Exploiting Externalities: Facebook versus Users, Advertisers and Application Developers

Transcription

1 Exploiting Externalities: Facebook versus Users, Advertisers and Application Developers Ernie G.S. Teo and Hongyu Chen y Division of Economics, Nanyang Technological University, Singapore March 9, 202 Abstract Social networking sites (SNS) like Facebook are fast becoming part of our daily lives. Increasingly, businesses and organizations are getting on this bandwagon as SNS have the ability to reach a large mass of the population and in uence consumer behavior. Facebook reported an annual revenue of US$3.7 billion in 20 and boosts an active user population of more than 845 million active users in 202. The emergence of the SNS industry has large in uence on every aspect of our society. Thus, it has attracted much academic attention, especially in social sciences. The business strategy aspect of SNS has yet been theoretically analyzed. This paper aims to study the market structure of the SNS industry and discuss the appropriate pricing strategies for the SNS rm. The market structure of the SNS industry is multi-sided; the service is provided by the SNS rm and three distinct user groups provide each other with network externalities. The three groups are the SNS users, application developers and advertisers. These externalities may be complementary or con icting. Analysis of the network externalities among these entities and the compromise adopted by the SNS Companies are presented in our study. Users do not pay to participate in the SNS but application developers and advertisers are charged a fee. Constructing a micro-founded model, we set up the utility functions of each entity by using heterogeneity assumptions and examine how the SNS rm can maximize pro ts within these constraints. We show that the SNS rm may allow too few advertisers and application developers into the market. This result is due to the interplay of externalities between and within each group. PRELIMINARY WORK, COMMENTS WELCOME JEL Codes: D2, L0, M37 Keywords: SNS Industry, Externality, Multi-sided Market, Market Structure, Pricing Strategy, Heterogeneity INTRODUCTION A Social Network Site (SNS) provides a social service where users can interact through an online platform. SNS mainly focus on constructing and maintain social networks or social relations among people. They also provide a variety of additional activities such as social games. Generally speaking, social network sites usually provide an individual-centered service which allows users to share ideas, activities, events and interests within their own individual network groups. We wish to acknowledge the support for this project from Nanyang Technological University under the Undergraduate Research Experience on Campus (URECA) programme. y Corresponding Author: Tel.: Address: Nanyang Technological University 36 Nanyang Crescent Hall 5, , Singapore address: chen074@ntu.edu.sg

2 Early versions of these online communities include Usenet, ARPANET and LISTSERV. In the late 990s, introduction of user pro les became a central feature of social networking sites, by allowing users to compile lists of friends and search for other users with similar interests. With this innovation, the SNS industry experienced unconceivable growth. Currently, the largest SNS is Facebook, which is estimated to have more than 845 million active users. It is self-evident that the potential power of the emerging SNS industries is beyond our imagination. SNS in uence nearly every aspect of our society, including communication, production, social ethics and even politics. A prominent and recent instance of this in uence is the 20 Egyptian Revolution, where Facebook groups were one of the main tools used in the call for mass demonstrations. Moreover, SNS is widely used as a mass media tool for political campaigns like presidential elections. Although considerable amount of research has been done on the media and information aspect of SNS, little contributions have been made on the economics of this emerging industry. The objective of our theoretical study is to construct the market structure and pricing strategies of an SNS rm using microeconomics foundations. We model the SNS market as a three-sided market with the SNS rm serving as a platform. The three entities in the SNS market are consumers, advertisers and application(app) developers. These entities exert direct and indirect network externalities among each other and within their own groups. This paper focus on the pricing strategies of the SNS rm in absence of competition and analyses the e ciency of the market. The supporting theories we referred to in this paper are mainly from the study of the bandwagon model for high tech industries by Rohlfs (2003), as well as the two sided market model introduced by Armstrong (2006). 2 LITERATURE REVIEW / BACKGROUND 2. NETWORK EXTERNALITIES Network externalities exists when one user of a good or service creates value to other users of the same good. When network e ects are present, the value of a product or service increases as more people use it. Due to its very nature, the presence of network externalities are particularly strong for SNS. In this section, we look at how network externalities were modelled by other scholars. 2.. Positive Network Externalities Positive network externalities mean that a consumer s utility increases when there are more users or subscribers. Hahn (200) has investigated how call and positive network externalities can a ect a monopolist s optimal nonlinear pricing behavior in the two-way telecommunication markets. The paper has shown that the existence of positive call externalities results in all types of subscribers (even the highest type) making suboptimal quantities of calls in the optimum. Due to call externalities, there may exist some subscribers who only receive calls without making any outgoing calls in equilibrium. Also, the rm may have incentives to subsidize some low-type consumers in order to take advantage of network e ects. There are two types of positive network externalities. The rst type of positive externality is exerted by the same group of entities. For example, a large amount of consumers would make it easier to form groups or nd friends on the SNS. Consumers derive higher utility if the user base of an SNS is bigger. Therefore, there exist positive externalities within users/consumers. The second type of positive externality used in our model is exerted by a distinct group. For instance, more applications available on the SNS platform would promote more consumers to use it. Therefore, positive externalities exist between third-party application developer and the SNS consumers. We use both within and between group externality in our model. 2

