Development costs and open source software. Xiaopeng Xu* University of California-Berkeley Abstract

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1 Development costs and open source software Xaopeng Xu Unversty of Calforna-Berkeley Astract Ths paper analyzes the effect of the development cost on an open source software enhancement that s developed y ndvdual programmers n an Internet communty. It consders a stuaton n whch each programmer's cost of development s common knowledge ut hs valuaton of the enhancement s hs prvate nformaton, wth other programmers knowng aout only ts dstruton. Dependng on the dstruton functons of programmers' valuatons of the enhancement, a programmer wth a lower development cost may have a less ncentve to develop. As the development cost decreases, the enhancement may e less lkely to e developed, and some programmers may e worse off. I thank Erc von Hppel for encouragement and Justn Johnson and Thomas Marschak for helpful comments.

2 1. Introducton The recent movement of open source software development has attracted a lot attenton not only of the computng communty, ut also of the meda, of economsts and, more recently, of poltcans. 1 In many countres, there are poltcal ntatves tryng to provde pulc support for open source, e.g., y payng drect susdes to open source projects or y requrng government agences and schools and unverstes to replace proprety software y open source software whenever possle (Schmdt and Schntzer, 2002). An open source process nvolves a large numer of programmers scattered across the world sharng the source code to develop and refne pre-exstng programs. 2 An experenced programmer who has access to the source code of a program s ale to fgure out exactly how the program works. He can then exert effort to mprove or enhance the pre-exstng program. Any developed software can e freely dstruted through the Internet to other programmers n the Internet communty and wll e dstruted f developed (Johnson, 2002). 3 An open source software enhancement or refnement developed y a programmer s thus a pulc good. To an economst, the ehavor of ndvdual programmers engaged n open source processes s startlng (Lerner and Trole, 2002). Why should thousands of top-notch programmers contrute freely to the provson of the pulc good? Ths s an mportant ssue to address. Lerner and Trole make an attempt toward answerng ths queston. They argue that career concerns can to some extent explan the ehavor of ndvdual programmers. Ths s echoed y some practtoners of open source software such as Erc Raymond (1999). Harhoff et al. (2000) consder other ndvdual motvatons related to the ncreased dffuson of freely revealed nnovatons. Bessen (2002) focuses on the partcpaton of consumers, oth ndvduals and frms, wth complex needs. He argues that complexty makes a dfference and that open source allows consumers to meet ther needs y customzng the code themselves. 1 The open source movement was started n 1998 y a numer of promnent computer "hackers" such as Bruce Perens and Erc Raymond (von Hppel, 2002). 2 Well-known examples of open source software are the GU/Lnux computer operatng system, Perl programmng language, and Internet e-mal engne SendMal (Raymond, 1999). 3 Kollock (1999) argues that the advent of the Internet has een a oon to the development of open source development projects. 2

3 Ths paper sdesteps the queston of group formaton, ut focuses on analyzng the ehavor of the ndvdual programmers who are n a gven open source group. 4 Specfcally, t addresses the followng questons. Ceters parus, wll a programmer who has a lower development cost e more lkely to develop a program? Wll a program e more lkely to e developed when all programmers have lower development costs? Fnally, wll all the programmers e etter off as ther development costs decrease? Conventonal wsdom may lead one to answer these questons affrmatvely. The paper challenges such wsdom. In a model where each programmer's valuaton of the open source development or enhancement s hs prvate nformaton, I show that the answers to the questons depend crtcally on the dstrutons of valuatons. Specfcally, I otan the followng results. Frst, a programmer wth a hgher development cost may e more lkely to develop. Second, the enhancement program may e less lkely to e developed when all programmers have lower development costs. Fnally, some of the programmers may e worse off, as ther development costs decrease. The reason for the results s smple. Developng the software enhancement s essentally an R&D actvty. The provson of the enhancement, a pulc good, thus s of "max" or "estshot" type. So long as there s one programmer who develops, the pulc good s provded; addtonal developers add nothng to the provson of the good. Smply put, only the est effort counts. And f every programmer can exert the est effort, then a programmer wll e less lkely to develop f he thnks that other programmers are more lkely to develop. In other words, everyone has an ncentve to e a free-rder. The free-rder prolem s well known n the pulc economcs lterature. The prolem s partcularly severe when the pulc good s of est-shot type (Hrshlefer, 1983). In the development of the open source software enhancement, programmers ehave strategcally, each programmer's decson to develop depends on hs elef aout the proalty that other programmers develop. The margnal eneft of development to a programmer s hs valuaton of the enhancement tmes the proalty that no one else develops. Therefore, there s no smple lnk etween the ncentve to develop and the cost of development for each programmer, ecause a programmer develops f and only f hs cost of development s no greater than the margnal eneft, rather than hs valuaton, of the enhancement. 4 For example, 15 programmers dd much of the ntal codng of Apache. And durng the frst three years of Apache, 388 developers contruted 6,092 feature enhancements and fxed 695 ugs (Mockus et al., 2000). 3

