Open Source Software, Competition and Potential Entry

Transcription

1 Open Source Software, Competton and Potental Entry Po Baake fo Insttut for Economc Research Thorsten Wchmann Berlecon Research GmbH May 2003 Prelmnary Verson Abstract We analyze a model wth two software frms, qualty mprovng codng expendtures and potental competton. The frms can publsh parts of ther software as open source. Publshng software mples postve spllovers and thus reduces the frms codng costs. On the other hand there exst two negatve effects. Frst, lower codng costs nduce hgher codng expendtures whch decreases the frms profts f ther programs are substtutes. Second, open source encourages entry and ncreases the expendtures requred to deter entry. The frms optmal open source decsons balance these opposte effects. JEL classfcaton number: L10 Keywords: Open Source, Spllovers, Potental Entry Poschngerstr. 5, D München, Germany. E mal: baake@fo.de Oranenburger Str. 32, D Berln, Germany. E-mal: tw@berlecon.de 1

2 1 Introducton Open source software (OSS), such as the Lnux operatng system or the Apache web server have recently found ncreasng nterest n the software ndustry as well as n the economc research communty. OSS s software, of whch the source code (.e. the code nstructons showng how the software works) s publshed and thus made open. In contrast, most software today s only avalable n bnary code, whch hdes the way the program works. 1 Typcally, ths source code can be accessed free of charge and compled nto a bnary program, whch can be executed on computers. However, OSS s not free n the sense of you can do wth t what you want. Rather, t s protected by copyrght just lke all other forms of software. Its usage requres acceptance of and adherence to the terms of the lcence under whch t has been publshed. Often, these lcenses pose certan condtons upon some usage forms, such as alterng the program or ntegratng t wth other applcatons. Nevertheless, OSS code s free for everyone to nspect and to derve from ths nspecton how the programmer has solved a certan problem. 2 At frst sght, the exstence of OSS seems to be a puzzle. Why should anyone take the effort to wrte a program, whch s then made avalable free of charge to the world and from whch everybody can steal deas about how a trcky problem can be solved? Obvously lesure or altrustc motves are able to explan these actvtes of programmers and therefore early work on OSS focussed on these explanatons. Most current work, however, emphaszes recprocty or ndvdual labor market consderatons. For nstance, Lerner/Trole (2002) argue that a programmer can sgnal her codng abltes by partcpatng n open source projects. Ths should rase her expected future wage or gve her access to programmng jobs, as Raymond (2000, Chapter 5) already ponted out, although he consders the latter as rare and margnal motvaton for most hackers. Although mportant for explanng the open source phenomenon, the focus on the ndvdual programmer neglects an mportant open source drver: frms. Part of the open source communty conssts of ndvduals employed explctly for developng open source software. Ghosh et al. (2002) surveyed OSS developers and 1 Just lke Coca-Cola does not publsh ts recpe on ts bottles. 2 Ths short descrpton only descrbes coarsely what OSS s. There are many varants of OSS as well as more extensve concepts lke free software. A short ntroducton to the dfferent ssues can be found n Wchmann/Spller (2002). 2

3 found that 16% of them were pad drectly for developng open source software. For another 13% the development of OSS s part of ther work. Thus, ther contrbutons to open source projects are the result of frms delberate decsons to fnance the development of OSS. In addton there exst several examples of companes that have made avalable formerly propretary software as open source software. 3 Despte these consderable actvtes the companes motvatons behnd ther OS engagement s not as well understood as the motvaton of ndvdual developers. Although dscussed n passng by, e.g., Lerner/Trole (2002a) or Schmdt/Schntzer (2002), much less attenton has been devoted to frms open source actvtes than to open source actvtes of ndvduals. If OS actvtes by frms are dscussed at all, they are often explaned as actvtes to sell complmentary goods: Frms gve away ther software for free n order to sell more of a complementary good,.e., hardware or other software programs. However, even these arguments mss a crucal part of the story. Snce open source software s avalable to everyone, the OSS developed by one frm can be ntegrated nto the products of another frmandvceversa.thus,theoss actvtes seem to resemble much more the frms engagements n basc research or n standardzaton actvtes (see Lerner/Trole (2002b) or Wchmann (2002)). Just lke n basc research there exst counteractng postve and negatve effects: Makng the source code of a program publcly avalable enables educated users to fnd flaws and errors n the code and thus ncreases the qualty of the software. Gven enough eyeballs, all bugs are shallow (Raymond, 1998)samantraof the open source communty. Addtonally, other programers may also contrbute to the solutons of certan problems and may thus decrease the frm s own codng costs. However, there also exst negatve effects from gong open source : OSS publshed by a frm s avalable to all other frms. Therefore, also the frm s rvals may drectly beneft from the frm s OSS, whch n most cases a frm would rather not happen. These counteractng forces form the startng ponts of our paper. We study the effects, of dfferent technology and market envronments on frms decsons to publsh ther software under an open source lcense. We dscuss a smple world, n whch frms sell software products to consumers. 3 OneexamplesNetscape,whchmadethesourcecodefortsbrowseravalableasOSS. The browser Mozlla has developed out of ths project. Another famous example s Sun, whch has publshed the source code to ts offce sute StarOffce leadng to the open source sute OpenOffce. 3

