Impact of Competition from Open Source Software on Proprietary Software

Transcription

1 Impact of Competition from Open Source Software on Proprietary Software Vidyanand Choudhary and Zach Z. Zhou This is a student authored paper. October 2007 Abstract The number and impact of open source projects is increasing. We examine the impact of competition from open source software on the quality of proprietary software and overall social welfare. We nd that in the case of duopoly competition between proprietary and open source software, there are three possible outcomes: (i) when the cost of quality for the proprietary rm is low, he covers the entire market, (ii) when the proprietary rm s cost of quality is high and open source quality is high then it is not pro table for the proprietary rm to enter the market and the market is covered by the open source solution, (iii) under remaining conditions, the market is shared by the two competitors with technically sophisticated users preferring open source software and other users buying proprietary software. Comparing the case of duopoly competition between open source and proprietary software with the case of the proprietary monopoly, we nd that the quality developed by the proprietary rm can be higher or lower than monopoly quality, whereas prior research showed that competition can never increase the quality of proprietary software. Another counter-intuitive nding is that competition from open source software can reduce social welfare. Whereas, prior research has shown this result in the context of network e ects, we show that this can occur in the absence of network e ects and o er a novel explanation. Key words: open source software, proprietary software, software quality, duopoly, monopoly, competitve strategy

2 1 Introduction Open Source refers to a way of producing information goods that exhibits certain common features such as large scale collaboration among volunteers, free access, and limited protection for intellectual property. The term was originally coined in the context of open source software and the term source refers to the source code that is compiled to create software. Large successful examples of open source software include the Linux operating system, and Apache web server. Open source products lead in terms of market share in certain product markets, and are a signi cant force in other markets. Apache is running more than 60% of the world s web servers most of whom also use Linux. Individual aspects of open source have been generalized and applied to broader classes of goods. For example Tapscott and Williams [13] use Wikinomics and Mass Collaboration to refer to products that are created through the joint e orts of lots of volunteers. They list several examples spanning many sectors of the economy including, Wikipedia a free online encyclopedia, and Zopa an online lending community. Other variations of the open source software model include hybrids such as MySQL which is owned by a pro t maximizing rm but uses the open source method for product testing [12]. For the purpose of this article we will focus only on those open source projects that are owned by not-for pro t communities. The increasing penetration of broadband and adoption of social networking is leading to tremendous growth in the number of open source projects. SpikeSource a company that has built an infrastructure for testing and certifying open source products for business use tracks about 150,000 open source projects. Many of these open source products are having a signi cant impact on competition in many markets. In this paper we seek to understand the impact of competition from open source software on the quality of software and on social welfare. We focus on open source software that is not owned or directed by pro t motivated rms. Therefore, we assume that open source communities are nonstrategic players that are not pro t maximizing entities. We decompose overall software quality in to two components: (i) Usability: includes ease of installation, documentation, user interface and level of technical support and (ii) Functional quality, which includes all other aspects of software quality including feature set, reliability, maintainability, security etc. This is because even though consumers 1

3 have the same tastes for the functional quality, they can have di erent preferences for usability. User surveys have shown that open source software is weak on usability relative to proprietary software [2]. Thus open source software is generally used by technically savvy users. We seek to understand how the presence of an open source competitor can impact the quality of a proprietary software vendor and the overall social welfare. To create a benchmark for this analysis we solve the case of a proprietary software vendor as a monopoly. We compare our ndings in the benchmark case to the case of competition between open source software and a proprietary software vendor. We nd a number of interesting results. In the case of duopoly competition between proprietary and open source software, we show that there are three possible outcomes: (i) when the cost of quality for proprietary vendor is low, he covers the entire market, (ii) when the proprietary vendor s cost of quality is high and open source quality is high then it is not pro table for the proprietary vendor to enter the market and the market is covered by the open source solution, (iii) under remaining conditions, the market is shared by the two competitors with technically sophisticated users preferring open source software and other users buying proprietary software. Comparing the case of duopoly competition between open source and proprietary with the case of proprietary monopoly, we nd that (i) the price charged by the proprietary rm is always lower when it is competing with open source relative to the monopoly price, (ii) the quality o ered by the proprietary rm can be higher or lower than monopoly quality; it is higher when the quality of open source software is su ciently low and the cost of quality for proprietary rm is moderate. (iii) the proprietary vendor does not enter the market in the presence of an open source competitor whereas entry is pro table in the monopoly case when the open source quality and cost of quality for proprietary vendor are high enough. Early literature on open source software focussed on understanding the motivation of programmers to contribute signi cant time and e ort to open source projects without an explicit compensation mechanism. Krogh and Hippel [6] provide an excellent overview of the various research streams in the open source area. They list three streams of research motivation of contributors, governance and organization and competitive impact. Lerner and Tirole [7] describe four popular open source software programs and hypothesize that programmers may be contributing e ort to signal their talent to potential employers. Roberts et al. [11] examine data from the Apache project to study several potential motivations for programmers include status and opportunity, use-value, intrinsic and extrinsic 2

4 motives. Eric Raymond [10] has written an in uential article detailing the di erences in the software development methodology between open source and proprietary software. Meng and Lee [8] examine competition between open source and proprietary software in a horizontal di erentiation model with network e ects where software quality is exogenously determined. They study the proprietary rm s compatibility strategy and nd that he will not choose incompatibility or outward compatibility. The rm may select inward compatibility or two-way compatibility depending on whether the market is fully or partly covered and the quality di erence between open source and proprietary software. Economides and Katsamakas [4] examine platform competition where each platform can support multiple applications. The platform can be either proprietary or open source whereas the applications are proprietary. They nd that when these two platforms compete, the proprietary platform and its applications obtain greater market share and larger pro ts relative to the system based on the open source platform. Raghunathan et al. [9] model some of these di erences and compare the optimal qualities under proprietary monopoly and duopoly competition between open source and proprietary software. They nd that the proprietary rm never increases quality under competition from open source. In contrast to Raghunathan et al. [9], our model captures a key di erence between the open source software and the proprietary software: the software usability (e.g. vendor support) of the open source software is lower than that of the proprietary software [2]. Using our model, we nd that the proprietary rm can increase quality under competition from open source. We specify the conditions under which this can happen and o er a novel explanation. Casadesus-Masanell and Ghemawat [1] examine a model of network e ects where the open source player is non-strategic while the proprietary rm is pro t maximizing. They develop a multi-period, multi-generational model and nd conditions under which both open source and proprietary software can coexist. They also nd that social welfare can be lower under the duopoly relative to proprietary monopoly. This occurs because the user base is split in to two sets of users thus potentially reducing network e ects whereas there is a single network in the monopoly case. 3

