Open Source Software Subsidies and Network Compatibility in a Mixed Duopoly

Transcription

1 Open Soure Software Subsidies and Network Compatibility in a Mixed Duopoly Thierry Pénard and Mourad Zeroukhi y niversity of Rennes, CREM - IDEC November 0 First version - For peer review Abstrat For many appliations, Open soure software (OSS) an o er a high-quality alternative to proprietary software (e.g. Linux, Apahe,...). But even if OSS is usually free of harge, its installation and use require some skills. Should the government intervene to promote the di usion of OSS and provide some learning support to potential users? This paper analyses whether publi subsidies towards open soure software is soially desirable and how the extent of ompatibility between open soure software and proprietary software an in uene the amount of subsidies. We onsider a mixed duopoly model with a proprietary software (PS) ompany that ompetes with an open soure software (OSS) ommunity. An important strategi deision is whether to make these software ompatible or inompatible. Four situations are possible: full (two-way) ompatibility, full inompatibility, and one-way inompatibility (either only OSS or only PS is ompatible). sers are heterogeneous in their ability to use OSS, and they are also sensitive to the number of users who are adopting the same software or a ompatible software (network externality). We show that if the government only takes are of onsumer surplus, publi subsidies are welfare-enhaning. But the optimal subsidies are larger with full ompatibility and PS ompatibility than full inompatibility and OSS ompatibility. These results suggest that government poliy towards OSS must be onditional to the degree of ompatibility between PS and OSS. However if the government is maximizing the total welfare (inluding the rm s pro t), subsidies towards open soure are not soially desirable regardless of the regime of software ompatibility. Introdution For many appliations, Open soure software (OSS) an o er a high-quality alternative to proprietary software (e.g. Linux, Apahe, Gimp, Sendmail, et.). They are usually Department of Eonomis-IDEC, niversity of Rennes y Department of Eonomis-IDEC, niversity of Rennes and Foundation of Rennes OSS is software for whih the soure ode is freely available, and that the liense under whih is distributed enables eah user not only to use the software, but also to opy it, to modify it, and to redistribute the original or modi ed version to other users.

2 more reliable and an be ustomized to meet spei needs. But even if OSS are usually free of harge, its installation and use require some skills. Proprietary software tends to be more user-friendly and o ers more help and support. A typial example is Linux, an open soure alternative to the proprietary operating system Windows. Despite its superior quality, Linux has a limited market share and has mostly been adopted by expert users. OSS advoates are pushing for government intervention to promote the di usion of OSS (Varian and Shapiro, 00; Benkler, 00; Smith, 00). Over the last deade, several (national and loal) governments around the world have atively enouraged the adoption of OSS, through training programs and publi prourement of OSS (for administrations and shools). This artile aims to examine whether publi poliy in favor of OSS an be e ient and how they impat users and proprietary software ompanies. Here we will fous on publi subsidies or support direted to users to redue their ost of OSS adoption (i.e. the ost of installing and using OSS): (i) What is the welfare impat of publi subsidies for OSS? (ii) How the extent of ompatibility between PS and OSS an in uene the amount of subsidies? The publi poliies towards open soure software have yet been studied in the open soure litterature. Shmidt and Shnitzer (00) develop a model of spatial ompetition between PS and OSS in the presene of heterogeneous users. They disuss the welfare impat of publi poliies that enourage administrations, shools and universities to adopt OSS. They show that an inrease in the open soure market share redues the proprietary software rm s inentives to improve the quality of its software, whih further redues the total welfare. Mustonen (00) shows that publi e orts to provide better information on open soure alternative are welfare inreasing. Comino and Manenti (00) examine subsidies for OSS users in a market where some potential users ignore the existene and/or harateristis of OSS. They show that OSS subsidies always redue soial welfare. In this paper, we develop a model that extends the model Comino and Manenti (00) by introduing a subjetive ost or disutility of using open soure software. We also assume that the proprietary and open soure software are vertially di erentiated (whereas in Comino and Manenti (00), the two produts have the same quality. Finally, we examine di erent regimes of software ompatibility between PS and OSS. More spei ally, our model onsiders a rm that sells PS, an OSS ommunity that o ers OSS free of harge and a government that subsidizes users that adopt OSS. An important strategi deision is whether to make these software ompatible or inompatible. Four situations are possible: full (two-way) ompatibility, full inompatibility, and oneway ompatibility (either only OSS or only PS is ompatible). OSS ompatibility means that the PS users an unilaterally aess the OSS ommunity and derive bene ts from it. For instane, they an use some programs or appliations developed by the OSS ommunity, or read and modify the les sent by the OSS users, whereas the OSS users annot aess the les or appliations of the proprietary software. With PS ompatibility, only OSS users an derive some utility from the PS users. We also assume that users are Network e ets an also hinder the entry of higher quality software. Network e ets arise both diretly from the number of onsumers who are using ompatible software and indiretly from the provision of omplementary servies. Suh network e ets may tip the market in favor of only one software produt. This an happen for any produt or tehnology with network externality. For instane, it an explain the dominane of QWERTY keyboard even if it is less performant than the DSKs of August Dvorak (David, 985).

