On oligopoly with positive network effects and incompatible
networks
Rabah Amir∗ and Adriana Gama†
November 13, 2015
Abstract
We model symmetric oligopolies with positive network effects where each firm has its own
proprietary network. That is, each firm’s network is incompatible with that of its rivals. We
provide minimal conditions for the existence of (non-trivial) equilibrium in a general setting; in
this model, the equilibria may be either symmetric or asymmetric. For the symmetric equilibria,
we analyze the comparative statics effects of entry. In addition, we compare the viability and
equilibrium outcomes of oligopoly markets with compatible and incompatible networks. We show
that viability and production are higher under compatible networks. However, the relationship
between equilibrium prices, profits and consumer surplus is ambiguous, but social welfare is
always higher in markets with completely compatible networks.
JEL codes: C72, D43, L13, L14.
Keywords and phrases: network effects, network goods, demand-side economies of scale,
complete compatibility, complete incompatibility, supermodularity.
∗
†
Department of Economics, University of Iowa (e-mail: [email protected]).
Centro de Estudios Económicos, El Colegio de México, Mexico (e-mail: [email protected]).
1
1
Introduction
It is very common to find industries with positive network effects, i.e., industries where the
willingness to pay of the consumers increase with the number of people that are purchasing the
good. A widely used example in the literature is telecommunications such as telephone and fax. A
consumer will be more interested in buying a cell phone if more people acquire a compatible cell
phone, among other reasons, the consumer will be able to communicate with more persons and in
the case of smart phones, the consumer expects more software to be developed if the good is more
popular.
In their seminal work about network effects, Katz and Shapiro (1985) identify three kind of
oligopolies with network effects; in all cases, the firms produce goods that are substitutes and
compete in quantity. The first model is when all the goods produced by the firms are completely
compatible; in the second one, any two goods produced by two different firms are incompatible, and
finally, there is the partial compatibility model, where there are groups of goods that are compatible
among them but incompatible with the goods outside of that group.
In this paper, we center our attention to the study of oligopolies with positive network effects
and complete incompatibility. An example of this industry is video games (Church and Gandal,
1992). Several firms in the market produce different video game consoles that are not compatible
among them, in the sense that a game that is designed for a particular console cannot be played
using a different one. Then, every firm possesses their own network and the consumers benefit
from the size of it, which is known in the literature as “demand-side economies of scale”. If more
consumers purchase a particular console, they expect to have a bigger variety of games because the
software developers might have more incentives to create more if the number of users increase, also,
the agents benefit with a bigger size of the network because they will have more people to play
with.
Other common examples of industries with incompatible networks come from diverse technologies in the 1980’s and 1990’s. For instance, personal computers, when IBM and Macintosh were
not compatible; video cassette recorders, that will be discussed with more detail later on, and digital music systems such as digital compact cassette and mini disc (see Church and Gandal, 1992;
Cusumano, Mylonadis and Rosenbloom, 1992 and Katz and Shapiro, 1986).
2
Specifically, this paper focuses in the study of the symmetric model, i.e., when the firms face the
same demand and same costs of production. One of the objectives of this study is to find minimal
conditions such that an equilibrium exists in these kind of symmetric industries. The concept of
equilibrium that we use is the one introduced by Katz and Shapiro (1985), the fulfilled expectations
Cournot equilibrium, where all the firms maximize their profits and the consumers’ expectations
are fulfilled, i.e., the output of each firm equals the expected size of the consumers.
An interesting feature of this symmetric model is that besides symmetric equilibria, asymmetric
equilibria in all their possibilities can easily arise, as first acknowledged by Katz and Shapiro (1985).
First, it can be the case that some firms stay out of the market by producing an output of zero
while the firms producing a positive output produce the same output among them, i.e., a symmetric
equilibrium with less than all the firms active. This equilibrium includes the natural monopoly
equilibrium, where only one firm is active producing a strictly positive output. The other possibility
is when at least two firms produce a strictly positive but different amount of output between them.
These asymmetric equilibria arise when some firms produce more than others because the consumers expect them to do so. An illustration of this fact follows from the competition between
Betamax and Video Home System (VHS) in the 1970’s-1980’s. Betamax and VHS were two different and incompatible formats of home video cassette recorders (VCRs). Not only the sizes of
the cassettes for the VCRs were different, also the tape-handling mechanisms and the technology
to read the tapes differed, which kept the two formats completely incompatible. Betamax was introduced in 1975 by the Sony Corporation and VHS, just one year after by the Victor Company of
Japan (JVC). Although the sales of the Betamax kept increasing until 1985, its market share was
below the VHS by 1978. Finally, Sony and the firms that adopted this format stopped producing
the Betamax (see Cusumano, Mylonadis and Rosenbloom (1992) for more details). Although in
this market more than one firm was producing the same format, this case helps us to illustrate the
emergence of asymmetric equilibria in the presence of firms that could be considered symmetric
given the similarities between the two technologies. At the beginning of the 1980s, VHS had a
higher market share, and by the end of the decade, Betamax was out of the market.
An important question in this industry is viability, i.e., whether an equilibrium with production
different from zero exists. This is a particularly important issue in our model since it allows for
3
pure networks good, i.e., for goods whose value for the consumers is zero if the expected size of the
network is zero, independently of the total production. Since pure network goods are included in
the model, it is possible that all or a subset of the firms do not produce anything at all; thus, in
addition to focus on the existence of equilibria, we also look for conditions such that non trivial
-symmetric- equilibria exist, which is as relevant as the existence question.
In their seminal work, Katz and Shapiro (1985) do not consider the viability problem, since their
separable demand function does not allow for pure network goods and trivial equilibria are not a
problem. In a later work, Katz and Shapiro (1986) discuss the viability problem for a dynamic
setting, where there are two periods and the existence of a network mainly depends on whether it
is sponsored or not. Other characteristics that dictate the viability of a network are the differences
in the technology and second-mover advantages.
Church and Gandal (1992) study the viability problem from a different perspective. They look
at viability as a consequence of the complementary goods suppliers decisions. In other words, we can
think of firms in the industry producing hardware, like video game consoles, that require software
to be enjoyed, video games; in this case, the software is the complement to the hardware. Church
and Gandal (1992) research the viability of the hardware producers given the software suppliers
decisions, i.e., which hardware they decide to support (which network they decide to join). In
contrast to previous works on viability, the present paper looks on conditions on the demand and
costs structure to predict whether all of the networks emerge or only some of them, for a single
period.
On a separate topic, we look at the effects on the extremal equilibria of adding firms to the
industry, including effects on the equilibrium production (total and individual), prices and individual
profits.
Finally, we compare the equilibria for the oligopolies with complete compatibility and complete
incompatibility. Recall that complete compatibility refers to demand-side economies of scale with
firms that share a unique common network. Several authors like Katz and Shapiro (1986), Economides and Flyer (1997) and Chen, Doraszelski and Harrington (2009) have addressed this question
which is closely related to viability. Typically, to study the standardization problem (the incentives
of a firm to adhere to an existing network or to differentiate itself), the game is modeled in two
4
stages. In the first one, the firms choose the technology, and in the second one, they compete in price
or quantity. Chen, Doraszelski and Harrington (2009) extend this model to a dynamic stochastic
setting. Economides and Flyer (1997) and Chen, Doraszelski and Harrington (2009) discuss the
relevance of the strength of the network effects in their results, particularly the former ones.
In this paper, we simplify the standardization problem by giving specific conditions on the
demand such that the firms, consumers and society are better off with compatible products. The
results are provided for a static game where the firms compete in quantity. We find out that the
oligopoly with complete compatibility produces more than the one with incompatible networks.