5 2 The Model We wish to examine the impact of open source software on the optimal design of proprietary software. We begin with a benchmark in section 3 by solving the case of the proprietary software provider as a monopoly. In section 4, we analyze the case of competition between proprietary software and open source software. We compare the solutions in the monopoly case with the solution from the duopoly (competition between open source and proprietary) case in section 5. It is generally known that open source software is less user friendly than proprietary software [2]. This occurs because open source software is designed and created by technically savvy programmers who are also potential users of the software. Given their expertise, they can easily perform certain tasks that can be fairly challenging for the average user. For example the installation of open source software may require downloading source code, linking libraries, setting environment variables for the operating system and compiling the source code. In contrast, most proprietary software requires just a few clicks and if that should prove di cult, technical support is usually available. In order to capture this di erence between open source and proprietary software, we model software quality as consisting of two components: (i) usability (v): includes ease of installation, documentation, user interface and level of technical support and (ii) functional quality (q), which includes all other aspects of software quality including feature set, reliability, security etc. In this paper we use the generic term quality to refer to functional quality. We model consumers as heterogeneous in their level of technical ability using a consumer type parameter x. Consumers with higher degree of technical capability have lower x, while those with lower level of technical skills have higher x. We assume that x is uniformly distributed in [0; 1]. We make the following assumption. Assumption 1 A consumer with lower technical expertise has higher willingness-to-pay for software usability than a consumer with higher technical expertise does. The implication of this assumption is that a tech-savvy consumer cares less about software usability relative to a non-tech-savvy consumer. For example, technical support of installation is quite useful to a user with low level of technical expertise whereas such support is less useful to an expert user. The consumers net surplus from using open source software and proprietary software can be 4

6 expressed as U o (x) = v o x + q o ; U p (x) = v p x + q p p p (1) where U o, v o is the utility and usability of open source and U p, and v p is the utility and usability of proprietary software. Assumption 2 Open source software quality and usability are not determined by strategic considerations and are therefore exogenous. Assumption 3 (Base Model Only) The usability of proprietary software is normalized to one (v p = 1). A survey (CIO, 2002) shows that low usability and lack of provider support is a weakness of open source software. Thus our base model assumes v o < v p = 1, where v o satis es 0 v o < v p so that usability of open source software is always less than that for proprietary software. Let v be the di erence in the level of usability between open source and proprietary, then v = 1 v o. Later in section 6 we examine a more general consumer utility function and relax the assumption on exogenous usability for proprietary vendor. We nd that our results are robust to changes in these assumptions so our ndings hold in the relaxed models. 3 Proprietary Software Vendor as a Monopoly (Case M) We begin with the case of the proprietary software vendor as monopoly and use the subscript m to represent the monopoly case: q m, p m, d m, and m. The marginal consumer type (x i ) who is indi erent between buying the proprietary software and not buying it is obtained by solving U p (x) = x + q m p m = 0 for indi erent x. x i = p m q m (2) Note that the demand or the market share for the proprietary software vendor is d m = 1 x i. Therefore x i must satisfy: 0 x i 1. The xed cost of developing quality q m is cq 2 m, where c is a scaling parameter for xed cost. Then 5

7 the pro t function of is m (p m ; q m ) = d m (p m ; q m ) p m cq 2 m = p m (1 x i ) cq 2 m Solving the proprietary software vendor s problem max pm;q m m yeilds Proposition 1 The monopolist software vendor s optimal quality, price, pro t, and demand are: (i) Interior solution:when c 1=2 then q m = 1=(4c 1), p m = 2c=(4c 1), m = c=(4c 1), d m = 2c=(4c 1), x i = 2c 1 4c 1. (ii) Boundary solution: when c < 1=2 then q m = 1=(2c), p m = 1=(2c), m = 1=(4c), d m = 1, x i = 0. Proposition 1 shows that when the cost of quality (c) is very small, it is optimal for the proprietary monopoly to develop a very high quality product and set prices to cover the entire market. When (c) is larger, the functional quality is smaller thus it is optimal for the monopolist to set prices so that consumers who put a greater value on usability will still purchase while other will not (interior solution). We would like to compare the solution for this case to the solution obtained from the case of duopoly competition between proprietary and open source vendors. 4 Duopoly Competition between Proprietary Software and Open Source Software (Case PO) In this section, we examine the case of competition between an open source software and a proprietary software. We use the subscript d to represent proprietary software in this case: q d, p d, d d, and d. Open source software is created by open source communities where individuals voluntarily contribute their e orts. Since open source software is freely downloadable, these communities do not earn a pro t. The competing proprietary rm is a pro t maximizing seller that determines its quality strategically with respect to the quality of the open source competitor. In contrast the software quality of open source is not strategic or pro t maximizing. These assumptions are consistent with prior research in this area (see e.g. Casadesus-Masanell and Ghemawat [1]). We do not model the complex process of software development (known as the bazaar style) for open source and instead assume that the quality of the open source software (q o ) is exogenously determined. Open source software can be freely download 6

