Department of Economics. Strategic Choice of Network Externality

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1 Department of Eonomis Working Paper Series Strategi Choie of Network Externality Yuanzhu u & Sougata Poddar 05/03

2 Strategi Choie of Network Externality Yuanzhu u Sougata Poddar China Eonomis and Management Aademy Department of Eonomis Central University of Finane and Eonomis Aukland University of Tehnology Beijing, China Aukland, New Zealand yuanzhulu@ufe.edu.n spoddar@aut.a.nz January 05 Abstrat In many produt markets, impat of network externality plays an important role to affet the overall uality of a produt. owever, the degree or the strength of network externality is assumed as a parameter in most of the literature. We propose a model of vertial produt differentiation with two ompeting firms where the strength of network externality is endogenized as a strategi hoie of the high uality firm. We show how the euilibrium market struture and market overage depend on the ost of hoosing the network strength and on the relative uality differene of the ompeting produts. We also show that the relationship between the optimal level of network externalities and the relative uality differenes of the produts an be monotoni or non-monotoni. Keywords: Vertial produt differentiation, Network externality, Market struture, Market overage, Investment ost JE Classifiations: D3, D43, 3, 86 We thank seminar and onferene partiipants at Nanyang University of Tehnology, Singapore, the 03 APJAE Symposium on Industrial Organization and Global Value Chains in ong Kong and Jadavpur University, India for helpful omments on earlier versions of this paper. Yuanzhu u thanks National Natural Siene Foundation of China (No. 707) for finanial support. Address for orrespondene.

3 . Introdution In the modern environment of digital produts market with strong features of network externalities, a leading firm in this industry is not only just interested to inrease the size of its users-base i.e. the network size of its produts, it is also interested to enhane the strength of the network effet by making appropriate investments, whih onseuently inreases the overall value of the produts. Adding new apps, features and supporting devies to its ore produt over time enhanes the usage and overall uality of the produt. This is an alternative way to enhane the effetive uality of an existing produt without investing to improve the intrinsi uality of the produt whih may not also be feasible in many irumstanes. In this paper, we study this partiular aspet through a model of vertial produt differentiation with two ompeting firms. There is one high uality and one low uality firm. The high uality firm strategially invests to improve the strength or the degree of network externality to inrease the overall value of its ore produt for its users. The low uality firm also benefits (and free rides) from the effetive inrease of the network strength for its produt. owever, due to intrinsi lower uality of its produt, the absorption apaity of the network effets is limited. A onseuene of this is, the high uality produt generates more value (apart from its inherent high uality) to its users ompared to the low uality produt. Thus, it also gives adeuate inentive for the high uality firm to invest in inreasing the strength of network externality. In this set up we study the strategi interation between the two firms in the produt market and see its impat on the euilibrium market struture and market overage. To apture the situation, we onsider a two-stage game where in the first stage, the high uality firm makes ostly investment to improve or enhane the impat of the network externality. In the seond stage it ompetes in prie with the low uality produer. We solve the two-stage game and find out the impat of investment ost on the network strength hosen by the high uality firm for any given level of relative uality differene between the produts. To get the main idea in a simplified framework we first onsider two levels of investment ost on the network, namely, low and high. We first find out what would be the optimal levels or strengths of the network externalities for the high uality firm under low and high osts environment. Then Most digital produts fall in this ategory where bringing new aessories, supporting devies and suitable appliations (apps) to the ore produt that onnet different users enhane the strength of the network effet and hene improve the effetive uality of the ore produt.