3 2..2 Negative Network Externalities In contrast, negative network externalities mean that consumer utility decreases when there are more users or subscribers. Shy (200) looked at this from a social welfare point of view and analyzed the pricing of a congestible resource such as the internet services. Other than congestion, there are many kinds of negative externalities between entities. For instance, there exists con ict between the level of advertisement and number of consumers. The more advertisement a SNS allows on their website, the more annoyed (negative externality) consumers become. In our three-sided market model, we incorporate this by having negative network externalities is exerted by advertisers on consumers (or end users). 2.2 MULTI-SIDED MARKET Multi-sided markets are economic platforms with two or more distinct user groups that provide each other with network externalities. Each group of users of the platform are heterogeneous, and the externalities they exert on each other could be either positive or negative Monopoly Platform Market For the simple economic value that created by interactions or transactions between pairs of end users, buyers (denoted as B) and sellers (denoted as S). They assume that buyers are heterogeneous in that their gross surpluses, b B ; associated with a transaction di er. Similarly, sellers gross surplus, b S ; from a transaction di er. Moreover, the buyers and sellers could only complete their trading by o ering and accepting a dealed price. In the absence of xed usage costs and xed fees, the buyer s (sellers ) demand depends only on the price p B (respectively, p S ) charged by the monopoly platform. Previous research of models a la Baye and Morgan (200) and Caillaud and Jullien (2003) has focused on the matching process between buyers and sellers, and also they studied the proportion of such matches that e ectively results in a transaction. If we make the simple assumption that the independence between b B and b S, the proportion of transactions is equal to the product D B (p B ) D S (p S ). Rochet and Tirole (2004) found that in the absence of xed usage costs and xed fees, there are network externalities in that the surplus of a buyer with gross per transaction surplus b B. (b B p B )N S ; depends on the number of sellers N S. The buyers quasi-demand function, N B = Pr(b B p B ) = D B (p B ) () is independent of the number of sellers. Similarly, the sellers quasi-demand for platform services is also independent of the number of buyers: N S = Pr(b S p S ) = D S (p S ) (2) Without loss of generality, we can assume that each such pair corresponds to one potential transaction. In our model, we assume a monopolistic market with one SNS rm. We also made the same simple assumption that b B and b S are independent. The price level that the SNS rm may charge from each entity is di erent, thus there may be an incentive to charge above the e cient level Competition in Two-Sided Markets Roson (2005) studied competition in two-sided markets. His study contains various types of competition that a ect two-sided markets. Inside competition occurs between subjects within the same platform, whereas outside competition occurs between two or more platforms. As far as inside competition is concerned, it should be noticed that belonging to a common platform does not rule out the emergence 3

4 of internal competition. For example, a shopping mall is a two-sided market, attracting both customers and shops. Shops may still compete among themselves, though. In this case, an especially interesting question is how platform access can occur and how access prices are set, Nocke, Peitz and Stahl (2004). Schi (2003) considers the possibility that open systems share access to one or both market sides, so that cooperation between platforms may coexist with competition. An example could be some real estate agencies, sharing directories of units for sale, or Internet backbones with peering interconnection agreements (Cremer, Rey and Tirole (2000); Little and Wright (2000); and Rochet and Tirole (2002)). Generalizing this case, one could easily conceive systems, in which access is sold to the other platforms at a price. This draws an interesting analogy between some two-sided markets and telecommunication networks, for which a wide literature on access pricing is available (for example, La ont and Tirole (2000)). Chakravorti and Roson (2004) compare the market equilibrium of a duopoly with the one of a cartel between di erentiated platforms. They show that, when switching from the monopolistic cartel to the duopolistic competition, the e ect of price reduction dominates the change on the price structure, with non-ambiguous positive e ects on welfare, unless the market power of the cartel was already restricted by the nature of the platform, Rochet and Tirole (2002), or by some other speci c characteristics of the market. In our model, we only discuss the monopoly scenario and assumed that there would be no competition between SNS Companies. However in real-life, the SNS industry could be considered as oligopoly, among which Facebook accounts for majority of the market. This is a limitation of our model which could be analyzed in future research. 3 THE THREE-SIDED MARKET MODEL The methodology we used in constructing the three-sided market model is to rst build the utility functions of the three sides, namely consumers, advertisers and application developers. We derive the demand of each group using heterogeneity assumptions. Using each group s demand, we set up the pro t function of the SNS rm and solve for the equilibrium set of prices and number of users from each group. First of all, we will de ne some important notations used in this model. We use numerals to represent each party (i) in the three-sided market, represent consumers, 2 represent advertisers and 3 represent application developers. a i;j represents the externalities exerted by party j on one member of party i, given that i 6= j. When i = j; a i;j represents the direct network externalities of each party exerts on its own individual member. u i;j represents utility of the jth member from party i, for i 2 ; 2; 3. Notations for the quantity of each party, namely consumers, advertisers and app developers, are represented by N, A and D respectively. The network externalities a i;j are generally assumed to be positive and its maximum is normalized to. The only special case for speci c a i;j is a ;2 2 [ ; 0]. It is assumed that advertisers will exert negative network externalities on consumers (Advertisement is purely an annoyance factor to consumers rather than a source of information to consumers such as that described in Shapiro (980).). Moreover, to simplify our model, we assume that network externalities are non-existent between and within advertisers and app developers. That is to say, a 2;2 = a 3;3 = a 2;3 = a 3;2 = 0. The number of other advertisers or app developers do not have an e ect on its own or each other s utility. 3. Utility Function for Consumers In the three-sided market, we make use of the heterogeneity of a ;2, the network externalities of advertisers exerted on consumers. Although a ;2 is negative, we may normalize it into the range of [0; ] and use 4