8 from web servers [7], [1] thus its price is zero. Consumer x would choose the proprietary software if x + q d p d v o x + q o, or the open source software otherwise. The marginal consumer x i is given by x i = (p d + q o q d )=(1 v o ) = (p d + q o q d )=v (3) The demand for proprietary software is d d = 1 x i, while the demand for the open source software is x i. Clearly x i must satisfy the following constraints: 0 x i 1. The pro t function for the proprietary software vendor can be stated as: d = (1 x i ) p d cq 2 d (4) Solving the proprietary software vendor s problem max pd ;q d d yeilds Proposition 2 The proprietary software vendor s optimal quality, price, pro t, and demand are given as follows. (i) Case 1 Interior Solution: If the open source software quality is low and proprietary vendor s cost of quality (c) is relatively high, the proprietary software and the open source software share the market. Speci cally, if q o < v and c > 1= [2(v + q o )], then qd = (v q o) = (4cv 1), p d = 2cv (v q o ) = (4cv 1), d = c (v q o) 2 = (4cv 1), d d = 2cv (v q o ) = (4cv 1). (ii)case 2: If the proprietary vendor s cost of quality (c) is low, the open source software is competed out of the market. Speci cally, if (a) q o < v and c 1= [2(v + q o )], or (b) q o v and c 1= (4q o ) then qd = 1= (2c), p d = 1= (2c) q o, d = 1= (4c) q o, d d = 1. (iii)case 3: If the open source software quality is high and proprietary vendor s cost of quality is high, then entry is not pro table for the proprietary software. Speci cally, if q o v and c > 1= (4q o ), then p d = q d = d = d d = 0. We want to examine how the optimal quality of proprietary software behaves with changes in the quality of open source software. Using the results in Proposition 2, it is can be shown that the proprietary software vendor does not increase its software quality (qd ) when it faces competition from higher-quality open source software d =@q o 0). The intuition is as follows. Open source 7

9 software is free, so proprietary software vendor faces pressure to reduce its software price (p d ) when the open source software is higher quality. It can be veri ed d =@q o < 0. By setting a lower price, the proprietary software vendor loses revenue which reduces his incentive to invest in software quality. Note that quality development is a costly strategy for the proprietary software vendor because cost of quality is a convex function. Hence, the proprietary software vendor reduces price and quality when it competes against the open source software. Examining the impact of cost of quality (c) on the proprietary software quality (qd ), we nd that if it is more costly to develop software quality, the proprietary software vendor tends to reduce its e orts towards quality =@c < 0). And the proprietary software vendor cuts its price as well d =@c < 0). Since the proprietary software usability is higher than open source software usability, it is possible that even though the proprietary software quality is lower than the open source software quality (q d < q o), the proprietary software vendor can still sell its software to consumers who value high usability. We are interested in nding conditions under which proprietary software quality is less than that of open source software. Examining the results of Proposition 2, we nd that (1) if the proprietary software drives the open source software out of the market, the proprietary software quality must be greater than the open source software quality, (2) if the proprietary software shares the market with the open source software, then the proprietary software quality is lower than open source software quality when quality cost (c) is su ciently large. Speci cally, given q o < v and 1= [2(v + q o )] < c (i.e. proprietary software and open source software share the market), then q d < q o when c > 1= (4q o ) > 1= [2(v + q o )] and q d q o otherwise. 5 Comparison of proprietary monopoly case with duopoly competition case (Case M versus Case PO) Using the results of Proposition 1 and Proposition 2, we can now examine the impact of open source competition on the optimal price and quality of proprietary software, consumer surplus and social welfare. It can be shown that the competition from open source software reduces the price of proprietary software relative to the optimal price charged by a monopoly (p d < p m). This result is intuitive. The open source software is free. Even though the open source software is competed out of the market 8

10 by the proprietary software, the potential threat of such free software forces the proprietary software vendor to cut its price. This may be the case with Microsoft which is facing a growing threat from Linux some analysts including those at Deutsche Bank securities have speculated that price cuts by Microsoft in 2003 were motivated by the increased adoption of Linux (Internetnews 2003). Proposition 3 Impact of competition with open source software on the optimal quality of proprietary software: (i) If the quality cost of proprietary software is su ciently low (i.e. c min(1=2; 1= [2(1 + q o v o )] ; 1= (4q o ))), then the competition from the open source software does not a ect the proprietary software quality (i.e. qd = q m). (ii) If open source software usability is su ciently high (i.e. v o > q o ), its quality is su ciently low (i.e. q o < 1=2) and its quality cost of proprietary software is moderate (i.e. 1=2 < c < min (1= (4q o ) ; (q o + v o ) = (4q o ))), then the competition from the open source software increases the proprietary software quality comparing with that in monopoly (i.e. qd > q m). (iii) when q o v and c > 1= (4q o ), proprietary software vendor nds it optimal to o er a product as a monopoly but entry is not optimal in the presence of the competition from the open source software. (iv) In other areas, the competition from the open source software reduces the proprietary software quality (i.e. qd < q m). The intuition of this proposition is as follows. For part (i) where the quality cost is su ciently low and the proprietary software covers the whole market in both cases, the quality cost is so low that the proprietary software vendor develops a high quality in the monopoly case. When the competiting open source software is low quality, then the proprietary software vendor does not need to increase its software quality because its quality is already very high. Proposition 3, part (ii) shows that the proprietary software quality can be higher under competition (relative to quality in the monopoly case) under certain conditions. It can be seen that this occurs in the interior solution when the usability level provided by open source is su ciently large (i.e. v o > q o ). Intuitively, the proprietary software is not di erentiated from the open source software on the usability dimension when the usability of the open source software is large. Note that the open source software is free, the proprietary software has to increase its di erentiation from the open source software on 9