3 heterogeneous in their ability to use OSS, and they are also sensitive to the number of users who are adopting the same software or a ompatible software. We show that if the government only takes are of onsumer surplus, publi subsidies are welfare-enhaning. But the optimal subsidies are larger with full ompatibility and OSS ompatibility than full inompatibility and PS ompatibility These results suggest that government poliy towards OSS must be onditional to the degree of ompatibility between PS and OSS. However if the government is maximizing the total welfare (inluding the rm s pro t), subsidies towards open soure are not soially desirable regardless of the regime of software ompatibility. The paper is organized as follows. The next setion presents the parameters of the model. Results of the two stage model are displayed in Setion and 4. In Setion 5, we ompare the prie, pro ts, market shares and subsidies under the four ompatibility regimes.poliy and managerial impliations are disussed in setion 6. Model We onsider a rm that sells a proprietary software (PS) of quality, V P S, at prie, p. But onsumers have also the alternative to use an open soure software (OSS) developped by an open soure ommunity. This software is free and has a level of quality, V OS. We assume that V OS > V P S meaning that the OSS has a superior quality (e.g. more features and better performane). This assumption is quite realisti. For instane, in the operating system market, Linux has less bugs and is updated more frequently than Windows (Raghunathan et al., 005). However, OSS require some skills or expertise to be set up and used. We de ne V OS V P S > 0:In the remainder of the paper, we suppose that V OS and V P S are su iently large to ensure that the market is fully overed. We assume that users are heterogeneous in their skills and bear a ost to use an OSS that diretly depends on their level of expertise. sers skills are uniformly distributed on (0; ): for high skilled users, is losed to 0 and for low skilled users is losed to : For a given level of expertise, the ost of using an OSS is equal to : The ost of using a proprietary software is assumed to be zero as PS are usually hararaterized by user-friendly interfae. For simpliity, the mass of users is equal to and users only adopt one software (no multi-adoption). ser s utility depends on the (intrinsi) quality of the software, but also on network e ets. We suppose that users are idential in their valuation of software quality. Moreover, as Katz and Shapiro (985) and Shy (00), we assume that network e ets are a linear funtion of the number of users who have adopted the same software or a ompatible software. More users mean that it is easier to share or exhange data and les or get support. As the number of software users is only known after users make their adoption hoie, they have to form expetations about the respetive number of OSS and PS users. We suppose that eah user orretly antiipates the size of eah software network (self-ful lling beliefs). The value of network externalities is times the expeted number Proprietary software is more user friendly than open soure software beause open soure software is developed by high skilled programmers who are also the potential users of these software. For example, the installation of open soure software require downloading soure ode, linking libraries, setting environment variables for the operating system and ompelling the soure ode. In ontrast, most proprietary software requires just a few liks and tehnial support is usually available.

4 of users who have adopted the same software or a ompatible software 4. We distinguish four situations, depending on whether the PS and the OSS are fully (two-way) ompatible, partially (one-way) ompatible or fully inompatible: Full inompatibility: if PS and OSS are fully inompatible, the value of network externality for PS users is N P S (with N P S the number of users who have adopted the PS) and the value of network externality for OSS users is N OS (with N OS the number of users who have adopted the OSS) Full ompatibility: if PS and OSS are fully ompatible, then the value of network externality for both users of OSS and PS is (N OS + N P S ) (as the market is fully overed and the total number of users is, we have N OS + N P S ) OSS-ompatibility: if OSS is unilaterally ompatible, PS users an aess the OSS ommunity and derive utility from it, but OSS users annot get any utility from the network of PS users. In this ase, the value of network externality for PS users is (N P S + N OS ), and the value of network externality for OSS users is N OS. PS-ompatibility 5 : if PS is unilaterally ompatible, only OSS users an aess PS users and the value of network externality for OSS users and PS users is respetively (N P S + N OS ), and N P S. For sake of simpliity, let II (resp. CC) denote the full inompatibility ase (resp. full ompatibility ase). Similarly, let CI (resp. IC) denote the OSS-ompatibility ase (resp. PS-ompatibility ase). We de ne k II; CC; CI; IC: The utility of type user under the di erent ompatibility senarii is given by: V P S + N P S p if the user buys a PS that is OS-inompatible (.a) V OS + N OS + s if the user downloads OS that is PS-inompatible (.b) V P S + p if the user buys PS that is OS-ompatible (.) V OS + + s if the user downloads OS that is PS-ompatible (.d) Assuming that the marginal ost to sell a PS is onstant and normalized to zero, the pro t of the software rm is given by: k p k N k P S with k II; CC; CI; IC () 4 Following Farrell and Saloner (99), we assume that the value of network externality is the same for both software. 5 This ase is less realisti as unilateral ompatibility from PS to OSS is seldom observed. 4