Similarly, the first one generates a higher social welfare. The comparison between consumer surplus
and industry profit is not clear, but in line with Economides and Flyer (1997), it strongly depends on
the size of the network effects. For instance, if the latter are big enough to offset the market effect,
the firms will prefer to have compatible networks because a bigger network benefits them through
a higher willingness to pay from the consumers. Since social welfare is higher under compatible
networks, we can conclude that at least one group (consumers or firms) is always better off under
this setting.
2
Oligopoly with positive network effects and complete incompatibility
2.1
The model and the assumptions
In this section we describe the model, which is a static game of oligopolistic competition with
positive network effects and complete incompatibility. This is, we describe a market situation
where the firms produce substitute goods and the consumers’ willingness to pay for any good is
increasing in the number of agents that purchase the good, also known in the networks literature
as demand-side economies of scale. In our model, the goods are substitutes but not compatible
among the firms; thus, every firm possesses its own network. This model is a generalization of the
oligopoly with network effects and complete incompatibility described by Katz and Shapiro (1985).
In equilibrium, every firm maximizes its profit given the production of the rest of the firms and
their expected network size, which is fulfilled by their production. This notion of equilibrium is
5
know as fulfilled expectations Cournot equilibrium (FECE) (Katz and Shapiro, 1985), and will be
formally defined later on.
According to the description above, in this market structure we have n symmetric firms that
produce substitute goods. Every firm in the market faces the inverse demand given by the function
P (z, s), where z denotes the total output in the market and s, the expected size of the network.
Since the goods produced by the firms are totally incompatible, every firm has its individual network
with expected size s, which is not necessarily the same among the firms. If every consumer purchases
at most one unit of the good, s accounts for the expected number of agents to consume the good.
We assume that the firms are identical, thus, they all face the same cost of production. By
simplicity, we assume that such cost is linear since its curvature does not play any crucial role in
the results. Hence, for every s given, firm i chooses the output that maximizes its profit given by
π(x, y, s) = xP (x + y, s) − cx,
where c ≥ 0, x is the firm’s output level and y is the joint output of the other (n − 1) firms. Notice
that the firm does not choose s, this is exogenous for the firm as well as y, i.e., the firm does not
have the power to influence the consumers’ expectations about the network size. Then, the firm’s
best reaction correspondence is given by
x(y, s) = argmax{π(x, y, s) : x ≥ 0}.
Alternatively, one can think of firm i as choosing total output z = x + y that maximizes
π̃(z, y, s) = (z − y)P (z, s) − c(z − y),
with best-reaction correspondence
z(y, s) = argmax{π̃(z, y, s) : z ≥ y},
given a particular y and s. Then, it should be the case that z(y, s) = x(y, s) + y.
As mentioned above, the equilibrium of this game is given by what Katz and Shapiro (1985)
called a “fulfilled expectations Cournot equilibrium” (FECE), i.e., the equilibrium is given by a
vector of individual outputs (x1n , x2n , ..., xnn ) and a vector of expected networks sizes (s1 , s2 , ..., sn )
such that:
6
(1) xin ∈ argmax{xP (x +
P
j6=i xjn , si )
− cx : x ≥ 0}, and
(2) xin = si ,
for all i ∈ {1, 2, ..., n}. Throughout the paper, the subindex n denotes that the variable in question is
under equilibrium, when the subindex i is added, it means that the variable in equilibrium is specific
for firm i. However, since we focus in symmetric equilibria, the subindex i is usually dropped for
simplicity in the notation.
The following assumptions will be assumed through all the paper:
(A1) P : [0, ∞) × [0, ∞) → [0, ∞) is twice continuously differentiable, P1 (z, s) < 0 and P2 (z, s) > 0.
(A2) c ≥ 0.
(A3) xi ≤ K, for each firm i.
(A4) ∆1 (z, s) , P (z, s)P12 (z, s) − P1 (z, s)P2 (z, s) > 0 on ϕ.
The first assumption is standard in the literature, in particular, P2 (z, s) > 0 reflects the positive
network effects or demand-side economies of scale, i.e., the increment in the willingness to pay of
the consumers when more people is expected to buy the good.
The second one is used by simplicity since the concavity of the cost does not play any crucial
role in the results. Notice that (A1) and (A2) imply that ∆(z, y, s) , −P1 (z, s) > 0, hence, every
selection of the total output best-response correspondence, z(y, s), is increasing in y for all s given.
For a detailed discussion, see Amir and Lambson (2000). On the other hand, the cost function is
increasing in the production and zero when there is no production.
The capacity constraint assumption (A3) is needed for technical reasons but the magnitude of
K does not affect the results.
The last assumption, (A4), implies that the inverse demand function is strictly log-supermodular
in (z,s) which implies that every selection of z(y, s) is increasing in s for every y given. The
latter result follows because (A4) guarantees that the alternative profit function π̃(z, y, s) has strict
increasing differences in (z,s) and by Topkis (1998). Economically, (A4) tells us that the demand is
more elastic when the expected size of the network is bigger, which reflects the demand-side scale
economies discussed in the network industries literature. See Amir and Lazzati (2011) for a deeper
discussion.
Notice that there is no special restriction on the value of P (z, 0), i.e., it can take values that
7
are greater or equal than zero. This characteristic of the inverse demand function allows the model
to account for pure and and mixed network goods. Pure network goods are those that do not have
stand-alone value, i.e., P (z, 0) = 0, meaning that if the expected size of the good’s network is zero,
no consumer will value this good. On the other hand, P (z, 0) > 0 reflects a mixed network good,
where the consumers value the product even if the expected size of the network is zero.
2.2
Existence of equilibrium and viability
In their pioneering work on network externalities, Katz and Shapiro (1985) noticed that the
symmetric oligopoly modeled in this paper can lead to three different kinds of equilibria: 1) symmetric equilibria where all the firms produce the same output; 2) m-active-firm symmetric equilibria
where m < n firms produce a positive output that is the same among them and the rest of the
n − m firms produce nothing, and 3) asymmetric equilibria where m ≥ 2 firms produce positive but
different equilibrium outputs.
Networks externalities are the only responsible for the existence of asymmetric equilibria. In a
standard Cournot symmetric model, where there are no network effects, asymmetric equilibria may
only arise in the presence of strong economies of scale. In other words, asymmetric equilibria in
standard Cournot oligopoly model may only arise when its corresponding ∆ is negative, which in
this paper is ruled out by the linearity of the production cost. Moreover, in standard Cournot model
it cannot be the case that two firms have positive but different production (Amir and Lambson,
2000).
The possibility of having asymmetric equilibria is not the only difference that we find between
Cournot oligopoly with and without network effects under similar assumptions (in particular, positive ∆). As we will see in Section 2.3, in the presence of networks externalities, entry of firms
may decrease total output, increase equilibrium prices and profits of the firms. According to Amir
and Lambson (2000), in standard Cournot oligopoly, production always increases with the entry of
firms and price and per-firm profits always decrease. Nonetheless, there are some important characteristics of the standard Cournot oligopoly model that survive to the presence of demand-side
economies of scale. Specifically, the conditions for the equilibrium individual output to increase or
decrease with the entry of firms are the same. As we will see later, these similarities between the
8
models are more frequent than the differences.
On the other hand, it is relevant to notice that having incompatible networks substantially
changes the results of having only one big compatible network, like in Amir and Lazzati (2011).
The results described above will be formalized and discussed in detail in Section 2.3.