11 the functional quality dimension and at the same time increase the value of its software to users. It can achieve both by raising its quality. Proposition 3, part (ii) also requires that the cost of quality for the proprietary software vendor is moderate. This can be understood by examining the impact of cost of quality on optimal quality for the monopoly and duopoly cases. When the cost of quality is too small, we return to part (i) where the proprietary software vendor only needs to reduce its price to compete the open source software out of the market. When the cost of quality is too high, It is too expensive for the proprietary software vendor to raise quality and he prefers to lower both price and quality. So the proprietary software vendor responds to competition from the open source software by raising its software quality only when the cost of quality is moderate. Proposition 3, part (iii) is interesting because it shows that the presence of open source software can make the market so unattractive to the proprietary rm that it prefers not to enter. This sometimes lead to lower overall social welfare, which will be discussed later. Proposition 3, part (iv) is an expected outcome. This occurs because competition from open source always reduces the duopoly price as shown above. This reduction in price reduces the proprietary software vendor s incentive to invest in its software quality leading to lower quality relative to the quality in the monopoly case. Raghunathan et. al. (2005) study the e ect of competition from open source software on the quality of proprietary software. They nd that the proprietary software quality is reduced or remains the same when competing with open source software. In contrast, we take software usability into consideration and nd that proprietary software quality can be higher in the duopoly competition case relative to the monopoly case. Next we analyze the impact of the open source competition on consumer surplus. If the proprietary software vendor is a monopoly, the total consumer surplus is CS m = R 1 x im (x + q m x im = p m p m ) dx, where q m. When the proprietary software vendor faces competition from open source software, the total consumer surplus turns out to be CS d = R 1 x id (x + q d p d ) dx + R x id 0 (v o x + q o ) dx, where x id = (p d + q o q d )=(1 v o ). Using the results of Proposition 1 and Proposition 2, we can compare the total consumer surplus in the two scenarios. It can be shown that the competition from open source software increases the consumer surplus (i.e. CS d > CS m ). There are two reasons. First, 10

12 those consumers who can not a ord the monopoly price of the proprietary software may now use the open source software and obtain positive net surplus. Second, any consumer can obtain a positive net surplus from using open source software because q o + v o x > 0 always holds. To compete against open source software, the proprietary software vendor must o er a positive net surplus to all of its consumers. That is, the marginal consumer also gets a positive net surplus. While in the proprietary monopoly case, the marginal consumer gets a net surplus of zero. Thus, competition from open source software bene ts consumers. Next, we consider the impact of competition from open source software on social welfare. Proposition 4 Impact of competition from open source software on social welfare: Given v o > 0, there exists a ^q o such that (i) When q o ^q o, competition from open source software does not reduce social welfare. That is, SW d SW m holds regardless of c. (ii) When q o > ^q o, competition from open source software can reduce social welfare. That is, SW d < SW m at some c. We use a numerical example to show the result of Proposition 4. When v o = 0:5, it can be veri ed that ^q o = 0:370. If q o = 0:4 > ^q o, there exist c l = 0:576 and c h = 0:687 such that SW d < SW m in c 2 (c l ; c h ). Proposition 4 is an interesting result because competition is generally understood to be socially desirable, and in particular competition from open source software should increase social welfare since the open source product is available to consumers for free. As seen in Figure 1, social welfare can be lower in the presence of the open source competitor relative to the case of a proprietary monopoly. Note that we do not model the xed cost of developing open source software which is borne by individuals in the open source community. Proposition 4 is a strong result since including the social 11

13 cost of developing open source software will further reduce the social welfare in duopoly case. SW, SW d m SW d SW m 2 2 WTP, cqm cqd ( v o, q o ) = ( 0.5,0.4) reduction in total willingness to pay reduction in quality cost 0.7 c = l c = h c c Figure 1: SW m vs. SW d, (v o ; q o ) = (0:5; 0:4) Figure 2: SW m > SW d, (v o ; q o ) = (0:5; 0:4) The key reason for the counter-intuitive result shown above is that competition from open source software can reduce proprietary software quality as shown in proposition 3 part (iv). This reduction in quality reduces the sum of the willingness-to-pay of all consumers and the cost of quality for the proprietary vendor as well. Under certain conditions, the reduction in total willingness-to-pay can be greater than the reduction in cost os quality. That is, it can lead to a reduction in realized social welfare. Figure 2 shows this scenario. Here the reduction in total willingness-to-pay is R 1 ^x m (x + q m ) dx h R 1 ^x d (x + q d ) dx + R i ^x d 0 (v o x + q o ) dx, while the reduction in cost of quality is cqm 2 cqd 2. The results of Proposition 4 are apparent in a special case where monopoly proprietary software covers the market, but competition from open source software makes it is no longer optimal for the proprietary software to enter. If quality cost c is a bit greater than 1= (4q o ), then it can be shown that the proprietary software quality is higher than that of the open source software (q m = 1= (2c) > q o ). Thus, the open source software competes the proprietary software out of the market but such competition generates less social welfare than that provided by the proprietary monopoly. Although the entry of open source software increases the consumer surplus, it could make the competition so intense that the reduction in revenue for the proprietary rm is greater than the increase in surplus obtained by consumers. Proposition 4 shows that competition from open source software is not always socially desirable. 12

14 6 Extensions and Discussions In this section we extend the consumer utility function and allow for endogenous determination of software usability on the part of the proprietary rm. Due to page limits, we will keep our comments brief. 6.1 Nonlinear Two-Attribute Utility Function We extend the consumer utility function by adding an interaction term between usability and quality. Speci cally, The new consumers utility function is: U o (x) = v o x + v o q o x + q o ; U p (x) = v p x + v p q p x + q p p p (5) Prior research ([5]) has used multiattribute utility functions that are either multiplicative or additive. We add a new multiplicative term vqx in the utility function used in the base model. Thus the new WTP of a consumer depends on: (1) the value of usability, (2) the value of quality, and (3) the interaction of usability and quality. Using the new utility function, we obtain Proposition 5. Proposition 5 If both softwares coexist in the duopoly competition, then the competition from the open source software can increase or reduce the proprietary software quality. Speci cally, if the usability of open source software is su ciently small, then the proprietary software quality in duopoly is lower than that in monopoly (i.e. qd < q m); if the usability of open source software is su ciently large, then the proprietary software quality in duopoly is higher than that in monopoly (i.e. qd > q m). The intuition is similar to that for proposition 3. If the proprietary software is not su ciently di erent from open source software in the dimension of usability and the open source quality is low, then the proprietary software vendor increases its quality to increase di erentiation. We obtain proprietary vendor s optimal strategies in monopoly case and in duopoly case (see Appendix). Since the analytical expression of social welfare is very complex, we use numerical methods to compare social welfare in both cases. We have veri ed that under certain conditions, competition from open source can lead to lower social welfare relative to the social welfare in the monopoly case. As explained in the dicussion following proposition 4, this occurs when competition from open source causes the proprietary rm to reduce quality relative monopoly quality. 13

