ),\alpha )+zP_{12}(z(\alpha ),\alpha )}{(n+1)P_{1}(z(\alpha ),\alpha )+zP_{11}(z(\alpha ),\alpha )}. \end{aligned}$$
(12)

Since \(p(\alpha )=P(z(\alpha ),\alpha )\), we have \(p^{\prime }(\alpha )=P_{1}(z(\alpha ),\alpha )z^{\prime }(\alpha )+P_{2}(z(\alpha ),\alpha )\). Substituting (12) yields upon collection of terms that (with all P terms evaluated at \((z(\alpha ),\alpha )\))

$$\begin{aligned} p^{\prime }(\alpha )=-\frac{-P_{1}P_{2}+z(P_{1}P_{12}-P_{2}P_{11})}{ (n+1)P_{1}+zP_{11}}=-\frac{\varDelta _{4}}{(n+1)P_{1}+zP_{11}}. \end{aligned}$$

The denominator \((n+1)P_{1}+zP_{11}<0\) by the stability (in the sense of best-reply Cournot dynamics) of the unique Cournot equilibrium here; therefore, \(p^{\prime }(\alpha )\) has the same sign as \(\varDelta _{4}\), and is thus \(\ge 0.\) \(\square \)

Proof of Proposition 5

  1. (a)

    The proof here follows the same idea as the previous proof [part (c)] of using the parametrized Cournot oligopoly with profit function given by (10). We show that the Cournot equilibrium profit \(\pi (\alpha )\) is increasing in \(\alpha \), under condition (5). This clearly implies our desired conclusion for reasons similar to those of the previous proof.

To this end, differentiate \(\pi (\alpha )=x(\alpha )[P(z(\alpha ),\alpha )-c] \) with respect to \(\alpha \) and then use the first- order condition for a Cournot equilibrium to simplify to

$$\begin{aligned} \pi ^{\prime }(\alpha )=-x\frac{ -P_{1}(2P_{2}+xP_{12})+z(P_{1}P_{12}-P_{2}P_{11})}{(n+1)P_{1}+zP_{11}}=-x \frac{\varDelta _{5}(z)}{(n+1)P_{1}+zP_{11}} \end{aligned}$$

Since the denominator \((n+1)P_{1}+zP_{11}<0\) by Cournot stability, the desired conclusion follows.

  1. (b)

    Consider the following inequalities

    $$\begin{aligned} CS_{n}^{C}-CS_{n}^{I}= & {} \int _{0}^{n{x}_{n}^{C}}[P(t,n{x}_{n}^{C})-P(n{x} _{n}^{C},n{x}_{n}^{C})]\;\;\mathrm{d}t\\&-\,\int _{0}^{n{x}_{n}^{I}}[P(t,{x}_{n}^{I})-P(n{x} _{n}^{I},{x}_{n}^{I})]\mathrm{d}t \\\ge & {} \int _{0}^{n{x}_{n}^{I}}[P(t,n{x}_{n}^{C})-P(n{x}_{n}^{C},n{x} _{n}^{C})]\;\;\mathrm{d}t\\&-\,\int _{0}^{n{x}_{n}^{I}}[P(t,{x}_{n}^{I})-P(n{x}_{n}^{I},{x} _{n}^{I})]\;\;\mathrm{d}t \\\ge & {} \int _{0}^{n{x}_{n}^{I}}[P(t,n{x}_{n}^{C})-P(n{x}_{n}^{C},n{x} _{n}^{C})]\;\;\mathrm{d}t\\&-\,\int _{0}^{n{x}_{n}^{I}}[P(t,{x}_{n}^{I})-P(n{x}_{n}^{C},{x} _{n}^{I})]\;\;\mathrm{d}t\ge 0. \end{aligned}$$

The first and second inequalities follow from the facts that \({x} _{n}^{C}\ge {x}_{n}^{I}\) and \(P_{1}(z,s)<0\). The last one is implied by the submodularity of P, i.e., the assumption that \(P_{12}(z,s)\le 0\). To see this, notice that \(t\in [0,n{x}_{n}^{I}]\) and \(n{x}_{n}^{C}\ge n{x} _{n}^{I}\) imply that \(n{x}_{n}^{C}\ge t\), so that \(P_{12}(z,s)\le 0\) and \(n {x}_{n}^{C}\ge {x}_{n}^{I}\) imply that \(P(t,n{x}_{n}^{C})-P(t,{x} _{n}^{I})\ge P(n{x}_{n}^{C},n{x}_{n}^{C})-P(n{x}_{n}^{C},{x}_{n}^{I})\) for all \(t\in [0,n{x}_{n}^{I}]\).

  1. (c)

    The social welfare function for any per-firm output x and expected size of the network s with n symmetric firms is given by

    $$\begin{aligned} V_{n}(x,s)=\int _{0}^{nx}P(t,s)\;\mathrm{d}t-ncx, \end{aligned}$$

    which is a concave function with respect to x since \(\frac{\partial ^{2}V_{n}(x,s)}{\partial x^{2}}=n^{2}P_{1}(nx,s)<0\), by (A1). Then, we have that at any regular equilibrium

    $$\begin{aligned} W_{n}^{C}-W_{n}^{I}= & {} \left\{ \int _{0}^{n{x}_{n}^{C}}P(t,n{x} _{n}^{C})\;\mathrm{d}t-nc{x}_{n}^{C}\right\} -\left\{ \int _{0}^{n{x}_{n}^{I}}P(t,{x} _{n}^{I})\;\mathrm{d}t-nc{x}_{n}^{I}\right\} \\\ge & {} \left\{ \int _{0}^{n{x}_{n}^{C}}P(t,n{x}_{n}^{C})\;\mathrm{d}t-nc{x} _{n}^{C}\right\} -\left\{ \int _{0}^{n{x}_{n}^{I}}P(t,n{x}_{n}^{C})\;\mathrm{d}t-nc{x} _{n}^{I}\right\} \\= & {} V_{n}({x}_{n}^{C},n{x}_{n}^{C})-V_{n}({x}_{n}^{I},n{x}_{n}^{C}) \\\ge & {} \frac{\partial V_{n}({x}_{n}^{C},n{x}_{n}^{C})}{\partial x}({x} _{n}^{C}-{x}_{n}^{I}) \\= & {} n[P(n{x}_{n}^{C},n{x}_{n}^{C})-c]({x}_{n}^{C}-{x}_{n}^{I})\ge 0. \end{aligned}$$

The first inequality follows by \(n{x}_{n}^{C}\ge {x}_{n}^{I}\) and \( P_{2}(z,s)>0\): the second one, by concavity of \(V_{n}(\cdot ,s)\), and the last one, because \(P(n{x}_{n}^{C},n{x}_{n}^{C})\ge c\) and \({x}_{n}^{C}\ge { x}_{n}^{I}\). This completes the proof of the Proposition. \(\square \)

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Amir, R., Evstigneev, I. & Gama, A. Oligopoly with network effects: firm-specific versus single network. Econ Theory 71, 1203–1230 (2021). https://doi.org/10.1007/s00199-019-01229-0

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Keywords

  • Network effects
  • Network industries
  • Demand-side economies of scale
  • Compatibility
  • Incompatibility

JEL Classification

  • C72
  • D43
  • L13
  • L14