4 The second result of the paper has mportant polcy mplcatons. Some frms pay ther employees to work on open source software, ecause they eleve that they wll eneft from the enhancement. Smlarly, the government and not-for-proft organzatons have supported open source projects ether y offerng drect susdes to the projects or y employng computer experts at unverstes or government agences to support t (Schmdt and Schntzer, 2002). The ntenton of such support, effectng to reduce programmers' development costs, s clearly to promote open source software developments. Unfortunately, the end result of the support may not promote ut rather hnder the open source program enhancement. Schmdt and Schntzer (2002) dscuss the mplcaton of drect susdes (as well as other forms of government support) for open source. They argue that the government should restrct tself to susdzng asc research, whch s a pulc good wth strong postve external effects that wll not e provded y the market. The development of most software projects, however, s appled R&D, whch can e provded y the market. They also rase another prolem wth pulc susdes to appled R&D n that t nvtes rent seekng actvtes. Ths leads well-ntended projects to e captured y large companes managng to acqure vast amount of pulc susdes as happened n the space, defense and nuclear ndustres. My analyss goes one step further. Even f there s no rent-seekng ehavor and an open source project has the feature of asc research, pulc support may, due to the strategc nteracton among programmers, have the unntended effect n that t may hnder rather than foster the enhancement of open source software. The paper s closely related to Johnson (2002), who ulds a model of open source software, n whch ndvdual user-programmers decde whether to nvest ther own tme and effort to develop a software enhancement that wll ecome a pulc good f so developed. Each of the programmers has a prvate cost of workng on the project and a prvate value of usng t; oth of whch are prvate nformaton. The man focus of hs paper s on the relatonshp etween the proalty of nnovaton and the numer of programmers who are symmetrc ex ante. I consder a modfed verson of hs model n whch each programmer has prvate nformaton aout hs valuaton of an open source program enhancement, ut hs development cost s common knowledge. Programmers may dffer n ther development costs, or ther valuatons of the enhancement may e dfferently dstruted. The focus of my paper s on the effect of development costs on each ndvdual's ncentve to nnovate, and on the proalty that the 4