15 6.2 Endogenous Software Usability In this subsection, we allow the proprietary rm to endogenously determine its software usability. All assumptions regarding consumers remain the same as those in the base model. We only change the proprietary software vendor s problem, which turns to be max pd ;q d ;v d d = d d p d c q 2 d s v 2 d in duopoly case, where s is cost parameter of software usability. Consider the duopoly competition between the open source software and the proprietary software, we have Proposition 6 (i) If the proprietary software covers the whole market at the equilibrium, then q d, v d, and p d satisfy (i.a) If 1= (2s) < v o, then q d = 1= (2c), v d = 1= (2s), p d = 1= (2c) + 1= (2s) q o v o. (i.b) If 1= (2s) v o, then q d = 1= (2c), v d = v o, p d = 1= (2c) v o. (ii) If the proprietary software and the open source software share the market at the equilibrium, then q d, v d and p d satisfy q d = 1= (4c) + 6s (1 4cq o) = 4c (c 2s 8csv o ) 1 + cos + p 3 sin, vd = 1= (4c)+v o +(c 2s 8csv o ) 1 + cos + p 3 sin = (24cs), p d = 1=2(q d q o + vd v o ), where h i = arccos 54cs 2 (1 4cq o ) 2 = (c 2s 8csv o ) 3 1. (iii) If the proprietary software is competed out of the market, then q d = v d = p d = 0. Based on the results in proposition 6, we have veri ed that there exists an equilibrium where the quality of the proprietary software under competition can be greater than its quality as a monopoly. Under di erent conditions, the quality of the proprietary software under competition can be less than its quality as a monopoly leading to reduced social welfare under competition relative to the monopoly case. 7 Conclusion This paper examines the impacts of open source competition on proprietary software quality, price, consumer surplus and social welfare. We compare two cases: (1) proprietary software vendor is a monopoly, and (2) proprietary software competes with open source software. Prior literature on competition between open source and proprietary software has focussed on the role of network e ects ([8], [1]). Economides and Katsamakas [4] have focussed on two-sided platform competition. Raghunathan et al. [9] do not model network e ects and nd that competition reduces proprietary software quality. 14

16 In contrast, we model quality as consisting of two attributes: usability and functional quality and show that functional quality can increases under competition. Whereas Raghunathan et al. nd that competition always improves social welfare, we show that competition from open source can reduce social welfare. While models of network e ects have previously shown that social welfare can fall in competition, this result is driven by the loss in welfare due to two incompatible networks in a duopoly whereas there is a single network in monopoly. Our paper does not model network e ects and thus we o er a novel explanation for this surprising outcome. When the quality of open source software is high, but less than monopoly quality, open source software captures a signi cant share of the market thus reducing the available revenues for the proprietary rm. This reduces the proprietary rm s incentive to invest in software quality leading to lower quality relative to the monopoly case and under some conditions lower social welfare as well. The results depend on the cost of quality parameter (c). When the cost of quality is very small, the monopoly quality of the proprietary rm is quite high and a boundary solution is obtained where the monopolist covers the market. Thus competition from low quality open source does not alter the optimal quality and instead, he chooses to lower price. When the cost of quality is very high, the proprietary software vendor o ers a lower quality software with a lower price. When the cost of quality is moderate, the proprietary software vendor has a choice between increasing quality to di erentiate itself from open source software or reducing prices to compete with open source software. It turns out that that loss of revenue from a price reduction is large relative to the cost of increasing quality to increase di erentiation and the proprietary rm choose to increase its quality. Our results must be understood within the limitations of this model. First, we assume that open source software is a non-strategic player. This is consistent with the approach of Casadesus-Masanell and Ghemawat [1]. Raghunathan et al [9] model the complex development process of open source software, thus endogenizing open source quality. They nd that open source quality is independent of proprietary rm s quality even when they compete against each other. There are many interesting extensions to the work presented in this paper for example comparisons can be made with additional benchmarks such as the case of duopoly competition between proprietary software vendors. Other extensions can examine sequential games where the proprietary rm is an incumbent and open source software is the entrant. 15

17 References [1] Casadesus-Masanell, R. and P. Ghemawat Dynamic mixed duopoly: A model motivated by Linux vs. Windows, Management Science, 52(7), [2] CIO Open Source Gains Momentum. [3] Internetnews Microsoft Alters Licensing Plan, Drops O ce XP Price, [4] Economides, N., E. Katsamakas Two-Sided Competition of Proprietary vs. Open Source Technology Platforms and the Implications for the Software Industry, Management Science, 52(7), [5] Keeney, R. L Multiplicative utility functions, Operations Research, 22(1), [6] Krogh, v. G., E. v. Hippel. 2006, The promise of research on open source software, Management Science, 52(7), [7] Lerner, J., J. Tirole Some simple economics of open source, Journal of Industrial Economics, 50(2) [8] Meng, Z., S. T. Lee Open source vs. proprietary software: competition and compatibility, Working Paper.. [9] Raghunathan, S., A. Prasad, B. K. Mishra, and H. Chang Open source versus closed source: software quality in monopoly and competitive markets, IEEE Trans. Systems, Man, Cybernetics: Part A: Systems and Humans. 35(6) [10] Raymond, E. S The Cathedral and the Bazaar, [11] Roberts, J. A., Hann, I., and Slaughter, S. A Understanding the motivations, participation, and performance of open source software developers: A longitudinal Study of the Apache projects, Management Science, 52(7), [12] Stanford Business Case MySQL open source database in [13] Tapscott, D. and A.D. Williams, Wikinomics: How mass collaboration changes everything, Portfolio Hardcover (Penguin Group), New York, NY. The Appendix is attached as a separate le 16