5 By de nition the OS ommunity has no revenues and no ost (i.e. proft equal to zero) 6. In this paper, we analyze the impat of subsidies that are direted to OSS users. These subsidies an take the form of training sessions, support,... to redue the ost of adoption of OSS. Let s be the amount of subsidies per user with s > 0: What should be the optimal level of subsidies or support by the government? Let S sn OS be the ost of subsidizing OSS users. The soial welfare is the sum of rm s pro t () and onsumers surplus (CS) minus the ost of subisidies: W k k + CS k S k with k II; CC; CI; IC () In our paper, we will onsider that the government only takes into aount the onsumers surplus. This an be justi ed by two reasons. Firstly, many software ompanies are operating abroad and their pro ts annot be part of the domesti soial surplus 7. Seondly in matters of market regulation and ompetition poliy, governments tend to put more weight on onsumers welfare. In this ase, the government will hoose the amount of subsidies that maximizes the following funtion: W k S k S k with k II; CC; IC; IC (4) In the Appendix, we also present the results when the government maximizes the soial welfare that inludes the pro t of the software rm (see ()). Table : Notation Notation V OS ; V P S p s N OS ; N P S S Desription The intrinsi quality of OSS and PS Prie set by the software rm Cost of using OSS Level of expertise Subsidy per OSS user Valuation of network externality Market share of OSS and PS Pro t of rm Tax burden of subsidizing OSS Di erene in the qualities of OSS and PS The timing of the strategi game is as follows. In the rst stage, the government hooses the amount of the subsidies to help uses to adopt the OSS. In the seond stage, the rm sets the prie of its software and then the users hoose to adopt either the PS or the OSS. In the following setion, we start to solve the seond stage of the model under the four ompatibility senarii. 6 Open soure software are developped by open soure ommunities whose members voluntarily ontribute. Sine open soure software is freely downloadable on the Internet, these ommunities do not earn a pro t. 7 The main sofware ompanies are S ompanies and this explains why many European government, but also China, want to enourage the adoption of OSS (espeially for publi admistrations and shools). 5

6 Prie deision and market shares. Full inompatibility When the software produts o ered by the rm and the OS ommunity are fully inompatible, network externalities for users of OSS and PS users are respetively N OS, and N P S. Then the utitity of type user is given by II VP S + N P S p if hoie of PS V OS + N OS + s if hoie of OSS Let b be the marginal user who is indi erent between adopting PS and OSS. Sine the users are uniformly distributed between (0; ), then N OS b and N P S b. Solving V P S + ( b ) p VOS + b b + s yields b (p+s +) : sers with a type < b (high skilled) will prefer OSS and users with > b will adopt PS.. It implies that market shares an be expressed as: N II N II OS p + s + P S p s (5.) (5.) To ensure that PS and OSS,have positive market shares at the equilibrium (i.e. to eliminate orner solution) we need to assume that > 0 8 and that < (see appendix). In other terms, the ost of OSS adoption must be su iently high or network externalities not too large. nder these two onditions and from (), the rm s pro t is p s II pnp II S () p (5.) Given the amount of subsidies s, the rm hooses the pro t-maximizing prie that is equal to p II (s) s (5.4) We observe that the prie dereases with the amount of subsidies. The e et of subsidies is to inrease ompetition between the two types of software and redue the market power of the software rm. After rearrangement the market shares are and the equilibrium pro t is OS (s) + s + ( ) N II N II P S (s) s ( ) (5.5) (5.6) 8 If network externalities are too strong, i.e. <, the rm exits the market and the OSS is the only sofware adopted by users. 6

7 II (s) ( s ) 4 ( ) (5.7) Note that the rm and the OS ommunity have a positive market share if and only if s > > s:. Full (Two-way) Compatibility When the software produts are two-way ompatible, any user is able to interat with both PS and OSS users. The value of network externalities is (N OS + N P S ) whatever the software adopted and the utility of a type is CC VP S + p if hoie of PS V OS + + s if hoie of OSS The user indi erent between the two software produts is haraterized by b (p+s+) and the market shares are respetively: N CC OS p + s + (6.) As the rm s pro t is the optimal prie is given by CC pn CC P S N CC P S p s p s () p (6.) (6.) p CC (s) s (6.4) As previously, the prie of PS dereases with subsidies Then the equilibrium market shares and pro t an be rewritten as follows N CC OS (s) + s + N CC P S (s) s (6.5) (6.6) CC ( s ) (s) (6.7) 4 Note that the rm and the OS ommunity have a positive market share if and only if > + s:otherwise only the OS ommunity will remain ative in the market. 7

8 . OSS (one-way) ompatibility If the omptatibility is only from OSS toward PS, then the PS users an use programs that are developped by the OS ommunity and enjoy more network externality. Then the utility of a type is CI VP S + p if hoie of PS V OS + N OS + s if hoie of OSS The user who is indi erent between adopting the OSS and the PS produt is haraterized by b p+s + assuming that > : Then, the market shares are given by: N CI OS N CI (p + s + ) P S ( p + s ) (7.) (7.) The opimal prie for the proprietary software is p CI (s) s, and the equilibrium market shares and pro t are as follows N CI OS (s) ( + s + ) ( ) N CI P S (s) s ( ) (7.5) (7.6) CI (s) ( s ) 4 ( ) (7.7) Note that the Firm and the OS ommunity have a positive market share if and only if s > > s..4 PS (one-way) ompatibility The last (but probably less realisti) senario is a PS ompatibility regime. OSS users an aess the network of PS users and derive some utility from it. The utility of type user is IC VP S + N P S p if hoie of PS V OS + + s if hoie of OSS The user who is indi erent between adopting the OSS and the PS produt is haraterized by b (p+s+) (assuming > ). Then, the market shares are given by N IC N IC OS p + s + P S p s (8.) (8.) The pro t-maximizing prie for the proprietary software is p IC (s) s, and the equilibrium market shares and pro t are as follows 8