5 nnovaton s made, as well as on the programmers' expected utlty. The two papers are thus complementary. The analyss of the paper has applcatons to other economc prolems. For example, a large multnatonal corporaton may have several research frms that smultaneously engage n an R&D actvty to carry out a corporaton-specfc technologcal nnovaton. The rule of the corporaton s that any nnovaton y one of the frms s shared fully wth other frms, whle the frm's R&D cost s not shared (Mas-Collel, Whnston and Green, 1995, pp ). To promote nnovaton, the headquarter of the corporaton may susdze the frms. Unfortunately, my analyss shows that the well-perceved polcy may lead to an unntended result of hnderng the prospect of nnovaton. Another applcaton s to the market for evaluatons (Avery, Resnck and Zeckhauser, 1999). A group of consumers shares product evaluatons whch may e treated as pulc goods; evaluatons are non-rval f the good eng evaluated has elastc supply (e.g., a ook or an applance), and each ndvdual can eneft from an evaluaton wthout reducng ts value to anyone else. Recent developments n computer networks have dramatcally drven down the cost of dstrutng nformaton. One may naturally thnk that these developments wll lead more nformaton or evaluatons to e dstruted. My analyss shows that the answer may not e so straghtforward as long as there s stll a postve cost of dstruton. The remander of the paper s organzed as follows. Secton 2 sets up and analyzes the model, and Secton 3 concludes. 2. The Model There are > 1 software programmers n an Internet communty, each decdng smultaneously whether to expend effort and tme to develop an enhancement or refnement of a pre-exstng software applcaton, the source code of whch s open. In other words, each programmer makes a nary choce: s {0, 1}, where 1 means "develop" and 0 means "don't develop". The enhancement, once developed, wll then ecome a pulc good, avalale to all programmers n the communty. Developng the software enhancement s essentally an R&D actvty. The provson of the enhancement, the pulc good, thus s of "max" or "est-shot" type. 5

6 So long as there s one programmer choosng to develop, the enhancement s made; addtonal developers add further value to the nnovaton. Programmers derve enefts from the enhancement; they may e drven to wrte software for ther own use, such as facltatng ther own work or deuggng (Johnson, 2002; von Hppel, 2002). The eneft of the enhancement to programmer s, whch s hs prvate nformaton. Other players know aout only the dstruton of. The cumulatve dstruton functon for s F ( ), and the correspondng densty functon s f ( ) > 0, for [, ], 0 <. Assume that programmers' enefts are ndependently dstruted. A programmer workng on a software development project ncurs an opportunty cost of hs tme; he cannot work on other projects whle he s workng on ths project. The development cost for programmer s c > 0, whch s common knowledge. Ths specfcaton dffers from that of Johnson (2002) n whch each programmer's valuaton of the enhancement and cost of development are hs prvate nformaton. Gven the est-shot feature of the enhancement, programmer 's utlty functon s U = max{s,, s } - s c. (1) If at least one of the programmers develops, programmer receves hs prvately known eneft,, ut he ncurs hs development cost, c, f and only f he s a developer. Assumpton 1. < c <, = 1,,. Ths assumpton mples that, f programmer were to exst n solaton, he would develop f and only f hs valuaton of the enhancement s no less than hs cost of development, c, and hs ex ante proalty to develop would e 1 - F (c ), whch s strctly etween 0 and 1. When he nteracts wth other programmers, programmer 's decson to develop depends on hs elef aout the proaltes that other programmers develop. Let p j e programmer 's elef aout the ex ante proalty that programmer j develops, j {1,, }\{}. If programmer develops, hs utlty or payoff s - c. If programmer does not develop, hs expected payoff s [1 - j1, j p j ( 1 )], whch s equal to the product of hs valuaton of the enhancement and the proalty that at least one of the other - 1 programmers develops. 6

