European Journal of Operational Research

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1 European Journal of Operaional Research 196 (29) Conens liss available a ScienceDirec European Journal of Operaional Research journal homepage: Compuaional Inelligence and Informaion Managemen Shor-erm and long-erm compeiion beween providers of shrink-wrap sofware and sofware as a service Ming Fan a, *, Subodha Kumar a,1, Andrew B. Whinson b,2 a Foser School of Business, Universiy of Washingon, Seale, WA 98195, USA b McCombs School of Business, Universiy of Texas a Ausin, Ausin, TX 78712, USA aricle info absrac Aricle hisory: Received 2 May 27 Acceped 16 April 28 Available online 26 April 28 Keywords: Pricing Economics Qualiy compeiion Sofware as a service Differenial game Sofware as a service (SaaS) has moved quickly from a peripheral idea o a mainsream phenomenon. By bundling a sofware produc wih delivery and mainenance service, SaaS providers can effecively differeniae heir producs wih radiional shrink-wrap sofware (SWS). This research uses a game heoreical approach o examine shor- and long-erm compeiion beween SaaS and SWS providers. We analyze he facors ha affec equilibrium oucomes, including user implemenaion coss, SaaS provider s operaion efficiency, and qualiy improvemen over ime. Bundling sofware wih service lowers sofware implemenaion cos for users, and our resuls sugges ha i increases equilibrium prices. In providing sofware services, SaaS providers have o incur significan operaion cos. In he long run, service operaion cos may significanly affec SaaS firm s abiliy o improve is sofware qualiy. Ó 28 Elsevier B.V. All righs reserved. 1. Inroducion An ougrowh of he applicaion service provider (ASP) model in he do-com era, sofware as a service (SaaS) has moved quickly o a mainsream phenomenon. Some well-known examples of SaaS are he sales force auomaion and cusomer relaionship managemen (CRM) applicaions provided by Salesforce.com and NeSuie. Companies have also unveiled SaaS applicaions for individual cusomers. Examples include Google s spreadshees and Microsof s OneCare service; he laer provides virus and spyware cleanup for personal compuers (Richmond, 25). I is esimaed ha SaaS marke will grow a abou 25% a year o a $9 billion marke by 29 (Pallao, 26). According o indusrial sudies, he bigges appeal of SaaS is is lower implemenaion and mainenance cos (Kaplan, 25). SaaS provider delivers sofware over he Inerne and can poenially eliminae he need for companies and individuals o implemen, and mainain complex sofware applicaions. Companies ha implemen business applicaions such as Enerprise Resource Planning (ERP) and CRM sysems ofen have o face unexpeced high implemenaion coss, pay big bills o consulans, and sill end up wih projecs overruns. SaaS can empower business unis and allow businesses o reduce he upfron coss of deploying business * Corresponding auhor. Tel.: ; fax: addresses: mfan@u.washingon.edu (M. Fan), subodha@u.washingon.edu (S. Kumar), abw@us.cc.uexas.edu (A.B. Whinson). 1 Tel.: ; fax: Tel.: ; fax: soluions. To individual cusomers, if sofware applicaions need o be periodically updaed such as ani-virus applicaions, SaaS is an appealing choice. Wih SaaS, cusomers do no need o rack he insalled version of ani-virus sofware, and can manage compuer securiy from one place and make PC care as simple as car mainenance (Richmond, 25). From an economics poin of view, SaaS essenially bundles sofware producs wih sofware delivery and mainenance service. Produc bundling has been an acive research field in economics (e.g., McAfee e al., 1989; Whinson, 199). A monopoly firm can use bundling o leverage is marke power in one marke o a second marke. In compeiive markes, a firm can use bundling o effecively compee wih oher firms by bundling a primary good wih complemenary goods and services. For example, credi-card issuers bundle he use of cards wih variey of goods and services. By providing sofware ogeher wih service, a SaaS provider can effecively differeniae is produc wih radiional shrink-wrap sofware (SWS) by lowering sofware implemenaion coss. The innovaive bundling of radiional sofware produc wih service is inconceivable wihou he rapid progress in Inerne and elecommunicaion echnologies. Those echnologies are fundamenally changing how sofware producs are developed, released, and markeed. One of he goals of his research is o apply he economic principles o invesigae he business of SaaS and analyze compeiions in he sofware marke. By bundling sofware wih service, SaaS providers are facing a number of challenging issues. Firs, providing sofware service could be cosly. In lae 25, Salesfoce.com cusomers experienced several service ouages. Since hen cusomers have become very /$ - see fron maer Ó 28 Elsevier B.V. All righs reserved. doi:1.116/j.ejor

2 662 M. Fan e al. / European Journal of Operaional Research 196 (29) concerned abou sysem availabiliy and reliabiliy (Herber, 26; Vara, 26). To saisfy cusomer demands, SaaS firms have o inves heavily on capaciies and processes o guaranee service qualiy. Second, in choosing sofware applicaions, users ofen have o face radeoffs of applicaion performance versus cos of deploymen. Alhough SaaS is easy o use and is implemenaion cos is lower, here remains performance gap beween he hin clien inerfaces of SaaS applicaions and he rich deskop of SWS. Some believe innovaions on deskop sysems and Windows Visa could even widen he performance gap beween rich deskop applicaions and HTML-based hin cliens (Zeie, 25). Thus, sofware qualiy such as funcionaliy and performance is an imporan facor affecing he compeiion beween SaaS and SWS providers. This research uses a game heoreical approach o examine shor- and long-erm compeiion beween SaaS and SWS. Our model has he following characerisics: (i) We recognize ha cusomers wih heerogeneous sensiiviy o implemenaion cos will reac differenly o SaaS, and we model cusomer choice based on price and implemenaion cos. (ii) We examine he benefis as well as he coss relaed o SaaS. In providing sofware service, SaaS companies have o incur significan operaion cos. We consider queueing delays in providing sofware service over he Inerne and examine he effecs of service cos on SaaS provider s sraegy and he equilibrium oucome. (iii) We analyze price compeiion in a saic game as well as dynamic qualiy compeiion using a differenial game approach. Bundling sofware wih service lowers sofware implemenaion cos, and our resuls sugges ha i increases equilibrium prices. In he long run, sofware qualiy plays an imporan role in firms compeiiveness, and service cos significanly affec wheher he SaaS firm can compee effecively. Our resuls provide imporan managerial implicaions on compeiive sraegies and operaion policies for sofware companies. The res of he paper is organized as follows: Secion 2 reviews relaed lieraure. We se up he model in Secion 3, and analyze a one-period price compeiion model wih service guaranee from SaaS providers in Secion 4. Secion 5 examines he qualiy compeiion in a finie horizon differenial game, and Secion 6 provides soluions in an infinie horizon. Secion 7 discusses he resuls, and we provide he conclusions in Secion Relaed work This research is relaed o economics lieraure on compeiion beween firms wih heerogeneous producs (Shaked and Suon, 1983). Many of he compeiion models have been inspired by he work of Hoelling (1929), who assumes ha consumers have