5 the minus sign in the utility function. To make things easy, we assume that a ;2 conforms to Continuous Uniform Distribution ranging from 0 to, denoted as a ;2 U(0; ). The graph of a (;2)i for various consumers is shown as below: Npotential N a(,2)i 0 a(,2)i a (;2)i is the network externality of advertisers exerted on the ith consumer. N potential here denotes the entire base of consumers who have potential to enter the SNS market. In another word, N potential is the maximum quantity of consumers the SNS market could possibly have, and it is normalized to. N will thus denote the actual number of consumers in the SNS market. Utility function of each individual consumer is derived as below: u i = g + ; N + ;3 D (;2)i A > 0 (3) g represents the constant utility gain, which is also normalized to [0; ]. Utility of certain consumer i is the summation of total network externalities obtained from every party in this market, as well as the constant gain. A simple corollary here is that any consumer with utility non-negative will join this SNS market. Moreover, as N potential is normalized to, the actual number of consumers N will exactly equals to (;2)i, for the ith consumer who is indi erent between join or not. Thus, the expression for N is solved by nding (;2)i for the indi erent consumer as below. u i = g + ; N + ;3 D (;2)i A = 0 (4) g + ; N + ;3 D NA = 0 (5) (;2)i = N = g + ;3D (6) This gives us the demand of consumers as a function of the number of app developers (D), the number of advertisers (A) and the externalities these groups impose on consumers. 3.2 Utility Function for Advertisers By the same token, we will make use of the heterogeneity of a 2; (the network externalities of consumers exerted on advertisers). a 2; also conforms to Continuous Uniform Distribution ranging from 0 to, denoted as a 2; U(0; ). The graph of a (2;)i for various advertisers is shown as below: A 0 a(2,)i a(2,)i One thing we should note here, is that A = a (2;)i, for the ith advertiser who is indi erent between joining or not. This is di erent from the a ;2 case, where a ;2 is negative and normalized to [0; ]. Utility for each advertiser is the total network externalities gained minus the price paid to the SNS rm, denoted 5

6 as p a. By our previous assumption that there is no externalities between or within advertisers and app developers, we will model the utility function for each advertiser as below: u 2i = (2;)i N p a > 0 (7) p a = (2;)i N = ( A)N (8) This is the inverse demand of advertisers which is also a function of N (the number of consumers). 3.3 Utility Function for App Developers The construction of an app developer s utility is quite similar to that of an advertiser s. In this case, a 3; is made heterogenous and conforms to the Continuous Uniform Distribution ranging from 0 to, denoted as a 3; U(0; ). The graph of a (3;)i for various app developers is shown as below: D 0 a(3,)i a(3,)i By the same logic in the advertiser case, D = a (3;)i, for the ith app developer who is indi erent between join or not. Utility for each app developer is the total network externalities gained minus the price charged by the SNS rm, denoted as p d. By the previous assumption that there is no externalities between or within advertisers and app developers, we will model the utility function for each app developers as below: u 3i = (3;)i N p d > 0 (9) p d = (3;)i N = ( D)N (0) This is the inverse demand of app developers which is also a function of N (the number of consumers). 3.4 Pro t Function for the SNS rm Using the demand functions found above, we can set up the pro t function for the SNS rm as below: = p a A + p d D () = N( A)A + N( D)D (2) = N A 2 + A D 2 + D (3) As consumers are not charged a fee to participate in the SNS, the pro t for the SNS rm contains two parts, namely revenue obtained from advertisers and app developers. The cost for the SNS rm is comparatively xed and small, thus normalized to zero. 6

7 In our three-sided market model, only p a and p d are endogenous variables. SNS Companies can only choose p a and p d to alter their pro t. However, the discussion of pro t by using p a and p d is much more complicated than using A and D as endogenous variables. In the latter part, we will demonstrate that it is equivalent to set p a, p d as endogenous variables or to set A, D as endogenous variables. The SNS rm is always intending to maximize their pro t by indirectly choosing the number of advertisers and app developers. However, nding the pro t-maximizing equilibrium is not straightforward. More sophisticated technics are presented and discussed in the following sections. 4 THE MONOPOLY PROFIT-MAXIMIZING ANALYSIS In this section, we will model the SNS rm as monopoly, who can choose p a and p d to maximize their pro t. However, before the pro t-maximizing analysis, we shall rst clarify the number of consumers N in di erent cases. By the normalization restriction stated above, the number of consumers in the SNS market N is normalized to be in the range [0; ]. Speci cally, N = means that all potential consumers shall join in the SNS market. In another words, any consumer who participates in the SNS has a non-negative utility. Nevertheless, the expression of N obtained from last section, which is N = g+;3d can sometimes violate our normalization restriction given certain externalities. There are basically three ranges for the value of g+;3d to be discussed, namely g+;3d 2 ( ; 0), [0; ] and (; +). When g+;3d belongs to the ranges ( ; 0) and (; +), the number of consumers equals to one; when g+;3d 2 [0; ], the number of consumers is exactly g+;3d. More intuitively speaking, when the constant gain (g) and externalities are su ciently high, all potential consumers will participate in the SNS market, even when advertising exerts negative externalities on consumers. For simplicity of the analysis, we shall use A and D to replace p a and p d as endogenous variables. It is always consistent for us to change the endogenous variables due to the nice property of their relationship functions. 2 of the pro t-maximizing equilibrium. More explicitly, the choice of endogenous variables will not a ect the outcome We can rewrite the pro t function of the SNS rm as below: = N (A 2 )2 (D 2 )2 + 2 We can separate into two parts: N and (A 2 )2 (D 2 ) It is obvious that A = D = 2 is the maximum solution for (A 2 )2 (D 2 )2 + 2, and N is normalized to have maximum value of one. We can easily consider the scenarios where N = for given externalities, and A = D = 2. Then, the pro t of the SNS rm is maximized and equals to 2. Therefore, we can divide the choice of pro t-maximizing A and D into two scenarios. (4) 4. Scenario One: Externalities are Su ciently Large When the externalities g, ; and ;3 are su ciently large, setting A and D to be 2 will result in N = (full participation of consumers). By our previous discussion on number of consumers, it can only be the case when g+;3d belongs to the range ( ; 0) or (; +). Generally speaking, either the constant utility gain g and ;3 are large enough to make the numerator larger than the denominator; or when ;, the network externality between consumers, is larger than 2 which makes the denominator negative. Proof to be found in Appendix 8. Preliminary Discussion of Numbers of Consumers 2 Proof to be found in Appendix 8.2 Consistency of The Model 7

