A Dynamic Model of Mixed Duopolistic Competition: Open Source vs. Proprietary Innovation

Transcription

1 A Dynamic Model of Mixed Duopolistic Competition: Open Source vs. Proprietary Innovation Suat Akbulut Murat Yılmaz August 015 Abstract Open source softare development has been an interesting investment model of production and innovation in recent years, especially in the last fe decades. Unlike the private investment model, open source innovators freely share the proprietary softare that they have developed at their private expense. Also, open source development is usually subject to certain licenses, one of hich being the General Public License GPL), the most popular open source license. In this thesis, e study the competition dynamics beteen a proprietary firm and an open source firm, Windos and Linux, for instance. We model the the competition beteen such to firms by incorporating the nature of the GPL, investment opportunities by the proprietary firm, user-developers ho can invest in the open source development, and a ladder type technology. We use a to period dynamic mixed duopoly model, in hich a profit- maximizing proprietary firm competes ith a rival, the open source firm, hich prices the product at zero, ith the quality levels determining Marie Curie Career Integration Grant, FP7PEOPLE-011CIG is acknoledged. Boğaziçi University, Department of Economics, Bebek, Istanbul, Boğaziçi University, Department of Economics, Bebek, Istanbul, muraty@boun.edu.tr. Phone: Fax: Web site: 1

2 their relative positions over time. We ask the folloing questions. Ho does the existence of open source firm affect the investment and the pricing behavior of the proprietary firm? Does the social elfare increase ith the existence of the open source development? Ho are the users/consumers affected from the open source firm being available? Is it possible that an open source firm, hich is behind the proprietary firm in the technology ladder, catches up ith the proprietary firm? We also discuss the limitations of our model and possible extensions.

3 1 Introduction A softare is called open source, if its source code is open in the sense that anyone has free access to it. Open Source movement aims to bring programmers not concerned ith proprietary onership or any financial gain together to produce a more useful and bug-free product for everyone to use. By revealing its source code, an open source can be refined by many independent developers all around the orld. The source code of an open source product is made available free of charge to the public. So, developers on the Internet read, redistribute and modify the source code, forcing an advantageous evolution of it. Although there are many licenses used to distribute an open source projects, is GU General Public License GPL)the most commonly used one as of late 014, by a share above 51% 1. GPL has to main features. The first feature that GPL has is that although every user has the right to use and modify the the code freely, the modifications must be distributed under the terms of the same license, if they are to be distributed at all. That is to say, GPL is a copy-lefted license. Although the rationale behind the open source movement is that larger group of programmers ho are not interested in the onership produces a better, faster, more useful and, bug-free softare, the second aspect of GPL allos the commercial exploitation of the program. Hence, the users have to sustain the free access to the source code, yet, as long as they maintain the free access, they are alloed to make profit. For example, according to its 015 income statement, Red Hat, the orld s largest commercial distributor of the Linux operating system, made a total net income of $ million in 014. Open source softare OSS), particularly Linux, has gained significant impact on softare industry, thereby it has attracted noticeable interest of the researchers, as ell. Having observed the provision of OSS and its ongoing developments are costly, and moreover it is almost alays publicly available at a price of zero,hich does not reflect the 1 See 3

4 economic costs of user-developers incur to enhance it, economists have tried to understand the motives that might encourage the user-developers to involve in such a costly activity. The success of open source softare has led a literature on it, hich has been flourishing since early 000s. Lerner and Tirole 00, 005) introduce a broad sense discussion on economics of open source development. They indicate to reasons that might lead the developers to contribute to open source evolution. First reason that might make developers involve in this costly activity is that they receive a direct benefit in the form of improved softare. Secondly, they get an indirect benefit by signaling their abilities in the job market. They also point out that the literature mostly considers individual incentives to adopt open source softare. Contributing to open source innovations brings the public good nature onto the surface. A considerable amount of literature focuses on open source development as public good in a static approach. Johnson 00) uses public good approach in a static environment, here private provisions of user-developers to a public good -the open sourcediminishes as the number of user-developers increases because of free riding problem, and presents some comparative statistics and elfare results. Modica 01) takes a to period oligopoly game using a circular city approach in order to model the open source innovations from a public good perspective. Some of the open source literature focuses on the competition beteen proprietary firm and open source firm. Casadesus-Masanell and Ghemaat 006) study the competition beteen proprietary firm and open source firm in a dynamic mixed duopolostic industry ith the demand side learning, and sho that it is better to have the proprietary firm as a monopoly hen the total elfare is considered. Casadesus-Masanell and Llanes 011) use a mixed duopoly structure, here a for-profit proprietary firm competes ith an open source firm, hich tries to maximize the value of its open softare. Our model differs from 4