First we look into the existence of equilibria; under our assumptions, at least one symmetric
equilibrium exists. In this paper, we focus on the symmetric equilibria since this part of the model
by itself gives a lot of interesting results. Hence, the existence of asymmetric equilibria is set aside
for further research. All the proofs are shown in the Appendix and assumptions (A1)-(A4) are
assumed throughout the paper.
Theorem 1 For each n ∈ N , The Cournot oligopoly with positive network effects and incompatible
networks has (at least) one symmetric equilibrium.
In industries with network effects, it is common to face equilibria where none of the networks is
viable, i.e., where in equilibrium, all the firms stay out of the market by producing nothing. The
following proposition characterizes this kind of equilibrium, moreover, it can be shown that the
existence of the trivial equilibrium in the n-oligopoly implies the existence of the trivial equilibrium for the (n + 1)-oligopoly with positive network effects and incompatible networks, as stated
in Corollary 1. The latter result implies that whenever a zero individual output (an thus, zero aggregated output) is an equilibrium for the industry with n firms, an oligopoly with m > n identical
firms (among them and among the industry with n firms) will have it. Notice that the converse
is also true, the n-firms oligopoly having the trivial equilibrium implies that any identical industry
differing only in having a less number of firms, say m < n, will have the trivial equilibrium as well.
Proposition 1 The trivial outcome is an equilibrium if and only if xP (x, 0) ≤ cx for all x ∈ [0, K].
Corollary 1 The trivial outcome is an equilibrium for n ∈ N firms if and only if the trivial outcome
is an equilibrium for (n + 1) firms.
A consequence of Corollary 1, is that if the trivial equilibrium is not an equilibrium for an
industry with n firms, it will not be an equilibrium for the industry with one more firm and
9
viceversa. In this case, a non-trivial equilibrium will exist for both industries, as a consequence of
the existence theorem, Theorem 1.
One natural question to ask is under what conditions of the primitives the industry is viable,
i.e., when can we find a non-trivial equilibrium. In other words, we look for conditions such that
at least one firm produces an output different from zero. Theorem 2 answers to this question, in
particular, it focuses on the existence of a non-trivial symmetric equilibrium. But before presenting
the result, we introduce Lemma 1 that leads to part (ii) in Theorem 2. In this paper, qn (s) denotes
the individual output symmetric equilibrium correspondence for each of the i = 1, ..., n firms when
there are n firms and s is exogenous.
Lemma 1 If 0 ∈ qn (0), then qn (0) = 0. If in addition P (0, 0) = c, then qn0 (0) exists, it is also
single-valued and right-continuous at 0, and
qn0 (0) = −
P2 (0, 0)
.
(n + 1)P1 (0, 0)
(1)
If P (0, 0) < c, then qn (0) = 0 and qn0 (0) = 0.
Theorem 2 A non-trivial symmetric equilibrium exists if
(i) 0 ∈
/ qn (0), i.e., zero is not an equilibrium individual output (or xP (x, 0) > cx for some x ∈
(0, K]);
A non-trivial symmetric equilibrium with n active firms exists if
(ii) 0 ∈ qn (0), P (0, 0) = c and P1 (0, 0) + P2 (0, 0) > −nP1 (0, 0); or
(iii) 0 ∈ qn (0) and P (z, s) + nz P1 (z, s) ≥ c for some s ∈ (0, K] and for all z ≤ ns.
The result in Theorem 2 part (i) is immediate because we know that at least a symmetric
equilibrium always exist (Theorem 1).
Theorem 2 is closely related to the viability conditions for firms with compatible networks in
Amir and Lazzati (2011). These results are very powerful since they are analogous for compatible
and incompatible networks.
Now that we have discussed the existence of equilibria and more importantly, of non-trivial
equilibria, we are ready to do some analysis on the effects of entry. The next section is of interest
because the inclusion of demand-side economies of scale in a standard Cournot model lead to
10
different results depending on whether the networks are compatible or incompatible. Unfeasible
results in the standard Cournot model under the same assumptions (specially, under linear cost)
might arise in the compatible networks model, for instance, the firms might be able to increase their
profits with the entry of competitors (Amir and Lazzati (2011), Theorem 10 (i)). In contrast, as
we will see in Theorem 6, though possible, this is much more unlikely with incompatible networks.
2.3
Entry of firms
In this section, we analyze what happens to the symmetric equilibrium when the number of
firms exogenously increases in the market, i.e., we look at the changes in the equilibrium variables
of interest when the equilibrium is symmetric. The results hold for the maximal and minimal
equilibria, we cannot tell the direction of change of all the points in the equilibrium sets when we
increase the number of firms since the comparative statics results in this paper are based on the
results due to Milgrom and Roberts (1990, 1994), that are valid only for the extremal equilibria of
the games.
The results are listed for the maximal equilibria, denoted by the corresponding equilibrium
variable (the variable sub-indexed by n) with an upper bar. The reader should keep in mind that
these results are true for the minimal equilibria as well, denoted by a lower bar.
The following definitions will be useful in doing the comparative statics analysis:
∆2 (z) = nz P1 (z, z/n)P2 (z, z/n) − [P (z, z/n) − c][P1 (z, z/n) + nz P12 (z, z/n)], and
∆3 (z) = P1 (z, z/n) + [1/n]P2 (z, z/n).
Notice that ∆3 (·) denotes the change in the market price with an increase in the aggregate
output along the fulfilled expectation path.
In this model, equilibrium output might increase or decrease with the entry of firms. Clearly,
if individual output increases in n, total output will do too; nonetheless, this is not a necessary
condition. A sufficient condition for firms to increase their own production with new competitors
is given in Theorem 5. By now, Theorem 3 provides additional sufficient conditions for industry
output to go up or down with the number of firms. Whether total output increases or decreases
in n plays an important role in the rest of the comparative statics, as we can see immediately in
Theorem 4.
11
Theorem 3 For any n ∈ N
(i) z̄n+1 ≥ z̄n if ∆2 (z) ≥ 0 for all z ∈ [0, nK]; and
(ii) z̄n+1 ≤ z̄n if ∆2 (z) ≤ 0 for all z ∈ [0, nK].
Theorem 4 For any n ∈ N , P̄n+1 ≤ P̄n if
(i) ∆3 (·) ≤ 0 and z̄n+1 ≥ z̄n , or
(ii) ∆3 (·) ≥ 0 and z̄n+1 ≤ z̄n .
P̄n+1 ≥ P̄n if
(iii) ∆3 (·) ≥ 0 and z̄n+1 ≥ z̄n , or
(iv) ∆3 (·) ≤ 0 and z̄n+1 ≤ z̄n .
Example 1 shows that if the inverse demand function is linear in both output and expected size
of the network, and production is costless, the equilibrium output may increase or decrease with
the entry of firms.
Example 1. Consider an oligopoly with inverse demand function P (z, s) = max{a + bs − z, 0},
with a ≥ 0, b > 0, and costless production (c = 0). The reader can easily verify that the symmetric
equilibrium is given by xn =
a
n+1−b
and hence zn =
na
n+1−b .
The production by firm decreases in n
for all the possible values of a and b; if b > 1, total output decreases too with n. Notice that P (z, s)
is log-concave in z; below, in Theorem 5, we will show that log-concavity of P (·, s) is sufficient for
the per-firm production to decrease with the entry of firms.
k
It is very interesting that the model studied in this paper shares an important result of the
standard Cournot oligopoly game: the sufficient conditions that characterize the direction of change
of the individual production when n changes are the same. Such conditions are independent of
whether total output goes up or down with the number of firms in the presence of incompatible
networks. Theorem 5 formalizes this result. Notice that the results are stated for the case where
the extremal equilibria are interior; if they are not interior, i.e., the firms produce zero or K in the
presence of n firms, the result becomes trivial.