9 OS (s) + s + ( ) N IC N IC P S (s) s ( ) (8.5) (8.6) IC (s) ( s ) 4 ( ) (8.7) Note that the rm and the OS ommunity have a positive market share if and only if s > > s. 4 Open soure subsidy deision of the government Now, we onsider the rst-stage of the model where the government sets the susbisidy per OSS users in order to maximize the onsumers surplus (net of the ost of subsidies). Solving the rst order ondition under the four senarii, the optimal subsidies are given by 9 s CC ; s CI ; s IC ; s II By replaing s by its optimal value, we an derive the equilibrium pries, market shares (or the users base for eah type of sofware) and the pro t that are summarized in Table. II CC CI IC p s NP S ( ) NOS 5+ ( ) ( ) 9( ) + ( ) 9 ( ) + ( ) ( ) 9( ) ( ) + ( ) ( ) 9( ) Table : Equilibrium solutions under the four senarii Full inompatibility (II), Full ompatibility (CC), OSS-ompatibility (CI) and PS-ompatibility (IC). 5 Impliations for publi poliy In this setion, we ondut some omparative statis analysis on the equilibrium outomes aross the four ompatibility regimes. These results provides some insights on the desirability of publi e orts to promote OSS. To allow omparisons aross the four senarii, we restrit.our analysis to the situations where PS and OSS have both positive market shares whatever the regime of ompatibility. The neessary onditions to have oexistene of PS and OSS are > + and > 9 In eah regime of ompatibility, the optimal subsidy is unique as W k s k < 0 for > + and >. 9

10 5. Comparaison of subsidies, publi de its, market shares, pries and pro ts The next proposition ompares the optimal subsidy under the four ompatibility regimes: Proposition Assuming that a sub-game perfet equilibrium exists where both rm and OS ommunity are ative in the market, i.e. implementation ost of the OSS is su iently higher with > + and >. Then, s CC s CI > s IC s II Proof. See Table. The government gives a larger subsidy per user in the ases of two-way ompatibility and OSS ompatibility. This result is quite intuitive. When the OSS is ompatible with the PS, then PS users derive some gain from an additional OSS user (through network externality). This is not the ase under full inompatibility and PS ompatibility. Clearly the government has more inentives to subsidize OSS users in the CC and CI situation as the return in terms of welfare will be larger. Proposition ompares the PS prie, market shares, and pro t under the four ompatibility situations. Proposition When the government gives welfare-maximizing subsidies to OSS users, then : (i) p CC ss CC p CI ss CI > p IC ss IC p II ss II ; (ii) NP II S ss II > NP CI S ss CI > NP CC S ss CC > NP IC S ss IC ; (ii) II ss II > CI ss CI > CC ss CC > IC ss IC : Proof. See Table and Appendix. The rm sets a higher prie when the utility of its ustomers inreases with the number of OSS users. As they are willing to pay more for the proprietary software, the rm is able to inrease its prie in the situations of full ompatibility or OSS ompatibility. However, the ranking of the four senarii in terms of prie and pro t is di erent. The rm is better under full inompatibility: it enjoys a larger market share and pro t. The seond best situation is OSS ompatibility in whih users are harged a higher prie for the PS and reeive more subsidies for the OSS ompared to the full inompatibility ase. The results is a lower market share for the proprietary sofware and pro t than under full inompatibity. The worst situation for the software rm (in terms of market share and pro t) is PS ompatibility beause users get more utility to adopt OSS. The rm has to be more agressive in its priing but it s not su ient to retain its onsumers given the subsidies distributed by the government. The situation of full ompatibility is between the OSS and PS ompatibility ases. The prie of the PS (and the subsidy per OSS user) under full ompatibility is the same as under OSS ompatibility, but its market share is lower beause the rm has no exlusive advantage in terms of network externality under full ompatibility. Its produt is less attrative than under PS ompatibility. If the rm has the possibility to hoose the ompatibility regime, it is inited to deny aess to its servies and ostumer base (by making its software inompatible for OSS users). Proposition ompares the sal burden of these subsidies. 0

