# When to Use the Open Business Model for Software Products under Network Effects?

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2 on the case when c h is high enough (c h >> c l ) so that entry is not profitable for firm 2 if the incumbent does not open its intellectual property. We will relax this assumption as an extension. After incurring the investment cost c l, the entrant is capable of developing a competing software product with quality q 2. With experience and reputation accumulated over time, the incumbent s product is superior to the new entrant s, i.e. q 1 > q 2. Without loss of generality, we let q 1 = 1 and q 2 = q (0, 1). The development costs to the incumbent are sunk and, given that software is a digital good, we assume negligible replication (marginal) costs for both firms. There is a continuum of consumers with total mass 1 in the market, and each individual consumer needs at most one of the software products. Consumers exhibit heterogeneity in their product valuation θ, where θ is uniformly distributed in the interval [0, 1]. We assume the software products to exhibit network effects - they are more valuable to each user if the installed base is larger. Such network effects could be due to the ability of users to use the software to collaborate or exchange information. If the incumbent opens its IP, and the entrant develops the product on top of the incumbent s IP, the two firms products are compatible, and the network effects are proportional to the sum of the network sizes of the two product. Specifically, the utility a consumer with valuation θ from making a selection s (where s {1, 2} represents a choice of one of the two products and s = 0 indicates that the consumer does not buy any of the products) is as follows: u (s = 0 θ) = 0, u (s = 1 θ) = (θ + γn) p 1, u (s = 2 θ) = (θ + γn) q p 2, where γ > 0 indicates the strength of the network effects, and N = N 1 + N 2 is the sum of both firms installed bases. Prices p 1 and p 2 are strategically determined by each firm. We consider the following timeline corresponding to a sequential Stackelberg game. First, the incumbent (firm 1) decides whether to open its IP or not. Next, contingent on the incumbent s IP being open, firm 2 enters the market and sets its price p 2. Third, the incumbent responds with price p 1. Finally, consumers make their purchase decisions after observing the prices. 3 Analysis and Preliminary Results We start the analysis with the case when the incumbent chooses not to open its IP to the potential entrant. As the entry cost is prohibitive, firm 2 stays out of the market, and the incumbent acts as a monopolist. In this scenario, we can derive the incumbent s equilibrium price, demand, and profit as follows. Proposition 1. In the case of monopoly under firm 1, the equilibrium can be summarized as follows: { } (i) Under weak network effects (i.e., γ < 1), we have 2 {p 1, N1, π1} 1 =, 1, 1 ; 2 2(1 γ) (1 γ) 2

3 (ii) Under strong network effects (i.e., γ 1 2 ), we have {p 1, N 1, π 1} = {γ, 1, γ}. Note that the equilibrium concept we use here is the fulfilled-expectations equilibrium (FEE), in which firms decide their optimal prices given the consumers expectations about the network sizes, and these expectations are rational and fulfilled in equilibrium (Katz and Shapiro 1985). Next, we consider the case in which the incumbent chooses to open its IP to the entrant, who subsequently enters the market. To derive the equilibrium outcome, we start with the incumbent s best response (in terms of pricing strategy) to the entrant s price. The incumbent sets p 1 (p 2 ) to maximize its profit such that p 1 (p 2 ) = arg max p 1 p 1 D 1 (p 1, p 2, N (p 1, p 2 )), where D 1 is the demand of firm 1 s product. Note that the demand depends not only on the prices but also on consumers expectations about the total network size N. By FEE, N = D 1 + D 2 which, in turn, also depends on the prices. In other words, consumers expectations about the total network size must be endogenously determined when deriving the equilibrium demand. It is also worth noting that some consumers may choose not to purchase from either firm if the prices are too high. Therefore, in deriving the demand functions, we analyze the boundary consumers who are indifferent between firm 1 s and firm 2 s products as well as those indifferent between purchasing firm 2 s product and not purchasing any product at all. As a result, in addition to the case of fully covered market, we also consider the scenario when the market is not fully covered. After deriving the incumbent s best response to the entrant s price, p 1 (p 2 ), the entrant s optimal price p 2 can be determined by solving the following optimization: p 2 = arg max p 2 p 2 D 2 (p 2, p 1 (p 2 ), N (p 1 (p 2 ), p 2 )). The equilibrium prices, demands, and profits for both players are summarized in the following result. Proposition 2. When the incumbent opens its IP to the entrant and the latter enters the market, the equilibrium outcomes can be summarized as follows: (i) If 0 < γ < 2 2 and 0 < q < 2γ2 γ+1, the equilibrium prices are 2 1 2γ { p 1 = (1 q)( q 3γ) p 2 = (1 γ)(2 q γ) q(1 q) 2(2 q γ) and the equilibrium demands and profits are N q 3γ 1 = (1 γ)(2 q γ) 2 = 1, (1 γ) N = π1 = (1 q)( q 3γ)2 16(1 γ) (2 q γ) 2 π2 = q(1 q) 8(1 γ)(2 q γ) 3 q 2γ 2(1 γ)(2 q γ) < 1 3

