Open Source Development with a Commercial Complementary Product or Service 1. Ernan Haruvy, Suresh Sethi, Jing Zhou. University of Texas at Dallas

Transcription

1 Open Soure Developent with a Coerial Copleentary rodut or Servie Ernan aruvy, Suresh Sethi, Jing Zhou University of Texas at Dallas Abstrat Opening the soure ode to a software produt often iplies that onsuers would not pay for the software produt itself. owever, revenues ay be generated fro opleentary produts. A software fir ay be willing to open the soure ode to its software if it stands to build a networ for its opleentary produts. The rapid networ growth is doubly ruial in open soure developent, where the users of the fir s produts are also ontributors of ode that translates to future uality iproveents. To deterine whether or not to open the soure, a software fir ust jointly optiize pries for its various produts while siultaneously anaging its produt uality, networ size, and eployent strategy. Whether or not potential gains in produt uality, networ size, and labor savings are suffiient to justify opening the soure ode depends on produt and deand harateristis of both the software and the opleentary produt as well as on the ost and produtivity of in-house developers relative to open soure ontributors. This paper investigates these ruial eleents to allow firs to reah the optial deision in hoosing between the open and losed soure odels. Key words: riing Researh; Optial ontrol; Open Soure; Networ Externalities We are grateful for insight and advie fro Sujoy Charavarty, Andy Nova, Ashutosh rasad, Fang Wu, and University of Texas at Dallas seinar partiipants. Corresponding author: Ernan aruvy, eharuvy@utdallas.edu, Shool of Manageent, JO 5.., The University of Texas at Dallas, O Box 83688, Rihardson, Texas , hone: , Fax:

2 Open Soure Developent with a Coerial Copleentary rodut or Servie. Introdution The open soure developent paradig has proven very suessful in reent years in eeting a wide variety of software needs. Open soure projets suh as Linux, Apahe, and erl have beoe enorously suessful, trapling bigger and better endowed losed soure opetitors. Their well publiized suess has inspired tens of thousands of open soure prograers worldwide and has given a boost to thousands of saller open soure projets Shiff,. In open soure developent, volunteersost often users of the software- ollaborate on the developent of the software by ontributing ode and bug reports and by sharing and disseinating the ode. Advantages resulting fro this for of developent inlude savings on labor osts, inreased reativity assoiated with freedo fro opany restritions, and greater developent speed arising fro rapid disseination and higher relevane to end user needs Rayond,. The ajor aveat is that the ode is freely distributed, aing it diffiult to sell that very sae ode. Given this harateristi of open soure developent, a fir wishing to pursue this developent odel ust first arefully forulate its revenue odel for the projet and weigh the produtivity benefits and labor savings against the loss of potential revenue for its software produt. Open soure progra developent ay not be appropriate for every fir. The deision to ebrae this paradig and the forulation of related strategies suh as prie, uality, and hiring ritially depend on the business odel that is used by the fir to generate revenues for its produts. Given that the open soure ode is free, a fir whih oordinates and invests its own resoures in an open soure projet generally has a oerially sold produt that would benefit fro the open soure ode by either nesting the ode within its own oerially sold produt, or by exploiting the opleentarities that are present between the ode and one of its oerially sold produts. aruvy, rasad and Sethi 3 evaluate the forer odel where the open Although we annot rule out altruis and soial onsiderations on the part of firs and any firs lai to be otivated by suh onsiderations, we feel it is safe to assue a profit otive. The publi relations aspet of open soure sponsorship ay be another otive, and the odeling of suh benefit is outside the sope of the present wor.

3 soure ode is sold as part of a oerial produt. In that senario, open soure ode ay beoe a oponent of losed soure ode and the software an be ade oerial. This an be done while eeping the soure open by providing paaging, distribution, servie or brand nae or by losing the soure, suh as Mirosoft s use of the BSD ode, or the BSDi operating syste derived fro Unix BSD. In this wor we exaine the latter odel in a onopoly setting, where the open soure ode is free but opleents another produt that is sold oerially. The open soure business odels disussed in the literature Rayond, ; Shiff, suggest several variations on this onept: The Loss-Leader Model uses the open soure ode to establish brand nae and lead users to the fir s other produts. Netsape s Mozilla web browser is an exaple for suh a pratie. Netsape s early business odel relied upon the sale of server software. The Copatible ardware odel has an open soure software opatible with the fir s oerially sold hardware. An exaple is the AppleMaxOS X software whih is opatible with Apple s hardware. The Servie and Support odel, of whih Red at is a proinent exaple, provides servie and support for fee to opleent the open soure ode. Sun s Star Offie falls under the Brand Copatibility odel wherein the open soure software opleents the fir s produt lines and brand iage. Finally, under the Inforation odel a fir harges for inforation assoiated with the open soure ode. Consulting servies often harge for inforation relating to open soure douentation, aintenane, et. Notie that the odels disussed here an also apply to a freeware software produt a software produt whih is free but whose ode is not neessarily open to the publi. owever, the ey differene lies in the developent efforts whih are less expensive and ore rapid in the open soure odel. For a disussion of when the freeware odel would be ost appropriate see aruvy and rasad 998;. 3