11 Proposition Assuming that a sub-game perfet equilibrium exists where both rm and OS ommunity are ative in the market, i.e. implementation ost of the OSS is su iently higher with > + and >. Then S CC > S CI > S IC > S II Proof. See Appendix 4. An open soure sofware poliy is more ostly when the PS and OSS are fully ompatible beause the number of subsidy bene iaries is higher under full ompatibility than under OSS ompatibility even if the susbsidy per user is the same in both situations. The full inompatibility situation is the least ostly for the government as the subsidy per user is lower than in the CC and CI regimes and the number of bene iaries is also lower. 5. Comparative statis aross software ompatibility situations We also analyze how the equilibrium outome hange when the ost of OSS adoption (), the gap between OSS and PS quality (); and the magnitude of network externality () vary. CC s CC + 0 p CC + 0 NP CC S + 0 NOS CC + 0 CC + 0 II s II + p II + NP II S + + NOS II + II + CI s CI + 0 p CI + 0 NP CI S + + NOS CI + CI + + IC s IC + p IC + NP IC S + NOS IC + + IC + Table : Impat of (ost of OSS adoption), (quality advantage of OSS) and (magnitude of network externality) on subsidies, prie, market shares and pro ts under Full ompatibility (CC), Full inompatibility (II), OSS-ompatibility (CI) and PSompatibility (IC). Interpretation: + means positive impat, means negative impat and 0 means neutral e et. Table shows that the rm s pro t, pries and market shares, as well as the subsidies, inrease with the ost of using OSS. Moreover, the subisidies and the prie harged by the rm, as well as its pro t, tend to derease with the di erential in quality between the OSS and the PSS. Howewer, the role of network externality di ers under the four regimes. It has no impat on market shares and pro ts under full ompatibility, but it has a positive impat on rm s market share under full inompatibility and OSS ompatibility and a negative impat under PS ompatibility. Moreover, when onsumers have a high valuation for network externality, the rm is better o when its onsumers an unilaterally bene t from OSS s users base. When it s not the ase, an inrease in the value of network externality will indue the rm to prie more agressively, and its pro t will be redued.

12 5. Comparaison of Welfare Levels Finally, we ompare soial welfare (exluding rm s pro t) under the four ompatibility situations. Welfare under full ompatibility, full inompatibility, OSS ompatibility and PS ompatibility are, respetively: W II (4V OS + V P S ) ; (9.) 6 W CC ( V OS + V P S 0V OS V P S + ) ; (9.) 6 ( ) W IC W CI (6 + 4V OS + V P S + 6V OS ) ; (9.) 6 ( ) (7 + 4V OS + V P S + 5 4V OS V P S ) : (9.4) 6 ( ) Proposition 4 When the government gives welfare-maximizing subsidies to OSS users, then : W k ss W k s0 > 0 for kii, CI, CC, IC and W CC Proof. See Appendix 6. ss CC > W IC ss IC > W CI ss CI > W II ss II The intervention of the government in favor of OSS is always welfare enhaning. But the welfare with subsidies is larger when the proprietary and OSS are perfetly ompatible. In this situation, the welfare impat of subsidies is to stimulate ompetition and push the rm to redue its prie (but subsidies have no impat on network externality). The worst situation in terms of welfare is full inompatibility. Even if subsidies intensify ompetition, onsumers surplus is lower as network externalities are only intra-network. The intermediate situations are one way ompatibility regime with PS ompatibility outranking OSS ompatibility. When the OSS users bene t from unilateral network e ets (PS ompatibility), onsumers are more likely to adopt the OSS that o ers superior quality and extended network externality. Subsidies an reinfore this trend and inrease the market share of OSS and the utility of OSS users. In the Appendix B, we have examined the situation in whih the government takes into aount the rm s pro t. When the government objetive is the sum of onsumers surplus and rm s pro t, then publi subsidies for OSS s users have negative impat on welfare regardless of the ompatibility regime and the best poliy is laissez-faire or publi neutrality. This nding is similar to the results obtained by Comino and Manenti (00) with horizontal produt di erentiation between PS and OSS. Our nal proposition analyzes how a hange in the ost of adopting OSS an impat the welfare gain of subsidies.

13 Proposition 5 ( W k j ss W k j s0 ) > 0 for all > + and >. Proof. See Appendix 8. The di erene in soial welfare with and without subsidies inreases with the (subjetive) ost of using OSS regardless of the ompatibility regime. It implies that subsidies are more e ient and more justi ed when there are a lot of obstales and osts to adopt OSS. 6 Conluding remarks Although the open soure litterature has extensively studied the issue of ompetition between open soure and proprietary software, little attention has been paid to the impat of publi poliy to promote open soure software in this ontext. The aim of this paper is to extend the literature. We analyze the impat of publi subsidies for OSS users in presene of network e ets and under di erent ompatibility regimes. Our main ndings are that publi subsidies push down the prie of proprietary software, inrease the market share of the OS software and may stimulate network externality when PS and OSS are partially inompatible (PS one-way ompatibility). When only users welfare is taken into aount and the adoption ost of OSS is su ient high, publi subsidies for OSS users are welfare-enhaning. However, the optimal poliy is to provide larger subsidies per user under full ompatibility and OSS one-way ompatibility than under full inompatibility and PS one-way ompatibility. We have also examined the optimal poliy when the government maximizes the total welfare (inluding the rm s pro t). In this ase, subsidizing OSS is not soially desirable regardless of the regime of software ompatibility. This result is similar to that obtained by Comino and Manenti (00) with a model of horizontal di erentiation and an be used as an argument in favor of a tehnology neutrality (meaning that a government should never intervene to sponsor a tehnology, but let the market hoose the best tehnologies. The theoretial model developed in our paper has several limitations. First, we onsider that the quality of OSS and PS is exogenous. It would be interesting to add a stage in whih the OS ommunity and the software ompany an invest in the quality of their software. Moreover, the hoie of ompatibility ould also be endogenized. Our results suggest the sofware ompany has strong inentives to make its produt inompatible with the OSS. Another limitation is that our model is stati and does not allow for intertemporal priing strategies. Instead we ould onsider two periods and two generations of potential users. In the initial period, the software ompany ould be more aggressive to get a ritial mass of users and obtain ompetitive advantage (through network externality) in the seond period. In this dynami setting, optimal publi subsidies ould learly be di erent over time. In addition to these results, we have shown that the optimal OSS subsidy is higher under full ompatibility and OSS ompatibility followed by PS ompatibility and full inompatibility.