18 8 Appendix 8.1 Proof of Proposition 1 Proof. We compute rst m =@p m m =@q m m = 1 2p m + q m m = p m 2cq m We solve the rst conditions to obtain a unique interior solution. This yields ^x i = 2c 1 4c 1. We check the constraints 0 x i 1 and nd that this solution is feasible when c 1=2. Second order conditions require that c > 1=4, which is weaker than the condition required for feasible interior solution (c 1=2). When c < 1=2, then the cost of quality for proprietary is low and the resulting quality is so high that proprietary covers the market. Thus x i = 0 and applying eq. 2, this yields p m = q m. Now the pro t function is reduced to m (p m ; q m ) = 1 q m cq 2 m (6) Solving rst order conditions (@ m =@q m = 0), we obtain the boundary solution as shown in proposition. 8.2 Proof of Proposition 2 Proof. Case 1: Interior Solution (P and O share the market) Vendor P s pro t function is d (p d ; q d ) = (1 ^x)p d cqd 2, where ^x = (p d + q o q d )=(1 v o ). Solving the rst-order d (p d ; q d )=@p d = 0 d (p d ; q d )=@q d = 0 yeilds p d = 2cv (v q o) = (4cv 1) q d = (v q o) = (4cv 1) where v = 1 v o > 0. It follows d = c (v q o) 2 = (4cv 1) and d d = 2cv (v q o ) = (4cv 2 d =@p 2 2 d =@p d The second derivative test requires that H () = > 0 2 d =@p 2 d =@qd 2 d =@p 2 d < 0. Solving these inequalities yeilds 4cv 1 > 0. The solution is meaningful only when qd > 0, p d > 0 and ^x 2 (0; 1). These inequalities together with 17

19 4cv 1 > 0 lead to v > q o and c > 1= [2 (v + q o )]. Case 2: Boundary Solution 1 (O is competed out of the market) In case 2, the demand for P is exactly 1, the size of the whole market. Solving ^x = (p d + q o q d )=(1 v o ) = 0 yeilds p d = q d q o. p d < q d q o is not an optimal strategy because the demand for P is still 1 even though a lower price is charged to consumers. p d > q d q o is impossible because O would not be competed out of the market. Inserting p d = q d q o in d (p d ; q d ) = p d cqd 2 and solving the rst order d(p d ; q d)=@q d = 0 yeilds p d = 1= (2c) q o; qd = 1= (2c) So d = 1= (4c) q o. Now, consider the conditions for the boundary solution. (1) v > q o and c 1= [2 (v + q o )]. As discussed in case 1, there is no interior solution in this region. That is, either P or O would be competed out of the market. It is straightforward to verify that d = 1= (4c) q o > 0 always holds. Thus, O is competed out of the market. (2) v q o. The boundary solution 1 is meaningful when d = 1= (4c) q o 0, which implies that c 1= (4q o ). Case 3: Boundary Solution 2 (P is competed out of the market) When v q o and c > 1= (4q o ), P is competed out of the market. There are two reasons: (1) The analysis of case 1 shows that P s optimal strategy is not an interior solution, so either P or O is competed out of the market, and (2) The analysis of case 2 shows that the maximum possible d is less than zero. Therefore, P is competed out of the market. 8.3 Impact of the Open Source Competition on Proprietary Software Price Proof. Let p m be the optimal price of P in monopoly, and p d be the optimal price of P in duopoly. Then 8 >< p p m1 = 2c= (4c 1) if c 1=2 m = >: p m2 = 1= (2c) if c 1=2 18

20 8 >< p d = p d1 = 2cv (v q o ) = (4cv 1) if c > 1= [2 (v + q o )] ; >: p d2 = 1= (2c) q o if c 1= [2 (v + q o )] ; 8 >< p d = p d3 = 0 if c > 1= (4q o ) ; >: p d2 = 1= (2c) q o if c 1= (4q o ) ; and and v > q o v q o Case 1: v > q o Subcase 1.1: c > max (1=2; 1= [2 (v + q o )]) In this subcase, p d = p d1, and p m = p m1. Then, p m p d = p m1 p d1 = 2c(a 1c+a o) (4c 1)(4cv 1), where a 1 = 4v (q o + v o ) > 0 and a 0 = v 2 v q o 1. Note that c > 1= [2 (v + q o )], we have a 1 c + a o > a 1 = [2 (v + q o )] + a o = (v q o ) q o v + v 2 o = (v + qo ) > 0. Thus, p m > p d (note that 4cv 1 > 0 holds because c > 1= [2 (v + q o )] > 1= (4v)). Subcase 1.2: 1= [2(v + q o )] < c 1=2 In this subcase, p d = p d1, and p m = p m2. Then, p m p d = p m2 p d1 = 4c2 vq o (1 2cv) 2 2c(4cv 1), which is a linear function of q o. And the coe cient of q o is positive. q o > 1= (2c) v because c > 1= [2 (v + q o )] is satis ed. Thus, p m p d > p m p d j q o=1=(2c) v = 1= (2c) v > 0. The last inequality holds because c 1=2 < 1= (2v) in this subcase. Subcase 1.3: 1=2 < c 1= [2(v + q o )] In this subcase, p d = p d2, and p m = p m1. p m p d = p m1 p d2 = q o + (2c 1) 2 = [2c (4c 1)] > 0. Subcase 1.4: c min (1=2; 1= [2(v + q o )]) In this subcase, p d = p d2, and p m = p m2. p m p d = p m2 p d2 = q o > 0. Case 2: v q o Apparently, if c > 1= (4q o ), then p m > p d. If c 1= (4q o), then p d = p d2 = 1= (2c) q o. Using the result in subcase 1.3 and subcase 1.4, we may conclude that p m > p d. 8.4 Proof of Proposition 3: Impact of the Open Source Competition on Proprietary Software Quality Proof. 8 >< q qm m1 = 1=(4c 1) if c 1=2 = >: q m2 = 1= (2c) if c 1=2 19