4 Table. Open soure Business Models Business Model Exaple Loss-leader Netsape s Mozilla Copatible ardware Apple MaxOS X Servie and Support Red at Brand Copatibility Sun s StarOffie Wide variety of vendors, Inforation inluding IBM, Orale, and onsulting firs Note that an open soure produt in the ategories disussed here need not be a full software produt. It an be a ore liited group of odules or lasses for existing open soure software. For exaple, IBM initiates and supports various Apahe-related projets and releases related ode even when it is developed in-house. IBM s ain soure of revenue in this ase is derived fro selling web servers, whih benefit greatly fro iproveents and additions to Apahe. Given the nown business odels, not every fir should pursue open soure as its software odel. When a fir deides to pursue a software projet, it will generally have an in-house developed prototype and soe assessent of the in-house developent potential. If the potential ontribution fro outside prograers is not pereived to be large, obviously the fir should not pursue open soure. Furtherore, when the benefit to the fir s opleentary oerial produt is sall relative to the value of the inhouse developed software to the onsuers, the fir should eep the software produt proprietary and extrat as uh of the onsuer surplus as it an. Our analysis is intended to unover the relationships between iportant produt harateristis and the projeted suess of open and losed soure initiatives. Our ain findings are that firs should pursue open soure developent, as opposed to losed soure developent, if user ontributions or prograers who are users are ritial to developent: e.g., bug reports speifi to the ode, the relative effetiveness of losed soure prograers is low, 3 in-house prodution is expensive, or 4 initial uality is low. In addition, other iportant insights are revealed: 5 in finite horizon 4

5 settings, software uality dereases towards the end of the produt s life in the losed soure but not in the open soure. This finding has iportant welfare ipliations. 6 When initial uality of the opleentary produt is high, the losed soure software produt ay be pried at zero initially for a long period of tie, aing it freeware. In suh a ase, open soure will doinate the losed soure. The solution approah taen here is optial ontrol. This approah is ideal here sine the fir is able to vary the prie of its oerial produts fro period to period and sine related variables state variables suh as uality and networ size vary over tie. Suh approah has been often taen in the literature to address dynai priing probles e.g., Chintagunta and Rao, 996, dynai uality probles Muller and eles, 988, and dynai advertising expenditures probles for a survey of that literature see Feihtinger, artl and Sethi, 994. The approah presented here allows for dynai priing in the presene of networ externalities e.g., Dhebar and Oren, 985 as well as for dynai uality, where the forer is a ontrol variable and the latter is a state variable. Also note the parallel between the present paper and soe wors in the innovation diffusion literature. Diffusion is generally defined as the proess by whih inforation about an innovation is ouniated through a soial syste Rogers, 983. In essene, diffusion an be thought of a speial type of networ effet 3 whih, when aounting for the dependene between deand and prie, translates to a dynai priing proble for a produt e.g., Kalish, 983; Kalish and Lilien, 983; Nasiento and Vanhonaer, 988. The approah we tae here expands on the above dynai priing literature by exaining a senario of two opleentary goods, one of whih an be developed through user ontributions ontingent on it being free. The paper is organized as follows: Setion lays out the basi odel and assuptions. Setion 3 develops the theory and its analytial ipliations as they relate to dynais and euilibriu in both finite and 3 A networ effet iplies that the larger the networ of existing users the ore liely non-adopters are to adopt. In diffusion odels, this is generally aptured by the oeffiient of iitation Bass, 969. At the individual level, diffusion an also be explained by dereased unertainty and ris on various attributes as the networ inreases Oren and Shwartz, 988; Chaterjee and Eliashberg, 99; Lattin and Roberts, 989; also see Mahajan, Muller, and Bass, 99, for a oprehensive survey. 5

6 infinite horizon settings. Setion 4 pursues nuerial siulations for oparative statis and further insights. Setion 5 onludes.. The odels and setting The fir has two produts, produt and produt, whih have different interdependent deands, D and D, respetively. In the ase of losed soure developent, all uality iproveents arise in-house as a funtion of the nuber of inhouse developers. In the ase of open soure developent, all uality iproveents oe fro the users, as a funtion of the size of the networ. The uality of the software at tiet. t t size of networ of users i.e., installed base at tiet. N t Nuber of in-house developers at tiet. The odel is as follows: The profit in a given period is the revenue D fro the software, plus the revenue D fro the opleentary produt, inus developent osts if appliable. In the open soure ase, is by definition zero, and so the revenue is only D. This loss of revenue ay be offset by the fat that developent osts are zero. In the losed soure ase, developent osts inrease in the nuber of in-house developers, N. We assue a ost funtion of wn in aordane with the eonoi priniple of inreasing arginal ost 4. In the losed soure ase, the in-house ontributors iprove the uality of the software over tie. That is, & N δ,, > The paraeter denotes the produtivity or effetiveness of the in-house losed soure prograers. When is high, losed soure prograers are very effetive and open 4 w is not unit wage but rather a ost paraeter. One an thin of a generalized ost funtion Given the law of inreasing arginal osts, γ is a reasonable exaple. γ wn. 6