7 Clearly, programmer develops,.e., s = 1, f and only f ( 1 ) c. In other j1, j words, programmer 's development decson oeys a cutoff rule: he develops f and only f hs valuaton of the enhancement s no less than a cutoff level, = c / ( 1 ). Ths s true for Equlrum condtons (2) are general, allowng dfferent development costs and dstrutons of valuatons of the enhancement among all the programmers. Wth restrctons on the dstrutons of valuatons or development costs, we can use (2) to address many mportant ssues. For nstance, one may ntutvely thnk that, ceters parus, a programmer wth a lower development cost may e more lkely to develop. Unfortunately, ths ntuton s generally ncorrect, as I wll show elow. Assume that all programmers' enefts of the enhancement are dentcally and ndependently dstruted. Denote the common dstruton functon F(), [, ], > 0. From (2), one can easly see that, for any two programmers and j, j, we have c F( )/ = c j F( j )/ j. (3) Let G() = F()/. Clearly, f G'() = [f() - F()]/ 2 < 0, then > j, when c > c j, mplyng that a programmer wth a lower development cost s more lkely to develop. On the other hand, f G'() > 0, then < j, when c > c j, mplyng that a programmer wth a lower development cost s less lkely to develop. The reason for the result s as follows. Each programmer 's margnal eneft of development s hs valuaton of the enhancement tmes the proalty that no one else develops. Hs margnal cost of development s smply hs cost of development. In equlrum, a programmer develops f and only f hs margnal eneft of development s no less than hs cost of development. Consequently, he develops f and only f hs valuaton of the enhancement s no 7 j1, j all programmers. Thus, the ex ante proalty that programmer develops s p = 1 - F ( ), = 1,,. ote that > c. In the presence of other programmers, each programmer s less lkely to develop than he would e n solaton, n hopng that others wll develop so that he can e a free-rder. In equlrum, we have, for = 1,,, j1, j F ( ) = c. (2) j j p j p j

8 less than a cutoff level. For two programmers wth dfferent development costs, there are two possltes. In the frst whch comples more wth ntuton, the programmer wth a lower (hgher) development cost has a lower (hgher) cutoff level, ndcatng that he s more (less) lkely to develop. In the second whch s n contrast wth ntuton, the programmer wth a lower (hgher) development cost has a hgher (lower) cutoff level, ndcatng that he s less (more) lkely to develop. Dependng on the dstruton functon of development costs, ether posslty can take place. It s easy to dentfy dstruton functons such that G'() s ether postvely or negatvely sgned. For example, let F() = k, [0, 1], k 1. Then, G'() > 0, when k > 1, ut G'() < 0, when k < 1. Indeed, one can easly fnd commonly used dstruton functons F() such that G'() can e ether postve or negatve. For example, f F() s unformly dstruted, for [, ], > > 0, then G'() > 0, mplyng that a programmer wth a lower development cost s less lkely to develop. If F() s exponentally dstruted over [0, ), then G'() < 0, mplyng that a programmer wth a lower development cost s more lkely to develop. Let H() = F()/[f()]. Because F()/f() s the hazard rate, I call H() the unt hazard rate. Proposton 1 summarzes the aove analyss. Proposton 1. Suppose that programmers' valuatons of the enhancement are ndependently and dentcally dstruted. If the unt hazard rate for the common dstruton s greater (smaller) than 1, then a programmer wth a lower development cost s more (less) lkely to develop. A successful open source project requres a credle leader, and most leaders of open source projects are the programmers who developed the ntal code for the projects (Lerner and Trole, 2002). The development cost for a leader may e hgher than that for other programmers, hs followers. Beng leaders, they may have etter outsde opportuntes and hence hgher opportunty costs of workng on open source projects. Proposton 1 provdes an explanaton for the leader's ehavor. Even f hs valuaton of the open source software development s dentcally dstruted wth other programmers, the leader may e more lkely to develop when the unt hazard rate of the dstruton s less than 1, even though he has a hgher development cost. 8