Theorem 5 At an interior equilibrium, per-firm outputs are such that
(i) x̄n+1 ≤ x̄n if P (·, s) is log-concave; and
(ii) x̄n+1 ≥ x̄n if P (·, s) is log-convex and c=0.
12
Theorem 5 part (ii) is a rare case, since it is more natural to face a log-concave inverse demand
function in total output, which leads to part (i). Hence, usually we will have that the individual
output decreases with the introduction of competition in this model. We find a similar implication
for the firms’ profits. As stated in Theorem 6 part (i), a decreasing individual output plus increasing output guarantees a decreasing individual profit. The profits are evaluated at the maximal
equilibria.1
Theorem 6 For any n ∈ N , per-firm profits are such that
(i) πn+1 ≤ πn if x̄n+1 ≤ x̄n and z̄n+1 ≥ z̄n ;
(ii) πn+1 ≥ πn if x̄n+1 ≥ x̄n and P (z̄n+1 , x̄n+1 ) ≥ P (z̄n , x̄n ).
Recall that in standard Cournot oligopoly model, total output is always increasing in the number
of firms given our assumptions; similarly, individual profit decreases with n (Amir and Lambson,
2000). The fact that it is more natural to face a log-concave inverse demand function, and hence,
a decreasing individual output, brings our model closer to standard Cournot oligopoly than to
oligopoly with compatible networks when looking at the change in individual profit driven by a
change in n. Amir and Lazzati (2011) show that in a Cournot model with a common network,
total output also increases with n, but the network effects make more feasible to increase individual
production with the number of firms and benefit firms with higher profits (Lemma 9 and Theorem 10
in Amir and Lazzati (2011) respectively). In our model, these results are as unusual as in standard
Cournot oligopoly (when total output is increasing). Incompatible networks do not have enough
influence in the model to change such comparative statics results.
To finish this section, we present an example that illustrates that total output and individual
output need not be monotonic in n, i.e., they may increase or decrease with n depending on how
many firms are there in the industry. In particular, the inverse demand function in Example 2 is
log-convex in z.
s
Example 2. Let P (z, s) = s/z 2 . Then, a firm maximizes its profit given by x (x+y)
2 − cx,
1
In Theorem 6, the conditions x̄n+1 ≥ x̄n and x̄n+1 ≤ x̄n can be replaced by log-concavity of P and log-convexity
of P plus c = 0 respectively, by Theorem 5; nonetheless, the second pair of conditions are more restrictive since they
are sufficient conditions for x̄n+1 ≥ x̄n and x̄n+1 ≤ x̄n respectively. Thus, we keep the conditions of the theorem in a
more general way.
13
which leads to the FOC s(y − x) = c(x + y)3 . Solving for the symmetric equilibrium we get
q
xn (s) = (n−2)s
.
cn3
Hence, the FECE is given by
xn =
n−2
cn
n−2
2
, zn =
, Pn =
and πn = 3 .
3
2
cn
cn
n−2
n
Notice that for one or two firms, the industry is non-viable, i.e., x1 = x2 = 0. On the other
hand, as n increases:
• xn increases for n ≤ 3 and decreases for n ≥ 3;
• zn increases for n ≤ 4 and decreases for n ≥ 4;
• Pn and πn decrease globally in n.
k
Now that we fully characterized the oligopoly with positive networks effects and incompatible
networks, we are ready to contrast it with the oligopoly whose network is unique. In other words, we
compare it to an oligopoly with compatible networks, for example, the mobile telephone industry.
There are several carriers, and the consumers benefit positively with a larger number of users, but
in this case, their networks are compatible, i.e., users can call other persons even if they receive
service from different companies.
3
Complete compatibility versus complete incompatibility
In this section, we compare viability and equilibria between the oligopolies with positive net-
work effects under complete compatibility and complete incompatibility. By viability, we mean the
existence of a non-trivial equilibrium. In particular, we are interested to see when it is easier for an
industry to emerge: if the networks are compatible or incompatible. We will soon show that if the
industry emerges when there is complete incompatibility, it will emerge as well when the networks
are compatible, but the reverse does not hold in general.
On the other hand, by comparing equilibria, we look at which kind of goods, compatible or
incompatible, induce higher equilibrium variables such as output, prices, profits, etc. Specifically,
when comparing equilibria, we refer to the extremal (highest and lowest) individual equilibrium
14
outputs and their corresponding market prices, profits, consumer surpluses and social welfare. The
main results, that we justify later on, are that the industry will produce more when the networks are
compatible and this scenario is more desirable in terms of social social welfare, since it maximizes
it.
Before presenting the results, we review the concept of oligopoly with compatible networks which
has been mentioned before in this study, but formalized in this section. For further details look at
Katz and Shapiro (1985) or Amir and Lazzati (2011). An oligopoly with positive network effects
and complete compatibility is defined in a similar way as the one with complete incompatibility,
described in this paper, except that in the first model, the goods produced by the firms are perfectly
compatible with each other, thus, there is only one network for the goods with expected size S.
Hence, each one of the firms in the industry maximize their profit given by
π(x, y, S) = xP (x + y, S) − cx,
where the primitives and variables of the previous equation are as in Section 2.1.
It is crucial that the reader keeps in mind that although the profit functions look identical for
both models, they are fundamentally different by the interpretation put on the parameters s and S,
which is the reason why we differentiate them by capitalizing our original parameter s. In oligopoly
with incompatible networks, s denotes the expected size of the firm’s own network2 ; in oligopoly
with complete compatibility, S denotes the size of the whole network, which in equilibrium matches
the industry production.
Amir and Lazzati (2011) provide a throughout characterization of the latter model. Under
similar assumptions on the primitives of the problems, there are several differences between the
models with compatible and incompatible networks. For example, Amir and Lazzati (2011) show
that an industry with compatible networks will never have an asymmetric equilibrium (Amir and
Lazzati (2011), Theorem 2), only symmetric equilibria are guaranteed. As we discussed earlier, the
model in this paper may have asymmetric equilibria, even if the firms are identical.
Besides the difference stated above, there are more that are worth studying. We first focus
in comparing the viability of the models. The next section shows that viability is higher under
2
One can think about it as being si for firm i, but we have dropped the sub-index along this paper for simplicity
in the notation.
15
compatible that under incompatible networks.
3.1
Viability
According to Amir and Lazzati (2011), an oligopoly with compatible networks only emerges or
is viable with a non-trivial symmetric equilibrium. In contrast, to say that an industry emerges
under complete incompatibility, it suffices to have at least one firm active.
But, what is the relationship between the viability of the oligopolies with compatible and incompatible networks? Is it easier to have one of the other? Do industries have the same chance to be
viable? In this section, we prove that viability is more feasible or higher in an industry with a unique
network. In other words, we state (in Theorem 7) that if the industry emerges with incompatible
networks, it will emerge with compatible networks as well.
Theorem 7 If the industry is viable for n ∈ N firms when the networks are completely incompatible,
the industry will be viable for n ∈ N firms with completely compatible networks.
The reverse of Theorem 7 does not hold in general: if an industry emerges when the networks
are compatible, it does not imply that it would emerge if the networks where incompatible.
Moreover, notice that with a monopolist the two models coincide. A crucial difference between
the models is that with compatible networks, there is minimal viability with one firm and viability
increases with the number of firms (Amir and Lazzati (2011), Theorem 7 part (ii)). In contrast,
viability is maximal with one firm when the networks are incompatible and it decreases with the
number of firms (Theorem 8).