8 In conclusion, when the externalities are su ciently large, the SNS rm is able to choose A = D = 2 to maximize their pro t. The pro t of the SNS rm is maximized to be 2 in this scenario. 4.2 Scenario Two: Externalities are Intermediate When the externalities g, ; and ;3, are intermediate, setting A and D to be 2 will result in N 2 [0; ]. N 2 [0; ] implies that N = g+;3d by the preliminary discussion of N from above. We propose that in order to maximize pro t, the SNS rm will choose A and D such that N =. First, we prove the feasibility of setting N =. Next, we will show that setting N = is pro t-maximizing for the SNS rm. Lemma Setting N = is always feasible under Scenario two Proof. By the second case of preliminary discussion of N, given externalities g, D, ;3 and A = D = 2, N = g+;3d 2 [0; ]. This means that A > ;, and g + ;3 D. Therefore, setting N = is equivalent to setting g + ;3 D =. It is obvious that the SNS rm can indirectly decrease A or increase D or apply both strategies. Therefore, when N 2 [0; ], it is always feasible for the rm to choose another combination of A and D other than ( 2 ; 2 ), such that N =. Lemma 2 Setting N = is pro t-maximizing for the SNS rm Proof. Since variables in the expression of N are all positive, by the logic from Linear Programming, we can add in a slack variable s 0, such that g + D ;3 + ; + s = A. Then we substitute A = g + D ;3 + ; + s to the original pro t function = g+d;3 A 2 + A D 2 + D and we get: = g + D ;3 g + D ;3 + s By taking the rst partial derivative of with respect to s: (g + D ;3 + ; + s) 2 + g + D ;3 + ; + s D 2 + = g + D ;3 (g + s + D ;3 ) 2 (g + s + D ;3) 2 D 2 + D 2 ; + ; (6) It is < 0, as D2 + D 0 and 2 ; + ; 0, given D; ; 2 [0; ]. Therefore, it is proved that is of decreasing relationship with s. This means that the smaller the slack variable s, the higher the pro t of the SNS rm. By feasibility property of Lemma, the SNS rm will choose such a combination of A and D such that s = 0, or equivalently N = g+d;3 = to maximize their pro t. Thus, we have shown that if the choice of A = D = 2 results in N 2 [0; ] the SNS rm would prefer to readjust A and D such that N = as it is pro t-maximizing. As a result, we can set N = and represent A as a function of D, (it would be the same if we represent D as a function of A) and substitute it back to the pro t function. From equation N = g+d;3 =, we have g + D ;3 + ; = A. Substituting the expression of A into the pro t function, we can get the pro t function of single endogenous variable D: = (g + D ;3 + ; ) 2 + g + D ;3 + ; D 2 + D (7) Simpli cation of the pro t function will give us: = ( 2 ;3 + )D 2 + ( ;3 + 2 ; ;3 2g ;3 )D (g + ; ) 2 + g + ; (8) 8