14 Referenes Benkler, Y., 00. Coase s Penguin, or, Linux and the nature of the rm. Yal Law Journal, (), Bessen, J., 00. What Good is Free Software? In Robert H. Hahn (ed) Government Poliy towards Open Soure Software (pp -). Washington D.C. : AEI-Brookings Joint Center For Regulatory Studies. Bonaorsi A., Rossi C., 00. Why open soure software an sueed? Researh Poliy (7) 4-5 Comino, S., Manenti, F., 005. Government poliies supporting open soure software for the mass market. Review of Industrial Organization, 6: Evans, D., 00. Politis and programming: government preferenes for promoting open soure software. In. R. Hahn (Ed.), Government poliy toward open soure software (pp. 4-49). Washington, DC: AEI-Brooking Joint Center For Regulatory Studies. Evans, D., Layne-Ferrar, A., 004. Software patent and open soure: the battle over intelletual property rights. Verginia journal of law and tehnology, 9 (0): -8. Evans, D. S. and Reddy, B. J., 00. Government Preferene for Promoting Open- Soure Software: A Solution in Searh of a Problem. Mihigan Teleommuniations and Tehnology Law Review, 9(), -94. Ghosh, R., Glott, R., 005. sage of and Attitudes towards Free/Libre and Open Soure Software in European Governments. Results from the Governments Survey, Free/Libre and Open Soure Software: Poliy Support». MERIT, niversity of Maastriht. Katz, M., Shapiro, C., 994. Systems Competition and Network E ets. Journal of Eonomi Perspetives, vol. 8, no., pp Katz, M., Shapiro, C., 985. Network externalities, ompetition, and ompatibility. Amerian Eonomi Review, Vol. 75, No., pp Lee, S.Y.T., Meng, Z., 005. Open Soure vs. Proprietary Software: Competition and Compatibility. Working paper, available at SSRN: Leibowitz J., Margolis E., 994. Network externality: an unommon tragedy. Journal of Eonomi Perspetives, Vol. 8, No., -50 Lee, J. A., 006. Government Poliy toward Open Soure Software: The Puzzles of Neutrality and Competition. Knowledge, Tehnology, & Poliy, Vol. 8, No. 4, pp. -4. Lerner, J., Tirole, J., 00. Some Simple Eonomis of Open Soure. The Journal of Industrial Eonomis, L,, Lerner, J., Tirole, J., 005. Eonomi Perspetives on Open soures in Perspetives on Free and Open Soure Software», edited by Feller J. J.,Fitzgerald S., Lakhani K., The MIT Press, Cambridge, Massahusetts. Lessig, L., 00. Open soure baselines: ompare to what?", In. R. Hahn (Ed.), Government poliy toward open soure software (pp ). Washington, DC: AEI- Brookings Joint Center For Regulatory Studies. Shmidt, K., Shnitzer, M., 00. Publi subsidies for open soure? Some eonomi poliy issues of the software market. Harvard Journal of Law and Tehnology, 6(), Smith, B. L., 00. The Future of Software: Enabling the Marketplae to Deide. In Robert H. Hahn (ed.) Government Poliy Toward Open Soure Software (pp ). Washington D.C.: AEI-Brookings Joint Center for Regulatory Studies. 4

15 Varian H., Shapiro C., 999. Information Rules: A Strategi Guide to the Network Eonomy. Harvard Business Shool Press. Boston, Massahusetts Varian H., Shapiro C., 00. Linux adoption in the Publi Setor: An Eonomi Analysis. Tehnial Report, C Berkeley, Deember. Zhu, X., Zhou, Z. Z., 0. Lok-In Strategy in Software Competition: Open-Soure Software vs. Proprietary Software. Information Systems Researh, Artiles in Advane, pp. 0, 0 INFORMS. 5

