Exploiting Externalities: Facebook versus Users, Advertisers
and Application Developers
Ernie G.S. Teo and Hongyu Cheny
Division of Economics, Nanyang Technological University, Singapore
March 19, 2012
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$3.71 billion in 2011 and boosts an active user population of more than 845 million active users in
2012. 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: D21, L10, M37
Keywords: SNS Industry, Externality, Multi-sided Market, Market Structure, Pricing Strategy,
Heterogeneity
1
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.: +6594673016. Address: Nanyang Technological University 36 Nanyang Crescent Hall 15,
70-1-1448, Singapore 637635 E-mail address: [email protected]
1
Early versions of these online communities include Usenet, ARPANET and LISTSERV. In the late
1990s, 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 2011 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
2.1
LITERATURE REVIEW / BACKGROUND
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.
2.1.1
Positive Network Externalities
Positive network externalities mean that a consumer’s utility increases when there are more users or
subscribers. Hahn (2001) 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
2.1.2
Negative Network Externalities
In contrast, negative network externalities mean that consumer utility decreases when there are more
users or subscribers. Shy (2001) 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.
2.2.1
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, bB ; associated with a transaction di¤er. Similarly, sellers’gross surplus, bS ;
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 pB (respectively, pS ) charged by the monopoly platform.
Previous research of models a la Baye and Morgan (2001) 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 bB and bS , the proportion of transactions is equal to the product DB (pB ) DS (pS ).
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 bB . (bB pB )N S ;
depends on the number of sellers N S . The buyers’“quasi-demand function”,
N B = Pr(bB
pB ) = DB (pB )
(1)
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(bS
pS ) = DS (pS )
(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 bB and bS 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.
2.2.2
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
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, 1 represent consumers, 2 represent advertisers and 3 represent
application developers. ai;j represents the externalities exerted by party j on one member of party i,
given that i 6= j. When i = j; ai;j represents the direct network externalities of each party exerts on its
own individual member. ui;j represents utility of the jth member from party i, for i 2 1; 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 ai;j are generally assumed to be positive and its maximum is normalized
to 1. The only special case for speci…c ai;j is a1;2 2 [ 1; 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 (1980).). Moreover, to
simplify our model, we assume that network externalities are non-existent between and within advertisers
and app developers. That is to say, a2;2 = a3;3 = a2;3 = a3;2 = 0. The number of other advertisers or
app developers do not have an e¤ect on its own or each other’s utility.
3.1
Utility Function for Consumers
In the three-sided market, we make use of the heterogeneity of a1;2 , the network externalities of advertisers
exerted on consumers. Although a1;2 is negative, we may normalize it into the range of [0; 1] and use
4
the minus sign in the utility function. To make things easy, we assume that a1;2 conforms to Continuous
Uniform Distribution ranging from 0 to 1, denoted as a1;2
U (0; 1). The graph of a(1;2)i for various
consumers is shown as below:
Npotential
N
a(1,2)i
a(1,2)i
0
1
a(1;2)i is the network externality of advertisers exerted on the ith consumer. Npotential here denotes
the entire base of consumers who have potential to enter the SNS market. In another word, Npotential is
the maximum quantity of consumers the SNS market could possibly have, and it is normalized to 1. N
will thus denote the actual number of consumers in the SNS market. Utility function of each individual
consumer is derived as below:
u1i = g +
1;1 N
+
1;3 D
(1;2)i A
>0
(3)
g represents the constant utility gain, which is also normalized to [0; 1]. 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 Npotential is normalized to 1, the actual number of consumers N will exactly
equals to (1;2)i , for the ith consumer who is indi¤erent between join or not. Thus, the expression for N
is solved by …nding (1;2)i for the indi¤erent consumer as below.