heerogeneous ases ha lie on a coninuum. In conras o Hoelling s horizonal differeniaion models, verical differeniaion suggess ha producs have differen feaures, which reduces price compeiion (Shaked and Suon, 1983). If wo companies have very similar producs wih lile differeniaion, he companies are ofen engaged in a Berrand compeiion and do no make posiive profis (Tirole, 1992). Bundling complemenary producs can serve as a produc differeniaion device in a compeiive marke and help firms avoid direc price compeiion (Chen, 1997). To bundle sofware produc wih service, companies have o incur capaciy cos. In modeling he cos of providing sofware service, we follow prior sudies in IS on pricing compuing services and modeling of queueing delays (e.g., Gupa e al., 1997; Tan and Mookerjee, 25). Tan and Mookerjee (25) model he cos of queuing delay as a nonlinear loss funcion. Anoher sream of research relaed o our sudy is produc qualiy compeiion. Fine (1986) refers produc qualiy as feaures, syling, and oher produc aribues ha enhance finess for use. In manufacuring applicaions, qualiy is ofen described as conformance o specificaions or as meeing sandards on he performance of he produc (Karmarkar and Pibalddo, 1997). Sofware qualiy has been sudied in sofware engineering economics (e.g., Boehm, 1981). The qualiy of sofware usually includes funcionaliy, reliabiliy, and usabiliy (Khoshgofaar and Allen, 21). Early game heoreic models in produc compeiion emphasized saic models. A dynamic model can add he imporan dimension of ime and recognize he compeiive decisions ha do no necessarily remain fixed (Fudenberg and Tirole, 1991). Models involving compeiion in coninuous ime are ypically reaed as differenial games, in which criical sae variables, e.g. demand or marke share, are assumed o change wih respec o ime according o specified differenial equaions (Dockner e al., 2). Differenial games have been widely applied in analyzing compeiion in dynamic adverising, pricing, and qualiy innovaion (Sehi, 1977; Erickson, 1995; Jørgensen e al., 23; Nair and Narasimhan, 26). For example, Piga (1998) analyzes dynamic adverising and pricing sraegies in duopolisic rivalry. Mukhopadhyay and Kouvelis (1997) examine design qualiy decision over he produc life cycle for wo compeing companies wih a similar produc. Bass e al. (25) sudy generic and brand adverising sraegies in a dynamic duopoly. In his research, we sudy he dynamic qualiy sraegies of SWS and SaaS providers in a differenial game. 3. The model We consider a compeiion model wih wo players: Firm 1 is an SWS provider and firm 2 offers SaaS over he Inerne. Cusomers, eiher corporae users or individuals, choose sofware producs based on heir values or uiliies. We assume ha he wo sofware producs have lile difference and cusomers have homogenous valuaion of v for boh producs. Le he implemenaion cos be c 1 for SWS and c 2 for SaaS. By bundling sofware wih service, SaaS company can lower he implemenaion cos of is produc, and we have c 2 < c 1. This assumpion is suppored by a sudy of over 6 companies ha SaaS indeed significanly reduced implemenaion cos and ime (PR Newswire, 26). Wih differen knowledge levels, users will have heerogeneous sensiiviy o he implemenaion cos. We denoe he cos sensiiviy as h, which is randomly drawn from a uniform disribuion wih suppor on [,1]. The price of SWS is p 1 while he price of SaaS is p 2. Cusomers choose he wo sofware producs based on price, implemenaion cos, and heir own sensiiviy o implemenaion coss. A cusomer s uiliy is u 1 = v p 1 hc 1 for using SWS, and u 2 = v p 2 hc 2 for using SaaS. A cusomer is indifferen beween he wo producs if u 1 = u 2,orv p 1 hc 1 = v p 2 hc 2 ; solving his equaion leads o h * =(p 2 p 1 )/(c 1 c 2 ), which is he indifferen poin for cusomers (see Fig. 1). When h > h *, he cusomer will choose SaaS because u 2 > u 1. When h < h *, he cusomer will choose SWS because u 2 < u 1. In his sudy, we are ineresed in a more general case when here are demands for boh SWS and SaaS. Here, we wan o find ha condiion. As shown in Fig. 1, when v = v * and h = h *, u 1 = v * p 1 h * c 1 = and u 2 = v * p 2 h * c 2 =. When v 6 v *, (i) u 2 6 if h P h *, and (ii) u 2 < u 1 if h < h *. Thus, he demand for SaaS will be zero under he condiion v 6 v *. Therefore, we need he condiion v > v * in order o have demands for boh SWS and SaaS, and we can easily find ou he value for v * as v * = p 1 + h * c 1 = p 2 + h * c 2. Thus, v * =(c 1 p 2 c 2 p 1 )/(c 1 c 2 ). In addiion, we consider he case ha he demands for boh SWS and SaaS do no cover he whole marke. This requires v < p 2 + c 2 ; oherwise, when v P p 2 + c 2, he whole marke will be

3 M. Fan e al. / European Journal of Operaional Research 196 (29) Price compeiion wih service guaranee from SaaS covered by SWS and SaaS firms. Therefore, cusomer valuaion should saisfy he condiion v < v < p 2 þ c 2 : ð1þ We denoe h ** as he value ha saisfies v p 2 h ** c 2 =. When a cusomer s sensiiviy h > h **, she does no use eiher SWS or SaaS. Thus, h ** =(v p 2 )/c 2 is he oal marke demand. Noe ha a duopoly problem covering he whole marke is a special case of he problem considered in his paper. As shown in Fig. 1, cusomers wih lower sensiiviy on implemenaion cos, i.e., h < h *, will choose he SWS. Cusomers wih higher sensiiviy on implemenaion cos, i.e., h > h *, will choose SaaS. Therefore, he demand funcion for SWS is D 1 ¼ h ¼ p 1 þ p 2 ð2þ c 1 c 2 c 1 c 2 and he demand funcion for SaaS is D 2 ¼ h h ¼ v þ p 1 1 p c 2 c 1 c 2 þ 1 : ð3þ 2 c 2 c 1 c 2 Since he marginal cos for SWS is usually very small, we normalize i o zero. Therefore, he SWS provider s profi funcion p 1 can be wrien as: p 1 ¼ p 1 D 1 : v * v p 2 p 1 D 1 For SaaS provider, we need o consider he queuing delay and he processing rae for sofware services. The Informaion Technology (IT) deparmen a SaaS firm is responsible for processing cusomer requess (Tan and Mookerjee, 25). We denoe he processing rae as l, which is also used o represen IT capaciy (Tan and Mookerjee, 25). A large processing rae requires a higher IT capaciy, and he cos for l includes a fixed cos c and marginal cos of c 1. Thus, he oal cos for capaciy l is c + c 1 l. Following prior lieraure, we model cusomer s arrival a SaaS as a Poisson process wih mean arrival rae k (Moe and Fader, 24; Tan and Mookerjee, 25). In an M/M/1 queue, he average delay for a cusomer can be represened as w ¼ 1. The SaaS provider s profi funcion p 2 can now be represened l k as: p 2 ¼ p 2 D 2 c c 1 l; 1 s:: l k 6 d; where d is he average delay guaranee of he SaaS. Our model has wo major pars. The firs par (Secion 4) analyzes a price compeiion game. The second par (Secions 5 and 6) analyzes dynamic compeiion beween SWS and SaaS providers on produc qualiies. D 2 θ = * ** θ θ θ = 1 Fig. 1. Demand model. p 1 + c 1 p 2 + c 2 ð4þ ð5þ In his secion, we consider he compeiion model where boh SWS and SaaS providers simulaneously decide produc prices. The game has wo sages: (i) SWS and SaaS providers simulaneously decide heir prices; and (ii) SaaS provider chooses is IT capaciy in order o mee he service delay guaranee. In his game, SaaS provider chooses price before capaciy because he price is deermined based on several exernal facors and i is hard o adjus. On he oher hand, he choice of capaciy is an inernal decision and may be adjused according o he price. This sequence of game is consisen wih he pas sudies (Wang and Gerchak, 23; Bernsein and DeCroix, 