9 Amng to foster the open source movement, the government and not-for-proft organzatons n many countres make susdes to open source projects, so do some prvate frms (Lerner and Trole, 2002; Schmdt and Schntzer, 2002). Such polcy effects to reduce development costs. The polcy s clearly ased on the elef that reductons n development costs are conducve to open source software development. But the mportant queston s: Does the elef make economc sense? I show that the answer to ths queston depends crtcally on heterogenety n programmers' valuaton dstrutons. To smplfy the analyss, assume that all programmers have the same development cost,.e., c = c > 0, = 1,,. It s clear from (2) that s a functon of c, = (c), for all. The ex ante proalty that the enhancement s not developed s Q(c) = F ( ( c)). 1 In the followng, I sometmes choose to suppress the argument of F ( (c)) and f ( (c)), as there should e no confuson, gven the context. Dfferentatng Q wth respect to c, we have Q'(c)/Q(c) = 1 f d /dc. (4) F Clearly, f d /dc > 0 for all, Q'(c) > 0, mplyng that a lower (common) development cost wll result nto a hgher proalty that the enhancement s developed. If F (.) = F (.) = F(.), for all and j. Then, n symmetrc equlrum, = j = (ths s guaranteed f G() s monotonc), and t follows from (2) that satsfes that F() - 1 = c. Ovously, d/dc > 0. Hence, dq/dc > 0. Proposton 2. Suppose that programmers are ex ante symmetrc n that they have the same development cost and ther valuatons of the enhancement are ndependently and dentcally dstruted. Then, n symmetrc equlrum, as the development cost decreases, the enhancement s more lkely to e developed. More generally, we have F ( (c))/ (c) = F j ( j (c))/ j (c). Dfferentatng oth sdes of ths equaton wth respect to c, one can readly check that [f ( (c)) (c) - F ( (c))]/ (c) 2 d /dc = [f j ( j (c)) j (c) - F j ( j (c))]/ j (c) 2 d j /dc. Thus, f H ( ) - 1 = F ( )/[ f ( )] - 1 s unformly and dentcally sgned, for all, then d /dc s unformly and dentcally sgned. Usng ths fact together wth (2), one can easly show that d /dc > 0, for all. Thus, we have 9

10 Proposton 3. If H ( ) - 1 s unformly and dentcally sgned, for all, then Q'(c) > 0. However, when H ( ) - 1 and H j ( j ) - 1 are unformly ut not dentcally sgned, then d /dc and d j /dc are oppostely sgned and one of them must e negatve, mplyng that one of the them s less lkely to develop, as the development cost decreases. Indeed, a lower development cost may result nto a lower proalty that the enhancement s developed! To rng out the result most straghtforwardly, I consder the case of two programmers, = 2, whose enefts are dstruted accordng to cumulatve functons F 1 ( 1 ) and F 2 ( 2 ), respectvely. It follows mmedately from (2) that the cutoff levels, 1 and 2, for the two programmers satsfy that 1 F 2 ( 2 ) = c and 2 F 1 ( 1 ) = c. Clearly, 1 and 2 are functons of c. Totally dfferentatng oth sdes of the two equatons wth respect to c, we have F f f F d = d 1 2 dc. (5) dc If F 1 F f 1 f 2 0, then one can solve (5) and otan d /dc = (F - f j )/(F 1 F f 1 f 2 ), =1, 2, j = 3 -. The ex ante proalty that the enhancement s not developed s Q(c) = F 1 ( 1 (c)) F 2 ( 2 (c)). Dfferentatng Q wth respect to c and pluggng nto d 1 /dc and d 2 /dc, we have Q'(c) = F 2 f 1 d 1 /dc + F 1 f 2 d 2 /dc = (F 1 F 2 f 1-1 f 1 f 2 F 2 + F 1 F 2 f 2-2 f 1 f 2 F 1 )/(F 1 F F 1 F 2 ). = [(H 1-1)F 2 / 2 + (H 2-1)F 1 / 1 ]/(H 1 H 2-1), (6) where H = F ( (c))/[ (c)f ( (c))]. ote that, n equlrum, F 1 / 1 = F 2 / 2. Hence, we have the followng result. Proposton 4. Sgn(dQ/dc) = Sgn((H 1 + H 2-2)/(H 1 H 2-1)). From Proposton 4, we know mmedately that, f H 1 = H 2, then dq/dc > 0, mplyng that a lower development cost wll result nto a hgher proalty that the enhancement s developed. Ths s, of course, not surprsng at all, gven Proposton 2. On the other hand, when H 1 H 2 < 1 and H 1 + H 2 > 2, dq/dc < 0. A necessary condton for ths to happen s that H > 1 and H j < 1, = 1, 2, j = 3 -. In ths case, a lower (common) development cost leads to a lower proalty that the enhancement s developed. 10