7 soure developent ay not be warranted. It is assued that the uality depreiates over tie at a onstant proportional rate 5 δ >. This depreiation an be thought of as obsolesene relative to exogenous tehnologial progress. It is assued that initial uality of the software is positive. No projet starts with zero uality. At the very least, an idea or onept exists, whih an be uantified as inisule uality. Sine the soure is losed to the publi, ontributions by users are not possible. In the open soure ase, on the other hand, there are no in-house developers and all ontribution is through users. ene, & α δ,, > The paraeter α represents the level of involveent by the open soure user ounity, whih inludes users of both the software and of the opleentary produt. The size of the networ of users inreases eah period by the nuber of new users of both produts. owever, the users of both produts need not be eually weighted. A user of the software ay be ore or less valuable to the networ than a user of the opleentary produt. The paraeter a > easures this relative weight. We also assue separability between deands for the two produts in the networ, though the two are opleents. That is, a user who uses both produts has a higher weight in the networ than a user who uses only one produt. The paraeter ε > is the rate of depreiation of the networ or the rate of exit. ene, & ad D ε, > 3 The deand of the opleentary produt D dereases as the prie of the opleentary produt inreases, and inreases as the size of the networ of users 5 Alternatively, we ould have a onstant depreiation eah period. That ould result in explosive solutions and no steady state. It is reasonable to assue that higher uality would anifest in ore rapid obsolesene relative to progressing tehnologies. 7

8 and the uality of the opleentary produt inrease. is assued to be positive and onstant. We assue the deand funtion of the opleentary produt as follows: D exp 4 We use the exponential deand funtion as in aruvy, rasad, and Sethi 3. Suh a funtion exhibits the desired properties of a deand funtion in prie, uality, and networ size. This funtional for has the additional advantage that it preludes the possibility of negative deand. The deand of the software D dereases as the prie of the software inreases, and inreases as the uality of the software inreases. We assue the deand of the software to be as follows: D D exp 5 The ultipliation of the exponential by D represents the effet of opleentarity. That is, the deand for the software produt inreases with the deand for the opleentary produt. The positive onstant in euation 5 is neessary to eep the nuerator stritly above zero in the open soure ase. Without the positive onstant, the deand would ease to be a funtion of in the open soure ase. Sine the prie for the open soure software produt is zero, in the open soure ase we have: D D exp 6 We suarize the two odels as follows: 8

9 Table. The odels Closed Soure Max e s.t. where ρt D D wn dt & N δ, & ad D ε, D D D exp exp Open Soure Max e s.t. where ρt D dt & α δ, & ad D ε, D D D exp exp At this point soe ore disussion is warranted on the relationship between D deand for software and D deand for the opleent. D and D affet eah other, but in different ways. D is affeted by D diretly, through opleentarities. That is, a portion of the opleent s users D users ay wish to use the software D. owever, the software users an use the software independently of the opleentary good they ould substitute any nuber of other produts. D is affeted by D indiretly, through the networ size whih is a funtion of purhases of both and. D is affeted by the networ beause of networ externalities. That is, it is affeted diretly by the size of the networ. eople derive utilities fro other people using the produt, though newsgroups, file sharing, inreased servie and opatible goods. D is not affeted by the networ diretly. owever, in the open soure ase, D is affeted by the networ indiretly through the ipat of the networ on the size of the ode,. 3. Analytial investigation 3.. Open Soure 9

10 The fir axiizes its disounted disount rate is ρ > profit strea, whih onsists of the prie for the oerial produt the prie for the open soure software produt,, is zero ultiplied by the deand for the oerial produt at eah period. s.t. Max exp ρ t exp dt, 7 & α δ,, & a exp exp exp ε,, 6 The prie annot be negative. We for the ailtonian: L exp a exp λ α δ exp Fro this we get the adjoint euations exp, 8 ε η & L a λ ρλ ρ δ λ exp exp, 9 & ρ L ρ ε αλ exp a exp, The optial ontrol ust satisfy L a exp exp η, and the ailtonian ultiplier η ust satisfy the opleentary slaness ondition 6 We assue the lower bound of prie is zero. It is possible to have a positive prie lower bound.