5 these studies in the ay that it combines the open source innovation and the competition beteen proprietary firm and open source firm in a dynamic environment. This study tries to examine the effects of the existence of an open source firm that is competing ith the proprietary firm on the proprietary firm s investment in innovation and production behavior, and ho it affects the total elfare in the market. We set up a dynamic model ith to periods, the first of hich has to stages: competition and investment. In the second period, being the last period of the model, there is no investment stage. In the to competition stages, proprietary firm and open source firm compete in a mixed duopolistic industry, here the former charges a price to maximize its overall expected profit, hereas, the latter is freely available. At the beginning of each period, a ne cohort of potential users enter into the model. At the beginning of the competition stage, they observe the quality levels and the price of proprietary firm s product, and they decide hich operating system to use during their life time of one period. As long as they have some valuation for the open source, everyone ill have an operating system, at least the free open source. In the next stage of the first period, the investment stage, hile proprietary firm invests in probability to increase its products quality level, user-developers incentives for involving this costly development activity is to signal their abilities. We find that under some circumstances, the proprietary firm supplies less and invests more in the presence of the open source rival, hich leads the proprietary firm to make less profit in the duopolistic industry compared to its monopoly, suggesting that a duopoly is likely to dominate the proprietary firm s monopoly in terms of total elfare generation. Hoever, this is not alays true, i.e it might be better for the total elfare hen there is only proprietary firm in the market. 5

6 The Model In order to model such a competition, here the competitors have heterogeneous objectives, e ill make use of the literature on mixed duopolies. Throughout this study, and l ill stand for Windos and Linux, respectively. There are to periods, the first of hich has to stages: competition and investment. There is only one stage in the second period, hich is competition stage. At the beginning of period t, for t = 1,, the quality level of an operating system OS) s {, l}, is denoted as kt s Z +. Although the initial quality levels k1 and k1 l ill be given, their levels at the beginning of the second period, k and k l, is determined endogenously by the investment decisions of both Windos and user-developers -Linux users. The evolution of quality levels follo a ladder type technology. For this reason, in second stage of the first period, investment stage, Windos invests in probability in order to climb one step on the technology ladder, hile, user developers involve in this costly development activity to signal their abilities to the job market. The realizations of the developments occur at the end of the period. Hence, if Windos invests i [0, 1], its quality level at the beginning of the second period ill be: k k ith prob. i = k1 ith the remaining prob. 1 i ) During the same stage, those user-developers, ho decided to get a free copy of Linux in the previous stage, simultaneously ith Windos investment decision, decide hether and ho much to invest in probability to get the exogenous bonus b 0, 1). Let i j be the user-developer j s investment level. Assuming the open source does not have the modular nature, if at least one user developer succeeds in development stage, because of the terms If a softare has modular nature, then it is possible to break up the large projects, such as developing it, into small modules. Improving the modules independently ill accomplish the project. Hoever, it 6