4 for eah optimal level of network externality under a given ost environment, we find out what would be the ensuing euilibrium market struture (i.e. whether market outome is monopoly or duopoly) and whether the market will be partially or fully overed in the euilibrium. Thus, to this end we endogenize these two important aspets of the market. We find when the ost of investment on the network is low and the uality differene between the two produts is high, the market is always monopolized and fully overed by the high uality firm, while it remains duopoly when the uality differene is low. On the other hand, when the ost of investment on the network is high, the market always remains duopoly, however, it may be partially or fully overed. From the omparative statis analysis, we find that under low ost environment, the optimal level of network externality is always inreasing in uality differene of the produts, and the relationship is monotoni. Under high ost environment even if the optimal level of network externality is always inreasing in uality differene pieewise, however, the overall relationship between the two is non-monotoni. We then generalize the analysis to find the optimal degrees of network externalities hosen by the high uality firm under all possible ost senrios. We find the euilibrium market strutures and market overage under low, low-medium, medium and high investment osts on the network that essentially aptures all possible ost senarios needed for a omplete euilibrium haraterization of the problem. We find that the market an be monopolized and fully overed by the high uality firm when the investment ost on the network is low or lowmedium and the uality differene between the two produts is high, otherwise the market mostly remains duopoly, whih may be partially or fully overed. Seondly, if the investment ost on the network is very low or very high, the optimal degree of network externality is always inreasing in uality differene of the produts, but for any other levels of investment ost the relationship is non-monotoni although the relationship is non-dereasing pieewise. The impat of network externality or network effets on various eonomi situations is studied in the literature in several ontexts. There is a sizeable researh disussing the effets of network externality in eonomi ontexts whih originated from the studies by Rohlfs (974), Katz and Shapiro (985), followed by Chou and Shy (990), Churh and Gandal (993) among many others. Grilo et. al (00) modelled soial pressures suh as onformity or vanity in terms Note that the low and high ost senarios in the simplified framework is a sub-ase of the general framework that orrespond to low and medium ost senarios respetively.

5 of onsumption externalities in a model of spatial duopoly and haraterized various possible euilibrium outomes and market strutures. ambertini and Orsini (005) onsidered a model of vertial produt differentiation with the feature of positive network externality and foused on the existene of uality-prie euilibrium. In another area, where the impat of network externality is getting a renewed interest is opyright violations or piray in the digital produts market (see the work done by Conner and Rumelt 99, Takeyama 994, Shy and Thisse 999, Banerjee 003, 03 among others). owever, the researh dealing with the impat of network externality on onsumer and produer behaviour in all the above studies assumes the degree or strength of the network effet as an exogenous parameter in the analysis. But it is apparent that from a firm s point of view, improving the network strength of its produt has to be a part of strategi deision making proess apart from the priing deisions. The rest of the paper is organized as follows. In setion, we present the basi model. Setions 3 and 4 analyze the prie ompetition between the two firms when the market is partially and fully overed respetively, assuming the degree of demand network externality as a parameter. In setion 5, we endogenize the degree of network externality under the low and high network ost environment. The high uality firm strategially hooses the optimal degree of network externality; and the market struture and market overage are endogenously determined in the euilibrium. In setion 6, we generalize the analysis under all possible osts environments and summarize the main findings. Setion 7 onludes.. The Model A two-stage game is onsidered in a vertially differentiated produt market with two firms. The high uality produt is denoted by and the low uality produt is denoted by. There is a ontinuum of onsumers distributed uniformly over a unit interval with heterogeneous preferenes towards produt uality. The produts also exhibit the feature of positive network externality. 3 owever, as explained before, the impat of the network externality is asymmetri between the users of the high uality produt and low uality produt. The users of the high uality produt enjoy a higher level of network externality ompared to the users of the low 3 We rule out the ase of any negative network externality in this model. 3

6 uality produt as the absorption apaity of the network effet of the high uality produt is more than that of the low uality produt. 4 In terms of utility, the onsumer who buys the high uality produt, first of all, gets all the intrinsi benefit from the produt due to its high uality; seondly she also enjoys the full extent of the network externality generated by those users who also buy the high uality produt, plus the (limited) network externality generated by the low uality produt users. The buyers of the low uality produt an enjoy all the value of the produt (intrinsi as well as network) subjet to limitation that the lower uality an permit. We normalize the uality of the high uality firm s produt to one. The low uality produt is indexed by, 0, aptures the uality depreiation. where Formally, the utility of a typial onsumer X ( X 0, ) is given as follows: 5 D, X D D p if buys high uality produt, U X D D p if buys low uality produt, 0 if buys none, p and D, where p are the demand and pries for the high uality and low uality produts respetively. 0 is the oeffiient whih measures the level or strength of network externalities. For example, higher implies stronger effet of network externality, whereas when is lose to zero, it implies almost no effet of network externality. In our model only the high uality firm an influene. The interpretation is hoosing a higher level of is same as hoosing more useful features, apps or supporting devies for the ore produt to enhane the strength of the network externality by improving onnetivity among the users. In stage, the high uality firm hooses how muh to invest to improve the strength of network externality, i.e. hooses a level of. In stage, the high uality firm and the low uality firm ompete in pries in the produt market for a given level of. The high uality firm an hoose the level of network externality by inurring a ost of k, where 4 igh uality produt has all latest features funtional to absorb all the network effet, while the low uality produt has limited funtionality and absorption apaity of the network effet. 5 The utility representation is borrowed from the standard model of vertial produt differentiation in the literature (see Shaked and Sutton, 98; Tirole, 988, see also Banerjee 003). The parameter an be interpreted as a uality index for the low uality produt. 4