4 As the prce for pure open source software for consumers s typcally zero, ths can best be thought of as a world of hybrd software. The best-known example of such software s probably the MacOS X operatng system by Apple. MacOS X s basedonanopensourceversonoftheoperatngsystemunxcalleddarwnand n fact ncludes an ncreasng number of OSS components, but t also ncludes components that are not open source. Thus, software used by the consumers s of the type of a package or bundle, whch conssts of several components. Some of them may be OSS. 4 Frms can decde to publsh some of ther software components under an open source lcense. In our model we assume a rather lberal lcense that bascally allows each user to use, modfy, ntegrate and dstrbute the software components wthout restrctons. Several OS lcense types are very close to ths deal scenaro, e.g. BSD-type lcenses (governng Apache or the FreeBSD Unx) or the Artstc Lcense (governng the programmng language Perl). Modellng other, more restrctve, lcense forms such as the General Publc Lcense (GPL) s left to future work. 5 Publshng software components under an open source lcense mples postve spllover effects and thus reduces the frms codng costs. However, there are also two negatve effects. Frst, lower codng costs nduce hgher codng actvtes, whch decrease the frms profts f ther software programs are substtutes. Second, publshed software components encourage entry and ncrease the nvestment requred to deter entry. We analyze the nteracton of these dfferent effects and show that some software wll be publshed as OSS even f the frms programs are substtutes and even f open source encourages market entry. Furthermore, we show that open source decsons can be nterpreted as ether strategc substtutes or complements dependng on whether the frms programs are substtutes or complements. In the next secton we set out our model. Secton 3 consders the optmal open source decsons. Solvng the model we frst present an overvew of the economc effects whch determne the soluton of the model. Wth respect to the formal solutons we do not provde a detaled dscusson; rather, we try to llustrate the man results graphcally. In secton 4 we provde a short summary. 4 For example, a word processor typcally conssts of a core program and many addtonal components, whch provde functonalty for spell checkng, drawng dagrams or mathematcal equatons. In the same way an operatng system conssts of many dfferent components. 5 For a comparson of the relatve mportance of the dfferent OS lcense types see Lerner/Trole (2002b). 4

5 2 The Model We consder a four stage game wth ntally two frms =1, 2 and potental market entry. We start by explanng the tmng of our model and then turn to the specfc assumptons on costs and demand. Tmng Two frms =1, 2 offer dfferent software programs composed by a varety of components. In the frst stage each frm decdes, whch of these components t publshes as open source software. In the second stage the frms choose ther codng expendtures n order to develop the qualtes q 1 and q 2 of ther software programs. Market entry takes place n the thrd stage. In our model entrants can beneft from the open source components revealed by frms 1 and 2. Werestrct entry such that for every market there s one potental entrant e =1, 2. Fnally, the two frms 1 and 2 and any actual entrant set ther prces. Summarzng, we have: t 0 : Frms 1, 2 decde on ther open source components. t 1 : Frms 1, 2 decde on ther qualtes q 1 and q 2. t 2 : Entrants decde whether to enter or not and on ther qualtes q e. t 3 : All frms set ther prces, demand and profts are realzed. Costs Both frms can not only decde how much they spend on codng,.e., n developng ther software programs, they can also decde whch of ther software components they publsh as open source programs. To smplfy the analyss of these decsons we use the followng reduced form approach: Let q denote the qualty of frm s software program and assume that q also measures the software components needed for ths program. Wth α [0, 1] denotng the fracton of open source components that each frm publshes, the frms costs for developng a qualty q are gven by (, j =1, 2, 6= j) 1 c (q,q j, α, α j )= q 2. (1) 2+α q + α j q j Accordng to ths cost functon, three factors nfluence costs: The costs are ncreasng n the own qualty chosen by each frm due to hgher codng expendtures. They are decreasng n the fracton of ther open source components due to bug fxng and mprovement of these components by users. And fnally, the costs of frm also decrease wth α j q j as t can use and learn from the OSS components of ts compettor. Turnng to the costs of potental entrants we have to take nto account that market entry takes place n stage 3. Hence, entrants can use the open source 5