24) Compeiion wih complee informaion We firs consider a game wih complee informaion. Using sandard backward inducion, we solve he second sage of he game firs. In he second sage, he prices are already given. Thus, he SaaS provider s problem is min l s:: c þ c 1 l 1 l k 6 d: Alhough he oal number of subscribers for SaaS is D 2, a any ime, only a small porion of all he subscribers use he sysem. Denoing he average usage rae of all he subscribers as k, we have k = k D 2. For a given p 2, i is sraighforward o show ha he soluion o SaaS provider s service guaranee problem is l ¼ kd 2 þ 1=d: ð6þ Subsiue Eq. (6) ino SWS and SaaS providers profi funcions, we can solve he equilibrium for he price compeiion game. The resuls are now summarized below. Proposiion 1. The SWS provider s opimal price p 1 and SaaS provider s opimal price and capaciy ðp 2 ; l Þ consiue a Nash Equilibrium, where p 1 ; p 2 ; and l* are given as follows: p 1 ¼ vðc 1 c 2 Þþc 1 kc 1 4c 1 c 2 ; ð7þ p 2 ¼ 2vðc 1 c 2 Þþ2c 1 kc 1 4c 1 c 2 ; and ð8þ l ¼ 2kðvc 1 c 1 Þ 3c 2 ð4c 1 c 2 Þ kc 1 ð4c 1 c 2 Þðc 1 c 2 Þ þ 1 d : For breviy, he proofs of all he proposiions and corollaries are relegaed o he Appendix. The comparaive saics for he equilibrium resuls are presened in Table 1. We firs examine he equilibrium prices. We can see ha he equilibrium prices for boh SWS and SaaS providers increase in SWS implemenaion cos (c 1 ) and decrease in SaaS implemenaion cos (c 2 ). Consisen wih produc differeniaion lieraure (Gabszewicz and Thisse, 1979), higher produc differeniaion helps firms reduce price compeiion and charge higher prices. By bundling sofware wih service, SaaS provider can lower is implemenaion cos and effecively differeniae is produc from SWS. This leads o a higher equilibrium prices for boh firms. Table 1 Comparaive saics resuls Variables c 1 c 2 k c 1 d p p l *??? + +, Increase;, decrease;, no effec;?, ambiguous. ð9þ

4 664 M. Fan e al. / European Journal of Operaional Research 196 (29) Equilibrium prices also increase in usage rae (k) and variable cos of IT capaciy (c 1 ) of he SaaS provider. An examinaion of SaaS provider s profi and capaciy consrain funcions (5) and (6) reveals ha boh variable IT capaciy cos (c 1 ) and usage rae (k) conribue o he marginal cos of he SaaS provider. Naurally, a higher marginal cos leads o a higher equilibrium price for SaaS. I is also ineresing o noe ha SWS provider can now charge a higher price as is compeior s cos rises. Comparaive saics resuls sugges ha lowering variable capaciy cos or increasing service qualiy guaranee will lead o higher SaaS provider s processing capaciy. As he cos for IT capaciy goes down, SaaS providers will uilize more IT processing capaciies. I is also inuiive ha an increase in service qualiy guaranee, as represened by a lower d value, will require higher IT capaciy from SaaS provider. The relaionship beween processing capaciy and sysem usage rae can be eiher posiive or negaive. We examine he relaionship numerically (Fig. 2). When he sysem usage rae is below a hreshold, IT capaciy increases in usage rae. This is due o he fac ha SaaS provider has o provide more capaciy as usage rae increases. However, if usage rae is oo high, hen he IT capaciy becomes lower. The reason is ha a very high usage rae can cause huge sysem congesion. In ha case, SaaS provider needs o raise is price, which reduces demand and can lead o lower sysem requess and lower IT capaciy Compeiion under asymmeric informaion We exend he analysis in Secion 4.1 and analyze he game under asymmeric informaion. SaaS provider knows is capaciy cos funcion, bu SWS provider only knows ha he SaaS provider s marginal cos for is capaciy is c 1H wih probabiliy h, and is c 1L wih probabiliy 1 h, where c 1L < c 1H. This seup follows a Bayesian game and is quie sandard in modeling games of incomplee informaion (Gibbons, 1992). Naurally, SaaS provider will choose a differen price depending on is marginal cos for he capaciy. If SaaS provider s capaciy cos is high, is profi funcion is p 2H D 2H c c 1H l H ; where p 2H and l H are SaaS price and SaaS provider s capaciy choice, respecively. D 2H = D 2 (p 2H ) is he demand funcion for SaaS when SaaS price is p 2H. Similarly, if SaaS provider s capaciy cos is low, is profi funcion is p 2L D 2L c c 1L l L ; where p 2L and l L are SaaS price and SaaS provider s capaciy choice, respecively. Also, D 2L = D 2 (p 2L ). SWS provider will opimize is price o maximize is expeced profi: Capaciy Usage Rae Fig. 2. Processing capaciy for differen usage rae. p 1 D 1H h þ p 1 D 1L ð1 hþ; where D 1H = D 1 (p 2H ) and D 1L = D 1 (p 2L ). Solving he above game of incomplee informaion, we have he following resul: Corollary 1. The soluion o he Bayesian Nash Equilibrium is as follows: p 1 ¼ vðc 1 c 2 Þþc 1 kðc 1H h þ c 1L ð1 hþþ 4c 1 c 2 p 2H ¼ 2vðc 1 c 2 Þþ2c 1 kc 1H c 2kðc 1H c 1L Þð1 hþ ; 4c 1 c 2 8c 1 2c 2 p 2L ¼ 2vðc 1 c 2 Þþ2c 1 kc 1L þ c 2kðc 1H c 1L Þh ; 4c 1 c 2 8c 1 2c 2 l H ¼ kc 1½4vðc 1 c 2 Þþkc 2 ðc 1H ð1 þ hþþc 1L ð1 hþþ 4kc 1 c 1H Š þ 1 2c 2 ð4c 1 c 2 Þðc 1 c 2 Þ d ; and l L ¼ kc 1½khðc 1H c 1L Þþ2k 4vŠ þ 1 2ð4c 1 c 2 Þðc 1 c 2 Þ d : We can see ha he SWS provider s equilibrium price under informaion asymmery case can eiher be higher or lower han his price under perfec informaion. When c 1H h + c 1L (1 h)>c 1, i.e. he expeced capaciy cos for SaaS is higher han c 1, SWS price will be higher han he perfec informaion case. 5. Dynamic qualiy compeiion 5.1. Modeling sofware qualiy compeiion In his secion, we model dynamic qualiy compeiion beween he wo sofware companies over a period of ime. In a longer horizon, sofware firms have o compee in produc feaures and funcionaliy (Gaes, 1998). Several pas sudies show ha he prices do no change very frequenly because of several facors such as he cos of changing price, inconvenience o cusomers, ec. (Carlon, 1986; Cecchei, 1986; Kashyap, 1995; Levy e al., 1997; Bils and Klenow, 24). Hence, in his model, prices of boh SWS and SaaS remain consan over ime and hey are exogenously deermined hrough he compeiion game in Secion 4. When he parameer values change significan enough o warran a change in price, hen we can easily deermine he opimal prices by re-solving he saic compeiion game wih new parameer values. The new prices can hen be used o solve he dynamic qualiy compeiion problem over ime. We use q i (), i = 1, 2, o represen he incremenal qualiy improvemen of firm i a ime. The cumulaive level of qualiy of he produc a is herefore Q i ðþ ¼ R q iðsþds þ Q i ðþ, i = 1,2, where Q i () is firm i s iniial qualiy. Similar o he models presened by Dixi (1979) and Banker e al. (1998), we assume ha he demand for a firm increases linearly wih is qualiy improvemen such as new feaures and funcionaliy and decreases linearly wih oher firm s qualiy improvemen. Meanwhile, cusomers also defec a a cerain rae (Sehi, 1973). Therefore he rae of change in demand for boh SWS and SaaS providers (hereafer also called sae equaions) can now be wrien as _D 1 ðþ ¼ dd 1ðÞ ¼ a 1 q d 1 ðþ b 1 q 2 ðþ u 1 D 1 ðþ; D 1 ðþ ¼D 1 ; and ð1þ _D 2 ðþ ¼ dd 2ðÞ ¼ a 2 q d 2 ðþ b 2 q 1 ðþ u 2 D 2 ðþ; D 2 ðþ ¼D 2 ; ð11þ where a i, i = 1, 2, is firm i s rae of demand increase due o is qualiy improvemen. The parameer b i, i = 1, 2, denoes demand decrease as a resul of qualiy improvemen of he compeior. We denoe cusomer defecion rae as u 1 for SWS and u 2 for SaaS. D 1 and D 2 are iniial demands for SWS and SaaS providers, respecively.