11 It s mportant to pont out that Proposton 4 s vald only for nteror equlrum (whch we have mplctly assumed n the analyss), n whch oth programmers develop wth postve proalty. If one of the programmers never develops, then, t s clear that the other programmer s more lkely to develop as the development cost decreases, leadng to a hgher proalty that the enhancement s developed. An example of ths s as follows. Let F 1 ( 1 ) = 1/j 1, 0 < j < 1, 1 [0, 1], F 2 ( 2 ) = 1/k 2, k > 1, 2 [0, 1], and 0 < c < 1. Smple calculaton shows that H 1 ( 1 ) j and H 2 ( 2 ) k. Let jk < 1 and j + k > 2. So, the condtons n Proposton 4 are satsfed. One can easly verfy that, for 2 c 1-1/j, there s a unque equlrum characterzed y 1 = 1 and 2 = c. 5 In equlrum, Q(c) = c 1/k. Clearly, Q'(c) > 0. I now gve an example n whch a smaller (common) development cost leads to a lower proalty that the enhancement s developed. Example 1. Let F 1 ( 1 ) = 1/2 1 /2, 1 [0, 4], and F 2 ( 2 ) = 2-1, 2 [1, 2], and 1 < c < 2. It s easy to verfy that H 1 ( 1 ) 2, and H 2 ( 2 ) = ( 2-1)/ 2. For 1 < 2 < 2, 0 < H 2 ( 2 ) < 1/2. Hence, we have H 1 H 2 < 1 and H 1 + H 2 > 2. Thus, n nteror equlrum, a lower development cost leads to a lower proalty that the enhancement s developed. All we need to check s whether there exsts nteror equlrum for some values of the development cost. If follows from (2) that 1 ( 2-1) = c and 2 1 1/2 /2 = c. Solvng 1 and 2, we have 1 (c) = c/[2c cc ( 1 ) ] and 2 (c) = 2c - 2 cc ( 1 ). In nteror equlrum, c < 1 (c) < 4 and c < 1 (c) < 2. One can easly verfy that, when 1 < c < 4/3, the aove two nequaltes are satsfed. Further, one can show that, for c > 1, d 1 /dc > 0 and d 2 /dc < 0. It follows from Proposton 4 mmedately that, for 1 < c < 4/3, a reducton n the development cost results nto a lower proalty that the enhancement s made. We thus see that, dependng on the dstrutons of programmers' valuatons of the enhancement, the nnovaton may e ether more or less lkely to e made avalale. Ths s the man result of the paper. Ths result has mportant polcy mplcatons. Prvate frms, the government, and not-for-proft organzatons, amng to promote open source development 5 When 2 < c 1-1/j, esdes the aove equlrum, there s another equlrum n whch 1 = c and 2 = 1. 11

12 processes, have made susdes to open source projects. My analyss ndcates that such polcy may not serve the ntended purpose, ut rather hnder the processes. Fnally, let us study the effect of the development cost on programmers' payoffs. Programmer 's expected payoff s EU = p EU + (1 - p )[1 - ( 1 p ) j ]EU j1, j = ( c) df ( ) + [1 - Fj( j) ] df ( ). (7) j1, j Dfferentatng EU wth respect to c, we have d(eu )/dc = (EU )/c + (EU )/ d /dc + j1, j ( EU ) d j /dc. (8) j It s easy to check that (EU )/ = 0, as programmer has optmally chosen to maxmze hs payoff. The frst term s the drect effect of the development cost on programmer 's payoff, whle the thrd term s the ndrect, strategc effect of the development cost on programmer 's payoff. Smple calculaton shows that (EU )/c = F ( (c)) - 1 < 0. The drect effect of the development cost on programmer 's payoff s always enefcal, as a lower development cost leads to a hgher payoff for programmer, gven other programmers' development choces. Dfferentatng EU wth respect to j, we have (EU )/ j = - f j ( j ) k k1, k, j F ( ) df ( ) < 0. If programmer j s more lkely to develop,.e., j decreases, programmer 's expected payoff ncreases. However, we have shown that, dependng on the dstruton functons of programmers' valuatons, d j /dc can e ether postvely or negatvely sgned. When d j /dc < (>) 0, programmer j's change of ehavor, n response to changes n the development cost, mposes a negatve (postve) external effect, - f j ( j ) k1, k, j Fk( k) df ( ) d j /dc, on programmer 's payoff. The ndrect effect s the sum of all the external effects mposed on programmer y the other - 1 programmers. From Propostons 2 and 3, we know that, f programmers are ex ante symmetrc, or f H ( ) - 1 s unformly and dentcally sgned for all, then d /dc > 0, = 1,,. Thus, each k 12