u1i = g +
g+
1;1 N
1;1 N
(1;2)i
+
+
1;3 D
1;3 D
=N =
g+
A
(1;2)i A
=0
(4)
NA = 0
(5)
1;3 D
(6)
1;1
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 a2;1 (the network externalities of consumers
exerted on advertisers). a2;1 also conforms to Continuous Uniform Distribution ranging from 0 to 1,
denoted as a2;1 U (0; 1). The graph of a(2;1)i for various advertisers is shown as below:
A
a(2,1)i
0
a(2,1)i
1
One thing we should note here, is that A = 1 a(2;1)i , for the ith advertiser who is indi¤erent between
joining or not. This is di¤erent from the a1;2 case, where a1;2 is negative and normalized to [0; 1]. Utility
for each advertiser is the total network externalities gained minus the price paid to the SNS …rm, denoted
5
as pa . 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:
u2i =
pa =
(2;1)i N
(2;1)i N
pa > 0
= (1
(7)
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,
a3;1 is made heterogenous and conforms to the Continuous Uniform Distribution ranging from 0 to 1,
denoted as a3;1 U (0; 1). The graph of a(3;1)i for various app developers is shown as below:
D
a(3,1)i
a(3,1)i
0
1
By the same logic in the advertiser case, D = 1 a(3;1)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 pd . 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:
u3i =
pd =
(3;1)i N
(3;1)i N
pd > 0
= (1
D)N
(9)
(10)
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:
= pa A + pd D
= N (1
=N
A)A + N (1
A2 + A
(11)
D)D
D2 + D
(12)
(13)
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
In our three-sided market model, only pa and pd are endogenous variables. SNS Companies can only
choose pa and pd to alter their pro…t. However, the discussion of pro…t by using pa and pd 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 pa , pd 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 pa and pd 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; 1]. Speci…cally, N = 1 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.
g+
D
Nevertheless, the expression of N obtained from last section, which is N = A 1;31;1 can sometimes
violate our normalization restriction given certain externalities. There are basically three ranges for the
g+
D
g+
D
g+
D
value of A 1;31;1 to be discussed, namely A 1;31;1 2 ( 1; 0), [0; 1] and (1; +1). When A 1;31;1 belongs
to the ranges ( 1; 0) and (1; +1), the number of consumers equals to one; when
g+
A
1;3 D
1
g+
A
1;3 D
1;1
2 [0; 1], the
number of consumers is exactly
. More intuitively speaking, when the constant gain (g) and
1;1
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 pa and pd as endogenous variables.
It is always consistent for us to change the endogenous variables due to the nice property of their
relationship functions.2 More explicitly, the choice of endogenous variables will not a¤ect the outcome
of the pro…t-maximizing equilibrium.
We can rewrite the pro…t function of the SNS …rm as below:
=N
(A
1 2
)
2
(D
1 2 1
) +
2
2
(14)
We can separate into two parts: N and
(A 12 )2 (D 12 )2 + 12 . It is obvious that A = D = 12
is the maximum solution for
(A 12 )2 (D 21 )2 + 12 , and N is normalized to have maximum value
of one. We can easily consider the scenarios where N = 1 for given externalities, and A = D = 21 .
Then, the pro…t of the SNS …rm is maximized and equals to 12 . Therefore, we can divide the choice of
pro…t-maximizing A and D into two scenarios.
4.1
Scenario One: Externalities are Su¢ ciently Large
When the externalities g, 1;1 and 1;3 are su¢ ciently large, setting A and D to be 12 will result in N = 1
(full participation of consumers). By our previous discussion on number of consumers, it can only be the
g+
D
case when A 1;31;1 belongs to the range ( 1; 0) or (1; +1). Generally speaking, either the constant
utility gain g and 1;3 are large enough to make the numerator larger than the denominator; or when
1
1;1 , the network externality between consumers, is larger than 2 which makes the denominator negative.
1 Proof
2 Proof
to be found in Appendix 8.1 Preliminary Discussion of Numbers of Consumers
to be found in Appendix 8.2 Consistency of The Model
7
In conclusion, when the externalities are su¢ ciently large, the SNS …rm is able to choose A = D =
maximize their pro…t. The pro…t of the SNS …rm is maximized to be 21 in this scenario.
4.2
1
2
to
Scenario Two: Externalities are Intermediate
When the externalities g, 1;1 and 1;3 , are intermediate, setting A and D to be 21 will result in N 2 [0; 1].
g+
D
N 2 [0; 1] implies that N = A 1;31;1 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 = 1. First, we prove the
feasibility of setting N = 1. Next, we will show that setting N = 1 is pro…t-maximizing for the SNS
…rm.