4 (ii) If 0 < γ < 2 2 and 2γ2 γ+1 < q < 1 γ, or 2 2 < γ < 1 and 0 < q < 1 γ, the 2 1 2γ 1+γ 2 1+γ equilibrium prices are { p 1 = 1+γ q 2 p 2 = qγ and the equilibrium demands and profits are N1 = 1+γ q, N 2(1 q) 2 = 1 γ q, N = 1 2(1 q) π1 = (1+γ q)2 (1 q) π2 = (1 γ q)qγ (iii) If 1 γ 1+γ 2(1 q) < q < 1, the equilibrium prices are { p 1 = 1 (1 q) (3 + γ) p 2 = 1 (1 q) (1 γ) 2 and the equilibrium demands and profits are N1 = 3+γ, N 2 = 1 γ, N = 1 π1 = 1 (1 q) (3 + γ)2 16 π2 = 1 (1 q) (1 γ)2 8 Figure 1 illustrates the equilibrium outcomes in Proposition 2. Figure 1: Equilibrium Outcomes When the Incumbent Opens its Platform Comparing the equilibrium profits of the incumbent in Proposition 1 and Proposition 2 over different parameter regions, we can derive the incumbent s optimal strategy contingent on the strength of network effects and the quality of the entrant s product, as illustrated in the next result.

5 Proposition 3. The incumbent prefers opening its IP to the entrant in the following parameter regions: (a) 0 < γ 2 1, and 0 < q < γ; (b) 2 1 < γ 1, and 2 0 < q < 1 ; (c) 1 16γ < γ < 1, and q < 1, as highlighted in the gray (1 γ)(3+γ) 2 2 (3+γ) 2 area in Figure 2. Figure 2: Equilibrium Outcomes When the Incumbent Opens its Platform When q > 2 1, the incumbent prefers monopoly to opening its IP to competition regardless of the strength of network effects. This is somewhat intuitive. As products become less differentiated, the price competition intensifies, hurting the profits of the incumbent. This adverse effect is too strong and it dominates any benefit from the increase in the willingness to pay of the consumers due to network effects (especially since both incumbent and entrant benefit from these effects). Thus, the incumbent prefers to keep its IP closed and enjoy a monopolistic position. On the other hand, things get interesting and nontrivial when products are differentiated enough (q < 2 1) as summarized by the following corollary to Proposition 3. Corollary 1. For every entrant product quality level q (0, 2 1), there exist thresholds 0 < γ 1 (q) < γ 2 (q) < 1 such that: (i) when network effects are of intermediate strength (i.e., γ 1 (q) < γ < γ 2 (q)), then the incumbent prefers to open its IP to the entrant; (ii) when network effects are either very week (i.e., 0 < γ < γ 1 (q)) or very strong (i.e, γ > γ 2 (q)) then the incumbent closes its IP and prevents the competitor from entering the market. Corollary 1 has interesting implications. First, it is intuitive that when the strength of network effects is weak, the incumbent software firm prefers the closed business model, given the limited value generated by the third party entrant through the incremental installed base. Second, it is somewhat surprising to find that the incumbent software firm also prefers the closed business model when the strength of network effects is strong. This is because the 5

6 market size is finite and once again, the value generated by the third party entrant through the incremental installed base is limited. We point out that this latter result contrasts with the result in Conner (1995). In a different context, Conner shows that, when the strength of network effects is strong enough, the incumbent would always prefer inviting imitators under the assumption of unlimited or infinite market size. Conclusions In this study, we provide a theoretical framework to help incumbent software firms choose between open vs. closed business models. We find that there is a region when open is preferred over closed: the software firm would like to share its IP and its market with a third party competitor, and forgo the monopolist power it has. On the other hand, we also show that there are regions where traditional closed business models are still favored over open business models. Furthermore, when the quality of the software product of the third party competitor is below a certain threshold, we uncover an interesting back-andforth switching pattern between optimal strategies (closed, open and then closed again) as the strength of network effects increases. We also contribute to the literature on markets for technology buyouts (Kremer 1998, Bessen and Maskin 2009) by identifying conditions when such markets are desirable and when they are not needed. References Bessen, J., E. Maskin Sequential innovation, patents, and imitation. RAND Journal of Economics Ceccagnoli, M., C. Forman, P. Huang, D.J. Wu Co-creation of value in a platform ecosystem: The case of enterprise software. MIS Quarterly 36(1) Conner, K. R Obtaining strategic advantage from being imitated: When can encouraging clones pay? Management Science 1(2) Katz, M. L., C. Shapiro Network externalities, competition, and compatibility. The American Economic Review 75(3) 2 0. Kremer, M Patent buyouts: A mechanism for encouraging innovation. Quarterly Journal of Economics Lessig, L The Future of Ideas: The Fate of the Commons in a Connected World. Random House, New York. Parker, G., M. W. Van Alstyne Innovation, openness, and platform control. Available at SSRN: 6

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