7 of GPL, Linux ill move up one step in the technology ladder. Hence, then Linux quality level at the beginning of the second period ill be: k k l 1 l + 1 = k l 1, if at least one user-developer succeeds, if no user-developers succeed Hoever, investment is a costly activity for both Windos and user-developers. In order to invest i, they must incur a cost of 1 i. Here e assume that users-developers skills are homogeneous. Yet, in Section 6.3, e introduce heterogeneity among user-developers in their development skills. To specify the demand side, in each period, a ne cohort of potential users enter into the market. They observe the quality levels of both proprietary softare and open source. Let k t denote the quality differences beteen Windos and Linux, thus k t = k t α s k t ) > 0 denote the OS s s value given by the cohort entering at time t. k l t. Let α s k t ) is called as OS s s technological trajectory, hich is a function of the quality level of OS s, kt s, and the quality level of the competing OS, kt s. Even though the initial levels of these technological trajectories are exogenously given in the model, ho they evolve hile moving from period 1 to period is endogenous. In the beginning of the first stage of period 1, and period, Windos charges a price of P t, here t {1, }, to attract ne customers. Let q t be the number of users in period t, ho buy Windos, then q t is the number of user-developers in the same period since Linux is freely available and α l ) is positive. Assumption 1 We assume linear demand function. In period τ, for tau = 1, : the softare does not have the modular nature, then its development cannot be divided into pieces, and consequently, there may not be sufficient sophisticated user-developers ho can customize the softare to their on needs. Section 6.3 tries to deal ith the question hen the open source softare has the modular nature by introducing a contribution dimension to the model. You can see Lerner and Tirole 00) for details about modules. 7

8 Windos value to a user q τ {1,,..., } is: α k τ ) q τ 1) and, similarly, let the value of Linux be: α l k τ ) q τ Figure 1 illustrates the Assumption 1. The demand is dran as a straight line as if as a real number, but the readers are ell aare of that it must be a set of points on that line instead. Figure 1: Demand of operating system s by cohort τ Assumption α s k t ) 0. Since e assume OS s value is never negative, technological trajectories are to be bounded belo, hich is required to have ell defined demand functions. 8

9 Assumption 3 α k t ) is increasing in k t, hereas, as k t increases, α l k t ) decreases. Holding the competent OS quality level constant, α j ), here j i, the value of OS i, α i ), ill increase as its quality level increases. Let β be defined as βk t ) = α k t ) α l k t ), representing the difference beteen the trajectories of and l. Assumption 4 βk t ) is assumed to be concave in k t. Assumption 4 is needed to be assumed to have a ell defined maximization problem for Windos. It ensures that the difference beteen the technological trajectories of Windos and Linux ould not explode. Otherise, after some level of the quality difference, Windos ould become a monopoly-like firm in the market. 3 Monopoly In a market, here there is no substitute for Windos, and every user of any cohort has positive illingness to pay, inverse demand function is directly obtained by Equation 1. Figure illustrates the timing of the events in a monopoly industry. 9

10 Figure : Timing of events in monopoly benchmark To solve the equilibrium of monopolistic structure, e employ the Backard Induction methodology in this to-period game. 3.1 Second Period Since Windos is the only operating system producer and this is last period of the game, having monopoly poer, it produces the profit maximizing amount of /, and sets its price to α k )/ in accordance ith the demand structure. As a result, it generates a profit of: π mon k ) = 4 α k ) ) 3. Second Stage of The First Period Ho much investment is optimal for Windos? Investment affects Windos profit through the quality level, hich determines the valuations of the users, ith trajectory function. An investment level i ill increase its quality level by 1 ith the probability i. Having knon it ill generate a profit of π mon k ) in the next period by Equation, the optimal monopoly investment strategy for Windos require it to choose an investment level, i mon be in the folloing set: i mon { argmax i i 4 α k1 + 1) + 1 i ) } 4 α k1 ) Since the above term is linear in i, the monopoly investment level ill be: i mon = 4 [α k 1 + 1) α k 1 )], if α k 1 + 1) α k 1 ) 4 1, otherise 3) 10