Theorem 8 If P (·, s) is log-concave, the viability of an industry with incompatible networks decreases with the number of firms n.
Let us revisit Example 1 in Amir and Lazzati (2011) to illustrate Theorem 8, and that the
reverse of Theorems 7 does not hold.
2z
Example 3. Consider the inverse demand function P (z, s) = exp(− exp(1−1/s)
) and no pro-
duction costs. Amir and Lazzati (2011) show that every firm has a dominant strategy given by
x(y, s) = (1/2) exp(1 − 1/s) and that if the networks are compatible, the industry emerges when
16
there are two or more firms in the market. A monopoly would choose not to produce since it is
not profitable to do it. Nonetheless, if the networks are incompatible, this demand implies that
the industry will not emerge for any number of firms. Since every firm has its own network and
their strategy is dominant (independent of the other players’ choice), every single firms behaves as
a monopolist in the compatible networks world, thus, every potential firm will decide not to enter
the market. Formally, this equilibrium is given by the fixed point of qn (s) = (1/2) exp(1 − 1/s),
which is unique an equal to zero.
k
After analyzing the relationship between the viability of both models, an important question
arises: which model leads to a higher production? As we will formalize in the following section, an
immediate consequence of the higher viability of the oligopoly with complete compatibility is that
such oligopoly produces more. Moreover, the following section compares other equilibrium variables
of interest, such as prices, profit, consumer surplus and social welfare.
3.2
Comparison of equilibria
An interesting question in the networks literature is which setting induces higher equilibrium
variables of interest, such as production, profits, prices, consumer surplus and social welfare. In this
section, we provide answers to these questions, focusing in the symmetric equilibria (with all firms
active) for the incompatible oligopoly, unless anything else is stated. But first, let us define ∆4 (·),
which will be helpful to present our results.
∆4 (z) = P1 (z, z) + P2 (z, z),
which denotes the change of the market price when aggregated output changes in an industry with
complete compatibility and along the equilibrium path.
Before presenting the results in this section, we introduce a slight addition to our notation, so
that we can distinguish the equilibrium variables between the two models under study. For the
n-firms symmetric equilibrium, we still use the variable sub-indexed by the number of firms but
now we add a super-index that denotes complete compatibility (C) or complete incompatibility (I).
Now we are ready to present the first result, which states that the oligopoly with complete compatibility always produces at least as much as the oligopoly with incompatible networks. Nonetheless, there is no clear relationship between the prices of the industries; it depends on the magnitude of
17
the network effect, as we will discuss later. Although the result is stated for the highest equilibrium
output, it can be checked from the proof that it also holds for the lowest one.
Theorem 9 At the highest (symmetric) equilibrium output and for every n ∈ N
I
(i) x̄C
n ≥ x̄n , and
(ii) PnC ≥ PnI if ∆4 (·) ≥ 0 on [nx̄In , nx̄C
n ].
The following example shows that equilibrium outputs are the same despite the networks being
compatible or incompatible. Nonetheless, the resulting price is higher in the compatible world,
which is obvious since the network is larger and consumers’ willingness to pay increases with the
size of the network.
Example 4. Let P (z, s) = se−z and c = 0. Each firm chooses x that maximizes xse−(x+y) ,
which leads to the FOC se−(x+y) (1 − x) = 0. Then, the symmetric FECE when the firms are
incompatible is
xn = 1, zn = n, and Pn = πn = e−n .
On the other hand, if there is only one common network, the symmetric FECE is
xn = 1, zn = n, and Pn = πn = ne−n .
Hence, production is the same for both models and individual output is independent of the
number of firms in the industry.
k
It is straightforward that Theorem 9 part (i) implies that an industry with complete compatibility will produce at least the same aggregated output than the industry with complete incompatibility
in equilibrium (just multiply by n the inequality in part (i)). This is a generalization of the result
by Katz and Shapiro (1985) who prove the same result for a linear and separable inverse demand
function with costless production.
Theorem 9 part (ii) says that if the network effect is sufficiently big to offset the competitive
effect, reflected by ∆4 (·) ≥ 0, the consumers have a higher willingness to pay if the goods are
compatible, since the size of the network is bigger in this case. This results in a higher equilibrium
price for the compatible networks. Theorem 9 implies that a firm prefers complete compatibility
when ∆4 (·) ≥ 0; as we will formalize later, the firm’s new revenue is high enough to offset the
18
higher cost of production (because price and production are higher under compatibility). This is
consistent with Economides and Flyer (1997), who argues that the welfare of a firm depends on the
strength of the network effect.
The previous result is summarized in the following theorem, together with the analysis of consumer surplus and social welfare. Notice that society is always better off under complete compatibility, by Theorem 10 part (iii), but we cannot conclude in general that the firms or the consumers
always prefer one kind of industry. Nevertheless, we can conclude that complete compatibility
always benefits at least one group, consumers or firms.
Theorem 10 At the highest equilibrium outputs and for every n ∈ N
(i) πnC ≥ πnI if ∆4 (·) ≥ 0;
(ii) CSnC ≥ CSnI if PnI ≥ PnC or P12 (z, s) ≤ 0, and
(iii) WnC ≥ WnI .
Theorem 10 part (ii) relates to Church and Gandal (1992) in the sense that consumer surplus
cannot be compared in a global sense. Which setting gives the consumers a higher surplus relies on
the characteristics of the market, specifically, of the demand. To see that consumer surplus can be
bigger under complete incompatibility than under complete compatibility at the highest equilibrium
individual output, let us consider Example 3 in Amir and Lazzati (2011).
Example 5. Let P (z, s) = max{a − z/s3 , 0} denote the inverse demand function of an industry
with n symmetric firms that face zero production costs. Assume that a ≥ n/K 2 and K > 1. As
shown by Amir and Lazzati (2011), the reaction function of a firm is given by


max{(as3 − y)/2, 0}, if (as3 − y)/2 < K,
x(y, s) =

K
if (as3 − y)/2 ≥ K.
If the networks are completely compatible, the FECE industry output set is given by znC =
p
C
{0, (n + 1)/na, nK}. Then, for the highest equilibrium z C
n = nK, the consumer surplus is CSn =
1/(2nK).
Now, if the networks are completely incompatible, the FECE industry output set is given by
p
znI = {0, n (n + 1)/a, nK}. Then, CSnI = n2 /(2K) for the highest FECE z In = nK. Thus, in
19
this setting, FECE consumer surplus is higher under complete incompatibility than under complete
compatibility.
Notice that PnC = a − 1/(nK)2 is greater than PnI = a − n/K 2 and P12 (z, s) = 3/s4 > 0, i.e.,
the sufficient conditions of Theorem 10 part (ii) are violated.
Finally, we have that πnC = K(a − 1/(nK)2 ) ≥ K(a − n/K 2 ) = πnI and WnC = anK − 1/(2nK) ≥
WnI = anK − n2 /(2K), as Theorem 10 parts (i) and (iii) predicts, given that ∆4 (z) = P1 (z, z) +
P2 (z, z) = 2/z 3 > 0.
k
From the example above, it appears that a sufficient condition for the consumers to prefer
incompatible networks is that the incompatible firms produce at their maximum capacity when the
inverse demand function exhibits increasing differences in (z, s). By Theorem 9 part (i), this implies
that the compatible firms also produce at their maximum capacity. As a more general result, it can
be shown that if both industries produce exactly the same output (including asymmetric equilibria
in the incompatible setting), the consumers will prefer incompatibility. The reason is that the
market output is the same, a smaller network decreases the price and by supermodularity of the
inverse demand function, the competitive effect is reinforced. The previous result is summarized in
the following proposition. It is valid for any equilibria, not only for the extremal ones.