9 From the expression of the pro t function, we can easily conclude that it is a concave parabola, as ( 2 ;3 + ) < 0. Thus, the pro t for the SNS rm is maximized when D takes the value at the axis of symmetry as below: D = 2( 2 ;3 + ) ( 2 ; ;3 2g ;3 + ;3 + ) (9) Generally, there are three positions of the axis of symmetry D, namely D belongs to ( ; 0), [0; ] and (; ). However, we can rule out the possibilities of D falling in the ranges ( ; 0) and (; ). The only possible range left for D is [0; ], and we can further prove that D 2 [ 2 ; ]. It intuitively means that the SNS rm will choose more app developers to o set the annoyance of advertisement when externalities are intermediate. 3 Next, we want to nd the pro t-maximizing equilibrium. solutions for pro t-maximizing A, p a and p d as below: A = 2( 2 +) 2 ;3 + ;3 + 2g + 2 ; ;3 D = 2( 2 ;3 +) ( 2 ; ;3 2g ;3 + ;3 + ) p a = 2( 2 +) ;3 2g 2 ; ;3 ;3 p d = 2( 2 +) ;3 + 2 ; ;3 + 2g ; ;3 + ;3 By substituting D back, we have the By the conditions set in Scenario Two, we can prove that A is also well-de ned and belongs to the range [0; 2 ]4. In this scenario, the SNS rm will choose fewer advertisers than in Scenario One to maximize their pro t. It is consistent with the intuition that when externalities are intermediate, the SNS rm will sacri ce their pro t from advertisers for a larger consumer base. As both A and D are well de ned to satisfy the normalization restriction, we can derive the pro t for the SNS rm in this case. Substituting A, D, p a and p d into the pro t function, we get: (20) = p a A + p d D (2) As 2 = 4( ;3)(g + ; ) + ( ; g + 2 ; )( ;3 + 2g 2 ; ) 4 2 (22) ;3 + is the supremum for the pro t of the SNS rm, we can prove that the pro t is bounded in the range [0; 2 ] in Scenario Two.5 This means that the pro t for the SNS rm in this scenario is always non-negative, but still less than the pro t in Scenario One. The outcome in this scenario is logical, as the SNS rm will always compromise a larger consumer base by decreasing N and increasing D. 5 COMPARATIVE STATICS In this section, we are only concerned about small changes of exogenous variables. Therefore, we shall ignore the discussion for Scenario One where N = given externalities g, ; and ;3, when both p a and p d are set to be 2. The reason for that, is given slight changes in exogenous variables, N still equals to when p a = p d = 2. Thus, the following discussion only applies to Scenario Two, where N 2 [0; ] given externalities g, ; and ;3, when both p a and p d are set to be 2. Moreover, as N = always hold in any scenarios that we have discussed, we have the following equations: 3 Proof to be found in Appendix 8.3 Discussion For Axis of Symmetry 4 Proof to be found in Appendix 8.4. Proof For the Range of A and 5 Proof to be found in Appendix Proof For the Range of A and 9

10 A = p a (23) D = p d Since we have proved the consistency of setting A and D as endogenous variables, we shall ignore the discussion of changes of p a and p d, as they are exactly opposite to the e ect of A and D. Another notation to be made here is that ; and g are symmetric in all functions, we shall do the comparative statics of ; and g simultaneously. 5. Interpretation For the E ect of ;, g 5.. The pro t-maximizing monopoly In the pro t-maximizing case, we shall take the rst partial derivative of each term in equilibrium with respect to ; @ = 2 ;3 + > = 2 ;3 + = 2 ;3 + (2g + 2 ; + ;3 ) 0 (24) The sign of Scenario ; is ; 0 is proved by using condition restrictions of It is easily concluded from above that when ; or g increases slightly, there will be more advertisers and less app developers in the SNS market. The intuition behind is that when network externality ; or g increase, it would be less strict for the SNS rm to keep N =. Therefore, the SNS rm has more freedom to maximize their pro t by approaching the equilibrium of A = D = 2. The pro t for the SNS rm is also positively correlated with ; and g, due to the reason that consumers are more tolerable with annoyance of advertisement. 5.2 Interpretation For the E ect of ; The monopoly pro t-maximizing For the same method as above, we can also take the rst partial derivative of each term in equilibrium with respect to ;3 = 4 ;3 (g + ; ) + 2 2( 2 ;3 2 ;3 > 0 ;3 = 2( 2 2( ;3)(g + ; ) ;3 + 2 ;3 ;3 = 4 ;3 (g + ; ) 2 + 2( 2 2( 2 ;3 2 ;3 )(g + ; ) 2 ;3 + 0 ;3 +)2 Signs ;3 are determined and the proved. However, the sign ;3 is indeterminable. When ;3 increases slightly, the SNS rm will choose a higher level of advertisement to maximize their pro t. This intuitively means that annoyance of more advertisement can be o set by consumers increase of utility from applications. However, the quantity of app developers is indeterminable, due to the reason that other network externalities are unknown. The pro t for the SNS rm is negatively correlated with ;3, simply because they will charge less from the app developers when ;3 is higher. 6 Proof to be found in Appendix 8.5. Proof For Comparative Statics 7 Proof to be found in Appendix Proof For Comparative Statics 0

11 6 THE SOCIAL WELFARE OPTIMIZER In the monopoly pro t-maximizing analysis, we can easily conclude that the SNS rm will always choose N = to maximize their pro t. However, such combinations of A and D might not be social welfare optimum. Thus, we are curious about the social welfare optimizer equilibrium. In this section, we shall derive the social welfare function by summation of each party s total utility. Then, nd the social welfare optimum where the social welfare function is maximized. 6. Total Utilities for Consumers Recall from the three-sided market model section, a ;2 is set to be heterogenous by our assumptions above. Moreover, we assumed that a ;2 conforms to Continuous Uniform Distribution ranging from 0 to, denoted as a ;2 U(0; ). The probability density function for a ;2 is always one. Utility function for each individual consumer participated in the SNS market is denoted as below: u i = g + ; N + ;3 D (;2)i A > 0 (26) To sum up the utilities of all consumers in the SNS market, we can use the method of integration as a ;2 is a continuos random variable. Since we have shown that (;2)i equals to N for the indi erent consumer, the integration of u i should be over the range of [0; N] as below: U = Z N 0 (g + ; N + ;3 D (;2)i A)d( (;2)i ) (27) Here, (;2)i denotes the negative externality exerted by advertisers on consumers. The result of the integration can be easily computed: U = gn + ; N 2 + ;3 DN 2 AN 2 (28) 6.2 Total Utilities for Advertisers By the same method as the derivation of total utilities for consumers, a 2; is set to be heterogenous by our assumptions above. Moreover, we assumed that a 2; conforms to Continuous Uniform Distribution ranging from 0 to, denoted as a 2; U(0; ). The probability density function for a 2; is always one. Utility function for each advertiser in the SNS market is denoted as below: u 2i = (2;)i N p a > 0 (29) To sum up the utilities of all advertisers in the SNS market, we can use the method of integration as a 2; is a continuos random variable. Since we have shown that (2;)i equals to the A for the indi erent advertiser, the integration of u 2i should be over the range of [ A; ] as below: U 2 = Z A ( (2;)i N p a )d( (2;)i ) (30) Here, (2;)i denotes the externality exerted by consumers on advertisers. The result of the integration can be easily computed: U 2 = N 2 ( A2 + 2A) p a A (3)