16 Appendix A A. Equilibrium values under the four ompatibility situations In this Appendix we present all the equilibrium values under the four ompatibility situations when the government only ares about the surplus of users. II CC CI IC p s NP S ( ) NOS 5+ ( ) ( ) 09( ) W B 7 + 4V OS +V P S + 5 0V OS V P S 6( ) n:: > + ; 5 C A + ( ) 9 0 4V OS + V P S A ( ) + ( ) ( ) 09( ) B 6 + 4V OS +V P S + 6V OS 6( ) > > 0 > ; C A ( ) + ( ) ( ) 09( ) B 7 + 4V OS +V P S + 5 4V OS V P S 6( ) > + ; C A Table 4: n.. neessary ondition for subgame-perfet equilibrium, di erenes between OSS et PS quality V OS V P S, Full ompatibility (CC), Full inompatibility (II), OSS- ompatibility (CI) and PS-ompatibility (IC) A.. Neessary onditions for the oexistene of OSS and PS A duopoly ( with positive market shares for OSS and PS) exists under the four ompatibility regimes if the following onditions are satis ed 0 < k < and k > 0 with k CC; II; CI; IC (A) The rst ondition ensuresan interior solution for the indi erent user between OSS and PS. The seond ondition ensures that the indi erent user gets a non-negative utility. Assume k > 0 holds. The rst ondition under full ompatibility, full inompatibility, PS ompatibility and OSS ompatibility implies respetively: 0 < CC < ) 0 < < (A.) 5 0 < II < ) > max 0 < CI < ) > max ; ; + ; ; (A.) (A.) 6

17 Conditions (A:) 0 < IC < ) > max (A:4) are equivalent to ; ; + (A.4) 0 < k < ) > + d and > (A.5) Condition (A:5) is su ient to ensure the existene of the mixed duopoly under the four ompatibility regimes. A.. Proof of proposition p CI P S Optimal pries As p CC P S > pic P S pii P S ; p CI P S ; p IC P S ; p II P S, it implies that p CC P S Optimal market shares As N CC P S ; N CI P S ( ) ; N IC P S ( ) ;N II + d and >, then it implies that N II P S > N CI P S P S ( ) > NP IC S : > N CC P S ; and sine > Optimal pro ts ( ) As CC ; CI ( ) ; 9 9( ) IC ( ) ; II ( ), then it implies 9( ) 9( ) that II > CI > CC > IC : (assuming that > + and > ). A.4. Proof of proposition As S CC ( )(+) 9 ; S CI ( )( +) 9( ) ; S IC ( )( +) 9( ) ; S II ( )( 5+) 9( ) ;it an be shown that S CC > S CI > S IC > S II (assuming > + ) beause A.5. Proofs of Table S CC S CI 9 ( ) > 0 (B.) S CI S IC ( + ) > 0 (B.) 9 S IC S II ( + ) 9 ( ) + > 0 (B.) Comparative statis on prie, market shares and pro t of the rm: 0 0 Sine values of prie and subsidy under the all ompatibility regimes are equal at the equilibrium, the veri ation of the omparative statis of the subsidy is ignored in this Appendix. 7

18 Impat of ; and on prie p CC > 0 ; pcc < 0 p IC p II > 0 ; pii < 0 ; p II < 0 p CI > 0 ; p IC > 0 ; p CI < 0 < 0 ; p IC Impat of ; and on market share of the rm < 0 N CI P S N IC P S N CC P S > 0 ; NP CC S < 0 ( ) ( ) > 0 ; NP CI S ( ) < 0 ; NP CI S IC NP S > 0 ; ( ) ( ) IC NP S < 0 ; ( ) > 0 ( ) < 0 N II P S ( ) > 0 ; NP II S ( ) II NP S < 0 ; ( ) > 0 i > where > d + :Thus, the duopoly ondition is satis ed. Impat of ; and on market share of the OS ommunity N CI OS N IC OS N II OS N CC OS ( ) < 0 ; N CC OS > 0 ( ) ( ) < 0 ; NOS CI ( ) > 0 ; NOS CI ( ) < 0 ; NOS IC ( ) > 0 ; NOS IC ( ) ( ) < 0 ; NOS II ( ) II NOS > 0 ; ( ) < 0 ( ) > 0 ( ) < 0 Impat of ; and on pro t CI CC ( ) 9 > 0 ; ( ) ( + ) 9 ( ) > 0 ; CI 8 CC ( ) 9 < 0 ( ) 9 ( ) < 0 ; CI ( ) 9 ( ) > 0

19 IC Finally, IC ( + ) 9 ( ) 0 i [ ; + ] d. Finally, IC > 0 when > + d IC ( ) < 0 8 > + d 9 ( ) IC ( + + ) 9 ( ) 0 i [ ; + ] d. < 0 when > + d II ( ) 9 ( ) 0 i [ + ; ] d. Finally, II > 0 when > + d II ( ) < 0 8 > + d 9 ( ) II ( ) ( ) 9 ( ) < 0 8 > + d A.6. Proof of proposition 4 Assume in the following that > + to enable omparisons of welfare levels aross all four ompatibility senarios. Given the level of welfare under the four regimes in Table, we an derive the following results: W j (CC;CI) W CC W IC 6 ( ) (C.) Sine >, the h sign of this expression i is given by the sign of the numerator whih is negative for (CC;IC) ; (CC;IC), where (CC;IC) p 4 + and (CC;IC) + p 4 +, with (CC;IC) < (CC;IC): However, the values of that satisfy the duopoly ondition are higher than (CC;IC) and (CC;IC):This an be demonstrated by showing that (CC;IC) and (CC;IC) don t violate the duopoly ondition d ( + ). Indeed, p p4 4 + ( + ) + < 0; and p + p ( + ) + () 4 + (+) < 0: Finally, W j (CC;IC) > 0 8 > d : W j (IC;CI) W IC ( + ) > 0 8 > and > (C.) 6 W CI 9