Lemma 1 Setting N = 1 is always feasible under Scenario two
Proof. By the second case of preliminary discussion of N , given externalities g, D, 1;3 and A = D = 12 ,
g+
D
N = A 1;31;1 2 [0; 1]. This means that A > 1;1 , and g + 1;3 D A
1;1 . Therefore, setting N = 1
is equivalent to setting g + 1;3 D = A
1;1 . It is obvious that the SNS …rm can indirectly decrease A
or increase D or apply both strategies. Therefore, when N 2 [0; 1], it is always feasible for the …rm to
choose another combination of A and D other than ( 21 ; 12 ), such that N = 1.
Lemma 2 Setting N = 1 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 1;3 + 1;1 + s = A. Then we substitute
g+D 1;3
A = g + D 1;3 + 1;1 + s to the original pro…t function
= A 1;1
A2 + A D2 + D and we
get:
g + D 1;3
(g + D 1;3 + 1;1 + s)2 + g + D 1;3 + 1;1 + s D2 + D
(15)
=
g + D 1;3 + s
By taking the …rst partial derivative of
@
=
@s
g+D
with respect to s:
1;3
(g + s + D
2
1;3 )
(g + s + D
2
1;3 )
D2 + D
2
1;1
+
(16)
1;1
2
0, given D; 1;1 2 [0; 1]. Therefore,
It is obvious that @@s < 0, as D2 + D 0 and
1;1 + 1;1
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 1, the SNS …rm will choose such
g+D 1;3
a combination of A and D such that s = 0, or equivalently N = A 1;1
= 1 to maximize their pro…t.
1
Thus, we have shown that if the choice of A = D = 2 results in N 2 [0; 1] the SNS …rm would prefer
to readjust A and D such that N = 1 as it is pro…t-maximizing. As a result, we can set N = 1 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.
g+D 1;3
= 1, we have g + D 1;3 + 1;1 = A. Substituting the expression of A
From equation N = A 1;1
into the pro…t function, we can get the pro…t function of single endogenous variable D:
=
(g + D
1;3
+
2
1;1 )
+g+D
1;3
+
1;1
D2 + D
(17)
Simpli…cation of the pro…t function will give us:
=
(
2
1;3
+ 1)D2 + (
1;3
+1
2
1;1 1;3
8
2g
1;3 )D
(g +
2
1;1 )
+g+
1;1
(18)
From the expression of the pro…t function, we can easily conclude that it is a concave parabola, as
( 21;3 + 1) < 0. Thus, the pro…t for the SNS …rm is maximized when D takes the value at the axis of
symmetry as below:
1
D =
2(
2
1;3
+ 1)
( 2
2g
1;1 1;3
1;3
+
1;3
+ 1)
(19)
Generally, there are three positions of the axis of symmetry D , namely D belongs to ( 1; 0), [0; 1]
and (1; 1). However, we can rule out the possibilities of D falling in the ranges ( 1; 0) and (1; 1).
The only possible range left for D is [0; 1], and we can further prove that D 2 [ 21 ; 1]. 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. By substituting D back, we have the
solutions for pro…t-maximizing A, pa and pd as below:
A=
D=
pa =
pd =
1
2(
2 +1
1;3
1
2(
2
1;3
)
2 +1)
1;3
( 2
1
+
1;3
)
1;3
2 +1
1;3
2(
2 +1
1;3
1
+ 2g + 2
1;1 1;3
)
2(
1;3
2g
+2
1;1
2g
1;3
+
2
1;1
+2+
1;1 1;3
1;3
+ 2g
+ 1)
1;3
(20)
2
1;3
+2
2
1;3
+1
By the conditions set in Scenario Two, we can prove that A is also well-de…ned and belongs to
the range [0; 12 ]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, pa and pd into the pro…t function, we get:
= pa A + pd D
=
4(1
1;3 )(g
+
1;1 )
+(
1;3
+ 1 + 2g + 2
4 21;3 + 1
(21)
1;1 )( 1;3
+1
2g
2
1;1 )
(22)
As 12 is the supremum for the pro…t of the SNS …rm, we can prove that the pro…t is bounded in
the range [0; 12 ] 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 = 1 given externalities g, 1;1 and 1;3 , when both pa
and pd are set to be 12 . The reason for that, is given slight changes in exogenous variables, N still equals
to 1 when pa = pd = 21 . Thus, the following discussion only applies to Scenario Two, where N 2 [0; 1]
given externalities g, 1;1 and 1;3 , when both pa and pd are set to be 12 .