12 6.3 Total Utilities for App Developers By taking the same steps, a 3; is set to be heterogenous by our assumptions above. Moreover, we assumed that a 3; conforms to Continuous Uniform Distribution ranging from 0 to, denoted as a 3; U(0; ). The probability density function for a 3; is always one. Utility function for each app developer in the SNS market is denoted as below: u 3i = (3;)i N p d > 0 (32) To sum up the utilities of all app developers in the SNS market, we can use the method of integration as a 3; is a continuous random variable. Since we have shown that (3;)i equals to the D for the indi erent app developer, the integration of u 3i should be over the range of [ D; ] as below: U 3 = Z D ( (3;)i N p d )d( (3;)i ) (33) Here, (3;)i denotes the externality exerted by consumers on app developers. The result of the integration can be easily computed: U 3 = N 2 ( D2 + 2D) p d D (34) 6.4 Social Welfare Optimization Social welfare is simply the summation of utilities from each party, namely the SNS rm, consumers, advertisers and app developers. From consumers point of view, N = is always social optimum. Therefore, we shall take N = into the utility functions and the social welfare is denoted as below: SW = + U + U 2 + U 3 (35) SW = 2 (A 2 )2 2 (D ;3) ( + ;3) 2 + g + ; + 8 According to the social welfare function, it is obvious that the social welfare optimum is A = 2 and D = + ;3. However, as D is normalized in the range [0; ], we can conclude that D = is social welfare optimum. In conclusion, the social welfare optimum is achieved when we set A = 2 (36) and D =. 7 CONCLUDING REMARKS From our analysis of the social optimum, we can conclude that the following points:. There is full participation of the consumer s market. Both the social welfare maximizer and the monopoly scenarios result in N=, thus the number of consumers (under monopoly) is e cient. 2. It is social welfare maximizing for full participation in the app developer s market. Both scenarios under monopoly results in D, thus there are insu cient app developers in the market. 3. The social welfare maximizing number of advertisers is A=/2. In scenario of the monopoly model, A=/2. Thus when externalities are su ciently large, the number of advertisers is e cient. In scenario 2 of the monopoly model, A =2. When externalities are intermediate, there are insu cient advertisers. 2

13 Since consumers create a positive externality on both advertisers and app developers, the SNS rm will be able to charge higher prices if consumers fully participate. However, as advertisers create negative externalities to consumers, the SNS rm would prefer less advertisers. Another reason why the SNS rm does not prefer more advertisers and app developers is the following: increasing quantity leads to lower prices and less than optimal pro ts. Some of our results may be due to the functional forms used in the model. As the e ect of externalities are strictly increasing, the social optimal numbers of consumers and app developers (which exert only positive externalities) end up being (full participation). Future research will explore the e ects of concavity on the externalities. Another extension which could be made is the introduction of competition, this could be in the form of competition with other SNS rms or competition among advertisers and app developers. The introduction of competition with other SNS rms may result in more pressure to keep the number of advertisers low and thus more app developers. Our results demonstrates that in the presence of inter-related externalities in a three-sided market may result in under-advertisement. This is an interesting result as it is contrary to traditional models of advertisement where rms tend to over-advertise. In order to balance the e ect of negative externalities of advertising on consumers, the SNS rm may choose to compromise on the number of advertisers. 8 APPENDIX 8. Preliminary Discussion of Number of Consumers By normalization restriction, the number of consumers N is normalized to be in the range [0; ]. Speci - cally, N = means that all the potential consumers join the SNS market. In another word, any consumer participating in the SNS market has a non-negative utility. However, the expression of N obtained from last section, which is N = g+;3d can sometimes violate our normalization restriction given certain externalities. Therefore, we should discuss the value of g+;3d case by case in three ranges, namely ( ; 0), [0; ] and (; +). 8.. When g+;3d 2 ( ; 0) When g+;3d < 0, it is equivalent to say that A < ;, because g, D and ;3 are all non-negative. When we go back to utility function of consumers: u i = g + ;3 D + ( ; A)N (37) If A < ;, then u i the utility of consumer i is positive de nite. This means that every potential consumers would join the SNS market and gain an additional utility. Thus, by the normalization restriction, N = under the case g+;3d < When g+;3d 2 [0; ] When 0 g+;3d, the expression for N exactly satis es our normalization restriction. This means that there does exist an indi erent consumer, who has utility zero after joining the SNS market. Therefore, we have N = g+;3d under the case 0 g+;3d. 3