21 8 >< qd = q d1 = (v q o ) = (4cv 1) if c > 1= [2 (v + q o )] ; >: q d2 = 1= (2c) if c 1= [2 (v + q o )] ; 8 >< qd = q d3 = 0 if c > 1= (4q o ) ; >: q d2 = 1= (2c) if c 1= (4q o ) ; Case 1: v > q o (or v o + q o < 1) Subcase 1.1: c > max (1=2; 1= [2 (v + q o )]) and and v > q o v q o In this subcase, q m = q m1 and q d = q d1. q d q m = vo+qo 4cqo (4c 1)(4cv 1). Thus, q d qm > 0 when c < (q o + v o ) = (4q o ). Further, we have (q o + v o ) = (4q o ) 1= [2 (v + q o )] = (v q o ) (v o q o ) = [(4c 1) (4cv 1)] and (q o + v o ) = (4q o ) 1=2 = (v o q o ) = (4q o ). It shows that (q o + v o ) = (4q o ) > max (1=2; 1= [2 (v + q o )]) when v o > q o. And v o > q o implies that 1= [2 (v + q o )] > 1=2. If v o q o, then c < (q o + v o ) = (4q o ) is impossible in subcase 1.1. Subcase 1.2: 1= [2 (v + q o )] < c 1=2 In this subcase, q m = q m2 and q d = q d1. q m q d = [2c (v + q o) 1] = [2c (4cv 1)] > 0. Subcase 1.3:1=2 < c 1= [2 (v + q o )] In this subcase, q m = q m1 and q d = q d2. q d q m = [2c 1] = [2c (4c 1)] > 0. And the condition 1=2 < 1= [2 (v + q o )] implies that v o > q o. Subcase 1.4:c min (1=2; 1= [2 (v + q o )]) In this subcase, q m = q m2 and q d = q d2. q m q d = 0. Case 2: v q o (or v o + q o 1) Subcase 2.1: c > max (1=2; 1= (4q o )), then q m = q m1 > 0 = q d. Subcase 2.2: 1= (4q o ) < c 1=2, then q m = q m2 > 0 = q d. Subcase 2.3: 1=2 < c 1= (4q o ), then q d q m = q d2 q m1 = [2c 1] = [2c (4c 1)] > 0. The condition 1=2 < 1= (4q o ) implies that q o < 1=2. Subcase 2.4: c min (1=2; 1= (4q o )), then q m q d = 0. To summarize, q d > q m when the following condition holds. v o + q o < 1 and v o > q o and 1= [2 (v + q o )] < c < (q o + v o ) = (4q o ) ; v o + q o < 1 and v o > q o and 1=2 < c 1= [2 (v + q o )] or or v o + q o 1 and q o < 1=2 and 1=2 < c 1= (4q o ) : 20

22 The above inequalities can be rewritten as v o + q o < 1 and v o > q o and 1=2 < c (q o + v o ) = (4q o ) or v o + q o 1 and q o < 1=2 and 1=2 < c 1= (4q o ) : q o v o = q o v = 1 q o o q = 1/2 o v o q < v and q + v < 1 q < 1/ 2 and q + v 1 o o o o o o o Figure A1: The (q o ; v o ) region for q d > q m Note that min [1; (q o + v o )] = 1 when v o + q o 1, and min [1; (q o + v o )] = q o + v o when v o + q o < 1, the above inequalities can be furthre simpli ed to q o < min (v o ; 1=2) and 1=2 < c < min [1; (q o + v o )] = (4q o ) q d = q m when the following condition holds. q o + v o < 1 and c min (1=2; 1= [2 (v + q o )]) ; or q o + v o 1 and c min (1=2; 1= (4q o )) : Note that 1= [2 (v + q o )] < 1= (4q o ) when q o +v o < 1, and 1= (4q o ) 1= [2 (v + q o )] when q o +v o 1, the above inequalities can be furthre simpli ed to c min (1=2; 1= [2 (v + q o )] ; 1= (4q o )). 21

23 8.5 Impact of the Open Source Competition on Consumer Surplus Proof. Denote CS m the consumer surplus for monopoly, and CS d the consumer surplus for duopoly. Using the results in Lemma 1, we have 8 >< CS m1 = 2c 2 = (1 4c) 2 if c 1=2 CS m = >: CS m2 = 1=2 if c 1=2 Using the results in Proposition 2, we have 8 >< CS d = >: 8 >< CS d = >: CS d1 = vo 2 + q o + (p d1) 2 2(1 v o) if c > 1= [2(v + q o )] ; CS d2 = 1=2 + q o if c 1= [2(v + q o )] ; CS d3 = v o =2 + q o if c 1= (4q o ) ; CS d2 = 1=2 + q o if c < 1= (4q o ) ; and and v > q o v q o where p d1 = 2cv (v q o ) = (4cv 1). Case 1: v > q o Subcase 1.1: c > max (1=2; 1= [2(v + q o )]) In this subcase, CS m = CS m1, and CS d = CS d1 = R ^x d 0 (v o x + q o ) dx + R 1 ^x d (x + q d p d ) dx = v o2 + q o + (p d) 2 2(1 v o). First, R ^x d 0 (v o x + q o ) dx > R ^x d 0 (x + q d p d ) dx. This is because x-type consumers with x 2 [0; ^x d ) use O rather than P, so v o x+q o > (x + q d p d ) in this area. Then it follows CS d > R 1 0 (x + q d p d ) dx. Second, CS m = CS m1 = R 1 ^x m (x + q m p m ) dx R 1 0 (x + q m p m ) dx. Apparently, if we can prove that q d p d q m p m, then CS d > CS m is satis ed. (q d p d ) (q m p m ) = (q d1 p d1 ) (q m1 p m1 ) = 1 2 is a linear function of q o. h i (4c 1)(4cv 1) v o + 2cv 1 4cv 1 q o = k 0 +k 1 q o First, k 0 0 because c > 1=2, c > 1= [2(1 + q o v o )] and v > q o. If k 1 0, then k o + k 1 q o 0. If k 1 < 0, then k o + k 1 q o > k o + k 1 (1 v o ) = 2c 1 4c 1 > 0. Therefore, q d p d q m p m and thus CS d > CS m. Subcase 1.2:1= [2(v + q o )] < c 1=2 In this subcase, CS m = CS m2 = 1=2, and CS d = CS d1. So CS d CS m = CS d1 1=2 = s 2 q 2 o +s 1q o+s 0 2(4cv 1) 2, where s 2 = 4c 2 v > 0, s 1 = 2 (1 2cv) (6cv 1), and s 0 = v (1 2cv) (6cv 1). 22