6 components that frms 1 and 2 have made publc to create a software clone for each market at lower costs than the ncumbent frms. The costs of an entrant n market are thus gven by where q e c e (q e,q,q j, α, α j )= 1 2+α q + α j q j (q e α q ) 2 for q e α q (2) denotes the qualty of the entrant s software program. Demand We start by characterzng demand n the case where only the two programs of frms =1, 2 are avalable. Wth p as the programs prces demand D M (p,p j,q,q j, β) for program s n our model gven by ½ 1 D M ( ) =max q p + β( q j p j ) ¾, 0 wth β 1 1+β 2, 1. (3) 2 Whle β < 0 mples that the programs are substtutes, the programs are complements f β > 0. The factor 1/ (1 + β) normalzes aggregate demand such that the sum of D 1 and D 2 only depends on qualtes and on prces but not on the magntude of β. Snce frms may publsh software components as open source and snce entrants may offer programs wth qualtes q e1 and q e2,wehavetomodfy(3)norder to derve the demand functons frmsfacencaseofopensourceorentry.wth respect to the frst pont note that the qualtes of the open source components are gven by α 1 q 1 and α 2 q 2 and that ther prces are equal to zero. Concernng market entry, assume that entry has taken place and that entrants offer programs wth qualtes q e and prces p e. We assume that the consumers decsons to ether buy q,buyq e or smply use the open source components can be traced back to a comparson of the consumers rents as mpled by (3). The alternatve that yelds the hghest consumer rent wll be chosen. Usng the assumpton that D M ( ) s lnear n prces, we get the followng demand functons D ( ) and D e ( ) for the frms =1, 2 and for the entrants e (formally, D ( ) and D e ( ) depend on p,p e,q,q e, β and α wth =1, 2) ( 1 1+β q p + βθ j f p p D ( ) = (4) 0 else wth : p := max mn q q e + p e, q ª ª α q, 0 n qj and : Θ j := max p j, α j q j, o q ej p ej h D e 1 qe 1+β p e + βθ j f p e < p ( ) = e (5) 0 else wth : p e := max mn qe q + p, q e ª ª α q, 0 6

7 Havng specfed the frms costs and ther demand functons we now turn to the soluton of the game, whch s solved by backward nducton. 3 Optmal Open Source Decsons Before we analyze the varous stages of the game n more detal let us brefly summarze the man effects whch determne the soluton of the model. The basc ncentves for frms 1 and 2 to publsh software components as open source are due to the nduced reductons of ther own codng costs. However, as consumers mght be satsfed usng the open source components, gong open source mples that the consumers reservaton prces for the (commercal) software programs decrease. Moreover, t also leads to postve spllover effects wth respect to the other frms costs. Open source thus strengthens the frms ncentves to develop hgher qualtes and more mportant encourages market entry. We frst dscuss how these effects nteract and how they affect the frms open source decsons. We then turn to the formal analyss of the model where we skp most of the detals. Rather, we llustrate the man results graphcally. 3.1 Entry Deterrence and Strategc Interdependences Potental entry combned wth Bertrand competton on the last stage of the game mples that entry deterrence s both feasble and optmal for frms 1 and 2. Entry deterrence s feasble f the frms open source fractons α are relatvely low. Snce the reservaton prces p e of the entrants demands decrease wth the qualtes q 1 and q 2, respectvely (see (5)), frms 1 and 2 can deter entry by choosng relatvely hgh qualtes. Furthermore, due to the postve spllover effects the entry deterrng qualtes are hgher, the hgher the number of software components the frms publsh as open source. Entry deterrence s optmal snce (proftable) entry n market leads to p =0. Usng our assumptons on costs and demand t turns out that the frms open source decsons are such that the frms are n fact forced to choose entry deterrng qualtes. Hence, the optmal open source decsons of frms 1 and 2 balance the postve effects due to cost reductons and the negatve effects due to the tghtened restrcton wth respect to entry deterrence. In order to characterze the strategc nterdependence between the frms open source decsons note that neglectng postve spllovers due to open source the frms qualtes are strategc substtutes (complements) f ther programs are 7