Moreover, as N = 1 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
to be found in Appendix 8.4.1 Proof For the Range of A and
5 Proof to be found in Appendix 8.4.2 Proof For the Range of A and
4 Proof
9
A=1
D=1
pa
pd
(23)
Since we have proved the consistency of setting A and D as endogenous variables, we shall ignore
the discussion of changes of pa and pd , as they are exactly opposite to the e¤ect of A and D. Another
notation to be made here is that 1;1 and g are symmetric in all functions, we shall do the comparative
statics of 1;1 and g simultaneously.
5.1
5.1.1
Interpretation For the E¤ect of
1;1 ,
g
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 1;1 and g:
@A
@ 1;1
@D
@ 1;1
@
@ 1;1
=
=
=
@A
@g
@D
@g
@
@g
=
1
2 +1
1;3
=
=
>0
1
1;3
2 +1
1;3
1
0
(2g + 2
2 +1
1;3
(24)
1;1
+
1;3
1)
0
The sign of @@A
and @@D
is trivial, and @ @ 1;1
0 is proved by using condition restrictions of
1;1
1;1
6
Scenario Two.
It is easily concluded from above that when 1;1 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 1;1
or g increase, it would be less strict for the SNS …rm to keep N = 1. Therefore, the SNS …rm has more
freedom to maximize their pro…t by approaching the equilibrium of A = D = 21 . The pro…t for the SNS
…rm is also positively correlated with 1;1 and g, due to the reason that consumers are more tolerable
with annoyance of advertisement.
5.2
5.2.1
Interpretation For the E¤ect of
1;3
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 1;3 :
@A
@
1;3
@D
@ 1;3
@
@
1;3
=
=
=
1
2(
2 +1 2
1;3
2(
2 +1 2
1;3
4
)
1
2(
2(1
)
1
2 +1 2
1;3
)
1;3 (g
4
1;3 (g
+
1;1 )
2
1;3 )(g
+
+
2
1;1 )
2
1;3
+
1;1 )
+ 2(
+
2
1;3
2
1;3
2
1;3
2
+2
1;3
1 >0
1;3
1
1)(g +
(25)
1;1 )
2
1;3
+1
0
Signs of @@A
and @ @ 1;3 are determined and the proved.7 However, the sign of @@D
is indeterminable.
1;3
1;3
When 1;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 1;3 , simply because they will charge less from the app developers when 1;3 is higher.
6 Proof
7 Proof
to be found in Appendix 8.5.1 Proof For Comparative Statics
to be found in Appendix 8.5.2 Proof For Comparative Statics
10
6
THE SOCIAL WELFARE OPTIMIZER
In the monopoly pro…t-maximizing analysis, we can easily conclude that the SNS …rm will always choose
N = 1 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.
6.1
Total Utilities for Consumers
Recall from the three-sided market model section, a1;2 is set to be heterogenous by our assumptions
above. Moreover, we assumed that a1;2 conforms to Continuous Uniform Distribution ranging from 0 to
1, denoted as a1;2 U (0; 1). The probability density function for a1;2 is always one. Utility function for
each individual consumer participated in the SNS market is denoted as below:
u1i = g +
1;1 N
+
1;3 D
(1;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 a1;2 is a continuos random variable. Since we have shown that (1;2)i equals to N for the indi¤erent
consumer, the integration of u1i should be over the range of [0; N ] as below:
U1 =
Z
N
(g +
1;1 N
+
1;3 D
(1;2)i A)d( (1;2)i )
(27)
0
Here, (1;2)i denotes the negative externality exerted by advertisers on consumers. The result of the
integration can be easily computed:
U1 = gN +
6.2
1;1 N
2
+
1;3 DN
1
AN 2
2
(28)
Total Utilities for Advertisers
By the same method as the derivation of total utilities for consumers, a2;1 is set to be heterogenous by
our assumptions above. Moreover, we assumed that a2;1 conforms to Continuous Uniform Distribution
ranging from 0 to 1, denoted as a2;1 U (0; 1). The probability density function for a2;1 is always one.