7 k 0 measures how ostly it is to invest in enhaning the level of network externality. For simpliity, we assume the osts of prodution for the firms are zero. The size of the market is normalized to. We look for the subgame perfet Nash euilibrium of this game and work bakward. Consider the prie ompetition stage. We will first deal with ases when the market is partially overed and when it is fully overed separately. The market overage aspet gets endogenized later when is hosen. 3. Partial Market Coverage The marginal onsumer uality produt, is given by This gives us X X m, who is indifferent between buying the high uality and low X D D p X D D p. m m m p p D buying the low uality produt and buying none, is given by We thus have Y D D p. m. The marginal onsumer Y m, who is indifferent between 0 Y D D p. m The demand for the high uality produt is given by D X D = p p The demand for the low uality produt is given by: D X m Ym = m ; we then obtain. () p p. 6 () The high and low uality firms ompete by hoosing pries strategially. The Nash euilibrium pries and demands are p p 6 To be preise, this demand system is orret only when. owever, it an be shown that in euilibrium we do have p p. For the sake of simpliity, here we omit the demand system when p p. 5

8 p 4, p 4, D 4 The profits of the high and low uality firms are respetively 4 4, Under this ase, the market struture is always duopoly. Next we deal with the ase of full market overage. 4, D Full Market Coverage Note that we have normalized the size of the market to be. There will be an upper bound of the level of network effet for whih D D. Denote that upper bound by ˆ. From the previous analysis, we find ˆ. So when ˆ, the market is always fully overed. 4 Now, when the market is fully overed, we have the following analysis. First, in the ase of full market overage, we an distinguish two ases: (i) both the firms are ative (i.e. duopoly); and (ii) only the high uality firm serves the whole market (i.e. monopoly) Case (ii) arises when is above a threshold value denoted by ~ (whih will be defined later), and ase (i) realizes when ˆ ~. When ase (ii) arises the market atually gets bak to monopoly from duopoly as we will see that the low uality firm annot sell its produt even if it sets its prie at marginal ost. Thus, market monopolization happens endogenously here as the strength of the network effet rosses a ertain threshold. Now we will analyze ase (i) in Setion 4. and ase (ii) in Setion 4. under the full market overage in detail. 4. ˆ ~ (Moderate level of network externality) ere, the demand for the high uality firm remains same as in the previous ase, i.e. euation (), while the demand for the low uality firm is D p p D. (3) 6

9 Note that in this ase, full overage of the market implies the onsumer with X=0 also obtains nonnegative surplus, whih an be expressed as p 0. It turns out we need to distinguish two subases (a) and (b). In subase (a) the onsumer with X=0 has no surplus while the onsumer gets positive surplus in subase (b). The Nash euilibrium pries, demands and eah firm s profit are given below. The detailed analysis is presented in Appendix A. Subase (a) When p, ˆ ˆ 4 p Subase (b) When p, ; D ˆ ~, p 3 9., 3 D, D,, 3 D 4, 3 9,, 4. ~ (igh level of network externality) When ~, from the expression of the prie and demand of the low uality firm (in the previous setion 4., subase (b)), we find that they go to zero (hene the profit as well). This means when reahes that threshold or beyond, the low uality firm is unable to ompete profitably, market beomes monopoly and all onsumers buy the high uality produt even if the low uality produt is free (stritly speaking, sold at marginal ost). To ensure the onsumer with the lowest X(=0) buys the high uality produt, the following ondition has to be satisfied: D p D D p p (sine D, D 0, p 0 ). It is not hard to see that the high uality firm harges a prie to maximize its profit and its total profit is 7

