Abstract
This paper investigates the competition between vertically differentiated platforms in two-sided markets. We assume the presence of two competing platforms producing either higher- or lower-quality devices for consumers. Each platform decides the price of its hardware device for consumers and the royalty rate for software developers. We find that, despite the existence of quality differences, the decisions by the platforms about royalty rates are symmetric and only hardware pricing is asymmetric. We also demonstrate that an equilibrium may exist in which a lower-quality platform can enjoy greater profit than a higher-quality rival when there are higher development costs associated with creating software to meet the needs of higher-quality devices.
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Notes
Following Caillaud and Jullien (2003), we can also interpret this as meaning that the total gross gain from trade between a consumer and a software developer is b + π, where the allocation is negotiated as a function of their bargaining power. We assume that because of bargaining, the consumer receives b and the software developer earns π.
Following Katz and Shapiro (1985), we assume that the parameter c j has no finite upper limit to avoid the need to consider corner solutions.
As described in the Introduction, in the video game industry, software developers are charged about $7–8 as a royalty fee by Nintendo, Sony, or Microsoft. There is little difference in the price of the game titles across these video game console manufacturers.
We can also discuss the case in which each platform charges a fixed lump-sum fee to software developers. In this case, it is shown that both platforms charge different fixed lump-sum fees. The proof is available from the author on request.
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Acknowledgments
I would like to thank the editor, anonymous referee, Masayoshi Maruyama, Makoto Okamura, Hideo Suehiro, Kenji Matsui, Yasuyuki Miyahara, Hisao Hisamoto, seminar participants at Kobe University, Nagoya University, Nanzan University, and Kyoto University, and participants at the 2014 annual conference of the Japan Economic Association in Fukuoka and the 2014 annual conference of the Japanese Association for Applied Economics in Tokushima for their very helpful and constructive comments on an earlier version of our paper. Responsibility for all remaining errors lies solely with the authors. The research for this study was supported by a Grant-in-Aid for JSPS Fellows no. 12J03397 from the Japan Society for the Promotion of Science and the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government.
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Appendix
Appendix
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1.
Proof of Proposition 1.
The first-order conditions for profit maximization are given by
$$\begin{array}{@{}rcl@{}} \hat {\Pi}_{i}&=&(q_{H}-q_{L})\left( \tilde D_{i} |_{n_{i}=\hat n_{i}} \right)^{2}, \\ \frac{\partial \hat {\Pi}_{i}}{\partial r_{i}}&=&2(q_{H}-q_{L})\tilde D_{i} \left( \frac{\partial \tilde D_{i}}{\partial r_{i}}+\frac{\partial \tilde D_{i}}{\partial n_{i}}\cdot \frac{\partial \hat n_{i}}{\partial r_{i}} \right)=0. \end{array} $$Here, we can derive the three derivatives (\(\frac {\partial \tilde D_{i}}{\partial r_{i}}\), \(\frac {\partial \tilde D_{i}}{\partial n_{i}}\), and \(\frac {\partial \hat n_{i}}{\partial r_{i}}\)) as follows.
$$\begin{array}{@{}rcl@{}} \frac{\partial \tilde D_{i}}{\partial r_{i}}&=& \frac{\pi \hat n_{i}}{3(q_{H}-q_{L})}\\ \frac{\partial \tilde D_{i}}{\partial n_{i}}&=& \frac{b+r_{i} \pi}{3(q_{H}-q_{L})}\\ \frac{\partial \hat n_{i}}{\partial r_{i}}&=& -\hat n_{i} \left[ \frac{1}{1-r_{i}} + \frac{\pi q_{i} \{ b+r_{i} \pi-(1-r_{i})\pi \}}{3q_{L} q_{H}(q_{H}-q_{L})-(1-r_{2})\pi (b+r_{2} \pi)q_{L}-(1-r_{1})\pi (b+r_{1} \pi)q_{H}} \right] \end{array} $$Using these derivatives, we can rewrite the first-order conditions and derive a unique equilibrium royalty rate as follows.