11 η, η, Fro and, we an have: a exp,, Fro 3, we get an euation of otion for. a exp a exp < / a / & & ae & e, a exp & 4, a exp < 3 Now the following uestion arises: In the forulation with infinite horizon, does the optial solution onverge to a steady state as tie approahes infinity? The pratiality of suh a onern is soewhat unertain. After all, neither software produt nor opany an last forever. Nevertheless, the onvergene uestion is entral to dynai probles sine the onvergene property is suggestive of the solution to the finite horizon proble. Sine we want to axiize the present value of the revenue streas, the prie ust be positive in steady state for any viable fir. Therefore, the steady state value of the ailtonian ultiplier is zero, i.e., η. We have the following five euations in the steady state:

12 Table 3. Steady State euations for the open soure odel & α δ 5 & a exp exp exp ε 6 & a λ ρ δ λ exp exp 7 & ρ ε αλ exp a exp 8 a exp 9 Though a losed for analytial solution is intratable, it is of iportane to deterine whether a uniue solution exists to this syste of euations. roposition : In the open soure ase: a The arginal benefit of inreasing the size of the networ is positive. b The arginal benefit λ of inreasing the software uality is positive. The optial profit inreases with α. roof: See Appendix A for detail. Corollary.: In the open soure odel, is an upper bound to. roof: The proof iediately follows fro roposition. Speifially, sine is always positive and the paraeter a is positive, by euation 9, is less than or eual to. roposition : There exists a solution to the open soure odel s syste of steady state euations. That is, there exists a long ter stationary euilibriu,,, λ,, and η. roof: See Appendix B for detail.

13 roposition 3: The solution to the open soure odel s syste of steady state euations is uniue. That is, there exists a uniue,,, λ,, and η. roof: See Appendix C for detail. 3.. Closed Soure The fir axiizes its disounted profit strea, whih is the revenue of the software and the oerial produt inus the developent osts at eah period. Max exp ρ t exp exp exp wn dt s.t. & N δ a exp exp exp ε &, 7, N The pries and the nuber of in-house developers is nonnegative We for the ailtonian: L exp exp exp wn λ N δ a exp exp exp ε η η η3n Fro this we get the adjoint euations & a λ ρ δ λ exp exp & ρ ε exp a exp 3 7 We assue the lower bound of prie is zero in order to have one less paraeter in the odel. A positive prie lower bound an easily be inorporated. 3

14 The optial ontrol ust satisfy L a exp exp η 4 L L N a exp exp η wn λ η3 and the ailtonian ultipliers η, η and η 3 ust satisfy the opleentary slaness 5 6 ondition η, η, η, 3 η η η N 3 7 Fro , we an have: a, a 8, a < a exp,, a exp a exp < 9 N λ, w, λ w λ < w 3 Sine we wish to axiize the present value of the net profit strea, pries and ust be positive in steady state for any viable fir. Therefore, the steady state values of the ailtonian ultipliers of pries are zero, i.e., η η. In addition, the euation of otion & N δ iplies that the steady state value of the nuber 4

15 of in-house prograers, N, is positive. Therefore, the steady state value of the ailtonian ultiplier of the nuber of in-house prograers ust be zero, i.e., η 3. The following seven euations ust hold in steady state: Table 4 Steady State euations for the losed soure odel & N δ 3 & a exp exp exp ε 3 & a λ ρ δ λ exp exp 33 & ρ ε exp exp a 34 a 35 a exp 36 λ N 37 w roposition 4: In the losed soure ase: a The arginal benefit of inreasing the size of the networ is positive. b The arginal benefit λ of inreasing the software uality is positive. The optial profit inreases in and dereases in w. roof: See Appendix D for detail. Corollary 4.: In the losed soure odel, is an upper bound to and is an upper bound to. roof: The proof iediately follows fro roposition 4. Speifially, sine is always positive and the paraeter a is positive, by euation 35, is less than or eual to. By euation 36 is less than or eual to. 5

16 roposition 5: There exists a solution to the losed soure odel s syste of steady state euations. That is, there exists,,,, N, λ and. roof: See Appendix E for detail. roposition 6: The solution to the losed soure odel s syste of steady state euations is uniue. That is, there exists a single,,,, N, λ and. roof: See Appendix F for detail. Finite horizon T with no salvage value Suppose the produts, both the software and the opleentary produt, have finite lives with a nown terinal period. That ould be due to an antiipated release of a new generation of produts or tehnologies or due to a nown date for the fir to ease operations. In that ase, we an set the salvage value to zero. When the tie horizon is suffiiently long and the disounting signifiant, profit oparisons in the finite and infinite horizons should be lose. The forulation of the finite horizon proble with zero salvage value is the sae as before, but with the addition of the boundaries λ T and T. roposition 7: In the losed soure odel, the nuber of in-house developers N delines over tie for a finite horizon proble with zero salvage value and eventually N goes to zero at T. On the other hand, N approahes steady state over tie in the infinite horizon proble. roof: See Appendix G for detail and Figure for nuerial results. 4. Nuerial Analysis We exaine different senarios nuerially. In both open soure and losed soure odels, and both finite horizon without salvage value and infinite horizon settings. We use Fortran ode and Exel spreadsheets to find nuerial solutions. Due to the oputational liitations, nuerial solutions in both the finite and infinite 6