10 also (reall that the size of the market is normalized to ). This is true irrespetive of the uality of the low uality produt. 5. Choie of Optimal evel of Network Externality under ow and igh Cost Environment The main purpose of this setion is to show what type of market struture and the market overage ome out endogenously in euilibrium when the level of network externality is hosen by the high uality firm strategially. As we are fousing on two ost environments (low and high) for the high uality firm in hoosing network strength, it suffies to hoose a lower value of k i.e. k= to represent low ost environment and a higher value k= for high ost environment. We will see later that these two parameters values of k an apture all possible market strutures and market overage that may arise in euilibrium. So it is suffiient to fous on these two ases without any loss of generality of the problem. Now aording to the analysis in Setions 3 and 4, the high uality firm s profit in the seond stage an be summarized as 4 4 if 0, 4 if, 4 4 if, 9 if. k. N In Stage, the high uality firm hooses to maximize its net profit N Note that is ontinuous in. Denote f 4 4 k for 4, g k 4 for, h 4 k 9 for j k, and 8

11 for. To find the optimal level of network externality, we need to find the maximum of f, g, h and j in the relevant ranges of and then ompare these maximums. 5. ow Cost Environment: k= Consider relevant range Next we examine f first. We an easily show that Therefore, f is maximized at. 4 4 g. It an be shown that relevant range and thus 4 We now examine relevant range Finally we examine f 0 in the g 0 in the 4 g is maximized at h. We an easily show that. Therefore, h is maximized at j. Sine j, 0 0.5, and is maximized at if h 0 in the. j is maximized at if Combining the above results, we an onlude that (i) the optimal level of network externality is if 0.5 and (ii) if 0 0.5, to find the optimal level of network externality, we need to ompare the high uality firm s net profits when and when. Straightforward alulation yields that when h j 0 when We thus have the following proposition.. 9

12 Proposition : Under low ost environment with k=, the optimal level of network externality hosen by the high uality firm is if k 0.6 if k 0.6 The market is always fully overed and the market struture an be monopoly or duopoly. When is relatively small (as defined above), the high uality firm is a monopolist and hooses a higher level of network externality (i.e. high level of ); whereas when is relatively big (as defined above) both the high uality and the low uality firm share the market (duopoly), and the high uality firm hooses a lower level of network externality. The intuition of the result is as follows. When the market is fully overed and monopoly, from the orresponding seond stage profit expression ( ), it is not diffiult to see that the high uality would like to hoose a relatively high value of that maximizes its profit. On the other hand, when it has to share the market and the low uality firm gets the benefit from the network (free riding effet), the high uality firm wants to hoose a lower value of to redue this negative impat of free riding and keep its ompetitive advantage. Under the low ost environment, sine the market is always fully overed and no sope of growth, keeping the respetive market shares is the driving fore. Figure plots the optimal level of network externality as a funtion of the uality index of the low uality produt. We find that the optimal level of network externality is dereasing in as it should be as the high uality firm wants to limit the free riding effet. 0

13 gamma Figure The optimal level of network externality as a funtion of when k= 5. igh Cost Environment: k= Consider f first. By examining maximized at the interior of the range f and f, we an easily show that 0, 4, denoted by f is. owever, a losed-form expression of annot be derived. Next we examine g. By examining help of simulation, 7 we an show that g and g g is maximized at and tedious alulation with the when , at 4 7 In the relevant range,, the funtion g 4 is monotonially dereasing when , inreasing when 0.69, while it is inreasing first, then dereasing and then inreasing again when Given the omplexity of the funtional form, we have to turn to simulation when to determine the optimal level of network externality in this relevant range. Simulation results are available upon reuest.