$$\begin{array}{@{}rcl@{}} && \frac{\partial \hat {\Pi}_{i}}{\partial r_{i}}=0 \\ &\iff& \frac{2}{3} \tilde D_{i} \hat n_{i} \left[\phantom{{}\frac{1}{1-r_{i}}} \pi-(b+r_{i} \pi)\right.\\ &&{\kern2pc}\times\left.\left( \frac{1}{1-r_{i}}{}+{}\frac{\pi q_{i} \{ b+r_{i} \pi-(1-r_{i})\pi \}}{3q_{L} q_{H}(q_{H}{}-{}q_{L}){}-{}(1{}-{}r_{2})\pi (b{}+{}r_{2} \pi)q_{L}{}-{}(1{}-{}r_{1})\pi (b+r_{1} \pi)q_{H}}\right)\right]=0 \\ &\iff& b+r_{i} \pi = (1-r_{i})\pi \\ &\iff& r_{i}^{\ast} =\frac{\pi-b}{2\pi} \end{array} $$(7)The second-order conditions are satisfied as follows.
$$\begin{array}{@{}rcl@{}} \frac{\partial^{2} \hat {\Pi}_{1}}{\partial {r_{1}^{2}}}(r_{1}^{\ast},~r_{2}^{\ast})&=&-\frac{16\pi^{2} {q_{L}^{2}} q_{H}(q_{H}-q_{L})\{ 4q_{H}(q_{H}-q_{L})-(b+\pi)^{2} \}^{2}}{\{ 12q_{L}q_{H}(q_{H}-q_{L})-(b+\pi)^{2}(q_{L}+q_{H})^{2} \}^{3}}<0 \\ \frac{\partial^{2} \hat {\Pi}_{2}}{\partial {r_{2}^{2}}}(r_{1}^{\ast},~r_{2}^{\ast})&=&-\frac{16\pi^{2} q_{L} {q_{H}^{2}}(q_{H}-q_{L})\{ 8q_{L}(q_{H}-q_{L})-(b+\pi)^{2} \}^{2}}{\{ 12q_{L}q_{H}(q_{H}-q_{L})-(b+\pi)^{2}(q_{L}+q_{H})^{2} \}^{3}}<0 \end{array} $$Using this equilibrium royalty rate, we derive other equilibrium values as shown in Table 1.
Finally, we consider the condition for an interior solution as follows. From Table 1, if 4q H (q H −q L )−(b + π)2>0 and 8q L (q H −q L )−(b + π)2>0 hold, both platforms can attract a positive number of software developers and consumers and earn a positive profit (\(n_{i}^{\ast }>0,~0<D_{i}^{\ast }<1,~{\Pi }_{i}^{\ast }>0\) for i = 1, 2). Solving these inequalities for q H (> 0) yields
$$\begin{array}{@{}rcl@{}} &&4q_{H}(q_{H}-q_{L})-(b+\pi)^{2}>0 \\ &\iff& q_{H}>\frac{q_{L}+\sqrt{{q_{L}^{2}}+(b+\pi)^{2}}}{2}\equiv f_{L}(q_{L}),\\ &&8q_{L}(q_{H}-q_{L})-(b+\pi)^{2}>0\\ &\iff& q_{H}>q_{L}+\frac{(b+\pi)^{2}}{8q_{L}}\equiv f_{H}(q_{L}). \end{array} $$Thus, a necessary and sufficient condition for the existence of an inner solution is given by \(q_{H}>\max \left \{ f_{L}(q_{L}),~f_{H}(q_{L}) \right \}\).
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2.
Proof of Proposition 2.
$$\begin{array}{@{}rcl@{}} &&{\Pi}_{1}^{\ast}-{\Pi}_{2}^{\ast} \\ &=&\frac{q_{H}-q_{L}}{\{ 12q_{L}q_{H}(q_{H}-q_{L})-(b+\pi)^{2}(q_{L}+q_{H}) \}^{2}}\\ &&\cdot \left[ {q_{L}^{2}}\{4q_{H}(q_{H}-q_{L})-(b+\pi)^{2}\}^{2}-{q_{H}^{2}}\{8q_{L}(q_{H}-q_{L})-(b+\pi)^{2}\}^{2} \right]\\ &=&\frac{q_{H}-q_{L}}{\{ 12q_{L}q_{H}(q_{H}-q_{L})-(b+\pi)^{2}(q_{L}+q_{H}) \}^{2}}\\ &&\cdot \left[ q_{L}\{4q_{H}(q_{H}-q_{L})-(b+\pi)^{2}\}+q_{H}\{8q_{L}(q_{H}-q_{L})-(b+\pi)^{2}\} \right]\\ &&\cdot \left[ q_{L}\{4q_{H}(q_{H}-q_{L})-(b+\pi)^{2}\}-q_{H}\{8q_{L}(q_{H}-q_{L})-(b+\pi)^{2}\} \right]\\ &=&\frac{(q_{H}-q_{L})^{2}\{(b+\pi)^{2}-4q_{L}q_{H}\}}{\{ 12q_{L}q_{H}(q_{H}-q_{L})-(b+\pi)^{2}(q_{L}+q_{H}) \}}>0\\ &\iff& q_{L}<\frac{(b+\pi)^{2}}{4q_{H}} \end{array} $$ -
3.