11 3.3 First Stage of The First Period At the beginning of the game, knoing its optimal strategies for the second phase and the next period, Windos chooses a price level, P mon 1 or, equivalently, quantity level, q mon 1 ) that maximizes its folloing overall expected profit: { max α k1 ) q 1) q 1 1 q 1 i ) + i 4 α k1 + 1) 1 i ) } 4 α k1 ) When e take the first order derivative ith respect to q 1, e obtain the folloing optimality condition since second order condition holds: 0 = α k 1 ) q 1) q mon 1 = and P mon 1 = α k 1 ) Hence, Windos produces is used by one half of the number of users in each period, and the users should pay a price that is equal to the one half of the maximum value given to itself by their cohort. 4 Duopoly When e introduce Linux into the market, Windos no longer has its monopoly poer. It has to consider the presence of Linux and the user-developers investment decisions hile deciding ho much to produce in each period, and to invest in the investment stage. The timing of events in duopoly industry is described in Figure 3. Figure 3: Timing of events in duopoly industry 11

12 To solve duopoly equilibrium, e use the same methodology, backard induction technique, as e utilize in the monopoly case. Since Linux can be donloaded freely and α l ) > 0, i.e. all users are illing to pay something even only a small amount) for a product that they can get ithout paying anything, it is guaranteed that every user ill get one operating system, at least Linux. Hence, if q τ is the number of users ho buy Windos, then the remaining users of cohort τ, q τ, obtain the Linux at no price, and they become user-developers. 4.1 Second Period When Windos is sold at price P duo at period, the indifferent user beteen Windos and Linux, q, is found by the folloing equation: α k ) qduo P duo = α l k ) qduo Remembering βk t ) = α k t ) α l k t ), the inverse demand function for Windos in period ould be the folloing = βk ) qduo P duo 4) 1

13 As a rational profit maximizer agent, Windos produces q duo such that { q duo argmax βk ) q ) } q q Taking the first order derivative ith respect to q, e get quantity and price levels for the second period as follos: 0 = βk ) q ) q duo = and P duo = βk ) As a result, Windos ill make a profit of π duo = P duo q duo = 4 βk ) in the second period. 4. Second Stage of The First Period In the investment stage, the actions of the user-developers have impact on Windos objectives. Hoever, Windos investment decision does not affect the user-developers investment strategies since they invest only for the purpose of getting the bonus, b. There ould be some other version of the model, in hich Windos and user-developers decisions affect both their investment strategies. Hoever, Section 6 ill be discussing the issues that such models could create and ho these issues could be handled. For no, e ill analyze the investment decisions of user-developers and Windos separately User-Developers Investment Decisions We have already mentioned that user-developers are only interested in the bonus, b, hen deciding hether and ho much to invest to Linux development. Hence, a user-developer 13

14 j, for j {q duo 1, q duo 1 + 1),..., }, chooses and investment level i j, hich maximizes her expected net benefit. Hence, i j solves the folloing maximization problem max {i j b 1 } i j i j [0,1] In order to find the optimal solution for the problem above, e take the first order derivative of it ith respect to i j. Hence, b i j = 0 i j = b for j = q duo 1, q duo 1 + 1),..., Since her expected net benefit, b, is positive, in the equilibrium, she ill choose to invest i j = b. Due to the symmetry, i j = b for all j {q 1 + 1,..., }. So, Linux ill be developed ith probability 1 1 b) qduo 1, hich is the probability that at least one user-developer succeeds. 4.. Windos Investment Decision In contrast to investment decision of user-developers, Windos takes into account hat the user-developers strategy is. Thus, it chooses the investment level i duo maximizes its expected future profit π duo k ). max i i 1 b) q 1) βk 1 + 1) b) q 1) ) βk 1 ) ) +1 i ) 1 b) q 1) βk 1 ) b) q 1) ) βk 1 1) ) 1 i so that i duo By taking the first order derivative ith respect to i, e obtain i duo = 4 [ ] 1 b) qduo 1 βk 1 + 1) + βk 1 1) βk 1 )) + βk 1 ) βk 1 1)) 14