14 the interior of the range, 4 denoted by when 0.6. owever, a losed-form expression of when , and at annot be derived. We now examine relevant range h. We an easily show that. Therefore, h is maximized at 9 h 0. in the Finally we examine j. Sine j 0, j is maximized at. Combining the above results, we an onlude that (i) the optimal level of network externality is if , (ii) if , to find the optimal level of network externality, we need to ompare the high uality firm s net profits when and when, and (iii) if 0.6, to find the optimal level of network externality, we need to ompare the high uality firm s net profits when and when. Sine the losed-form expressions of and annot be derived, we have to turn to simulation to find the optimal level of network externality. Simulation yields () is always dominated by, i.e., g f, and () g f when , while g f when ene, we have the following result. Proposition : Under high ost environment with k=, the optimal level of network externality hosen by the high uality firm is if k if k The market struture is always duopoly and it an be partially or fully overed.

15 gamma When is small, the high uality firm hooses a low level of network externality and the market is partially overed; when beomes bigger, the high uality firm hooses a higher level of network externality and the market is fully overed. The intuition of the result is as follows. When is small, the market is not very ompetitive, hene the high uality firm does not need to hoose a high level of, given the hoie of is ostly. Whereas, when gets bigger and the market beomes more ompetitive, the high uality firm would like to hoose a higher level of, beause of the opportunity of growth in the market (note here the market overage moves from partial to full) ignoring some of the negative impat of free riding of the low uality firm. In other words, the positive impat of revenue growth dominates the negative impat of free riding for the high uality firm here. This is also evident from the figure below (Figure ) as we see a big upward jump in optimal gamma, when rosses the low threshold. Thus, although the optimal level of network externality is still dereasing in pieewise, the overall relationship between the two beomes non-monotoni Figure The optimal level of network externality as a funtion of when k= 3

16 6. Optimal evel of Network Externality under All Cost Environments In this setion, we go for a general analysis and find the optimal degree of network externality hosen by the high uality firm under all possible ost environments. Sine it is impossible to derive a losed-form expression of the optimal level of network externality for general values of k and, we turn to simulation with the help of numerial examples. ike before, our starting point is high uality firm s net profit: N 4 4 k if, 4 k if, 4 4 k if, 9 k if. et us refer the above four ranges of as the first, seond, third, and fourth range respetively. g, To find the optimal level of network externality h and, we need to find the maximum of j (all of them as defined before in setion 5) in the relevant ranges of. f, From more numerial analysis, we find that there are three ritial values of k, k (in between.5 and.8), k (in between.8 and ), k 3 (in between and 6), whih separate k into four ranges, eah orresponding to one pattern of the range distribution of. 8 When k k, as inreases from lose to 0 to lose to, moves from the fourth range to the ritial value separating the seond range and the third range, i.e.,. When k k k, as inreases from lose to 0 to lose to, moves from the fourth range to the first range and then to. When k k k3, as inreases from lose to 0 to lose to,. When k k3, for any 0,, always lies in the first range. moves from the first range to 8 We perform further numerial analysis for k =.5,.8 and k = 6 apart from k=, whih we have done in Setion 5. The results for the new numerial analysis are summarized in Appendix B. The details are available upon reuest. 4

17 gamma gamma For the omparative stati analysis, we find that the optimal level of network externality is dereasing in when k k and k k3, while the relationship is non-monotoni when k k k and when k k k3. Figures 3 and 4 plots the optimal level of network externality as a funtion of the uality index of the low uality produt when k=.8 and k=6. Note that to learly show that the optimal level of network externality lies in the fourth range for very small values of when k=.8, we also insert a small box (within figure 3) where the maximum is eual to Figure 3 The optimal level of network externality as a funtion of when k=.8 5

18 gamma Figure 4 The optimal level of network externality as a funtion of when k=6 Simulation verifies the results obtained from the numerial analysis. Figure 5 presents the main result from the simulation, in whih three olour-shaded (blue, green, brown) areas represent the first range, the ritial value between the seond range and the third range ( ), and the fourth range respetively. More simulation analysis refines the results and shows that the three ritial values of k are around.784,.805 and The interval between k=.784 and k=.805 is very narrow, whih makes it diffiult to distinguish the seond pattern and the third pattern. owever, if we look at Figure 5 more losely, we an distinguish these two patterns: The brown-shaded area intersets the k-axis at a point above the point at whih the brown-, green-, and blue-shaded areas interset. Alternatively, we an draw another figure to show this differene more learly. Figure 6 presents the range distribution of for.6 k and