5 M. Fan e al. / European Journal of Operaional Research 196 (29) Similar o Moorhy (1988) and Banker e al. (1998), he marginal cos of increasing qualiy is considered o be quadraic. Thus, he cos of increasing qualiy by q 1 () isd 1 q 1 () 2 for SWS provider, and he cos of increasing qualiy by q 2 () isd 2 q 2 () 2 for SaaS provider. In order o discoun all he fuure sreams of coss and revenues relaive o he presen, we consider he exisence of a consan discoun rae r P. For a given planning horizon T, he SWS provider s problems can be wrien as max q 1 ðþ Z T e r fp 1 D 1 ðþ d 1 q 1 ðþ 2 gd: Similarly, he SaaS provider s problem is max q 2 ðþ Z T e r fp 2 D 2 ðþ d 2 q 2 ðþ 2 c c 1 lðþgd: ð12þ ð13þ SWS and SaaS providers maximize heir respecive oal profis over he whole ime horizon by choosing he opimal qualiy improvemen over ime. In addiion o he cos for qualiy improvemen, SaaS provider incurs capaciy cos of c + c 1 l() o provide sofware as a service. We assume ha he capaciy a ime is a funcion of demand and saisfies l()=kd 2 ()+1/d. Noe ha he capaciy (i.e., he processing rae) is also opimized in his model and is opimal level varies wih ime. The opimal capaciy level l() a ime is a funcion of D 2 (), which is obained from he opimal values of q 1 () and q 2 (). Hence, he maximizaion funcions over q 1 () and q 2 () in Eqs. (12) and (13) also opimize l() Soluions In he discouned opimal conrol problem, i is usually convenien o form he curren-value Hamilonian (Arrow and Kurz, 197). The curren-value Hamilonian funcions H 1 and H 2 for he opimal conrol problems of SWS and SaaS providers are H 1 ðþ ¼p 1 D 1 ðþ d 1 q 1 ðþ 2 þ k 1 ðþða 1 q 1 ðþ b 1 q 2 ðþ u 1 D 1 ðþþ; and H 2 ðþ ¼p 2 D 2 ðþ d 2 q 2 ðþ 2 c c 1 lðþþk 2 ðþða 2 q 2 ðþ b 2 q 1 ðþ u 2 D 2 ðþþ; where k 1 () and k 2 () are curren-value adjoin variables corresponding o sae variables D 1 () and D 2 (), respecively (Sehi and Thompson, 2). These adjoin variables a any ime can be inerpreed as he per uni change in he objecive funcion for a small change in he corresponding variable a ime. In oher words, k 1 () is he marginal value from each SWS cusomer a ime, and k 2 () is he marginal value from each SaaS subscriber a ime. The adjoin variable a ime is also referred o as he shadow price of a uni of he corresponding variable a ha ime (Sehi and Thompson, 2). Using he curren-value definiions, he Ponrayagin necessary condiions for a pah o be opimal (Arrow and Kurz, 197) are oh 1 ðþ oq 1 ðþ ¼ a 1k 1 ðþ 2d 1 q 1 ðþ ¼; oh 2 ðþ oq 2 ðþ ¼ a 2k 2 ðþ 2d 2 q 2 ðþ ¼; ð14þ ð15þ _k 1 ðþ ¼ dk 1ðÞ ¼ rk 1 ðþ oh 1ðÞ d od 1 ðþ ¼ p 1 þðu 1 þ rþk 1 ðþ; k 1 ðtþ ¼; ð16þ and _k 2 ðþ¼ dk 2ðÞ ¼ rk 2 ðþ oh 2ðÞ d od 2 ðþ ¼ p 2 þ kc 1 þðu 2 þ rþk 2 ðþ; k 2 ðtþ¼: ð17þ Solving he differenial equaions for k 1 () and k 2 () wih boundary condiions, we can ge k 1 ðþ ¼ p 1 u 1 þ r ð1 eðu 1þrÞð TÞ Þ; and ð18þ k 2 ðþ ¼ p 2 kc 1 u 2 þ r ð1 eðu 2þrÞð TÞ Þ: ð19þ Now we solve for he equilibrium qualiies for he differenial game as follows. Proposiion 2. The finie horizon differenial game (1) (13) has a unique Nash equilibrium soluion for he wo firms. The opimal sraegies are given by: q 1 ðþ ¼ a 1 p 1 2d 1 ðu 1 þ rþ ð1 eðu 1þrÞð TÞ Þ; and ð2þ q 2 ðþ ¼a 2ðp 2 kc 1 Þ 2d 2 ðu 2 þ rþ ð1 eðu 2þrÞð TÞ Þ: 5.3. Analysis ð21þ > ; i ¼ 1; 2). If cusomers are sensiive o sofware qualiy, hen each incremenal qualiy improvemen will have a large effec on he demand. As a resul, boh firms will have sronger incenives o raise produc qualiies, which will lead o higher sofware qualiy improvemens over ime. From he soluions of he differenial game, we can analyze he effecs of various parameers on equilibrium qualiy improvemens for boh SWS and SaaS providers. I is undersandable ha each firm s incremenal qualiy improvemen increases in he effeciveness of is qualiy on demand (i.e. oq i ðþ oa i < ; i ¼ 1; 2, which sugges ha each firm s qualiy improvemen decreases in is qualiy developmen cos. This resul provides srong suppor o he economic benefis derived from coninuous improvemen in sofware engineering process. If a firm can improve sofware developmen process and reduce coss in sofware research and developmen by employing maure or innovaive sofware engineering process, hen he firm can have a higher level of sofware qualiy improvemen in equilibrium. We can also prove he following proposiion on he effec of cusomer defecion rae and discoun facor on qualiy improvemen: We also have oq i ðþ od i Proposiion 3. (a) A decrease of a firm s cusomer defecion rae u i (i = 1,2) leads o higher level of qualiy improvemen for he firm. (b) A decrease of discoun facor r i leads o higher level of qualiy improvemen for boh firms. From he firs order condiions for curren-value Hamilonian (Eqs. (14) and (15)), we have q 1 ðþ ¼ a 1k 1 ðþ 2d 1 and q 2 ðþ ¼ a 2k 2 ðþ 2d 2. Examining he values of adjoin variables in he opimal soluion Eqs. (18) and (19), we can see ha he effecs of cusomer defecion rae and discoun facor on equilibrium qualiy improvemen are mediaed hrough he adjoin variables. In our model, adjoin variable is he shadow price for each cusomer. A higher cusomer defecion rae simply means a higher probabiliy ha a cusomer will defec in he nex ime period. I is clear ha a higher cusomer defecion rae lowers he marginal value, or shadow price, of each cusomer. If cusomers marginal values are low due o high defecion raes, i is naural ha firms will have less incenive on qualiy improvemen. Similarly, a higher discoun rae lowers he marginal value of fuure cusomers and leads o lower qualiy levels for boh firms. On he oher hand, a lower cusomer defecion rae or discoun facor will lead o higher marginal value for fuure cusomers. Thus, boh firms will have sronger incenives o improve qualiy in order o arac cusomers, which resuls o a higher equilibrium qualiy improvemen for boh firms. One of he advanages of using differenial game approach is ha i allows us o look a he dynamic naure of compeiion. Nex

6 666 M. Fan e al. / European Journal of Operaional Research 196 (29) proposiion examines opimal qualiy improvemen rajecories over ime. Proposiion 4. Qualiy improvemen of boh firms decreases over ime. Proposiion 4 suggess ha he qualiy improvemen is higher in early sages and becomes lower as produc maures. Therefore, he overall qualiy for produc i, Q i ðþ ¼ R q iðsþds þ Q i ðþ, i = 1,2, increases a a seeper rae in he early sages and flaens ou oward he end of he ime horizon. This proposiion provides imporan managerial guidelines for qualiy design sraegy over produc life cycle. I suggess ha sofware companies should concenrae heir qualiy improvemen in he early sages of produc inroducion. In he mauriy sages, he sraegy should be o mainain he high level of qualiy wih sligh modificaion. The cos srucure for SWS and SaaS providers are quie differen. Compared o SWS, SaaS provider has a much higher operaion cos. Now we analyze how i affecs he equilibrium oucome. Proposiion 5. A higher sysem usage rae (k) or a higher variable capaciy cos (c 1 ) leads o lower qualiy improvemen for SaaS provider, bu higher qualiy improvemen for SWS provider. Proposiion 5 suggess ha he operaion cos of SaaS affecs he equilibrium oucome significanly. SaaS