14 8..3 When g+;3d 2 (; +) When g+;3d >, we have the following two inequalities: g + ;3 D > and > 0. By the same method, we look back to the consumer s utility function. It will give us the inequality as below: As N is always bounded in the range [0; ], and A u i > ( )( N) 0 (38) ; > 0 under condition restriction in this case. Utility of consumer is always positive, thus all potential consumers will join the SNS market. By the normalization restriction, N = under the case g+;3d >. 8.2 Consistency of The Model In this part, we are going to prove that it is equivalent in this model by choosing A and D as endogenous variables or choosing p a and p d as endogenous variables. By the same logic, we shall divide the discussion into two scenarios: 8.2. Scenario one Given externalities g, ; and ;3, when both p a and p d are set to be 2, N = under this scenario. When the SNS rm set p a = p d = 2, we can derive A and D by using the equations derived in the Three-Sided Model section: p a = ( p d = ( A)N D)N (39) In this scenario, N = when both p a and p d are set to be 2. Substitute p a = p d = 2 back to the equations, we can obtain the unique solution A = D = 2. This is consistent with our discussion when we choose A and D as endogenous variables, which give rise to a pro t of = Scenario two Given externalities g, ; and ;3, when both p a and p d are set to be 2, N 2 [0; ] under this scenario. By Lemma 2 proved above, the SNS rm will always choose such a combination of p a and p d, that the number of consumers N equals to one. Then, we want to prove that by setting p a and p d as endogenous variables will give us exactly the same result when we choose A and D as endogenous variables. As N is set to be one in this case, we can represent A and D as functions of p a and p d as below: If we substitute the equations into the pro t function, we can get: A = p a (40) D = p d = p a A + p d D = p a ( p a ) + p d ( p d ) (4) From the equation N = g+d;3 =, we can denote p a as a function of p d, and substitute back into the pro t function: p a = (p d ) ;3 ; g + = ((p d ) ;3 ; g + )(g (p d ) ;3 + ; ) + p d ( p d ) (42) By taking rst order partial derivative of with respect to p d will give us: 4

15 d = 2 ; ;3 2p d 2p d 2 ;3 + 2g ; ;3 ;3 + (43) By setting FOC equals to zero, we can get the pro t-maximizing p d and p a as below: p d = 2( 2 +) ;3 + 2 ; ;3 + 2g ; ;3 + ;3 (44) p a = 2( 2 +) ;3 2g 2 ; ;3 ;3 This exactly equals to the outcome when we set A and D as the endogenous variable. Thus, we have proved that the model is consistent no matter we choose p a and p d as endogenous variables or choose A and D as the endogenous variable. 8.3 Discussion For Axis of Symmetry Condition restrictions we have in Scenario Two is that 0 N, when A = D = 2. This means that 0 g+ 2 ;3, and it can be expanded into two inequalities: 2 ; < ; 2 and g + ; 2 ( + ;3). Then, we want to prove that D belongs to the ranges ( ; 0) and (; ) contradicts with our condition restrictions Contradiction of D 2 ( ; 0) First of all, we shall prove that D < 0 contradicts with our condition restrictions in Scenario Two. In another word, we shall prove 2 ; ;3 2g ;3 + ;3 + is positive semi-de nite. Proof. By our condition restriction g + ; 2 ( + ;3), we can infer that: 2 ; ;3 2g ;3 + ;3 + 2 ;3 0 (45) As ;3 2 [0; ], we can conclude that never be negative. 2 ;3 0. Thus, it is proved that the axis of symmetry can Contradiction of D 2 (; ) By the same logic, we shall prove that D > contradicts with our normalization assumptions. Proof. D > can be simpli ed as below: 2 ; ;3 2g ;3 + ;3 + > 2( 2 ;3 + ) (46) ; + g < ;3 2 ;3 + 2 (47) By Cauchy Schwarz inequality, we have ;3 + 2 ;3 > p 2. Substituting back and we can get: ; + g < p < 0 (48) This contradicts our assumptions that both ; and g are non-negative. Therefore, it is proved that the axis of symmetry can never be larger than one. 5

16 8.3.3 D is de ned in the range [ 2 ; ] Proof. As we have already ruled out the possibilities for D to lie in the ranges ( ; 0) and (; ), the only possible range left is [0; ]. If we subtract D by 2, we can get the following expression: D 2 = 2 2 ;3 + ;3 (2g + 2 ; + ;3 ) 0 (49) It is easily proved by applying the condition restrictions. Thus, the proof for D 2 [ 2 ; ] is done. 8.4 Proof For the Range of A and 8.4. A is well de ned in the range [0; 2 ] Proof. A 0 is easily seen, as both the numerator and the denominator of A are non-negative. Thus, we are going to prove the other half, which is A 2. By the same method as above, we can take the di erence and get the inequality as below: A 2 = 2 2 ;3 + (2g + 2 ; + ;3 ) 0 (50) It is easily proved by applying the condition restrictions. Thus, the proof for A 2 [0; 2 ] is done is well de ned in the range [0; 2 ] Proof. First of all, we want to prove the rst half 0. By our normalization and condition restrictions, ;3 and g + ; 2 ( ;3). The following inequality is derived: ;3 + 2g 2 ; 2 ;3 0 (5) Therefore, it is proved that all terms in the numerator of are non-negative, 0 is explicit. Next, we shall prove the second half of the inequality 2. By taking the di erence between and 2 will give us: 2 = 4 2 ;3 + (2g + 2 ; + ;3 ) 2 (52) It is explicit that 4( 2 ;3 +) (2g + 2 ; + ;3 ) 2 0, thus Proof For Comparative Statics 8.5. Partial e ect of ; and g is proved. As the e ect of A and D are trivial, we shall only prove that partial e ect of is positive. Proof. By taking rst order partial derivative, we have the equation as = = 2 ;3 + (2g + 2 ; + ;3 ) (53) By the condition restriction of Scenario Two, we have g+ ; 2 ( into the partial derivative, we can easily get: ;3). Substitute 0 (54) 6