17 horizon settings are the result of approxiations e.g., Sethi and Thopson,, pp We use the following paraeters: δ.3, ε., ρ., a w, α, Due to networ externalities, pries for both produts in the losed soure odel and the prie for the opleentary produt in the open soure odel begin low in order to uily establish the networ s installed base and gradually inrease as the networ size inreases, as figure deonstrates. <Insert Figure here> We an see that prie in the initial periods, for both software and opleentary produt is zero. This is beause of the networ externality effet. In order to build the networ and extrat the axial surplus, the produer is willing to sarifie short-ter profits for larger future profits. In the losed soure odel, hiring is assive early on due to the need to rapidly inrease uality in order to build up the networ As a result, we see uality inreases rapidly in the initial tie period. In finite horizon proble, hiring eases up over tie and uality begins dereasing towards the end of the life of the produts. This is uite intuitive sine given an antiipated produt ease, it is no use to hire people and iprove the uality at the end of the life of the produt. In the infinite horizon proble, hiring approahes steady state over tie and so does the uality. In the open soure odel, uality inreases ontinuously sine in this senario uality inreases as the size of networ of users inreases. The differene between finite and infinite horizon probles is that uality goes to steady state in infinite horizon proble and it is not the ase in finite horizon proble, as figure deonstrates. <Insert Figure here> 7

18 An iportant differene to note between the open soure and losed soure in the finite horizon setting is that in the losed soure ase software uality dereases towards the end of the life of the software, whereas in the open soure ase it ontinues to inrease. This will always be universally true and not just in the nuerial exaple shown here. This is beause the fir will not find it optial to invest in uality when the produt nears its death, whereas the open soure ounity, not otivated by profit, will ontinue to iprove. This presents interesting welfare ipliations to open soure developent, partiularly if the benefit to soiety fro inreased uality is substantial. Next, we illustrate the idea that when a fir has to hoose between open soure and losed soure, that hoie ay depend on the values of the paraeters α,, w, and the initial uality. Fro proposition and proposition 4 it is lear that threshold values exist for the paraeters α,, w, above whih one odel doinates and below whih the other odel doinates. The nuerial analysis learly onfirs that thresholds exist. There appears to exist a threshold open soure ounity involveent, α, above whih open soure will be preferred to losed soure. See Figure 3 <Insert Figure 3 here> Siilarly, there sees to exist a threshold in-house prograer produtivity,, above whih losed soure will be preferred to open soure. See Figure 4 <Insert Figure 4 here> In onsidering whether to develop in-house software, the fir ust onsider wages as well as in-house produtivity. As for in-house produtivity, we see a threshold wage fator, w, below whih open soure will be preferred to losed soure. See Figure 5 <Insert Figure 5 here> 8

19 Finally, initial uality is a ritial onsideration. When initial uality is high, the fir has little to benefit fro opening the soure ode to the open soure ounity. As suh, there exists a threshold initial uality,, below whih open soure will be preferred to losed soure. See Figure 6 <Insert Figure 6 here> In addition to oparative statis, we attept to exaine soe of the orollaries nuerially. First, we attept to nuerially verify orollary.. Aording to that result, in the open soure odel, is an upper bound to. Figure 7 suggests that is not only an upper bound on but that losely tras in agnitude and slope. <Insert Figure 7 here> Next, we illustrate orollary 4. nuerially. Speifially, we wish to verify that in the losed soure odel, is an upper bound to and is an upper bound to, and to see how losely pries tra their upper bound. Surprisingly, while losely tras uality in slope and agnitude, slowly diverges fro. <Insert Figure 8 here> Moreover, we find that in the losed soure ase, the higher the initial uality of the opleentary produt,, the longer the prie of the software will reain at zero initially. In addition, the higher the, the higher the and the lower the at the end of the tie horizon. <Insert Figure 9 here> 9

20 5. Conlusions Whereas the speed and reativity assoiated with open soure developent are well aepted, the profit potential is not always reognized. As opanies beoe inreasingly involved in open soure projets, the profit ipliations of open soure for opanies ust be better understood. In this artile we argue that the profitability of open soure software ay oe fro a produt whih is a opleent to the open soure ode. A fir onsidering open soure developent as an alternative to losed soure developent would need to arefully review the relationship between the software in uestion and the fir s other produts. If the software is found to enhane the usefulness or uality of opleentary produts and/or if the users of the software and the users of the opleentary produt belong to the sae networ, the opleentarity an be exploited. By dynaially and siultaneously anaging prie, produt uality, networ size, and hiring for both produts, the fir will be able to best exploit the opleentarity. This effort ay or ay not benefit fro opening the soure of the produt. Without a lear odel of how to exploit freeware, open soure developent as a substitute to in-house losed soure developent ay be detriental to a fir s profitability. Our analytial derivation and siulations deonstrate that under various onditions open soure ay not be benefiial to a fir. On the other hand, there are senarios where the opposite is the ase. Speifially, open soure ounity involveent is ritial to the suess of open soure initiative. Only above a ritial level of ounity involveent, open soure beoes a viable alternative to losed soure. To the extent that ounity involveent an be influened by the fir s efforts or influene, suh efforts ust opleent the priing deisions evaluated here. In-house prograer produtivity was shown to be the opposite side of the sae oin. More produtive and effiient in-house prograers result in less reliane on open soure. owever, wage is ritial in that respet. If prograer produtivity is high but wage is higher than soe threshold, our results show that open soure would be preferred. Finally, if the initial uality of the software is high, developent beoes a less ritial onsideration and extration of