provider s operaion cos could be considerable as cusomers are demanding qualiy sofware service. Two major facors could conribue o changing operaion cos for SaaS. Firs, as cusomers ge more familiar wih he sysem, heir usage rae may increase. In order o guaranee service qualiy, SaaS company has o provide and mainain higher level of sysem processing capaciy. Second, SaaS provider has o manage he cos relaed o processing capaciy. Alhough hardware cos goes down over ime, sysem cos including sysem sofware and human capial cos is more uncerain. Therefore, an increase of sysem usage or a higher variable IT capaciy cos could pu SaaS provider in a disadvanageous posiion in compeing wih SWS provider. As qualiy improvemen is cosly, i is difficul for SaaS provider o compee effecively wih a higher cos srucure. Proposiion 5 presens ineresing scenarios in he dynamic compeiion game beween SWS and SaaS providers. SWS provider can focus is business on sofware developmen. In comparison, SaaS has o compee on wo frons: sofware developmen and sofware service. By bundling sofware wih service, a SaaS firm has he benefis of lowering he implemenaion cos for is sysem and appeal o cerain cusomer segmens. A he same ime, he SaaS provider has o incur a high operaion cos. Therefore, i is criical for SaaS provider o balance hose wo effors. An inefficien operaion model could deplee is resources on sofware developmen and qualiy improvemen. Therefore, i is criical for SaaS provider o operae efficienly in order o compee wih SWS provider. For furher analysis, we define ðkc 1 Þ h ¼ vðc 1 c 2 Þð2a 2 d 1 a 1 d 2 Þ a 2 d 1 ð2c 1 c 2 Þþa 1 d 2 c 1 and prove he following proposiion: ð22þ Proposiion 6. (a) When u 1 > u 2 and kc 1 6 (kc 1 ) h, q 1 ðþ < q 2 ðþ; 8 < T. (b) When u 1 < u 2 and kc 1 > (kc 1 ) h,q 1 ðþ > q 2ðÞ; 8 < T. Corollary 2. When u 1 = u 2,ifkc 1 < (kc 1 ) h, hen q 1 ðþ < q 2ðÞ8 < T; if kc 1 P (kc 1 ) h, hen q 1 ðþ P q 2ðÞ; 8 < T. Proposiion 6 esablishes he condiions when qualiy improvemen of one firm always dominaes he oher. When SaaS provider has a lower cusomer defecion rae and is operaion cos, represened by kc 1, is below a cerain hreshold, SaaS has a higher produc improvemen han SWS over ime. On he oher hand, if SWS provider does no have a higher cusomer defecion rae and is operaion cos exceeds he hreshold, SWS will have a higher qualiy improvemen over ime. Corollary 2 presens a simpler resul by assuming ha he wo firms have he same cusomer defecion rae. I suggess ha he qualiy improvemen difference beween SWS and SaaS depends on he operaion cos of SaaS. In order o illusrae furher, a numerical example is shown in Fig. 3. We fix he parameer values as v =1, c 1 =.2, c 2 =.11, a 1 = a 2 =.3, u 1 = u 2 =.1, d 1 = d 2 =1, and r =.5 for all he numerical examples unless specified oherwise. Wih he above parameer values, we find ha (kc 1 ) h =.18. In Fig. 3a, kc 1 =.12 < (kc 1 ) h. Since he operaion cos of SaaS provider is below he hreshold, SaaS has a higher qualiy improvemen han SWS for < T. InFig. 3b, kc 1 =.3 > (kc 1 ) h. Since SaaS provider s operaion cos exceeds he hreshold, he qualiy improvemen for SWS is higher. I is also clear ha boh firms qualiy invesmen decrease over ime. As discussed earlier, accumulaed produc qualiy is he inegral of he qualiy improvemen over ime. We here examine he scenario ha SaaS has an iniial lead in sofware qualiy, i.e. Q 1 () < Q 2 (). When kc 1 <(kc 1 ) h, we have q 1 ðþ < q 2ðÞ. Thus, SaaS produc qualiy is always higher han ha of SWS (Fig. 4a). When kc 1 >(kc 1 ) h, alhough SaaS iniial qualiy is higher, SWS could cach up wih a higher produc qualiy in laer periods (Fig. 4b). Similarly, we can analyze he scenario when SWS has an iniial lead in sofware qualiy. Under ha scenario, he cos of SaaS provider, as represened by kc 1, sill plays an imporan role in affecing SaaS provider s abiliy o improve qualiy over ime. Qualiy Improvemen q 1 () q 2 () Qualiy Improvemen q 1 () q 2 () (a) k γ 1 < (k γ 1 ) h (b) k γ 1 > (k γ 1 ) h Fig. 3. Qualiy improvemen over ime.

7 M. Fan e al. / European Journal of Operaional Research 196 (29) Accumulaive Qualiy Q 1 () Q 2 () Accumulaive Qualiy Q 1 () Q 2 () * * (a) q 1 () < q 2 () * * (b) q 1 () > q 2 () Fig. 4. Accumulaive qualiy level over ime. Demand D 1 () 1 D 2 () * * (a) q 1 () < q 2 () Demand * * (b) q 1 () > q 2 () D 1 () D 2 () Fig. 5. Demand over ime. The analysis on dynamic qualiies for SWS and SaaS provides ineresing implicaions. Despie of iniial qualiy differences, he abiliy o improve qualiy coninuously is more imporan in he long run. Alhough SaaS has he advanage of lower implemenaion cos and ease of use, he long erm success of SaaS depends on is sofware feaures and funcions. To achieve a higher qualiy, he SaaS provider has o operae is service efficienly. Oherwise, he SaaS provider will no have adequae resources o innovae since qualiy improvemen is cosly. Therefore, providing an efficien sofware service and lowering is operaion cos is criical for he SaaS provider. I is clear ha Fig. 4a offers a good scenario for SaaS company. If SaaS can improve qualiy a a higher level han SWS, i is simply difficul for SWS o compee wih SaaS. However, since SaaS incurs higher operaion cos, is success depends on he operaion efficiency of he firm. Wih a business ha focuses on pure sofware developmen and qualiy improvemen, i is possible ha SWS provider can innovae and improve is qualiy more efficienly (Fig. 4b). Now we examine demand dynamics as a resul of qualiy compeiion. If SaaS firm can manage higher qualiy improvemen (i.e. q 1 ðþ < q 2ðÞ; 8 < TÞ, SaaS can mainain a larger marke share over ime (Fig. 5a). However, if SWS can susain a lead in qualiy innovaion over ime (i.e. q 1 ðþ > q 2ðÞ; 8 < T), SWS can compee effecively wih SaaS despie SWS has he disadvanage of higher implemenaion cos. Alhough SaaS provider could have a larger marke share a beginning, is marke lead posiion could erode if is operaion cos is high and canno susain produc qualiy lead over he long run (see Fig. 5b). This is no an ideal scenario for he SaaS provider. For SWS company, wih a low operaion cos and a large developmen budge, he long-erm sraegy should focus on produc feaures and funcionaliy. 6. Infinie horizon soluions Till now, we have sudied he problems wih finie ime horizon. In oher words, we obained he opimal soluion for a given period of ime. However, in cerain siuaions, an infinie ime horizon soluion is desirable if we wan o examine he long-run equilibrium resuls. Therefore, in his secion, we consider he problem where he opimizaion period is infinie, i.e., T = 1. The infinie horizon discouned objecion funcions for he SWS provider is max q 1 ðþ Z 1 e r fp 1 D 1 ðþ d 1 q 1 ðþ 2 gd: The SaaS provider s problem is max q 2 ðþ Z 1 e r fp 2 D 2 ðþ d 2 q 2 ðþ 2 c c 1 lðþgd: ð23þ ð24þ Clearly, he sae equaions for he infinie horizon problem are he same as hose for he finie horizon problem (see Eqs. (1) and (11)). Le us denoe he curren-value Hamilonian funcions corresponding o he opimal conrol problems of SWS and SaaS providers in he infinie horizon problem by H 1(inf) and H 2(inf), respecively. These curren-value Hamilonian funcions are now given below.