20 Indeed, W j (CI;II) W CI W II ( + + ) (C.) 6 + W j (CI;II) (CI;II) 0, where We an easily hek that W j (CI;II) (CI;II) p p (CI;II) and (CI;II) , with (CI;II) < (CI;II). Note that W j (CI;II) ()! when tends to. Then W j (CI;II) () 0 when h i (CI;II) ; (CI;II) : However, (CI;II) and (CI;II) do not satisfy the duopoly ondition. p + p ( ) < 0. Finally, W j (CI;II) > 0 for > d. ( + ) p < 0; ( + ) + p 4 8+5, ( ) ( + ) Aording to these results, we an easily hek that W CC ss CC > W IC ss IC > W CI ss CI > W II ss II 8 > + d and > A.7. Equilibrium results without publi subsidy under the four ompatibility regimes The equilibrium solutions for the proprietary rm and the OS ommunity in the absene of subsidies are readily obtained from N k P S (s), N k OS (s), pk (s) and k (s) by setting s 0, summarized in Table 5. II CC CI IC p j s0 NP S j s0 ( ) + ( ) NOS j s0 j s0 ( ) 04( ) + ( ) 4 ( ) ( +) ( ) ( ) 4( ) ( ) + ( ) ( ) 04( ) B 4 (V OS + V P S ) C B + + C B 4 (V A A OS + V P S ) C B +4 ( ) W j V OS 4V P S 4 (V OS + V P S ) 8V OS A (V OS + V P S ) s0 8( ) 8 8( ) 8( ) n:: > + and > > > > + Table 5: n.. neessary ondition for subgame-perfet equilibrium, di erenes between OSS et PS quality V OS V P S, Full ompatibility (CC), Full inompatibility (II), OSS-ompatibility (CI) and PS-ompatibility (IC) C A The alulation of the welfare is omitted for brevity. All aluls are available on request 0

21 A.8. Proof of proposition 5 From Table 4 (Row 7) and Table 5 (Row 6), we derive the following results: W CC W CC ss W CC s0 ( ) 4 (D.) W II W II ss W II s ( ) (D.) W IC W IC ss W IC s D 4 ( ) (D.) W CI W CI ss W CI s ( ) We are now in the position to proof the proposition 5. (D.4) The derivative of W CC with respet to is learly positive. The derivative of W II with respet to an be written as W II ( ) 4 ( ) (E.) This expression is negative when [ II ; II ] where II p 5 + and II + p 5 +, with II < II :Remark that only II satis es the duopoly ondition. Indeed, + p 5 + ( + ) + p 5 +, ( )+ > 0; and Finally, p 5 + W II < 0 when > + p 5 + : The derivative of W IC ( + ) p 5 + < 0: h + ; + p i 5 + and > 0 when with respet to an be written as W II W IC ( 5 + ) 4 ( ) (E.) This expression is negative when [ that only 5 + satis es the duopoly ondition. Thus, and W IC 5; 5 + ] > + d :We easly hek W IC < 0 for [ + ; 5 + ] : The derivative of W CI with respet to an be written as W CI > 0 when > 5 + ( 5 + ) 4 ( ) (E.)

22 h This expression is negative when + p 5 + ; p 5 i + > + d :We easly hek that only + p 5 + satis es the duopoly ondition. Indeed, + p 5 + ( + ) p 5 +, 4 + > 0; and p 5 + ( + ) p 5 + < 0. Finally, W CI > 0 8 > + p 5 + and Appendix B W CI < 0 for h + ; + p 5 + i : B.. Equilibrium values when the government maximizes the total welfare (inluding rm s pro t) We present the equilibirum outome when the government maximizes the total welfare (inluding the rm surplus). By using the welfare funtion given in (), and maximizing for s, we obtain the equilibrium results summarized in Table 6: CC II CI IC p N P S N OS ( ) s ( ) W +V P S + ( ) + +V P S ( ) ( ) + +V P S V OS n:: > > + and > > > + ( ) V P S ++ V P S ( ) Table 6: n..: neessary ondition for subgame-perfet equilibrium, di erene in qualities V OS V P S, Full ompatibility (CC), Full inompatibility (II), OSS-ompatibility (CI) and PS-ompatibility (IC) Remark From Table 6, we onlude that a positive publi subsidy to OSS users generates a negative total welfare surplus regardless of the ompatibility regime. It implies that publi subsidies to OSS users are not desirable when the government maximized the total welfare (inluding the rm s pro t).