8 substtutes (complements),.e., f β < 0 (β > 0). Combnng these observatons wth the result that the entry deterrng qualtes ncrease wth the number of software components publshed as open source, we fnd that the open source decsons are ether strategc substtutes or complements. If the frms programs are substtutes ther open source decsons tend to be strategc substtutes. The frms margnal profts from ncreasng ther own qualtes are lower the hgher the qualty of the other frm. Snce each frm s (entry deterrng) qualty ncreases wth t s level of open source, we fnd that the frms ncentves to provde open source are lower the more open source components the other frm publshes. Wth software programs that are complements the converse holds. The frms margnal proft from ncreasng ther own qualtes are hgher the hgher the other frm s qualty. Thus, the frms ncentves to provde open source ncrease wth the other frm s open source level. In other words, wth complementary programs open source serves as a commtment devce for choosng hgh qualtes. 3.2 The Prce Subgame Combnng revenues and costs yelds the followng proft functonsπ ( ) and π e ( ) for the frms 1 and 2 and for the entrants e 1 and e 2 : 6 π ( ) =p ( )D ( ) c ( ) and π e ( ) =p e D e ( ) c e ( ). (6) Consder frst the optmal prces of frm and of entrant e. Usng (4) and (5) and maxmzng π ( ) and π e ( ) wth respect to p and p e, respectvely, yelds the followng prce reacton functons ½ ¾ ½ ¾ 1 1 p r ( ) =mn 2 [ q + βθ j ], p and p r e ( ) =mn qe + βθ j, pe (7) 2 Usng 0.5 < β < 0.5 and (7) shows that there exsts a unque prce equlbrum n pure strateges. The equlbrum prces p ( ) and pe ( ) on both markets are characterzed n the followng Result 1 The equlbrum prces p ½ ½ 1 p ( ) = max mn 2 ( ) and pe ( ) satsfy (for, j =1, 2, 6= j)7 h q + βθ j, q q e, q α q ¾, 0 ¾, (8) 6 To shorten the notaton we often omt the arguments of the functons. Clearly, the frms as well as the entrants profts depend on all prces, on all qualtes, on the frms open source decsons α wth =1, 2 as well as on β. 7 Note that the equlbrum prces p ( ) and p e ( ) only depend on the qualtes q and q e wth =1, 2. 8

9 p e ( ) = ½ ½ 1 h qe max mn + βθ j 2 wth : n Θ qj j := max p j, o q ej p e j, q e q ¾ ¾, 0 (9) Proof. Wth. 0.5 < β < 0.5 and (7) we have 1 < p r ( )/ p j < 1 and 1 < p r e ( ) p j < 1. These propertes guarantee the exstence of a unque pure strategy equlbrum. Usng (7) and consderng both markets 1 and 2 leads to p ( ) and p e ( ). Result 1 reveals p e ( ) >p ( ) =0ff q e >q.thats,entrycanbeproftable ff the entrant s qualty s hgher than the qualty q.vceversa,fentryoccurswth q e >q frm wll ncur losses, whch mmedately mples that entry deterrence s optmal for frm. Furthermore, even wthout entry,.e., even n a stuaton wth q e1 = q e2 =0,the equlbrum prces of frms 1 and 2 canberestrctedbytheconsumers alternatve to use the open source components. Usng q e1 = q e2 =0tocharacterze the stuaton wthout entry (8) yelds 8 ½ 1 h p q ( ) =mn + β( q j p 2 j), q ¾ α q. (10) Note that the restrcton p q α q tends to be more severe, the larger β,.e., the hgher the complementary between the frms programs. 3.3 Market Entry and Entry Deterrence Although t turns out that the frms 1 and 2 wll deter entry, we have to specfy the entrants optmal qualty decsons. Usng these qualtes yelds the entrants reduced proft functons, whch ndcate whether entry s proftable or not. Thus, we frst characterze the equlbra n the entry game. We also ntroduce an equlbrum selecton crteron and we show that entry deterrence s proftable for each frm 1 and 2 as long as the number of software components publshed as open source s not too hgh. The equlbrum prces (9) obvously mply that the entrants can earn postve profts π e ( ) only f q e > q holds. Furthermore, wth π e ( ) > 0 the entrants h prces p e ( ) are ether determned by the nteror soluton,.e., by qe + βθ j,orbythecornersolutonp e ( ) = q e q. Consderng the 1 2 solutonofthecompletegamewefnd that the qualty decsons of frms 1 and 2 8 Wth q e1 = q e2 =0we have p e 1 = p e 2 =0.. 9