8 668 M. Fan e al. / European Journal of Operaional Research 196 (29) H 1ðinfÞ ðþ ¼p 1 D 1 ðþ d 1 q 1 ðþ 2 þ k 1ðinfÞ ðþða 1 q 1 ðþ b 1 q 2 ðþ u 1 D 1 ðþþ; and H 2ðinfÞ ðþ ¼p 2 D 2 ðþ d 2 q 2 ðþ 2 c c 1 lðþþk 2ðinfÞ ðþða 2 q 2 ðþ b 2 q 1 ðþ u 2 D 2 ðþþ; where k 1(inf) () and k 2(inf) () are curren-value adjoin variables corresponding o sae variables D 1 and D 2 respecively, in he infinie horizon problem. These curren-value adjoin variables can be inerpreed in a similar manner as he adjoin variables in he finie horizon problem. There are wo imporan soluions o an infinie horizon differenial game (Arrow and Kurz, 197). The opimal long-run saionary equilibrium is relaed o he auonomous sysem where he moion ceases and he soluion is no ime dependen. On he oher hand, non-saionary infinie horizon soluion describes he opimal rajecories of he conrol variables over ime and depics he process ha leads o he saionary equilibrium (Arrow and Kurz, 197; Sehi and Thompson, 2). In some cases, he nonsaionary soluion may no be dependen on ime eiher. We firs examine he non-saionary soluion. By seing T = 1 in he finie horizon soluion and checking he addiional sufficiency condiions for infinie horizon, we can obain he following equilibrium in an infinie horizon problem. Proposiion 7. The infinie horizon differenial game (1),(11),(23) and (24) has he following unique Nash equilibrium soluion: q 1ðinfÞ ðþ ¼ a 1 p 1 2d 1 ðr þ u 1 Þ q 2ðinfÞ ðþ ¼a 2ðp 2 kc 1 Þ 2d 2 ðr þ u 2 Þ 8; and ð25þ 8; ð26þ where q 1ðinfÞ ðþ and q 2ðinfÞðÞ are he opimal rajecories for incremenal qualiy for SWS and SaaS, respecively. We can see ha opimal rajecories for qualiy improvemen, as well as he adjoin variables, are consan over ime for boh SWS and SaaS in he infinie horizon problem. The following corollary shows ha hese soluions are also he long-run seady-sae equilibrium soluions, which are also called he urnpike (Sehi and Thompson, 2). Corollary 3. Infinie horizon opimal soluions are also he long-run seady-sae equilibrium soluions in he differenial game (1),(11), (23) and (24). We observe ha opimal qualiy improvemens under infinie horizon are always higher han heir counerpars under finie horizon. This is inuiive o undersand. I is naural for firms ha plan for a very long horizon o inves more on qualiy han firms ha only ac myopically. Similarly, we can obain condiions o compare he qualiy improvemens beween he wo firms. Corollary 4. When u 1 = u 2,ifkc 1 < (kc 1 ) h, hen q 1ðinfÞ ðþ < q 2ðinfÞ ðþ; 8 < T; if kc 1 P (kc 1 ) h, hen q 1ðinfÞ ðþ P q 2ðinfÞðÞ; 8 < T. Corollary 4 suggess ha he operaion cos hreshold (kc 1 ) h saed in Eq. (22) is robus. I applies in boh finie horizon and infinie horizon problems. 7. Discussion We here discuss he main feaures of he model, including he benefis and cos of SaaS, radeoff beween implemenaion cos and sofware qualiy and performance for cusomers, and shorand long-erm equilibriums The benefis and coss of bundling sofware wih service By bundling sofware wih service, SaaS provider offers more choices for sofware cusomers. Wih lower implemenaion cos and easy o use sysems, SaaS differeniaes wih radiional SWS and reduces price compeiion. However, SaaS companies have o incur significan operaion coss in providing he service operaion. As cusomers rely on exernal companies o deliver sofware soluions, sysem availabiliy and reliabiliy are heir major concerns. In order o guaranee service qualiy, SaaS provider has o inves on server and sysem capaciy. We can see ha he business of bundling sofware produc wih service is quie differen from he radiional develop-and-ship sofware business. As he cos of service operaion increases, i can hur he boom line of he SaaS firm. Our resuls indicae ha sysem usage rae and variable IT capaciy cos are imporan conribuors of he operaion cos. When he service cos, represened by kc 1 parameer, exceeds a criical hreshold, i is hard for SaaS firms o compee effecively wih SWS companies. This is simply because qualiy innovaion and improvemens are cosly. As operaion cos rises, SaaS canno mach he qualiy improvemen wih SWS in he long-run. The resuls of he sudy provide imporan managerial insighs for boh SaaS and SWS companies. SaaS firm has o achieve operaion efficiency in order o compee effecively wih SWS company. Founders of many SaaS companies worked for and gained heir experiences from SWS companies. The culure and many operaions of SaaS companies resemble hose of SWS companies. SaaS firms have o realize ha in order o succeed in a service-oriened business, hey have o go hrough significan ransformaions and achieve service excellence wih grea efficiency. Meanwhile, hey also have o realize ha sysem implemenaion cos and ease of use are only par of he facors ha drive cusomer choice. SaaS companies have o arge a higher design qualiy and achieve excellence in produc innovaion and qualiy improvemen in order o compee wih SWS in he long run. In order o do ha, SaaS company has o work hard o reduce he marginal operaion cos and qualiy improvemen cos. For SWS companies, he sraegy is simple: hey need o focus on produc qualiy. Wih coninuous qualiy innovaion, SWS firms can pu compeiive pressure on SaaS provider and ge payoff in he long run Implemenaion Cos versus Qualiy Cusomers ofen face radeoffs in choosing he righ sofware applicaions. On one hand, online SaaS applicaions are easy o use and implemenaion coss are lower. On he oher hand, SaaS hin-clien applicaions lack he full funcionaliy and performance of rich deskop sysems. In addiion, i is challenging o inegrae SaaS applicaion wih oher back-end sysems. Our analyical resuls sugges ha boh facors, i.e., implemenaion cos and performance qualiy, affec equilibrium oucomes. Implemenaion cos is a significan facor affecing cusomer s adopion decision. Over ime, sofware qualiy is a more imporan facor in reaining and aracing cusomers. Fig. 6 plos he marke share rajecory of SWS for he infinie horizon game. As shown in Fig. 6a, implemenaion cos difference beween SWS and SaaS affecs marke share in early periods. The impac of qualiy sraegies, however, is more robus over ime (Fig. 6b) Shor- and long-erm equilibrium We can gain several imporan and ineresing insighs by comparing shor-erm and long-erm equilibrium soluions. The longrun equilibrium soluion obained in he infinie horizon problem is higher han he myopic soluion obained in he finie horizon