13 programmer s etter off, as the (common) development cost decreases. I summarze the result y Proposton 5. Proposton 5. When programmers are symmetrc ex ante, or when the dfference etween the unt hazard rate of valuaton and 1 s unformly and dentcally sgned for all programmers, a lower common development cost leads to a hgher expected payoff for all programmers. When H ( ) - 1 s not dentcally sgned for all programmers, then the ndrect effect of reductons n the development cost s detrmental for some programmer, say, programmer. Hence, d(eu )/dc can e ether negatvely or postvely sgned, dependng on whether the drect effect domnates or s domnated y the ndrect effect. The followng example llustrates ths. Example 1 (contnued). Recall that d 2 /dc < 0. Hence, (EU 1 )/ 2 d 2 /dc > 0. When (EU 1 )/ 2 d 2 /dc > - (EU 1 )/c, d(eu 1 )/dc > 0, ndcatng that programmer 1 s worse off ex ante, as the development cost decreases. Smple calculaton shows that (EU 1 )/c = F 1 ( 1 (c)) - 1 = c/[2c - 2 cc ( 1 ) ] - 1, and 1 (EU 1 )/ 2 d 2 /dc = - f 2 ( 2 (c)) df 1 1( 1) d 2 /dc = - [2 - (2c - 1)/ cc ( 1 ) ]{c/[2c cc ( 1 ) ]} 3/2 /6. Wth some algera, one can show that, for c > 1, (EU 1 )/ 2 d 2 /dc > - (EU 1 )/c. Hence, d(eu 1 )/dc > 0, for 1 < c < 4/3, mplyng that, n nteror equlrum, a reducton n the development cost makes programmer 1 strctly worse off. 3. Concluson Ths paper attempts to answer the followng questons n open source software development. Ceters parus, wll a programmer wth a lower development cost more lkely to develop? Whether more of the enhancement wll e developed as programmers' development costs decrease? Wll all programmers e etter off as ther development costs decrease? Conventonal 13

14 wsdom may lead one to answer these questons affrmatvely. However, I show n the paper that the answers to the questons are not so straghtforward. There are two salent features of an open source software development or enhancement. Frst, t s a pulc good. The enhancement enefts all programmers n a group n a non-rvalrous way. Second, t s a pulc good of "max" or "est-shot" type. The enhancement can e essentally developed y one programmer; addtonal developers, whle ncurrng development costs, add no further value to the enhancement. The est-shot pulc good feature of the open source software enhancement leads to the well-known free-rder prolem; each programmer wants to e a free rder, hopng that someone else wll develop. The free-rder motve s the underlyng force drvng the results of the paper. The man result of the paper s that, as programmers' development costs decrease, the open source enhancement may e less lkely to e developed. It has mportant polcy mplcatons, as some prvate frms and government agences as well as not-for-proft organzatons, amng to promote open source software development, make susdes to open source projects. Such susdes effect to reduce programmers' development costs. The analyss of paper, however, ndcates that the susdes may hnder rather than foster, as ntended, the nnovaton of open source software programs. The reason for the result s that programmers nvolved n the open source development act strategcally. If a programmer perceves that other programmers are more lkely to develop owng to lower development costs, he s less lkely to develop. The end result may e that the program s less lkely to e developed even when all programmers' development costs decrease. 14

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