Utility function for each advertiser in the SNS market is denoted as below:
u2i =
(2;1)i N
pa > 0
(29)
To sum up the utilities of all advertisers in the SNS market, we can use the method of integration
as a2;1 is a continuos random variable. Since we have shown that (2;1)i equals to the 1 A for the
indi¤erent advertiser, the integration of u2i should be over the range of [1 A; 1] as below:
U2 =
Z
1
(
(2;1)i N
pa )d(
(2;1)i )
(30)
1 A
Here, (2;1)i denotes the externality exerted by consumers on advertisers. The result of the integration
can be easily computed:
U2 =
N
( A2 + 2A)
2
11
pa A
(31)
6.3
Total Utilities for App Developers
By taking the same steps, a3;1 is set to be heterogenous by our assumptions above. Moreover, we assumed
that a3;1 conforms to Continuous Uniform Distribution ranging from 0 to 1, denoted as a3;1 U (0; 1).
The probability density function for a3;1 is always one. Utility function for each app developer in the
SNS market is denoted as below:
u3i =
(3;1)i N
pd > 0
(32)
To sum up the utilities of all app developers in the SNS market, we can use the method of integration
as a3;1 is a continuous random variable. Since we have shown that (3;1)i equals to the 1 D for the
indi¤erent app developer, the integration of u3i should be over the range of [1 D; 1] as below:
U3 =
Z
1
(
(3;1)i N
pd )d(
(3;1)i )
(33)
1 D
Here, (3;1)i denotes the externality exerted by consumers on app developers. The result of the
integration can be easily computed:
U3 =
6.4
N
( D2 + 2D)
2
pd D
(34)
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 = 1 is always social optimum.
Therefore, we shall take N = 1 into the utility functions and the social welfare is denoted as below:
SW =
SW =
1
(A
2
1 2
)
2
1
(D
2
+ U1 + U2 + U3
1
2
1;3 )
1
+ (1 +
2
(35)
2
1;3 )
+g+
1;1
+
1
8
(36)
According to the social welfare function, it is obvious that the social welfare optimum is A = 12 and
D = 1 + 1;3 . However, as D is normalized in the range [0; 1], we can conclude that D = 1 is social
welfare optimum. In conclusion, the social welfare optimum is achieved when we set A = 21 and D = 1.
7
CONCLUDING REMARKS
From our analysis of the social optimum, we can conclude that the following points:
1. There is full participation of the consumer’s market. Both the social welfare maximizer and the
monopoly scenarios result in N=1, 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 1, thus there are insu¢ cient app developers in the market.
3. The social welfare maximizing number of advertisers is A=1/2. In scenario 1 of the monopoly
model, A=1/2. Thus when externalities are su¢ ciently large, the number of advertisers is e¢ cient.
In scenario 2 of the monopoly model, A
1=2. When externalities are intermediate, there are
insu¢ cient advertisers.
12
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 1 (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
8.1
APPENDIX
Preliminary Discussion of Number of Consumers
By normalization restriction, the number of consumers N is normalized to be in the range [0; 1]. Speci…cally, N = 1 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
g+
D
last section, which is N = A 1;31;1 can sometimes violate our normalization restriction given certain
externalities. Therefore, we should discuss the value of
( 1; 0), [0; 1] and (1; +1).
8.1.1
When
g+
g+
A
1;3 D
1;1
g+
A
1;3 D
1;1
case by case in three ranges, namely
2 ( 1; 0)
D
When A 1;31;1 < 0, it is equivalent to say that A <
When we go back to utility function of consumers:
u1i = g +
1;3 D
1;1 ,
+(
because g, D and
1;1
A)N
1;3
are all non-negative.