9 M. Fan e al. / European Journal of Operaional Research 196 (29) Marke Share c 1 /c 2 = q* c 1 /c 2 = 2.2 1(inf) /q* 2(inf) =.85 q* 1(inf) /q* 2(inf) = c 1 /c 2 = q* 1(inf) /q* 2(inf) = * * (a) q 1(inf) / q 2(inf) = 1.1 Marke Share Fig. 6. SWS marke share over ime. (b) c 1 / c 2 = 1.8 problem. This resul is consisen wih he resuls obained from prior opimal conrol problems (e.g., Sehi and Bass, 23), and i has srong managerial implicaions. Naurally, firms ha plan for a longer horizon end o inves a a higher level and anicipae payoffs in he fuure. In conrac, firms ha plan for a shor horizon will ac more myopically and end o reduce invesmen oward he end of he planning period. In pracice, finie and infinie horizon differenial games are applicable in differen business seings. For producs wih shorer life cycle and a clear mauriy sage, a finie horizon soluion is more appropriae. The opimal sraegy is o inves highly on qualiy during marke growh period and lower qualiy improvemen during mauriy sages. For producs wih a very long life cycle, such as operaing sysems, an infinie horizon game is more suiable and firms have o susain qualiy improvemen over a long period of ime. We can also show ha, as he ime horizon (i.e., T) increases, he shor-run equilibrium soluion for he finie horizon problem (Eqs. (2) and (21)) approaches he long-run saionary equilibrium soluion saed in Corollary 3 excep for some final ime (i.e., for? T). This resul is similar o he asympoic properies of he opimal pah, ofen referred o as urnpike properies (Cass, 1966; Fershman and Kamien, 199). Hence, for a long enough ime horizon, he equilibrium pah of a finie-horizon problem moves away from he long-run equilibrium soluion only owards he end of he ime period. Therefore, if he ime horizon is long enough, we can approximae he equilibrium pah in finie differenial game by he long-run saionary equilibrium excep for some final ime. The advanage is ha long-run saionary equilibrium is relaively easier o compue and operae. Hence, he managers can use simpler decision rules wihou sacrificing oo much of he gains obained from he dynamic decision. 8. Conclusions and fuure research direcions This research uses a game heoreical approach o examine shor- and long-erm compeiion beween SaaS and SWS. We analyze a one-period price compeiion model as well as he long erm qualiy compeiion beween he wo firms. By bundling sofware wih service, SaaS company can effecively differeniae is produc by lowering sysem implemenaion cos. Our resuls show ha sofware implemenaion cos affecs equilibrium significanly in he shor run. The effec of qualiy improvemen is more robus. The service operaion cos of SaaS firm significanly deermines wheher he firm can compee effecively wih SWS company. We esablish condiions ha SaaS can have higher (or lower) incremenal qualiy han SWS over ime. We find ha in a finie horizon problem, firms choose higher qualiy improvemen a beginning and incremenal qualiy decreases in ime. In an infinie horizon, opimal qualiy sraegies are consan over ime and qualiy improvemens are higher han ha in he finie horizon problem. The model has several limiaions. Firs, our equilibrium resuls are based on open loop soluions. Second, we assume prices are exogenous afer he firs period. In realiy, firms could adjus prices along wih qualiies in he longer erm. For fuure research, we could aemp o solve he close loop equilibrium and compare he resuls wih open loop soluions. I would be ineresing o examine models in which firms dynamically opimize boh qualiies and prices. Furher, rapid progress of echnologies provides never-ending opporuniies for companies o innovae and develop new producs and services. SaaS is such an innovaion by bundling radiional sofware produc wih service over he Inerne. Companies are experimening new ways o uilize he Inerne and offer producs and services o saisfy diversified cusomer groups. For example, Google is offering free spreadshees applicaion ogeher wih adverising; open source providers such as Redha and IBM provide free sofware bu charge cusomers for services. Fuure research could invesigae he sraegic implicaions of hose business pracices. Acknowledgemens We hank he edior and he anonymous reviewers for heir helpful suggesions. We also hank paricipans of research seminars a Georgia Insiue of Technology and Universiy of Washingon for heir valuable commens. Appendix Proof of Proposiion 1. From Eqs. (3) and (6), we have l ¼ kv þ kp 1 1 kp c 2 c 1 c 2 þ 1 þ 1 2 c 2 c 1 c 2 d : Subsiuing he resul ino SaaS s profi funcion (5) and solving he firs order condiions for (4) and (5) simulaneously, we can ge he equilibrium resuls. h Proof of Corollary 2. Taking FOCs of he following maximizaion problems: p 2H D 2H c c 1H l H, p 2L D 2L c c 1L l L, and p 1 D 1H h +