10 as well as ther open source decsons are such that we have π e (q e, ) < 0 for all h q e > 2 2. q +2βΘ j 9 We therefore restrct the followng analyss to the case n whch p e ( ) = q e q holds. Usng q e >q and p e ( ) = p q e q, the reduced proft functonπ e ( ) of entrant e can be wrtten as π 1 e ( ) = hqq e 1+β q [ q + βψ ] (q e α q ) 2 (11) 2+α q + α j q j n qj h max 1 qj 2 + β q, o α j q j f q ej =0 wth : Ψ := qj f q ej q j. Dfferentatng (11) wthrespecttoq e leads to the followng frst order condton (assumng q e q ) π e ( ) q e = and : β 2 [ q + βψ ] 2(q e α q ) 0 (12) q e 2+α q + α j q j π e ( ) q e q e =0. (13). Snce we also have 2 π e ( ) qe 2 < 0, (12) and (13) mplctly defne the entrants optmal qualty eq e r (q, ) q. Takng nto account that the entrant s proft must be postve, we can specfy the entrant s proft maxmzng qualty q r ( ) as (note that q r ( ) depends on q e e,q j,q ej, α, α j and β) q r e ( ) := ( eq r e (q, ) f π e (eq r e (q, ), ) > 0 0 else. (14) Consderng entry n both markets (11) and(14) reveal that the entrants reduced profts and hence ther optmal qualty decsons q r ( ) depend on whether e entry n the other market has taken place or not. Wth β < 0,.e., wth software programs that are substtutes, π e ( ) decreases f entry occurs n market j. Wth complementary software programs entry n market s more proftable f entry takes also place n market j. Therefore, the entry game may have multple equlbra. Result 2 The equlbra of the entry game are characterzed by the followng 9 Ths result s due to the fact that n order to deter entry frms 1 and 2 wll choose relatvely hgh qualtes q 1 and q 2. Snce the entrants costs are strctly convex n q e, relatvely hgh qualtes q 1 and q 2 mply that the entrants optmal qualtes satsfy q e < 2 q +2βΘ j 2. 10

11 qualtes q e (q,q j, α, α j, β) :=q r e (q e j ( ), ) q e ( ) =0 f qe r ( ) q =0or ej =0 qr e j ( ) =0 q ej >q j >q f qe r ( ) q > 0 or ej =0 qr e ( ) > 0. q ej >q (15) Proof. (11) and(12) lead to ether qe r ( ) qej = =0 qr e ( ) or to q r e qej >q ( ) j qej 6= =0 qe r ( ). Furthermore, (11) and(14) mply q r e qej >q ( ). q ej =0,.e., j qej >q j gven that entry takes place n both markets the entrants optmal qualtes do not depend on each other. Therefore, the solutons of equaton (15) fully characterze all possble equlbra. Usng result 2 we could turn to the next stage of the game,.e., the specfcaton of the qualty decsons of frms 1 and 2. However, consderng all possble equlbra would lead to a rather complex analyss. We therefore assume qe r ( ) >q q r e qej >q ( ) j qej =0 =0 q e ( ) =qe j ( ) =0. A In other words, we assume that the entrants can not coordnate ther decsons such that they both enter even f entry n each of the markets yelds negatve profts. Takng nto account that entry deterrence s optmal for frms 1 and 2, (A) does not restrct the analyss f β 0 holds. Entry n both markets s deterred as long as entry n each sngle market s not proftable. (A) mples that the same result holds for β > 0. Furthermore, analyzng the (entry deterrng) qualty decsons of frms 1 and 2, we can concentrate on the entrant s profts gven that entry n the other market does not occur. If qe r (q,q j, ) =0holds for both entrants qej =0 the frms qualtes q 1 and q 2 are such that entry n nether market occurs. Therefore, let us characterze the propertes of π e (qe r ( ), ) and let us also qej =0 consder the proft π r (q,q j, α, α j, β) of frm gven qe r (q,q j, ) qej and q e =0 j =0 (p ( ) s defnedn(8)): π r (q,q j, α, α j, β) :=p ( )D ( ) c ( ) wth q e = qe r ( ) qej and q e =0 j =0 (16) Analyzng π e (qe r ( ), ) qej and =0 πr ( ) we obtan 11