19 Figure 5 Range distribution of the optimal level of network externality (k 7 and 0 ) 7

20 Figure 6 Range distribution of the optimal level of network externality (.6 k and 0 0.) From the above analysis, we find that three possible optimal values of will atually be hosen by the high uality firm in euilibrium: (medium level of ), and (low level of ), 4 (high level of ), where k k.805. k M Now orresponding to eah optimal value of an euilibrium market struture (monopoly or duopoly) is emerged and the market overage (full or partial) is also determined endogenously. When the optimal value of is whih orresponds to a low degree of network externality, the market struture is duopoly i.e. shared by the high uality and the low uality firm and the market is partially overed. 8

21 When the optimal value of is M whih orresponds to a medium degree of network externality, the market struture is duopoly i.e. shared by the high uality and the low uality firm and the market is fully overed. When the optimal value of is whih orresponds to a high degree of network externality, the market is monopolized by the high uality firm and the market is fully overed. In the table below, we list all possible euilibrium outomes, where k are ritial values of in eah relevant ase. k, k and Table Euilibrium outomes for different investment osts and relative uality differenes 9 Investment ost (k) ow (k<.784) ow-medium (.784<k<.805) Medium (.805<k<5.05) ow uality Index () k Optimal degree of network externality Euilibrium Outome Market Struture Market overage igh ( ) Monopoly Full k Medium ( ) Duopoly Full M k igh ( ) Monopoly Full k k ow ( ) Duopoly Partial k Medium ( ) Duopoly Full k M ow ( ) Duopoly Partial k Medium ( ) Duopoly Full M igh ( k>5.05) 0 ow ( ) Duopoly Partial Below we summarize our main ualitative result from the above analysis. 9 The relative uality differene is defined to be -. 9

22 Theorem In a model of vertial produt differentiation with demand network externalities, when the degree of the network externality is a strategi hoie of the high uality firm, (i) the market an be monopolized and fully overed by the high uality firm when the investment ost on the network is low or low-medium and the uality differene between the two produts is high, otherwise the market mostly remains duopoly whih may be partially or fully overed. (ii) if the investment ost on the network is very low or very high, the optimal degree of network externality is always inreasing in uality differene of the produts, but for any other levels of investment ost the relationship is non-monotoni. 7. Conlusion In the paper, we address issues assoiated with produts with positive network externalities. A firm is not just interested to inrease the user-base or network size of its produt, it also wants to improve the strength or the degree of network externality to provide more value to its users and gain a strategi advantage over its ompetitor. 0 We apture this in a model of vertial produt differentiation where the high uality firm strategially invests to improve the strength of the network and ompetes with a low uality firm in the produt market. The low uality firms also benefits from the network strength in a limited manner. We solve for the optimal degrees of network externalities for the high uality firm under various network investement osts and haraterize endogenous euilibrium market strutures and market overage. In the omparative statis analysis, we show if the investment ost on the network is very low or high, the relationship between the optimal level of network externality and relative uality differene of the produts is monotoni, but for any other levels of investment osts, the relationship is nonmonotoni. 0 In the situations, where the produts do not have the expliit feature of network externality, the produers as part of their marketing strategy an invest to reate a network for the existing and potential users of the produt through an interative digital platform. For example, a firm an invest to reate a digital forum of its ore produts where the existing users an disuss and share their experienes about the produts and thereby intensify the effet of network externality of the relevant produts. A produt developer an also pay to use a popular soial networking site, (say like Faebook) to promote its produt as this would engage the potential buyers to ommuniate more effetively and seamlessly about the produt through that platform. 0