10 67 M. Fan e al. / European Journal of Operaional Research 196 (29) p 1 D 1L (1 h), and solving he FOCs simulaneously we can easily ge he equilibrium prices and capaciies. h Proof of Proposiion 2. We have already esablished he necessary condiions for opimaliy. Here, we show he sufficiency condiions. Firs, he Hamilonian equaions are concave and differeniable in (D i (),q i ()), i = 1,2. Second, expressions for dod 1 () and D _ 2 ðþ given by Eqs. (1) and (11) are linear and differeniable in (D i (), q i ()), i = 1,2. Furher, since e ðu1þrþð TÞ 6 1, we have k 1 () P from Eq. (18). Nex we prove k 2 () P. From he Condiion (1), we have v > v *. Subsiuing he equilibrium prices of (7) and (8), we ge v ¼ ð2c 1 c 2 Þðc 1 ðvþkc 1 Þ c 2 vþ ð4c 1 c 2 Þðc 1 c 2. Thus, v > ð2c 1 c 2 Þkc 1 Þ 2ðc 1 c 2 > kc Þ 1, and we have p 2 kc 1 ¼ 2vðc 1 c 2 Þ ð2c 1 c 2 Þkc 1 4c 1 c 2 >. Wih e ðu2þrþð TÞ 6 1, we have k 2 () P. Therefore, according o Arrow sufficiency heorem (Arrow and Kurz, 197), (14) (17) are sufficien condiions for opimaliy. From condiions (14) and (15), we ge q 1 ðþ ¼ a 1k 1 ðþ 2d 1 and q 2 ðþ ¼ a 2k 2 ðþ 2d 2 Subsiuing (18) and (19), we can now easily ge equilibrium qualiy improvemen funcions. h Proof of Proposiion 3. From he opimal qualiy sraegies in (2) and (21), we have oq 1 ðþ 1 e ou 1 ¼ a 1 p ðu 1þrÞðT Þ ½1þðu 1 þrþðt ÞŠ 1 ¼ a 2d 1 ðu 1 þrþ 2 1 p 1 e ðu 1 þrþðt Þ ½1þðu 1 þrþðt ÞŠ. Using Taylor s expansion, we have 2d 1 ðu 1 þrþ 2 e ðu 1 þrþðt Þ e ðu1þrþðt Þ ½ ¼ 1 þðu 1 þ rþðt Þþ ðu 1 þ rþðt ÞŠ 2 2! ½ þ ðu 1 þ rþðt ÞŠ n þ n! Thus, e ðu1þrþðt Þ > 1 þ u 1 ðt Þ, and subsequenly oq 1 ðþ ou 1 <. We can also show ha oq 1 ðþ <. Similarly, we can prove ha or oq 2 ðþ ¼ oq 2 ðþ ou 2 or ¼ oq 1 ðþ ou 1 ¼ a 2 ðp 2 kc 1 Þ 1 e ðu 2þrÞðT Þ ð1 þðu 2 þ rþðt ÞÞ 2d 2 ðu 2 þ rþ 2 < : Proof of Proposiion 4. From he soluions of he differenial game, we direcly have oq 1 ðþ ¼ a ðu 1p 1 e 1þrÞð TÞ o 2d 1 <, and oq 2 ðþ ¼ o a 2ðp 2 kc 1 Þe ðu 2 þrþð TÞ 2d 2. From he proof of Proposiion 2, we have p 2 kc 1 >. Thus, oq 2 ðþ <. h o Proof of Proposiion 5. From he opimal qualiy improvemen oq sraegy, i is sraigh forward o show ha 2 ðþ ¼ ok a 2c 1 ð1 e ðu 2 þrþð TÞ Þ 2ðu 2 þrþd 2 <, and oq 2 ðþ oc 1 ¼ a 2kð1 e ðu 2þrÞð TÞ Þ oq 2ðu 2 þrþd 2 <. Also, 1 ðþ ¼ ok a 1 c 1 c 1 ð1 e ðu 1 þrþð TÞ Þ 2ð4c 1 c 2 Þðu 1 þrþd 1 >, and oq 1 ðþ oc 1 ¼ a 1c 1 kð1 e ðu 1þrÞð TÞ Þ 2ð4c 1 c 2 Þðu 1 þrþd 1 >. h Proof of Proposiion 6. We compare he slope of q 1 ðþ wih q 2 ðþ by examining: oq 1 ðþ o = oq 2 ðþ ¼ a 1d 2 e ðu 1 u 2 Þð TÞ ðvðc 1 c 2 Þþc 1 kc 1 Þ o a 2 d 1 ð2vðc 1 c 2 Þ kc 1 ð2c 1 c 2 ÞÞ : When u 1 > u 2, we have e ðu 1 u 2 Þð TÞ < 1. If kc 1 6 (kc 1 ) h, hen we can a prove ha 1 d 2 ðvðc 1 c 2 Þþc 1 kc 1 Þ < 1. Thus, we have oq 1 ðþ a 2 d 1 ð2vðc 1 c 2 Þ kc 1 ð2c 1 c 2 = oq 2 ðþ < 1 ÞÞ o o when boh u 1 > u 2 and kc 1 <(kc 1 ) h hold. Since q 1 ðþ and q 2ðÞ cross a = T (i.e.,q 1 ðtþ ¼q 2 ðtþ ¼Þ and he slope of q 1ðÞis always less seep han ha of q 2 ðþ, q 1 ðþ and q 2ðÞ only cross a one poin. Therefore, " < T we have q 2 ðþ > q 1ðÞ. Similarly we can prove par (b) of he proposiion. h Proof of Corollary 2. Wih u 1 = u 2,e ðu 1 u 2 Þð TÞ ¼ 1. If kc 1 <(kc 1 ) h, hen we can have oq 1 ðþ = oq 2 ðþ < 1. If kc o o 1 >(kc 1 ) h, hen oq 1 ðþ = o oq 2 ðþ > 1. Res of he proof follows similar o ha of Proposiion o 6. h Proof of Proposiion 7. The Ponrayagin necessary condiions for a pah o be opimal in he infinie horizon case are oh 1ðinfÞ ðþ ¼ a 1 k 1ðinfÞ ðþ 2d 1 q oq 1 ðþ 1 ðþ ¼; ða1þ oh 2ðinfÞ ðþ ¼ a 2 k 2ðinfÞ ðþ 2d 2 q oq 2 ðþ 2 ðþ ¼; ða2þ _k 1ðinfÞ ðþ ¼ dk 1ðinfÞðÞ ¼ rk 1ðinfÞ ðþ oh 1ðinfÞðÞ d od 1 ðþ ¼ rk 1ðinfÞ ðþ p 1 þ u 1 k 1ðinfÞ ðþ; and ða3þ _k 2ðinfÞ ðþ ¼ dk 2ðinfÞðÞ ¼ rk 2ðinfÞ ðþ oh 2ðinfÞðÞ d od 2 ðþ ¼ rk 2ðinfÞ ðþ p 2 þ kc 1 þ u 2 k 2ðinfÞ ðþ: ða4þ Arrow and Kurz (197) propose he following se of ransversaliy condiions which esablish he sufficiency for he opimal soluion in he infinie horizon case: lim!1 e r k 1ðinfÞ ðþ ¼; and ða5þ lim!1 e r k 2ðinfÞ ðþ ¼: ða6þ In he infinie horizon problem, i is someimes possible o obain he opimal soluion by seing T = 1 in he finie horizon problem. From Eqs. (18) and (19), i is easy o see ha seing T = 1 for adjoin variables in he opimal soluion of finie horizon problem will give us k 1ðinfÞ ðþ ¼ p 1 ; 8; and ða7þ u 1 þ r k 2ðinfÞ ðþ ¼ p 2 kc 1 ; 8: ða8þ u 2 þ r I is easy o show ha he adjoin variable soluions saisfy he ransversaliy condiions (A5) and (A6), which are he sufficien condiions for opimaliy in he infinie horizon problem. Now, from he firs order condiions of he curren-value Hamilonian funcions (A1) and (A2), we can easily ge he equilibrium soluions (25) and (26). h Proof of Corollary 3. Le hese long-run saionary equilibriums be k 1ðinfÞ and k 2ðinfÞ, which can be easily obained by seing _k 1ðinfÞ ðþ ¼ and k _ 2ðinfÞ ðþ ¼ in Eqs. (A3) and (A4), respecively. These values are now given by k 1ðinfÞ ¼ p 1 rþu 1, and k 2ðinfÞ ¼ p 2 kc 1 rþu 2. From he firs-order condiions Eqs. (A1) and (A2), we can easily ge he long-run saionary equilibrium values for qualiy improvemens, q 1ðinfÞ ¼ a 1 k 1ðinfÞ 2d 1 ¼ a 1p 1, and q 2d 1 ðrþu 1 Þ 2ðinfÞ ¼ a 2 k 2ðinfÞ 2d 2 ¼ a 2ðp 2 kc 1 Þ 2d 2 ðrþu 2, which are Þ same as he opimal qualiy improvemen rajecories given in Eqs. (25) and (26), respecively. h Proof of Corollary 4. We compare he q 1ðinfÞ ðþ wih q 2ðinfÞðÞ when u 1 = u 2 by examining q 1ðinfÞ ðþ ¼ a 1d 2 ðvðc 1 c 2 Þþc 1 kc 1 Þ. q 2ðinfÞ ðþ a 2 d 1 ð2vðc 1 c 2 Þ kc 1 ð2c 1 c 2 ÞÞ If kc 1 <(kc 1 ) h, we have a 1 d 2 ðvðc 1 c 2 Þþc 1 kc 1 Þ a 2 d 1 ð2vðc 1 c 2 Þ kc 1 ð2c 1 c 2 ÞÞ < 1: If kc 1 P (kc 1 ) h, we have a 1 d 2 ðvðc 1 c 2 Þþc 1 kc 1 Þ a 2 d 1 ð2vðc 1 c 2 Þ kc 1 ð2c 1 c 2 ÞÞ P 1: References Arrow, K.J., Kurz, M., 197. Public Invesmen, he Rae of Reurn, and Opimal Fiscal Policy. The John Hopkins Press, Balimore. Banker, R.D., Khosla, I., Sinha, K.K., Qualiy and compeiion. Managemen Science 44 (9), Bass, F.M., Krishamoorhy, A., Prasad, A., Sehi, S.P., 25. 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