15 or in a more proper ay i duo [ ] ) = min 1 b) qduo 1 C +, 1 4 here C = βk 1 + 1) + βk 1 1) βk 1 ) and = βk 1 ) βk 1 1). 4.3 First Stage of The First Period In the competition stage of the first period, in order for the user q 1 to be indifferent beteen Windos and Linux, her net benefit from buying Windos and donloading a free copy of Linux must be equal. In the equilibrium, choosing to get a free copy of Linux ensures a user to get an expected benefit of b in the investment stage. So, hen Windos is sold at price P duo 1 at period, the indifferent user beteen Windos and Linux, q 1, is found by the folloing equation: α k 1 ) q 1 P 1 = α l k 1 ) q 1 + b Hence, the inverse demand for Windos in period 1 ould look like the folloing 1 = βk 1 ) qduo 1 P duo b 5) The optimal pricing/quantity strategy for Windos must be a solution of the folloing maximization problem, hich simply is Windos overall expected profit hen it chooses 15

16 to produce q 1. max q 1 P 1 q 1 1 i ) + i 1 b) q 1) 4 βk 1 + 1) + i 1 1 b) q 1) ) 4 βk 1) + 1 i ) 1 b) q 1) 4 βk 1) + 1 i ) 1 1 b) q 1) ) 4 βk 1 1) The first order condition of above maximization problem is βk 1 ) qduo 1 + βk 1) b 16 C 1 b) qduo 1 ) ln1 b) 1 b) qduo 1 ln1 b) 1 + C) 4 4 Although this condition does not have a closed form analytical solution in q duo 1, e are capable of comparing it ith the first period quantity in the monopoly case, q mon 1 since e kno that the overall expected profit function is concave in q 1, and maximized at q duo 1. When e evaluate the above first order condition at q mon 1 = /, e obtain the folloing function: fb) = b C 16 1 b) ln1 b) 4 1 b)/ ln1 b) ) C Proposition 1 For large enough bonus b, proprietary firm produces less in the first period of the duopolistic competition as opposed to the case in hich it is a monopoly. Proof. fb) is continuous in [0, 1). f0) = 0, and lim b 1 f1 b) < 0. Therefore, ˆb [0, 1) such that fb) < 0, b [ˆb, 1). Because the first order condition is negative at point q 1 = / for large b s and, the overall expected profit function is concave in q 1, q duo 1 < q mon 1 = 6) 16

17 Proposition 1 shos that the existence of an open source rival affects the monopoly production as existence of any other rivalry for-profit firm, in the sense that the monopoly firm decreases its production level. Proposition Proprietary firm makes more investment in the duopoly industry competition as opposed to the case here it is a monopoly. Proof. Let 1 b) q 1) = x. Observe that x 0, 1). Since β ) is concave, C = βk 1 + 1) + βk 1 1) βk 1 ) is negative, and = βk 1 ) βk 1 1) is positive. Hence, 4 x 1) βk 1 + 1) + βk 1 1) βk 1 )) > 0 4 x C + βk 1 + 1) βk 1 )) ) > 0 4 x C + ) > 4 βk 1 + 1) βk 1 )) i duo > i mon As Proposition suggests, competition results in Windos to increase its investment level. Hoever, this result is not special to having a open source rival. 5 Welfare Comparison Proposition 1 & concludes that the proprietary firm makes less profit in the duopoly industry, hich suggests that a duopoly is likely to dominate proprietary firm s monopoly in terms of total elfare generation. In this section, e analyze the elfare implications of the to industry structure that e studied above. Instead of finding the absolute level of total elfare in the duopoly industry, e ill compare the total elfare levels under the assumptions that α ) and α l ) are linear ith slope γ and γ l, respectively. α ) 17