Proposition 2 For every n ∈ N , if znI = znC and P12 (z, s) ≥ 0, then, CSnI ≥ CSnC .
Under the hypotheses of Proposition 2, and by Theorem 10 part (iii), one may conclude that
the profits the industry are higher under complete compatibility, since social welfare is higher for
compatible networks. Nevertheless, one has to be careful in stating this result, because Theorem 10 holds for a symmetric equilibrium in the incompatible setting and the extremal equilibria,
assumptions that would become additional to those in Proposition 2.
4
Conclusions
This paper generalizes the theory in Katz and Shapiro (1985) on oligopolies with positive network
effects and complete incompatibility. In particular, it allows for pure network goods, where the
viability question becomes relevant since now, trivial equilibria where none of the firms produce is
a possibility. A symmetric setting of this model, where all the firms face the same demand and
20
costs of production, becomes particularly interesting because asymmetric equilibria may arise in
any shape: it could be that a subset of the firms is active and produce an equal amount of goods
among them or that at least two firms produce a positive and different level of output.
The present study focuses on the characterization of symmetric equilibria, since they are interesting enough to develop important theory around them. In particular, we focus on the existence
of non-trivial equilibria, also called viability, since the possibility of having trivial equilibria, where
none of the firms produce, is a very common outcome in the model under study. The reason is that
we consider pure network goods, that do not posses any value by their own.
A challenge in the literature regarding this model has been to characterize asymmetric equilibria,
specially when two or more firms produce a positive and different output. Future research may look
towards this direction. This topic is particularly relevant because it may explain the success of some
networks over their rivals, for example, the triumph of VHS over Beta in the late 80’s, despite their
high level of substitution and Beta being the precursor of VCRs and the leader in the market by
the middle 70’s.
A striking finding is that the model in this paper can be considered closer to standard Cournot
symmetric oligopoly than to the oligopoly with positive network effects and compatible networks, in
a comparative statics way. The conditions for which the rare result of having increasing individual
output with the number of firms arises are the same for both standard Cournot and the oligopoly
with incompatible networks. This is not the case for oligopoly with compatible networks, where
such rare result might be driven by the network effect.
The present paper also provides some interesting public policy implications. It is shown that in
a general setting with symmetric firms, social welfare is higher when the networks are compatible;
similarly, there is higher production. The former result implies that at least the consumers or the
firms are better off with complete compatibility, but it is not certain that both groups simultaneously are. Sufficient conditions such that the firms and the consumers prefer compatibility over
incompatibility are provided.
Intuitively, when the networks effect is high enough that it offsets the demand effect, a higher
network generated by the complete compatibility of the firms’ networks allows the industry to
charge a sufficiently high price to obtain higher profits than under complete incompatibility (since
21
we already know that compatibility increases production). On the contrary, if the firms charge a
higher price under incompatibility, the consumers will prefer the cheaper product in the compatible
setting with a higher output, that clearly leads to a higher consumer surplus. This is only one
sufficient condition that provides the consumers with a higher surplus under compatibility; in fact,
this paper presents an example were the consumers are better off with complete incompatibility.
Summarizing, this paper provides a general framework for symmetric oligopolies with positive
network effects and complete incompatibility. It gives minimal conditions for the existence of
equilibrium with particular interest in non-trivial symmetric equilibria. Similarly, it studies the
effects of more competition in the market and compares the equilibrium outcomes of this model
with that when the networks are totally compatible.
5
Appendix
First, we introduce the following lemma that will be useful in the proof of Theorem 1.
Lemma 2 Every selection of the best-response correspondence z(y, s) increases in both y and s.
Proof of Lemma 2. The proof follows by Amir and Lazzati (2011), Lemma 1.
Proof of Theorem 1. Let the expected size of the network be s for each firm. By Amir and
Lambson (2000), a symmetric Cournot equilibrium for all n (and s) exists and comes from a fixed
point of the correspondence
Bs : [0, (n − 1)K] → 2[0,(n−1)K] ,
y→
n−1
z(y, s).
n
Now we need to prove that qn (s), the set of symmetric Cournot equilibrium individual outputs
for each firm i = 1, ..., n, has fixed points. By Lemma 2, every selection of z(y, s) is increasing in s.
Then, by Milgrom and Roberts (1990), the maximal and minimal fixed points of Bs (y), ȳ(s) and
y(s) respectively, increase in s. Hence, by symmetry and by Lemma 2, the extremal selections of the
correspondence qn : [0, K] −→ [0, K], q̄n (s) = ȳ(s)/(n − 1) and q n (s) = y(s)/(n − 1), are increasing
in s. Thus, by Tarski’s fixed point theorem (1955), q̄n (s) and q n (s) have fixed points that can be
22
seen as symmetric FECE’s.
Proof of Proposition 1. By definition, an individual (and hence, industry) output of 0 is a
symmetric FECE if 0 ∈ x(0, 0). This holds if and only if π(0, 0, 0) ≥ π(x, 0, 0) ∀x ∈ [0, K], i.e.,
0 ≥ xP (x, 0) − cx ∀x ∈ [0, K].
Proof of Corollary 1. By proof of Proposition 1, 0 ∈ qn (0) if and only if π(0, 0, 0) ≥ π(x, 0, 0)
∀x ∈ [0, K] if and only if 0 ∈ qn+1 (0).
The following lemma will be used to prove Lemma 1.
Lemma 3 Let q̃n (s) be an increasing selection of qn (s). Then q̃n (s) is differentiable for almost all
s, and, if q̃n (s) ∈ (0, K) for s > 0, its slope is given by (q̃n stands for q̃n (s))
∂ q̃n (s)
q̃n P12 (nq̃n , s) + P2 (nq̃n , s)
=−
.
∂s
(n + 1)P1 (nq̃n , s) + nq̃n P11 (nq̃n , s)
(2)
Proof of Lemma 3. If q̃n (s) is interior, it satisfies the first order condition
q̃n (s)P1 (nq̃n (s), s) + P (nq̃n (s), s) − c = 0.
(3)
Since q̃n (s) is increasing, it is differentiable almost everywhere w.r.t. Lebesgue measure. Hence,
differentiating both sides of equation (3) with respect to s on a subset of full Lebesgue measure and
reordering terms gives us the result.
Proof of Lemma 1. To see that qn (0) is single-valued and equal to zero, notice that 0 ∈ qn (0)
imply, by Proposition 1, that xP (x, 0) ≤ c for all x ∈ [0, K]. Therefore, by (A1), xP (x + y, 0) < c
for all x, y > 0. This tell us that 0 is a strictly dominant strategy with s=0 and thus, qn (0) = 0.
Now suppose that P (0, 0) = c, i.e., zero is an interior equilibrium, and that q̃n (s) is an increasing
selection of qn (s). Take a sequence sk ↓ 0 such that a selection q̃n (s) is differentiable at sk for all
k, this is possible because q̃n (s) is an increasing function, moreover, limk→∞ q̃n (sk ) exists. By
23
Fudenberg and Tirole (1991), qn (s) is upper hemi-continuous, therefore, limk→∞ q̃n (sk ) ∈ qn (0) =
{0}, i.e., limk→∞ q̃n (sk ) = 0.