24 When q o = (1 2cv) (6cv 1) = 4c 2 v, s 2 q 2 o + s 1 q o + s 0 achieves a minimum, which is (1 2cv) (6cv 1) (4cv 1) 2 = 4c 2 v 0. This minimum point is no less than zero because (1) 1 2cv 0, noting that c 1=2 1= (2v), (2) 6cv 1 > 0, noting that v > q o leads to c > 1= [2(v + q o )] > 1= (4v) > 1= (6v). The equality holds when c = 1=2 and v = 1. Thus, s 2 qo 2 + s 1 q o + s 0 > 0. The strict inequality holds because when c = 1=2 and v = 1, then s 2 qo 2 + s 1 q o + s 0 = qo 2 > 0. It follows that CS d > CS m. Subcase 1.3:1=2 < c 1= [2(v + q o )] In this subcase, CS m = CS m1 = 2c 2 = (1 4c) 2, and CS d = CS d2 = 1=2 + q o. It can be shown that CS d > 1=2 > 2c 2 = (1 4c) 2 = CS m. Subcase 1.4:c min (1=2; 1= [2(v + q o )]) In this subcase, CS m = CS m2 = 1=2, and CS d = CS d2 = 1=2 + q o. Apparently, CS d > CS m. Case 2: v q o In this case, CS d min (v o =2 + q o ; 1=2 + q o ) > 1=2 max 1=2; 2c 2 = (1 4c) 2 j c>1=2 CS m. 8.6 Proof of Proposition 4 Proof. First, we prove the following Lemma. Lemma 1: If q o 1=2, then SW d < SW m at some c. If v o > 0 and q o! 0, then SW d SW m holds regardless of c. Proof: Denote SW m the social welfare for monopoly, and SW d the social welfare for duopoly. Using the results in Lemma 1, we have 8 >< SW m1 = c (6c 1) = (1 4c) 2 if c 1=2 SW m = >: SW m2 = 1=2 + 1= (4c) if c 1=2 23

25 Using the results in Proposition 2, we have 8 >< SW d = >: 8 >< SW d = >: SW d1 = CS d1 + d1 if c > 1= [2(v + q o )] ; SW d2 = 1=2 + 1= (4c) if c 1= [2(v + q o )] ; SW d3 = v o =2 + q o if c > 1= (4q o ) ; SW d2 = 1=2 + 1= (4c) if c 1= (4q o ) ; and and v > q o v q o where CS d1 is de ned above, and d1 = c (v q o ) 2 = (4cv 1). It can be veri ed that SW d1 = v o2 + q o + c (6cv 1) qd1 2 and SW m1 = c (6c 1) qm1 2. If q o 1=2, then there are two possible cases. Case A1: q o < v. We claim that 1= [2 (q o + v)] < 1=2 holds. If not, then q o + v 1, which implies q o v o. Using three conditions q o v o, q o < 1 v o, and 0 v o < 1, we have q o < 1=2, violating the assumption q o 1=2. Given 1= [2 (q o + v)] < 1=2, we claim that SW d SW m = SW d1 SW m2 < 0 at c 2 (1= [2 (q o + v)] ; 1=2). Let = q o =v and = 1= (cv). That is, q o = v, c = 1= ( v). Since q o < v, we have < 1. Since 1= [2 (q o + v)] < c < 1=2, we have 1= [2v (1 + )] < 1= ( v) < 1=2, or 2 2=v < < 2 (1 + ) < 4. Inserting q o = v and c = 1= ( v) in SW d1 SW m2 yeilds SW d1 SW m2 = h v [2 (1 + ) ] h (; ) = 4 (4 ) 2i, where h (; ) = 4 2 (6 ) + (4 ) is a linear function of. It can be veri ed that h (0; ) = 4 + (4 ) > 0, and h (1; ) = (4 ) ( 2) > 0. Thus, SW d1 SW m2 < 0. Case A2: q o v. If q o > 1=2, then let c = 1= (4q o ) + " ("! 0, " > 0), we have SW d SW m = SW d3 SW m2 = v 2 < 0. If q o = 1=2, then let c = 1=2+" ("! 0, " > 0), we have SW d SW m = SW d3 SW m1 = v 2 < 0. Now, consider the case v o > 0 and q o! 0. 0 < v o < 1 leads to 0 < v < 1. We can always choose a q o! 0 such that v + q o < 1 and q o < v. Then 1= [2(v + q o )] > 1=2. It is straightforwad that SW d SW m = 0 at c 1= [2(v + q o )]. When c > 1= [2(v + q o )], SW = SW d SW m = SW d1 SW m1. Again, let 24