17 8.5.2 Partial e ect of First of all, we want to ;3 > 0, and the proof is as below. Proof. We have the following partial derivative of A with respect to ;3 = 2 2 ; ;3(g + ; ) + 2 ;3 2 ;3 (55) By the same token, we can substitute the condition restriction g + ; 2 ( derivative and get: ;3) into ;3 2 2 ;3 + 2 > 0 Then, we want to prove ;3 0, and the proof is shown below. Proof. We have the following partial derivative of with respect to ;3 = As both g + ; and 2 2 ; ;3(g + ; ) 2 + 2( 2 ;3 2 ;3 )(g + ; ) 2 ;3 + (57) ;3 are nonnegative, we can take the square and the inequality is still preserved. By substituting the condition restriction g + ; 2 ( can get the follows: ;3) into the partial ;3 0 (58) References Armstrong, M., 998, Network Interconnection, Economic Journal, 08: Armstrong, M. and Wright, J., 2004, Two-Sided Markets, Competitive Bottlenecks and Exclusive Contracts, mimeo, University College, London, and National University of Singapore. Armstrong, M., 2006, Competition in Two-sided Markets, RAND Journal of Economics, 2006(V0.37,No.3): Baye, M., and Morgan, J., 200, Information Gatekeepers on the Internet and the Competitiveness of Homogenous Product Markets, American Economic Review, 9, pp Caillaud, B. and Jullien, B., 2003, Chicken & Egg: Competition among Intermediation Service Providers, RAND Journal of Economics, 24: Chakravorti, S., and Roson, R., 2004, Platform Competition in Two-Sided Markets: The Case of Payment Networks, Federal Reserve Bank of Chicago Emerging Payments Occasional Paper Series, Cremer, J., Rey, P., and Tirole, J., 2000, Connectivity in the Commercial Internet, Journal of Industrial Economics, 48: Hahn, J.-H, 200, Nonlinear Pricing of Telecommunications with Call and Network Externalities, Department of Economics, Keele University. 7

18 Hagiu, A., 2004a, Two-Sided Proprietary vs. Non-Proprietary Platforms, in: Platforms, Pricing, Commitment and Variety in Two-Sided Markets, Ph.D. Dissertation, Princeton University. Hagiu, A., 2004b, Optimal Pricing and Commitment in Two-Sided Markets, in: Platforms, Pricing, Commitment and Variety in Two-Sided Markets, Ph.D. Dissertation, Princeton University. La ont, J., and Tirole, J., 2000, Competition in Telecommunications, Cambridge: MIT Press. Little, I., and Wright, J., 2000, Peering and Settlement in the Internet: An Economic Analysis, Journal of Regulatory Economics, 8: Nocke, V., Peitz, M., and Stahl, C., 2004, Platform Ownership in Two-Sided Markets, mimeo, University of Pennsylvania and University of Mannheim. Rohlfs, J. H., 2003, Bandwagon E ects in High Technology Industries, MIT Press, Cambridge, Massachusetts, United States Roberto, R., 2005, Two-sided Markets: A Tentative Survey, Journal of Review of Network Economics, 4: Rochet, J., and Tirole, J., 2002, Cooperation among Competitors: The Economics of Payment Card Associations, RAND Journal of Economics, 33: Rochet, J, and J., Tirole, 2003, Platform Competition in Two-Sided Markets, Journal of European Economic Association, : Rochet, J., and J., Tirole, 2004, Two-Sided Markets: An Overview, mimeo, IDEI University of Toulouse. Roson, R., 2003, Incentives for the Expansion of Network Capacity in a Peering Free Access Settlement, Netnomics, 5: Roson, R., 2005, Platform Competition with Endogenous Multihoming, in Dewenter, R., Haucap, J. (eds.), Access Pricing: Theory, Practice, Empirical Evidence. Amsterdam: Elsevier Science, forthcoming. Rysman, M., 2004, An Empirical Analysis of Payment Card Usage, mimeo, Boston University. Schi, A., 2003, Open and Closed systems of Two-sided Networks, Information Economics and Policy, 5: Shapiro, C., 980, Advertising and Welfare: Comment, Bell Journal of Economics vol., no. 2, Autumn 980, pp Shapiro, C., 983, Optimal pricing of Experience Goods, Bell Journal of Economics vol. 4, no. 2, Autumn 983, pp Shy, O., 996, Technology Revolutions in the Presence of Network Externalities, International Journal of Industrial Organization, 4: Shy, O., 200., The Economics of Network Industries, University of Haifa. Cambridge University Press. Wright, J., 2003, Optimal Card Payment Systems, European Economic Review, 47,