18 and α l ) being linear ith slope γ and γ l causes β ) to be a linear function, as ell, ith slope γ γ l, that is, C = 0 and = γ γ l. Assumption 3 ensures that γ > 0 and γ l < 0. Thereby, is positive. Proposition 3 If 4 γ l + γ ) < 1 and 4 γ > 1 b) /, then total elfare is higher in proprietary firm s monopoly than the total elfare in duopoly industry. Proof. To prove Proposition 3, e divide the total elfare into pieces and compare them piece-ise instead of measuring them as holes. And, hen comparing the to elfare levels, e interpret the absence of Linux in the monopoly industry as k l t and α l k t ) being zero. Therefore, βk t ) = α k t ). Expected total elfare in the Windos monopoly, W m is / W m = j=1 α k 1 ) j a mon c mon ) imon ) + i mon / j=1 α k 1 + 1) j ) + 1 i mon / ) { }} ){{}}{{ 3 = α k 1 ) imon ) ) }} ){ i mon α k 1 + 1) + 1 i mon ) α k 1 ) }{{}}{{} first period elfare second period elfare d mon j=1 α k 1 ) j ) Expected elfare in the first period of the duopoly industry, W d : W d 1 = q 1 j=1 α k 1 ) j ) + j=q 1 +1 α l k 1 ) j ) c duo a duo e { }}{ = βk 1 ) q 1 1 ) {}}{{ q 1 q 1 + 1) + α l k 1 ) 1 }} ) { i duo ) i duo 18

19 And the expected total elfare generated in the second period of the duopoly ill be: W d = i 1 b) q 1) +i 1 1 b) ) q 1) +1 i ) 1 b) q 1) +1 i ) 1 1 b) ) q 1) βk1 + 1) ) 3 βk1 ) ) 3 8 βk1 ) ) βk1 1) 3 8 ) ) α l k 1 + 1) +1 α l k 1 ) +1 α l k 1 ) +1 ) ) α l k 1 1) +1 ) Hence, d duo {}}{ W d 3 [ = i duo + 1 b) ) q 1) γ γ l ) + βk 1 1) ] [ i duo + 1 b) ) q 1) γ l + α l k 1 1) ] } {{} f o let us start to compare the pieces marked by loer case letters. For q 1 <, ) d q 1 1 q 1 q 1 +1) > 0 dq 1 Thus, βk 1 ) q1 duo 1 q duo 1 q duo ) 1 + 1) < βk 1 ) 1 + 1) ) = βk 1 ) 3 8 hich implies a duo < a mon 7) As a consequence of Proposition, e have i duo 1 imon ), hich in turn implies > i mon hich implies 1 ) i duo < c duo < c mon 8) 19

20 Since i mon > 1 b) q1) and, i duo cannot be more than 1, 3 8 i duo i duo + 1 b) q 1) 1 < i mon + 1 b) q 1) 1 ) βk 1 ) βk 1 1)) < i mon βk 1 ) βk 1 1)) [ i duo + 1 b) ) q 1) γ γ l ) + βk 1 1) ] < 3 [i mon βk 1 ) βk 1 1)) + βk 1 )] 8 hich implies d duo < d mon 9) Combining e and f e get: e + f < 1 = 1 [ αl k 1 ) i duo + 1 b) q 1) ) γ l α l k 1 1) ] 1 i duo 1 b) q 1) ) γ l ote that 1 i duo 1 b) q 1) ) is positive due to the assumptions 4 γ l + γ ) < 1 and 4 γ > 1 b) /. To see this, note 4 γ l + γ ) < 1 4 γ l + 4 γ + 4 γ < 1 4 γ β + 4 γ < 1 4 γ β + 1 b) / < 1 4 γ β + 1 b) qduo 1 0

21 Thus, the summations of the terms including α l in the duopoly elfare is negative, that is, e + f < 0 10) Combining Equations 7), 8), 9) & 10), e conclude that the total elfare that the monopoly proprietary generates is higher than the total elfare in the duopoly industry. Proposition 3 shos that the competition does not necessarily increase the elfare in an oligopoly industry hen compared to the monopoly market. This is because the presence of a rival induces the proprietary firm to set loer prices and those users ho do not buy the proprietary firm s product are not left empty handed; they can get the open source freely, hich increases the total surplus. Hoever, the decrease in proprietary firm s and its users surpluses do not, alays, need to be compensated by the increase in user-developers surpluses. 6 Discussion This paper is about the alternative models that could have been used to capture the effects of an open source firm s presence on the behavior of a proprietary firm. We also tried to examine changes in our results hen the question is modeled in different ays, and summarized the reasons behind the fact that hy e end up ith not using them. As our future ork, e ill improve the last alternative in order to increase the period number to finitely or even infinitely many because it might be useful to have infinitely many periods in order to study long-run behavior of the to firms and to question the lifespan of proprietary firm, hose faith might be releasing its source codes and becoming an open source, as ell. 1