Proof of Proposition 3.
Now we can use the results of Section 3.1. Thus, here we investigate the platforms’ decisions regarding their royalty rates in Stage 1, where software developers rationally forecast consumer demand \(D_{i}=\tilde D_{i}\) for Stage 2. We use this to calculate the marginal type of software developer that is indifferent between developing software for device i and not.
$$\begin{array}{@{}rcl@{}} \pi_{ij}&=&(1-r_{i}) \pi \tilde D_{i}-c_{j} \geq 0 \\ &\iff& c_{j}<(1-r_{i}) \pi \tilde D_{i} = n_{i}~~~(i=1, 2) \end{array} $$Solving these equations (i = 1, 2), we derive the range of software developed for device i as follows:
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{llllll} \hat n_{1}=\frac{(1-r_{1}) \pi \{ q_{H}-q_{L} -(1-r_{2})\pi(b+r_{2} \pi) \}}{3(q_{H}-q_{L})-(1-r_{1})\pi (b+r_{1} \pi)-(1-r_{2})\pi (b+r_{2} \pi)},\\ \hat n_{2}=\frac{(1-r_{2})\pi \{ 2(q_{H}-q_{L})-(1-r_{1})\pi(b+r_{1} \pi) \}}{3(q_{H}-q_{L})-(1-r_{1})\pi (b+r_{1} \pi)-(1-r_{2})\pi (b+r_{2} \pi)}. \end{array}\right. \end{array} $$Substituting this range of software \((\hat n_{1},~\hat n_{2})\) in \((\tilde {\Pi }_{1},~\tilde {\Pi }_{2})\), we have
$$\begin{array}{@{}rcl@{}} \hat {\Pi}_{1}&=&\tilde {\Pi}_{1}(\hat n_{1},~\hat n_{2}) \\ &=&\frac{\{ q_{H}-q_{L}+b(\hat n_{1}-\hat n_{2})+r_{1} \pi \hat n_{1}-r_{2} \pi \hat n_{2} \}^{2}}{9(q_{H}-q_{L})} \\ &=&(q_{H}-q_{L}) \left( \tilde D_{1} |_{n_{i}=\hat n_{i}} \right)^{2}, \\ \hat {\Pi}_{2}&=&\tilde {\Pi}_{2}(\hat n_{1},~\hat n_{2}) \\ &=&\frac{\{ 2(q_{H}-q_{L})+b(\hat n_{2}-\hat n_{1})-r_{1} \pi \hat n_{1}+r_{2} \pi \hat n_{2} \}^{2}}{9(q_{H}-q_{L})} \\ &=&(q_{H}-q_{L}) \left( \tilde D_{2} |_{n_{i}=\hat n_{i}} \right)^{2}. \end{array} $$Similar to the proof of Proposition 1, the first-order conditions for profit maximization are given by
$$\begin{array}{@{}rcl@{}} \frac{\partial \hat {\Pi}_{i}}{\partial r_{i}}&=&\frac{2}{3} \tilde D_{i} \hat n_{i} \left[ \pi-(b+r_{i} \pi) \left( \frac{1}{1-r_{i}}+\frac{\pi \{ b+r_{i} \pi-(1-r_{i})\pi \}}{3(q_{H}-q_{L})-(1-r_{1})\pi (b+r_{1} \pi)-(1-r_{2})\pi (b+r_{2} \pi)} \right) \right], \\ &\iff& b+r_{i} \pi = (1-r_{i})\pi , \\ &\iff& r_{i}^{\ast} =\frac{\pi-b}{2\pi}. \end{array} $$Therefore, if the development cost for software and the platform’s quality are independent, both platforms charge the same royalty rate.
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Zennyo, Y. Competition between Vertically Differentiated Platforms. J Ind Compet Trade 16, 309–321 (2016). https://doi.org/10.1007/s10842-016-0223-2
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DOI: https://doi.org/10.1007/s10842-016-0223-2