(37)
If A < 1;1 , then u1i 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
g+
D
restriction, N = 1 under the case A 1;31;1 < 0.
8.1.2
When
g+
g+
A
1;3 D
1;1
2 [0; 1]
D
1;3
1, the expression for N exactly satis…es our normalization restriction. This
When 0
A
1;1
means that there does exist an indi¤erent consumer, who has utility zero after joining the SNS market.
g+
D
g+ 1;3 D
Therefore, we have N = A 1;31;1 under the case 0
1.
A
1;1
13
8.1.3
When
g+
g+
A
1;3 D
1;1
2 (1; +1)
D
When A 1;31;1 > 1, we have the following two inequalities: g + 1;3 D > A
1;1 and A
1;1 > 0. By
the same method, we look back to the consumer’s utility function. It will give us the inequality as below:
u1i > (A
1;1 )(1
N)
0
(38)
As N is always bounded in the range [0; 1], and A
1;1 > 0 under condition restriction in this case.
Utility of consumer is always positive, thus all potential consumers will join the SNS market. By the
g+
D
normalization restriction, N = 1 under the case A 1;31;1 > 1.
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 pa and pd as endogenous variables. By the same logic, we shall divide the discussion
into two scenarios:
8.2.1
Scenario one
Given externalities g, 1;1 and 1;3 , when both pa and pd are set to be 12 , N = 1 under this scenario.
When the SNS …rm set pa = pd = 21 , we can derive A and D by using the equations derived in the
Three-Sided Model section:
pa = (1
pd = (1
A)N
D)N
(39)
In this scenario, N = 1 when both pa and pd are set to be 21 . Substitute pa = pd = 12 back to the
equations, we can obtain the unique solution A = D = 12 . This is consistent with our discussion when
we choose A and D as endogenous variables, which give rise to a pro…t of = 12 .
8.2.2
Scenario two
Given externalities g, 1;1 and 1;3 , when both pa and pd are set to be 12 , N 2 [0; 1] under this scenario.
By Lemma 2 proved above, the SNS …rm will always choose such a combination of pa and pd , that the
number of consumers N equals to one. Then, we want to prove that by setting pa and pd 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 pa and pd as below:
A=1
D=1
pa
pd
(40)
If we substitute the equations into the pro…t function, we can get:
= pa A + pd D
= pa (1 pa ) + pd (1
From the equation N =
the pro…t function:
= ((pd
g+D
A
1)
1;3
1;1
1;3
(41)
pd )
= 1, we can denote pa as a function of pd , and substitute back into
pa = (pd 1) 1;3
g+1
1;1
g + 1)(g (pd 1) 1;3 +
1;1
By taking …rst order partial derivative of
1;1 )
+ pd (1
with respect to pd will give us:
14
pd )
(42)
@
=2
@pd
2pd
1;1 1;3
2
1;3
2pd
+ 2g
+2
1;3
2
1;3
1;3
+1
(43)
By setting FOC equals to zero, we can get the pro…t-maximizing pd and pa as below:
pd =
1
2(
2 +1
1;3
pa =
1;3
)
+2
1
2(
2 +1
1;3
1;1 1;3
2g
1;3
)
+ 2g
1;3
2
+2+
1;1
+2
2
1;3
+1
2
1;3
(44)
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 pa and pd 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 1, when A = D = 12 . This means that
g+ 21 1;3
1
0
1, and it can be expanded into two inequalities: 1;1 < 12 and g + 1;1
1
2 (1 + 1;3 ).
1;1
2
Then, we want to prove that D belongs to the ranges ( 1; 0) and (1; 1) contradicts with our condition
restrictions.
8.3.1
Contradiction of D 2 ( 1; 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 1;1 1;3 2g 1;3 + 1;3 + 1 is positive semi-de…nite.
1
Proof. By our condition restriction g + 1;1
2 (1 + 1;3 ), we can infer that:
2
2g
1;1 1;3
1;3
2
1;3
As 1;3 2 [0; 1], we can conclude that 1
never be negative.