Now, by (A1), (A2) and limk→∞ q̃n (sk ) = 0, the right-hand side of equation (2) is rightcontinuous in s at 0. Doing s = sk and taking limit as k → ∞ in the right-hand side of (2),
it follows that limk→∞ ∂ q̃n∂s(sk ) exists and it is given by the right-hand side of equation (1). Because
this argument is valid for all sequences (sk ) taken from the subset of full Lebesgue measure of the
domain [0, K], ∂ q̃n (s)/∂s |s=0 exists, is continuous at 0 and given by (1).
Notice that the previous argument holds for the extremal selections of qn (s), q̄n (s) and q n (s),
since they are increasing in s by the proof of Theorem 1. Thus, their derivatives at s = 0 are
equal and given by the right-hand side of equation 1, moreover, we have that max{∂qn (s)/∂s |s=0
} = ∂ q̄n (s)/∂s |s=0 = ∂q n (s)/∂s |s=0 = min{∂qn (s)/∂s |s=0 }, then, ∂qn (s)/∂s |s=0 exists, is singlevalued, continuous and given by the right-hand side of equation (1).
Finally, assume that P (0, 0) < c. Then, by (A1), P (0, s) < c for s sufficiently small and in
consequence, qn (s) = 0 for all such s. Hence, qn0 (0) = 0.
Let Π(z, s) ,
n−1
n
R z
0
P (t, s)dt − cz +
1
n [zP (z, s)
− cz], a weighted average of welfare and
industry profits when s is the same for all firms. Thus, we have the following result.
Lemma 4 If z ∗ ∈ argmax{Π(z, s), 0 ≤ z ≤ nK}, then, x∗ ≡
z∗
n
∈ qn (s), for all n ∈ N and
s ∈ [0, K].
Proof of Lemma 4. Since the cost function is linear, it is convex. By Amir and Lazzati (2011),
Lemma 14, for any n ∈ N and s ∈ [0, K], if z ∗ ∈ argmax{Π(z, s), 0 ≤ z ≤ nK}, then, z ∗ ∈ Qn (s),
where Qn (s) is the total output equilibrium correspondence for a given s. Then, by symmetry,
z ∗ ∈ Qn (s) implies that x∗ ≡
z∗
n
∈ qn (s).
Proof of Theorem 2.
(i) If the trivial outcome is not part of the equilibrium set, Theorem 1 guarantees there is a symmetric FECE with strictly positive individual output.
(ii) Parts (ii) and (iii) use the following argument. By the proof of Theorem 1, the maximal selection of qn (s), q̄n (s), is increasing in s. Suppose that there exists s0 ∈ (0, K] such that q̄n (s0 ) ≥ s0 .
24
The facts that q̄n (·) is increasing and q̄n (s0 ) ≥ s0 (by assumption) imply that for all s ∈ [s0 , K],
q̄n (s) ∈ [s0 , K]. Then, q̄n (s) is an increasing function that maps [s0 , K] into itself. By Tarski’s fixed
point theorem (1955), there exists s00 ∈ [s0 , K] such that q̄n (s00 ) = s00 , which is a strictly positive
FECE. Therefore, we only need to show that there exists s ∈ (0, K] such that q̄n (s) ≥ s to prove
that there is a non-trivial symmetric equilibrium for n firms. To this end, it suffices to show that
qn0 (0) > 1 which implies that there is a small > 0 for which q̄n () > . By hypothesis, 0 ∈ qn (0)
which implies by Lemma 1 that 0 ∈ qn (0) for all n. By Lemma 1, qn0 (0) > 1 if P (0, 0) = c and
P1 (0, 0) + P2 (0, 0) > −nP1 (0, 0), which implies that a non-trivial symmetric equilibrium exists for
n firms.
(iii) The condition in this part implies that Π1 (z, s) ≥ 0 for some s ∈ (0, K] and for all z ≤ ns, i.e.,
there exist s ∈ (0, K] and z 0 ≥ ns such that Π(z 0 , s) ≥ Π(z, s) for all z ≤ ns. Hence, the largest
argmax of Π(z, s), say z ∗ , must be grater or equal than ns, i.e., z ∗ ≥ ns and z ∗ /n ≥ s. By Lemma
4, z ∗ /n ∈ qn (s) so there is an s ∈ (0, K] such that an element of qn (s) is greater or equal than s.
By the argument in part (ii), it follows that a non-trivial symmetric equilibrium exists for n firms.
Proof of Theorem 3.
(i) and (ii) At any interior equilibrium, z̄n must satisfy the first order condition
P (z̄n , z̄n /n) +
z̄n
P1 (z̄n , z̄n /n) − c = 0,
n
which implies that it is the maximal fixed point of the function
F (z; n) = −
n(P (z, z/n) − c)
.
P1 (z, z/n)
Notice that
∂F (z; n)
∆2 (z)
= 2
.
∂n
P1 (z, z/n)
Thus, by Milgrom and Roberts (1990), the direction of change in z̄n when n increases is given
by the same sign of ∆2 (z).
Proof of Theorem 4.
The result follows immediately from ∆3 (z) = dP (z, z/n)/dz and the fact that total output may
25
increase or decrease with respect to n.
Proof of Theorem 5.
(i) From Amir and Lambson (2000), Theorem 2.3, q̄n (s) is decreasing in n under linear cost,
P1 (z, s) < 0 and P log-concave, for all s. Consequently, so is its fixed point x̄n .
(ii) Similarly, Theorem 2.4 in Amir and Lambson (2000) shows that under our assumptions, q̄n (s)
is increasing in n for all s; so is its fixed point, which completes the proof.
Proof of Theorem 6.
(i) Consider the following
πn = x̄n P (x̄n + ȳn , x̄n ) − cx̄n
≥ x̄n+1 P (x̄n+1 + ȳn , x̄n ) − cx̄n+1
≥ x̄n+1 P (x̄n+1 + ȳn , x̄n+1 ) − cx̄n+1
≥ x̄n+1 P (x̄n + ȳn , x̄n+1 ) − cx̄n+1
≥ x̄n+1 P (x̄n+1 + ȳn+1 , x̄n+1 ) − cx̄n+1 = πn+1 .
The first inequality follows by equilibrium. The second one, by (A1), P2 (z, s) > 0, and the
assumption that x̄n+1 ≤ x̄n . The third one, by P1 (z, s) < 0 and x̄n+1 ≤ x̄n ; the last one, by (A1)
and z̄n+1 ≥ z̄n .
ii) Consider the following inequalities
πn+1 = x̄n+1 P (x̄n+1 + ȳn+1 , x̄n+1 ) − cx̄n+1
≥ x̄n P (x̄n + ȳn+1 , x̄n+1 ) − cx̄n
≥ x̄n P (x̄n+1 + ȳn+1 , x̄n+1 ) − cx̄n
≥ x̄n P (x̄n + ȳn , x̄n ) − cx̄n = πn .
The first inequality follows by equilibrium.
The second one, by (A1), P1 (z, s) < 0, and
x̄n+1 ≥ x̄n . The last one, by the assumption that P (x̄n+1 + ȳn+1 , x̄n+1 ) ≥ P (x̄n + ȳn , x̄n ).
Proof of Theorem 7. First, notice that a FECE of the oligopoly with compatible networks
is a fixed point of nq̄n (s). On the other hand, a FECE of the oligopoly with incompatible networks
is a fixed point of q̄n (s).
26
Since q̄n (s) ≤ nq̄n (s), a non-trivial FECE of the industry with incompatible networks imply that
there is a non-trivial FECE of the industry with compatible networks; hence, the viability of the
first one implies the viability of the second one.
Proof of Theorem 8. This result follows immediately from the proof of Theorem 5 part (i).
Since q̄n (s) decreases with n, so does the viability of the industry (since the critical mass increases).