21 surplus an begin iediately. In suh a ase, the fir would prefer to harge a positive prie for the software and lose the ode. We also haraterized prie, uality, and hiring paths for firs under both the open soure and losed soure odels. We find that due to networ externalities, pries for both produts in the losed soure odel and for the opleentary produt in the open soure odel will begin low in order to uily establish the networ s installed base and gradually inrease as the networ size inreases. We find that in the losed soure ase, the higher the initial uality of the opleentary good, the longer the prie of the software will reain at zero initially. That is, when ore surplus an be extrated fro the opleentary good, it ay be optial to have the software as a freeware for a long period of tie, even when the soure is losed. In the losed soure odel for produts with finite lives, uality will rapidly inrease early on to build the networ size and begin dereasing towards the end of the life of the produts. Siilarly, hiring will be assive early on and will ease up over tie. This is due to the need to rapidly inrease uality in order to establish the networ. This is not the ase in infinite horizon losed soure odel and not in the finite or infinite horizon open soure odels. As suh, it ay be argued that for finitely lived produts, open soure iproves soiety s welfare in ters of both uality and produtivity. Future researh should exaine other fors of revenue extration fro open soure as well as the proble fro a soial planner s perspetive. We would further lie to exaine the effet of opetitive pressures on the fir s deision to use open soure. When the fir opens it soure, it ay ipliitly provide advantages to opetitors. Copatibility and legal protetion will beoe paraount onsiderations when opetition is onsidered.

22 Referenes Bass, F. M. 969, A New rodut Growth Model for Consuer Durables, Manageent Siene, 5, 5-7. Bro, W. A., and Sheinan, J. A. 977, The global asyptoti stability of optial ontrol with appliations to dynai eonoi theory, Appliations of Control Theory to Eonoi Analysis. Ed. By J.D. ithford and St. J. Turnovsy. Asterda New Yor Oxford 997, Essay8, 73-5 Chaterjee, R. and J. Eliashberg 99, The Innovation Diffusion roess in a eterogeneous opulation: A Miroodeling Approah, Manageent Siene, 36, Chintagunta,. K., and Rao, V. R riing strategies in a dynai duopoly: A differential gae odel, Manageent Siene 4,, Dhebar, A. and S. Oren 985, Optial Dynai riing for Expanding Networs, Mareting Siene, 4, Feihtinger, Gustav; Rihard F. artl; Suresh. Sethi 994, Dynai Optial Control Models in Advertising: Reent Developents, Manageent Siene, 4., aruvy, E. and rasad, A. 998, Optial rodut Strategies in the resene of Networ Externalities, Inforation Eonois and oliy, aruvy, E. and rasad, A., Optial rodut Strategies in the resene of Networ Externalities: An Evolutionary Gae Theoretial Approah, Journal of Evolutionary Eonois, aruvy, E., A. rasad, and S. Sethi 3, arvesting Altruis in Open Soure Software Developent, forthoing in Journal of Optiization Theory and Appliations. su, J. and A. Meyer 968, Modern ontrol priniples and appliations MGraw-ill, New Yor 968 Kalish, S. 983, Monopolist riing with Dynai Deand and rodution Cost, Mareting Siene,, Kalish, S. and G. Lilien 983, Optial rie Subsidy for Aelerating the Diffusion of Innovation, Mareting Siene, 47-4 Lattin, J. M. and J.. Roberts 989, Modeling the Role of Ris Adjusted Utility in the Diffusion of Innovations, woring paper 9, Graduate Shool of Business, Stanford University. Mahajan, V., Muller, E. and F. M. Bass 99, New rodut Diffusion Models in Mareting: A Review and Diretions for Researh, Journal of Mareting, 54, - 6. Muller, Eitan and Yora eles 988, The Dynai Adjustent of Optial Durability and uality, International Journal of Industrial Organization, 6, pp

23 Nasiento, F. and W. Vanhonaer 988, Optial Strategi riing of Reproduible Consuer roduts, Manageent Siene, 34, Oren, Shuel S. and Ri G. Shwartz 988, Diffusion of New roduts in Ris- Sensitive Marets. Journal of Foreasting 7, Rayond, E. S., The Cathedral and the Bazaar: Musings on Linux and Open Soure by an Aidental Revolutionary. O Reilly, Sebastopol, CA. Rogers, Everett M. 983, Diffusion of Innovation, 3 rd ed. New Yor: The Free ress. Shiff, A., The Eonois of Open Soure Software: A Review of the Early Literature, Review of Networ Eonois,, Sethi, S.. and Thopson, G. L., Optial Control Theory: Appliations to Manageent Siene and Eonois, Seond Edition, Kluwer Aadei ublishers, Boston. 3