23 The other aspet we address here is, typially in the models of produt differentiation with network externality, the degree or the strength of the network effet is mostly assumed as a parameter. We depart from this and endogenize the level of network externality. We would also like to emphasize the fat that in models of vertial produt differentiation more often the extent of the market overage whether full or partial is assumed to be exogenous. ere we show that overing or not overing the market is at the heart of strategi problem and a onseuene of the strategi game between the firms. We onlude our disussion with a partiular senario where the findings of this analytial model an be useful. These days piray of digital goods beame uite prevalent due to the easy availability of opying tehnology. It is understood that rampant piray of digital goods is not desirable from the innovators as well as soiety s point of view. In this ontext, onsider our high uality firm as the original firm or the opyright holder of the onerned produt and the low uality firm as the ommerial pirate. Then our model will suggest that inreasing the degree of network effet strategially an eliminate the pirate from the market in some situations (see Table ). Thus, in the markets where ommerial piray is a serious threat, we propose piray an be fought by influening the demand network externality as an effetive instrument apart from other regular instruments (like monitoring and/or imposing lump-sum fine to the pirate whih are already disussed in detail in the literature of stopping piray). We think this is an additional instrument, whih an be used by the opyright holder or monitoring authority to effetively limit or stop rampant piray. Referenes Banerjee, D. S. (003), Software Piray: A Strategi Analysis and Poliy Instruments, International Journal of Industrial Organization, Vol., Banerjee, D. S (03), Effet of Piray on Innovation in the Presene of Network Externalities Eonomi Modelling, Vol.33, Chou, C. and O. Shy (990), Network Effets without Network Externalities, International Journal of Industrial Organization, Vol. 8, In another rare instane, Wauthy (996) onsidered a two stage duopoly game of uality-prie ompetition and emphasized the endogenous aspet of market overage.

24 Churh, J. and N. Gandal (993), Complementary Network Externalities and Tehnologial Adoption, International Journal of Industrial Organization, Vol., Conner, K. and R. Rumelt, (99), Software Piray: An Analysis of Protetion Strategies, Management Siene, Vol. 37, Grilo, I., Shy, O. and J. F. Thisse (00), Prie Competition when Consumer Behavior is Charaterized by Conformity or Vanity, Journal of Publi Eonomis, Vol. 80, Katz, M.. and C. Shapiro, (985), Network Externalities, Competition, and Compatibility, Amerian Eonomi Review, Vol. 75, ambertini,. and R. Orsini, (005), The Existene of Euilibrium in a Differentiated Duopoly with Network Externalities The Japanese Eonomi Review, Vol. 56, Rohlfs, J. (974), A Theory of Interdependent Demand for a Communiation Servie, Bell Journal of Eonomis, Vol. 5, Shaked, A. and Sutton, J., (98), Relaxing Prie Competition through Produt Differentiation, Review of Eonomi Studies, Vol. 49, 3-3. Shy, O. and J. F. Thisse, (999). A Strategi Approah to Software Protetion, Journal of Eonomis & Management Strategy, Vol. 8, Takeyama,. N., (994), The Welfare Impliations of Unauthorized Reprodution of Intelletual Property in the Presene of Network Externalities, Journal of Industrial Eonomis, Vol. 4, Tirole, J. (988) The Theory of Industrial Organization, Cambridge, MA, MIT Press. Wauthy, X. (996), Quality Choie in Models of Vertial Differentiation, Journal of Industrial Eonomis, Vol. 65, Appendix A p p The high-uality produer maximizes p D p. The first-order ondition is The low-uality produer maximizes p p 0. (A)

25 p p p D p p p p subjet to p 0. Define the agrangian onditions are and If p p p p p. The first-order and slakness p p 0, p 0, p 0, p 0., then we must have p p 0. Given p (A), it follows that p p / /, (A3) p p /. Thus, 0 ˆ If p 0, then we must have 0 and The solution to the system of euations (A.) and (A.4) is It follows that Thus, p p 0. (A4) 3, p p p 3 3. p 0 ˆ. 3. (A5) Note that for (A5) to be the solution, must not exeed : ~ ; so that the prie of the low-uality produt is nonnegative: p 0. The demands and profits for both firms are then obtained straightforwardly. The expressions are given in Setion 4. and will not be repeated here. Appendix B 3

26 Summary of the results for k =.5,.8 and 6. Details are available upon reuest. When k=.5, the optimal level of network externality hosen by the high uality firm is if k if k When k=.8, the optimal level of network externality hosen by the high uality firm is 5 if < if. 4 if 0.03 When k=6, the optimal level of network externality hosen by the high uality firm is < 4. 4

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