Proof of Theorem 9.
(i) The proof of this part follows immediately from the proof of Theorem 7. Since q̄n (s) ≤ nq̄n (s),
the highest fixed point of nq̄n (s) is higher than that of q̄n (s). Such fixed points correspond to the
highest FECE’s of the oligopoly with compatible and incompatible networks, respectively. The
proof for the lowest FECE’s follows analogously.
(ii) Consider the following inequalities
C
PnC = P (nxC
n , nxn )
≥ P (nxIn , nxIn )
≥ P (nxIn , xIn ).
The first inequality follows by the assumption that ∆4 (·) ≥ 0 and Theorem 9 part (i); the second
one, by the assumption that P2 (z, s) > 0.
Proof of Theorem 10.
(i) Consider the following inequalities
C
C
C
C
C
πnC = x̄C
n P (x̄n + ȳn , x̄n + ȳn ) − cx̄n
C
I
≥ x̄In P (x̄In + ȳnC , x̄C
n + ȳn ) − cx̄n
C
C
C
I
≥ x̄In P (x̄C
n + ȳn , x̄n + ȳn ) − cx̄n
≥ x̄In P (x̄In + ȳnI , x̄In + ȳnI ) − cx̄In
≥ x̄In P (x̄In + ȳnI , x̄In ) − cx̄In
= πnI .
The first inequality follows by Cournot property; the second one, by (A1), P1 (z, s) < 0, and
27
I
C
I
Theorem 9 part (i), x̄C
n ≥ x̄n ; the third one is true by the assumption that ∆4 (·) ≥ 0 and x̄n ≥ x̄n ,
and the last one, by (A1), P2 (z, s) > 0.
(ii) Consider the following inequalities
R nx̄In
R nx̄C
C
C
[P (t, x̄In ) − P (nx̄In , x̄In )] dt
CSnC − CSnI = 0 n [P (t, nx̄C
n ) − P (nx̄n , nx̄n )] dt − 0
R nx̄In
R nx̄I
C
C
[P (t, x̄In ) − P (nx̄In , x̄In )] dt
≥ 0 n [P (t, nx̄C
n ) − P (nx̄n , nx̄n )] dt − 0
R nx̄In
R nx̄I
I
C
C
[P (t, x̄In ) − P (nx̄C
≥ 0 n [P (t, nx̄C
n , x̄n )] dt ≥ 0.
n ) − P (nx̄n , nx̄n )] dt − 0
The first inequality follows by Theorem 9 part (i) and P1 (z, s) < 0.
Under the assumption that PnI ≥ PnC at the highest equilibrium, the second line becomes positive
I
given that nx̄C
n ≥ x̄n and P2 (z, s) > 0, which proves the first part of the result.
I
For the second part, notice that the second inequality follows from nx̄C
n ≥ nx̄n and P1 (z, s) < 0,
and the last one by the assumption that P12 (z, s) ≤ 0. To see this, notice that t ∈ [0, nx̄In ] and
C
I
nx̄C
n ≥ nx̄n imply that nx̄n ≥ t, thus, adding the assumption that P12 (z, s) ≤ 0 and the result that
I
C
I
C
C
C
I
I
nx̄C
n ≥ x̄n imply that P (t, nx̄n ) − P (t, x̄n ) ≥ P (nx̄n , nx̄n ) − P (nx̄n , x̄n ) for all t ∈ [0, nx̄n ], which
gives us the result.
(iii) First notice that the social welfare function at any production level x and expected size of
the network s, when there are n symmetric firms is given by
Z
nx
P (t, s) dt − ncx,
Vn (x, s) =
0
which is a concave function with respect to x since
∂ 2 Vn (x,s)
∂x2
= n2 P1 (nx, s) < 0, by (A1). Then, we
have that at the highest equilibria
o
o nR I
nR C
nx̄n
nx̄
I ) dt − ncx̄I
C
P
(t,
x̄
WnC − WnI = 0 n P (t, nx̄C
n
n
n ) dt − ncx̄n −
0
nR C
o nR I
o
nx̄n
nx̄n
C
C
I
≥ 0 P (t, nx̄n ) dt − ncx̄n − 0 P (t, nx̄C
n ) dt − ncx̄n
C
I
C
= Vn (x̄C
n , nx̄n ) − Vn (x̄n , nx̄n )
≥
C
∂Vn (x̄C
n ,nx̄n )
(x̄C
n
∂x
− x̄In )
C
C
I
= n[P (nx̄C
n , nx̄n ) − c](x̄n − x̄n ) ≥ 0
I
The fist inequality follows by nx̄C
n ≥ x̄n and P2 (z, s) > 0; the second one, by concavity of Vn (·, s),
C
C
I
and the last one, because P (nx̄C
n , nx̄n ) ≥ c and x̄n ≥ x̄n , by Theorem 9 part (i).
Proof of Proposition 2.
28
Amir and Lazzati (2011) show that under our assumptions, no asymmetric equilibria exist in the
Pn
I
compatible industry. Thus, the assumption that znC = znI implies that znC = nxC
n =
i=1 xin and
znC ≥ xIin for all i = 1, ..., n. Let us order the incompatible firms by output, i.e., xI1n ≥ xI2n ≥
... ≥ xInn , and define xI0n ≡ 0. Notice that by (A1), P (t, xI1n ) ≥ P (t, xI2n ) ≥ ... ≥ P (t, xInn ) for all
0 ≤ t ≤ znI . Then, R nxC
R P i xI
P
CSnI − CSnC = ni=1 xI j=1 jn [P (t, xIin ) − P (znI , xIin )] dt − 0 n [P (t, znC ) − P (znC , znC )] dt
(i−1)n
P
R ij=1 xIjn
Pn
I
C
I
I
C
C
[P (t, xin ) − P (t, zn ) − P (zn , xin ) + P (zn , zn )] dt ,
= i=1 xI
(i−1)n
which is positive by the assumption that P12 (z, s) ≥ 0.
6
References
Amir, R. and V. E. Lambson, 2000, On the effects of entry in Cournot markets, Review of
Economic Studies, 67:235-254.
Amir, R. and N. Lazzati, 2011, Network effects, market structure and industry performance,
Journal of Economic Theory, 146:2389-2419.
Cusumano, M. A., Mylonadis, Y. and R. S. Rosenbloom, 1992, Strategic maneuvering and
mass-market dynamics: the triumph of VHS over Beta, The Business History Review, 66:51-94.
Chen, J., Doraszelski, U. and J. E. Harrington, 2009, Avoiding market dominance: product
compatibility in markets with network effects, RAND Journal of Economics, 40:455-485.
Church, J. and N. Gandal, 1992, Network effects, software provision, and standarization, The
Journal of Industrial Economics, 40:85-103.
Economides, N. and F. Flyer, 1997, Compatibility and market structure for network goods.
Katz, M. L. and C. Shapiro, 1985, Network externalities, competition and compatibility, American Economic Review, 75:424-440.
Katz, M. L. and C. Shapiro, 1986, Technology adoption in the presence or network externalities,
Journal of Political Economy, 94:822-841.
Milgrom, P. and J. Roberts, 1990, Rationalizability, learning and equilibrium in games with
strategic complementarities, Econometrica, 58:1255-1278.
29
Milgrom, P. and Roberts, J., 1994, Comparing equilibria, American Economic Review, 84: 441459.
Fudenberg, D. and J. Tirole, 1991, Game Theory, The MIT Press.
Tarski, A., 1955, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of
Mathematics, 5:285-309.
Topkis, D., 1998, Supermodularity and Complementarity, Princeton University Press.
30