22 6.1 T 3 Periods When e tried to set up a model, here the number of periods is three or more, or infinitely many, e end up ith technical problems of solving the first order condition of proprietary firm s maximization problem. This problem occurs because there is no analytical solution, for sure, to the number of proprietary users at period t, q t, hen the number of potential users,, exceeds three. Employing the knon methods to solve the Bellman Equation that captures the recursive nature of the dynamic game problem is not helpful since transition matrix that should govern the evolution of the state variables are determined by the choice variables in each period, i.e the transition matrix is not stationary. 6. Endogenous Bonus ith OLG One other possibility could be utilize the first investment incentive for the user-developers that Lerner and Tirole 00) mention, i.e user-developers involve in the development activity because they receive a direct benefit in the form of improved softare. We set up a model, here users lived to periods. They could buy an operating system only hen they are young. User-developers could develop the open source hen they are young, and enjoy the appreciation of its quality hen they are old, if at least one of them succeeds due to General Public License. When e model the user-developers investment incentives in this frameork, ith alloing the investment levels to be in [0, 1] interval, e faced difficulties hile solving the optimal investment levels of user-developers since the optimal decisions include th order equations. To overcome such difficulties, one could think of forcing the possible investment level choices of the user developers to be binary, i.e they ould be either 0 or 1. Hoever, there occurs a free rider problem that Johnson 00) finds, too. Since it is guaranteed for the open source to be improved hen one userdeveloper chooses to invest in 1, it is optimal for every user-developer to let someone else

23 do it. 6.3 Contribution Game ith Infinitely Many Users When Lerner and Tirole 00) explain the favorable characteristics for an open source production, they mention about its modularity, hether the overall project is divided into smaller and ell-defined tasks modules) that individuals can handle independently from other modules. Sufficiently modular nature of an open source softare, hose different portions can be improved by independent user-developers, might turn the investment stage to a contribution game for open source user-developers. To do so, one other helpful ay could be having infinitely many users distributed on [0, 1]. Although in our original model, that ould create some compatibility problems hile finding the open source firm s development probability, since it has a multiplication part, hich is not a good ay to use hen there are infinitely many users, that ould provide a ell defined demand, and is a better ay to model the investment stage as a contribution game, here the probability of open source firms development is affected by a fraction of the measure of user-developers that contribute or all users. Such a model might also capture the direct benefit incentives of the user-developers, hich ould result in having different optimal investment strategies for different user-developers. To incorporate the direct benefit, a successful development of a user-developer could be rearded by enjoying the appreciation of her on operating system before the quality increase become public. 7 Conclusion It is impressive that a costly investment based upon not having the property rights has produced such a useful and reliable softare. In this study, a simple to-period model of open source innovation has been presented to understand the difference of the behavior 3

24 of the proprietary firm s production, pricing and investment strategies and to facilitate elfare comparisons beteen the presence of it and the traditional, profit driven method of development, here the quality levels of the to follo a ladder type technology frameork. It has been shon that the proprietary firm decreases its production level hen there is an open source rival, and in order to better compete ith the open source firm, it invests more. Hoever, that the proprietary firm loosing some of its profit cannot be concluded as a duopoly is likely to dominate the proprietary firm s monopoly in terms of total elfare generation because it has been shon that for some levels of the linear formed technological trajectory functions slopes, the total elfare is higher in proprietary firm s monopoly than the total elfare in duopoly industry. 4

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