24 Figures Figure : ries over tie in finite and infinite horizon senarios Closed Soure Finite, w 3 5 rie Tie Closed Soure Infinite, w 5 45 rie Tie 4

25 Open Soure Finite α rie Tie Open Soure Infinite α rie Tie 5

26 Figure : uality and hiring path for both finite and infinite horizon Closed Soure -- Finite, w N N Tie Closed Soure Infinite, w N N Tie 6

27 Open Soure Finite α uality Tie Open Soure Infinite α 5 uality Tie 7

28 Figure 3: rofit over different values of α for open soure and losed soure Open Soure rofit 35 3 Closed Soure, w Alpha Figure 4: rofit over different values of for open soure and losed soure rofit Closed Soure w Open Soure α

29 Figure 5: rofit over different values of w for open soure and losed soure 6 5 Open Soure α 4 rofit 3 Closed Soure w Figure 6. rofit over levels of initial uality Finite Closed Soure, w rofit Open Soure α

30 Figure 7: Illustration of Corollary.. Open Soure Infinite orizon α * / * 5 5 Tie Figure 8: Illustration for Corollary 4. versus Closed Soure - Infinite * versus Closed Soure - Infinite, w, w / 5 5 * / * Tie 5 5 Tie 3

31 Figure 9: The effets of on and for losed soure Finite 8 rie Tie 8 rie Tie 3 rie Tie 3

32 Appendix A: roof to proposition a Due to networ externalities, the prie for the opleentary produt in the open soure odel starts at zero in order to uily establish the networ s installed base and gradually inreases as the networ size inreases. Aording to euation 3 a exp,, a exp a exp < So at the very beginning, a exp <, that is > > a exp >. At ertain tie point, say, a exp. Tae the derivative of respet to at tie, we have a e t < also beause of ontinuity < and >. Using the sae arguent, we an prove that >, for all t of the networ is always positive.. So the arginal benefit fro inreasing of the size b By ontradition. Suppose at ertain tie point, λ <. We have & a λ ρ δ λ exp exp then λ <, for t. So λ t, as t <. That eans the inreasing of the uality would derease the profit infinitely, whih is ipossible. So λ >, for all t. Suppose that the profits for open soure and losed soure interset for soe value of α. Sine open soure profit is onotonially inreasing in α Use Envelop Theore and 3

33 dv L euation 8, we have λ > and losed soure profit is unhanging in α, dα α there will be a single point of intersetion a threshold value α. Appendix B: roof to proposition We try to apply the results by Bro and Sheinan 977 to our odel to prove the existene of the steady state. Lea : The profit funtion in open soure proble is onave in,. roof: Denote U, D exp as the profit funtion, The essian of U, is U U U U e 3 e e e Sine <, U <, U and U U U U The onavity follows. < Let V ρt, ax e U, Lea : If U, is onave in,, then V is onave. dt roof: By the definition of onavity. 33

34 We denote by,, the vetor of the state variables and, λ, are the vetor of the adjoint variables or shadow pries. Bro and Sheinan 977,.9 prove under the assuption of onavity of funtion U, & the following Lea 3: The shadow prie an be written as and the atrix / sei-definite and syetri at eah where it exists. It exists alost everywhere. is negative Now we onsider the syste of ordinary differential euations in the plane: &,,,, F &,,,, F * The Jaobian atrix of F is V J F V where and ;. su and Meyer 968, p.64 proved the following theore: Lea 4: If the trae does not hange sign for all, then there are no liit yle for *. J F It an now be proven using the result of Bro and Sheinan 977,.9-93: Lea 5: Let B negative definite, then i trae AB <. A, B be two real atries that are syetri; A positive definite and If only sei-positive definiteness and sei-negative definiteness holds, then ii trae AB 34

35 35 If neither A nor Bis and trae AB, then iii A and B are both singular. Theore : Assue that there is a bound M suh that / M t For all t. Also assue '' V, positive definite, and trae, then t t, /, when is a rest point of *. Now we apply these results to our open soure odel under the assuption >. We have δ α λ & &, exp, exp exp a ε & & So λ λ λ e ae e ae ε α δ e ae e ae e a We need to derive λ,,. We have

36 a exp So, a exp < and λ We have ae e >, so is syetri and positive sei- V definite. In addition V is onave so negative sei-definiteness and syetry follows. By lea 5, V J F V trae J trae trae trae Copute trae δ ae e ε δ ae e ε < Theore : In open soure ase, if a at the steady state, trae < and b there is a losed urve Γ ontaining the steady state suh that &, point inward on Γ, then, the steady state is loally asyptotially stable exept for hairline ases. 36

37 The ondition b is saying that & i > if i is sall enough and & i < if i is large enough. In our ase, & > if is sall enough and & < if is large enough; and & > if is sall enough and & < if is large enough. This an be easily shown. Appendix C: roof to proposition 3 We redue euation 5 9 to one euation and show that it has a uniue solution. Fro 7, a λ exp C ρ δ Fro 9, Fro 6, a exp C a exp ln C3 ε Let C C3, a exp ln ε C4 a exp Fro 5, δ C5 α ut 9, C, C4 and C5 into 8, we an get f ρ ε aεδ ρ δ a e ρ ε aεδ ρ δ a e α ae ln εδ 37