8.3.2
+
1;3
+1
2
1;3
1
0
(45)
0. Thus, it is proved that the axis of symmetry can
Contradiction of D 2 (1; 1)
By the same logic, we shall prove that D > 1 contradicts with our normalization assumptions.
Proof. D > 1 can be simpli…ed as below:
2
1;1 1;3
1;1
2g
1;3
+
1;1
1;3
+ 1 > 2(
1
+g <
By Cauchy–Schwarz inequality, we have
1;3
1;3
+
+g <
1
2
1;3
p
+
2
1;3
>
p
2+
2
1;3
+ 1)
1
2
(46)
(47)
2. Substituting back and we can get:
1
<0
2
(48)
This contradicts our assumptions that both 1;1 and g are non-negative. Therefore, it is proved that
the axis of symmetry can never be larger than one.
15
8.3.3
D is de…ned in the range [ 12 ; 1]
Proof. As we have already ruled out the possibilities for D to lie in the ranges ( 1; 0) and (1; 1),
the only possible range left is [0; 1]. If we subtract D by 12 , we can get the following expression:
1
=
2
D
1
2
2
1;3
1;3
+1
(2g + 2
1;1
+
1)
1;3
0
(49)
It is easily proved by applying the condition restrictions. Thus, the proof for D 2 [ 21 ; 1] is done.
8.4
8.4.1
Proof For the Range of A and
A is well de…ned in the range [0; 12 ]
Proof. A 0 is easily seen, as both the numerator and the denominator of A are non-negative. Thus,
1
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:
1
=
2
2
A
1
2
1;3
(2g + 2
+1
1;1
+
1)
1;3
0
(50)
It is easily proved by applying the condition restrictions. Thus, the proof for A 2 [0; 21 ] is done.
8.4.2
is well de…ned in the range [0; 21 ]
Proof. First of all, we want to prove the …rst half
0. By our normalization and condition restrictions,
1
1 and g + 1;1
1;3 ). The following inequality is derived:
1;3
2 (1
1;3
+1
2g
2
1;1
2
0
1;3
(51)
Therefore, it is proved that all terms in the numerator of are non-negative,
0 is explicit. Next,
1
we shall prove the second half of the inequality
.
By
taking
the
di¤erence
between
and 21 will
2
give us:
1
1
2
=
(2g + 2 1;1 + 1;3 1)
(52)
2
4 21;3 + 1
It is explicit that
8.5
8.5.1
1
4(
2 +1
1;3
)
(2g + 2
1;1
2
+
1)
1;3
1
2
0, thus
is proved.
Proof For Comparative Statics
Partial e¤ect of
1;1
and g
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 below:
@
@
=
1;1
@
=
@g
1
(2g + 2
+1
2
1;3
By the condition restriction of Scenario Two, we have g +
into the partial derivative, we can easily get:
@
@
=
1;1
@
@g
16
0
1;1
1;1
+
1;3
1
2 (1
1)
1;3 ).
(53)
Substitute the inequality
(54)
8.5.2
Partial e¤ect of
1;3
> 0, and the proof is as below.
First of all, we want to prove @@A
1;3
Proof. We have the following partial derivative of A with respect to
@A
=
@ 1;3
1
2
1;3
2
+1
4
2
1;3 (g
+
1;1 )
+
1;3 :
2
1;3
2
By the same token, we can substitute the condition restriction g +
derivative and get:
@A
@ 1;3
1
2
2
1;3
+2
1;3
1;1
1
1
2 (1
(55)
1;3 )
into the partial
>0
(56)
Then, we want to prove that @ @ 1;3
0, and the proof is shown below.
Proof. We have the following partial derivative of with respect to 1;3 :
@
@
1
=
1;3
2
2
1;3
+1
2
4
1;3 (g
+
2
1;1 )
+ 2(
2
1;3
2
1;3
1)(g +
1;1 )
2
1;3
+1
(57)
As both g + 1;1 and 1
1;3 are nonnegative, we can take the square and the inequality is still
1
preserved. By substituting the condition restriction g + 1;1
1;3 ) into the partial derivative, we
2 (1
can get the follows:
@
0
(58)
@ 1;3
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