12 Result 3 If entry occurs only n market the entrant s proft π e (qe r ( ), ) qej =0 ncreases n the number of software components publshed as open source π e (qe r ( ), ) > π e (q r e ( ), ) > 0 for q r e α α ( ) j qej =0>q. (17) =0 qej =0 qej =0 Wth moderate open source decsons,.e., wth α, α j < 0.375, thereexstsa qualty q d(q j, α, α j, β) such that π e (qe r ( ), ) > 0 q <q d π e (qe r ( ), ) qej =0 q q d =0, (18) qej =0 q d(q j, α, α j ) α > 0 and (19) π r (q d ( ),q j, ) > 0. (20).. Proof. The sgn of π e (qe r ( ), ) α and π e (qe r ( ), ) α j wth q ej =0can be determned by usng the envelope theorem. Restrctng the analyss to α, α j < 0.375, employng (11) and(12) we obtan π e (qe r ( ), ) qej > 0 for q =0and =0 lm q π e (q e, ) < 0 for all q e >q.dfferentatng π e (qe r (q ), ) qej wth =0 respect to q reveals that π e (qe r (q ), ) qej s strctly concave n q whch also =0. mples q d( ) α > 0. Evaluatng π r (qd ( ),q j, α, α j, β) for all α, α j < and β [ 0.5, 0.5] confrms π r (qd ( ),q j, α, α j, β) > 0. Obvously, postve spllover effects due to open source mply that the entrants profts are hgher, the more software components frms 1 and 2 have publshed as open source. On the other hand, the entrants prces,.e., p e ( ) = q e q, decrease n q but ther costs are strctly convex n ther qualtes q e. Hence, moderate open source decsons mply that entry n market can be deterred f the frms qualtes q are hgh enough. Furthermore, the entry deterrng qualtes q d(q j, α, α j, β) ncrease n α and entry deterrence s proftable as long as the number of software components publshed as open source s not too hgh. 3.4 Qualty Decsons of Frms 1 and 2 Turnng to the qualty decsons of frms =1, 2 we already know that the frms qualty decsons must be such that entry n ther own markets does not take place. Result 3 mples that we can also restrct the analyss to α 1, α 2 < 0.375,.e., to the range of open source decsons, n whch entry deterrence s feasble and proftable. 12

13 Employng (A) we determne the equlbrum qualtes of frms 1 and 2 by the followng procedure: Usng the reduced proft functonoffrm gven that entry can only occur n market,.e., π r (q, ) (see (16)), assume that maxmzng π r (q, ) wth respect to q leads to a unque soluton q r(q j, ). Assumefurther that q1(q r 2, ) and q2(q r 1, ) have a unque fxed pont (q1(α 1, α 2, β),q2(α 2, α 1, β)). If nether frm has an ncentve to devate from q ( ) n order to nduce entry n the other market, (q1(α 1, α 2, β),q2(α 2, α 1, β)) also consttutes a pure strategy equlbrum of the complete game n whch entry n both markets s possble. Now, analyzng (16) shows that there exsts a unque proft maxmzng qualty q r(q j, ) whch satsfes π r ( ) 0, πr ( ) (q q d (q j, α, α j, β)) = 0 and q q d (q j, α, α j, β). (21) q q Wth (21) the optmal qualty q r(q j, ) s ether determned by the nteror soluton,.e., by the qualty that maxmzes the frm s proft f entry deterrence s not bndng, or by the entry deterrng qualty q r(q j, ) =q d(q j, α, α j, β). Furthermore, smple comparatve statcs reveals q r( ) > qr ( ) > 0 for q r (q j, ) q d (q j, α, α j, β). (22) α α j That s, an ncrease n the number of software components the frms publsh as open source reduces the frms costs and thus ncreases ther optmal qualtes. Fgure 1 shows q r(q j, ) for β = The left pcture s based on α 1 = α 2 =0 whch leads q r(q j, ) >q d(q j, ). The rght pcture shows q r(q j, ) =q d(q j, ) for α 1 = α 2 =0.2. It also depcts the frm s optmal qualty eq r(q j, ) f entry s dsregarded q r (q j, 0, 0, ) q d (q j, 0.2, 0.2, ) r q (q j, 0.2, 0.2, -0.25) q j q j Fgure 1: Frm s optmal qualty q r(q j, ) wth β = 0.25, α 1 = α 2 =0and α 1 = α 2 =0.2. Fgure 2 depcts q r(q j, ) for β =0.25. Agan, α 1 = α 2 =0leads to q r(q j, ) > q d(q j, ) (see the left pcture of Fgure 2). Wth α 1 = α 2 =0.2 we get q r(q j, ) = q d(q j, ) (see the rght pcture of fgure 2). 13