38 38 When, li < f, li > f > f For > So there exists a uniue. Therefore, there exist uniue,, λ,. Appendix D: roof to proposition 4 a By ontradition. Suppose at ertain tie point, <. Fro euation 9, < exp, exp, exp a a a i If exp a, then exp a. ut into 3, we have exp < ε ρ & D We have < t, for t. So t, as t. That eans the inreasing of the size of the networ of users would derease the profit infinitely, whih is ipossible. ii If exp < a, then. ut into 3, we have < ε ρ & D

39 Sae as the previous ase, t <, for t. So t, as t. That eans the inreasing of the size of the networ of users would derease the profit infinitely, whih is ipossible. So t >, for all t. This eans that the arginal benefit fro inreasing of the size of the networ is always positive. b By ontradition. Suppose at ertain tie point, λ <. We have & a λ ρ δ λ exp exp < D3 Sine t and t > for all t, then λ t <, for t. So λ t, as t. That eans the inreasing of the uality would derease the profit infinitely, whih is ipossible. So λ t >, for all t. Suppose that the profits for open soure and losed soure interset for soe value of. Sine losed soure profit is onotonially inreasing in Use Envelop Theore and dv L euation, we have λ N > and open soure profit is unhanging in, d there will be a single point of intersetion a threshold value. Suppose that the profits for open soure and losed soure interset for soe value of w. Sine losed soure profit is onotonially dereasing in w Use Envelop Theore and euation, we dv L have N < and open soure profit is unhanging in w, there will be a dw w single point of intersetion a threshold value w. Appendix E: roof to proposition 5 We apply the sae ethod as in Appendix to prove the existene of the steady state. We denote by,, the vetor of the state variables and by x,, the vetor of ontrol variables,, λ, are the vetor of the N adjoint variables or shadow pries. 39

40 4 Let, x U exp exp exp wn, ax dt x U e V t x ρ Lea 6: The profit funtion, x U in losed soure proble is onave in, x. roof: Apply the siilar proof as the proof of Lea. Lea 7: If, x U is onave in, x, then V is onave. roof: By the definition of onavity. Under the assuption,, > N We have, N δ λ & &, exp, exp, exp a ε & & So λ λ λ λ λ e ae e a e ae e a N N

41 4 e ae e a e a N δ ε e ae e a e ae N We need to derive λ N, λ, λ, N,, N,,, N,, Fro euation 35 37, we have a f exp a f 3 w N f λ Using the ipliit funtion theore we obtain: > e a N f f f N f f f N f f f J Sine a And > w N λ, N λ, < a λ, < ae

42 N, N >, e <, > We have so > w, a e ae e > V is syetri and positive definite. In addition V is onave so negative sei-definiteness and syetry follows. By lea 5, V J F V trae J trae trae trae Copute trae δ ε ae e < Theore 3: In the losed soure ase, if a at the steady state, trae < and b there is a losed urve Γ ontaining the steady state suh that &, point inward on Γ, then, the steady state is loally asyptotially stable exept for hairline ases. 4

43 The ondition b is saying that & i > if i is sall enough and & i < if i is large enough. In our ase, & > if is sall enough and & < if is large enough; and & > if is sall enough and & < if is large enough. Appendix F: roof to proposition 6 We redue euation 3 37 to two euations. Fro 3, δ N F Let 37 F, wδ λ F ut 35 and F into 33, we get ρ δ wδ exp exp F3 and ρ δ wδ a ln a F4 Let 36 F4, a exp ρ δ wδ a ln a F5 ρ δ wδ a ln a F6 ut 36 into 34, & ρ ε exp F7 43

44 ut 35, F3 and F6 into F7, F, ρ ε ρ δ wδ a ρ δ wδ ln a a exp a ut F3 and F6 into 3, a ρ δ wδ a F, a exp ln a ρ δ w δ a a exp ε exp a Given, we an alulate the value of, whih satisfies F, and the value of, whih satisfies F,, onditioned on, > and, > At steady state, pries are positive. For the graph below, we plot the points on the range of whih eets the positive prie ondition. 6 Uniueness Closed Soure, w Fro this graphial illustration, dereases fro a positive nuber as inreases, whereas inreases fro a negative nuber as inreases. So there exist uniue and satisfy both F, and F,. Therefore,,,, λ, N are uniue. 44

45 Appendix G: roof to proposition 7 λ Condition N need to be satisfied for all t. w In the finite horizon without salvage value proble, λ T. ene, λ starts positive and delines to zero by tie T. So does N. Note that λ T eans that uality has no salvage value by tie T. ene, it is intuitive that the fir would not wish to spend oney to hire developers towards the end of the life of the produt. In the infinite horizon proble, the existene and the uniueness assure that N is onstant after a suffiiently long period of tie. 45