14 q r (q j, 0, 0, 0.25 ) q d (q j, 0.2, 0.2, 0.25 ) q r (q j, 0.2, 0.2, 0.25 ) q j q j Fgure 2: Frm s optmal qualty q r(q j, ) wth β =0.25, α 1 = α 2 =0and α 1 = α 2 =0.2. Turnng to the equlbrum qualtes of frms 1 and 2, nspectonofq1(q r 2, ) and q2(q r 1, ) yelds that there exsts a unque fxed pont (q1(α 1, α 2, β), (q2(α 2, α 1, β)). To verfy that (q1(α 1, α 2, β), (q2(α 2, α 1, β)) s a pure strategy equlbrum of the complete game, n whch entry n both markets s possble, we addtonally have to show that any devaton from q (α, α j, β), whch would nduce entry nto market j, snotproftable. Entry n market j would occur wth q ej >q j and would furthermore lead to p e j ( ) = q ej q j. Wth β < 0,.e., wth software programs that are substtutes, a hgher qualty n market j decreases the proft of frm. On the other hand, wth β > 0 entry n market j would not only ncrease the proft offrm but would also make entry n market more proftable. Hence, nducng entry n market j would force frm to ncrease q n order to deter entry n ts own market. Evaluatng these effects shows that any devaton from q (α 1, α 2, β) decreases frm s proft. Summarzng and usng (22) we obtan Result 4 If the frms open source decsons are such that entry deterrence s proftable, there exsts a unque pure strategy equlbrum wth qualtes q (α, α j, β) whch obey q ( )/ α > 0. To analyze q (α, α j, β) further and to specfy whether the frms qualtes are determned by entry deterrence let us start wth α 1 = α 2 = 0.Calculatng q (0, 0, β) for all β [ 0.5, 0.5] we fnd that the equlbrum qualtes do not nduce entry,.e., we have q ( ) α1 =α 2 =0 >qd (qj ( ), ) α1 =α 2 =0 β [ 0.5, 0.5]. (23) Snce open source decreases the entrants costs, t turns out that there exsts a crtcal level of α d at whch entry deterrence becomes a bndng restrcton. 14

15 o Defnng α d (α j, β) :=max nα q ( ) qd (q j ( ), ) we obtan 1 < αd (α j, β) α j < 1 whch mples that α d (α, β) =α has a unque soluton αd (β). Fgure3shows the graphs of α d (β) and the correspondng qualtes q ( ) = qd (q j ( ), ) and qe r (q ( ),q j ( ), ) qej. = α d ( β ) 0.5 q e r ( ) 0.14 q * ( ) β β Fgure 3: α d (β) and q ( ), qr e (q ( ),q j ( ), ) qej =0 wth α 1 = α 2 = α d (β) 3.5 Open Source Decsons Fnally, usng the equlbrum qualtes q (α, α j, β) let π (α, α j, β) denote the frms reduced proft functons. The frms proft maxmzng open source decsons are then characterzed by α r (α j, β) :=argmax α π (α, α j, β). (24) Evaluatng (24) yelds α r (α j, β) > α j for all α j α d (β). Confnng the analyss to α, α j > α d (β) shows that π (α, α j, β) has a unque maxmum n α. Hence, α r (α j, β) s unquely defned for α j > α d (β). Fgure 4 depcts α r (α j, β) and the correspondng equlbrum qualtes q ( ) for β =

16 q j * ( ) α r ( α j, -0.25) q * ( ) α j α j Fgure 4: Frm s optmal open source decson α r (α j, β) and equlbrum qualtes q ( ) for β = Fgure 5 shows α r (α j, β) and q ( ) for β = α r ( α j, 0.25) q * ( ) q j * ( ) α j α j Fgure 5: Frm s optmal open source decson α r (α j, β) and equlbrum qualtes q ( ) for β =0.25. Analyzng the frms mutual best responses α r 1 (α 2, β) and α r 2 (α 1, β) have a unque fxed pont. Therefore, we get Result 5 There exsts a unque pure strategy equlbrum α 1(β), α 2(β) whch satsfes α (β) :=α 1(β) =α 2(β) > α d (β). (25) Fgure 6 shows the graph of α (β). 16

17 0.2 α α ( β ) β β Fgure 6: Equlbrum open source decsons α (β). The knk at β 1 =0.263 s due to the fact that the frms equlbrum prces p ( ) satsfy (see (10)) h 1 q p 2 + βθj for β β ( ) = q 1 q ( ) q α d (β)qj ( ) for β β 1. Wth software programs that are relatvely strong complements the frms open source decsons do not only force them to choose entry deterrng qualtes. They also mply that the frms prces are bounded by consumers alternatve to use the open source software nstead of buyng the frms (commercal) software programs. Whle fgures 4 and 5 already ndcate that the frms open source decsons tend to be strategc substtutes (complements) f ther programs are substtutes (complements), we now analyze ths strategc nterdependence more carefully. For ths purpose t suffces to consder the open source decsons that would maxmze the frms jont profts,.e., α C (β) :=argmax α 2X =1 π (α, α, β). Comparng α C (β) wth α (β) (see Fgure 7) we fnd that as long as the programs are strong substtutes,.e., as long as β < 0.11 holds, the jont proft maxmzaton open source decson s lower than α (β). Thus,thefrms open source decsons are strategc substtutes for all β < Forβ > 0.11 the frms open source decsons are strategc complements,.e., α C (β) > α (β). 17

18 α α C ( β ) α ( β ) 0.05 β β β Fgure 7: Equlbrum open source decsons α (β) and optmal cooperatve decsons α C (β). Furthermore, wth β < β 2 = we have α C (β) =α d (β). That s, whle ncreasng α above α d (β) would reduce the frms costs, the frms would also be forced to ncrease ther qualtes n order to deter entry. Wth software programs that are strong substtutes ths second effect s negatve and domnates thepostveeffects from cost reductons. On the other hand, the hgher β the lower the negatve effects due to ncreased qualtes. Therefore, α C (β) > α d (β) and α C0 (β) > 0 for all β > β 2.Notethatα C0 (β) > 0 also holds f the equlbrum prces are restrcted by the consumers alternatve to use the open source software. Whle β > β 3 =0.07 mples p qq ( ) = ( ) q α d (β)qj ( ), we stll have α C0 (β) > 0. 4 Concluson We started wth the assumpton that open source reduces the codng costs of software frms developng new or qualtatvely enhanced programs. The users ncentves to detect bugs and the ncentves of programers to sgnal ther codng abltes by contrbutng to open source are two reasons why software frms may beneft from publshng parts of ther software as open source. Gven these cost reducng effects we analyzed how frms determne the degree up to whch they publsh ther software as open source. We set up a model wth only two frms and potental competton from new entrants. Consderng postve spllovers due to open source t turned out that the frms open source decsons balance the postve effects from costs reductons and the negatve effects due to the tghtened restrctons wth respect to entry deterrence. Furthermore, snce consumers may use the open source software nstead of buyng the commercal software pro- 18

19 grams, the frms prces can ultmately be bounded by the restrcton that the demand for ther commercal programs s postve. The strategc nterdependence between the frms open source decsons s determned by whether the frms programs are substtutes or complements. Programs whch are (strong) substtutes mply that the open source decsons are strategc substtutes, complementary programs lead to open source decsons whch are strategc complements. Ths result s due to the fact that the frms qualtes ncrease n the number of software components they publsh as open source. Therefore, the strategc nterdependence between the frms open source decsons mrrors the strategc nterdependence between the frms qualty decsons. The model n ths paper shows that not only the obvous techncal effects of open source software (e.g. cost reductons) have to be taken nto account when dscussng the decson of frms to publsh some of the software they have developed under an open source lcense. Rather, strategc consderatons takng nto account actual and potental compettors affect ths decson as well. These mght help to explan certan ssues nvolved n the open source actvtes of frms better than an analyss that purely focuses on the techncal effects of OSS. One of these ssues s the queston, why we see a lot of open source actvty by frms n the feld of operatng systems such as Lnux and not so much for many other types of software. The model, llustrated by fgure 6, suggests that ths may be due to the strong competton among operatng systems, especally among the dfferent flavors of Unx. Ths competton drves prces down, and consumers contnue to buy the commercal software (e.g. the Lnux dstrbuton) for ts better qualty nstead of usng the cheaper but qualtatvely nferor open source components. Knowng ths, frms can publsh more of ther software components as open source to beneft from the cost savngs. Ths does not work as well f software prces are hgh, such as n the scenaro wth two complementary goods. In ths stuatons the ncentves for consumers to use the (free) open source components alone s rather large and frms have an ncentve to restrct ther OSS publcatons. Thus, accordng to the model we should observe more open source actvtes by frms n areas where competton s large. Ths would support those n the software ndustry who see an ncreasng mportance of OSS as an answer to commodfcaton n areas lke offce sutes or applcaton servers. 19

20 References [1] Ghosh R., Glott R, Kreger B.; Robles G. (2002): Free/Lbre and Open Source Software: Survey and Study (FLOSS), Fnal Report, Part IV: Survey of Developers, [2] Lerner J., Trole J. (2002a): Some Smple Economc of Open Source, Journal of Industral Economcs, 50, [3] Lerner J., Trole J. (2002b): The Scope of Open Source Lcensng, [4] Raymond E. (2000): Homesteadng the Noosphere: An Introductory Contradcton, [5] Schmtz K., Schntzer M. (2002): Publc Subsdes for Open Source? Some Economc Polcy Issues of the Software Market, mmeo [6] Wchmann, T. (2002): Free/Lbre and Open Source Software: Survey and Study (FLOSS), Fnal Report, Part II: Frms Open Source Actvtes - Motvatons and Polcy Implcatons, [7] Wchmann, T., Spller D. (2002): Free/Lbre and Open Source Software: Survey and Study (FLOSS), Fnal Report, Part III: Bascs of Open Source Software Markets and Busness Models, [8] Wlcox J. (2000): IBM to spend $1 bllon on Lnux n 2001, CNET news.com, , 20