Strategic Investment in Networks*
Nalin Kulatilaka
nalink@bu.edu
and
Lihui Lin
lhlin@bu.edu
January 2004
Abstract
We examine how the presence of network effects influences investment decisions. Building a
network requires significant upfront investment but benefits carry tremendous uncertainty. This
creates an incentive to defer the commitment of irreversible investments. However, such investments
may also create the opportunity to convince the consumers about the network’s size, establish a
network standard, and preempt future competitors. Our models account for the tradeoff between
these countervailing forces to obtain the investment rules for building networks. First, we study the
investment decision faced by a monopolist in both the investment opportunity and product market.
By investing prior to the resolution of uncertainty, the monopolist convinces the consumers of the
network size. We solve for the threshold level of expected demand which must be exceeded in order
to commit the investment. This threshold is lowered by an increase in the intensity of the network
effect but the effect of uncertainty on the investment threshold is ambiguous. Our second model
allows for future competitor entry where the entrant may either adopt the monopolist’s standard or
build its own network. We find that the optimal licensing fee may be lower than the highest level
that the entrant would accept. When future competition is anticipated, the investment threshold is
monotonically decreasing in both the intensity of network effects and the level of uncertainty.
*
We would like to thank John Henderson, George Wyner, Chi-Hyon Lee, Kathleen Curley and other participants at
the IS seminar series at Boston University for their insightful comments. Funding was provided by the Boston
University Institute for Leading in a Dynamic Economy. All errors and omissions are ours.
0
1
Introduction
Networks are increasingly dominating our life by changing the very nature of how we work
and how we spend our leisure time. Advances in communications and information technologies
connect large populations of people, machines, and sensors to form a host of valuable networks. The
digitized content coupled with this pervasive connectivity has spawned a menagerie of new services
that touch our day-to-day life. Unlike in the case of normal goods, the value of network goods
extends beyond the standalone use to the consumer (autarky value). Consumers of network goods
reap additional utility from the ability to connect and collaborate with other users (network value).
As a result, the value of a network to each user increases with the number of other users in the
network.1
This tremendous value potential can present very attractive investment opportunities to firms
that build networks. Rushes to build new networks, however, have repeatedly been followed by
dramatic failures.2 It is, in fact, the very source of network effects that introduce the greatest
uncertainties to network investments. Builders of networks must commit large and irreversible
investments well ahead of widespread customer adoption of unproven goods and services. In the
early stages of a network’s evolution, it will have few users and each user will realize only low levels
of network benefits. As a result, consumers will remain unconvinced about the full value of the
network good until the network reaches maturity. The adoption decision of consumers introduces
tremendous uncertainty regarding the potential demand for the network good.3
1
Aggregating this phenomenon to the whole network has been articulated by Robert Metcalfe in the well known
Metcalfe’s Law, which states that the value of a network increases with the square of the number of users (nodes) on
a network (Gilder 2000, pp73, 151).
2
There are strong parallels between the investment booms and the subsequent busts in the railroad and telegraph
industries with the recent Internet bubble. See “The Victorian Internet” (Standage 1998) for an enlightening
historical case study of the telegraph industry.
3
The uncertainty is exacerbated when multiple firms compete to establish a network standard and when multiple
components in a complementary network system must be developed in order to deliver the network good.
1
Firms making investments in networks face not only uncertainties around the creation of
value but also added uncertainties around the appropriation of value. When users decide whether to
adopt a network good, they not only take into account the current size of the network but also form
expectations about the future growth of the network. Without a mechanism to influence consumers’
expectation, even a monopolistic producer is unable to capture values that reflect the full extent of
the network effect.4 We examine the early commitment of an irreversible investment as a credible
way to convince consumers about the future size of the network and thus internalize the network
externality.
The value appropriation becomes further complicated in the presence of competition. In such
cases, the commitment of investment preempts potential competitors and yields various competitive
advantages. While the early commitment of investment often allows firms to gain cost advantages
(e.g., due to learning) that can be used to dissuade potential entrants, networks introduce an
additional benefit by allowing early investors to establish standards.5 Subsequent entrants to the
market must either license the standard by paying a fee or develop a new standard. Since the
network effect increases with the size of the network, having multiple standards will result in smaller
network effects and act as a further incentive to first movement in network investments.
In this paper, we examine the implications of network effects on a firm’s investment
decision. This inherently calls for a treatment of the inter-temporal effects (invest now and receive
benefits later) in the presence of uncertainty. On the one hand, committing to an irreversible network
investment kills the option to defer that investment.6 This option can be very valuable in the face of
uncertainty about the future market demand and lead to the postponement of network investments.
4
There is a substantial economics literature on network effects addressing issues of optimal network size, pricing of
network goods, welfare implications of networks, and network externalities. See Katz and Shapiro (1994),
Leibowitz and Margolis (1994) and Economides (1996) excellent surveys of the literature. These models tend to be
set in a static setting and focus on equilibrium conditions. They also do not explicitly treat the effects of uncertainty.
5
For example, see Dixit (1980) and Spence (1984) for models of entry dissuasion through cost advantages.
Kulatilaka and Perotti (1998 and 2000) and Grenadier (1996) have found that the presence of such strategic benefits
offset the postponement incentives introduced by the option to defer investment.
6
McDonald and Siegel (1985) first modeled this investment-timing problem in the context of real options. A review
of the ensuing literature is in Dixit and Pindyck (1994).
2
On the other hand, the immediate commitment to a network investment can have strategic benefits
that act as an incentive to accelerate investment decisions. We consider two specific mechanisms for
such strategic effects.
Our model has two time points. At the current time, a single firm has a monopoly over an
investment opportunity but is uncertain about the demand for the network good at some future time.
We first consider the case where there are no potential competitors even at the future date when
uncertainty about the potential number of consumers is fully resolved. The price they pay, however,
depends on whether or not the firm commits an immediate investment prior to the resolution of
demand uncertainty. If the firm makes such an investment, then potential consumers adjust their
expectations of the network size and the network value of the good accordingly. The monopolist can
thus charge a price that fully internalizes the network effect. We use a functional form that
generalizes the Metcalfe’s Law to introduce network effects with varying intensity. If the firm does
not commit the investment, it may avoid regret of having committed an investment into a market
with low demand.
We show that the resulting prices, quantities, and profits are largest when early investment is
made. We solve for the investment threshold as the expected demand that must be exceeded for the
firm to invest immediately. We find that the investment threshold monotonically decreases with the
intensity of the network effect. The impact of increased uncertainty on the threshold is ambiguous,
similar to the case when the strategic benefits were introduced via cost or timing advantages.
We next study the strategic network investment problem in an imperfect competition setting.
We maintain that a single firm (M) has a monopoly over the investment opportunity but allow for a
potential competitor (N) at a future time. Early investment allows M to establish a network standard
that can be licensed to N. Then N has the choice of adopting M’s standard by paying a licensing fee
or investing in a new standard. By adopting M’s standard, N eliminates the investment needed to
develop a new standard, and contributes to a large single network and consequently earns greater
3
profits. If N does not accept the licensing proposal, it can invest in a different standard leading to
two smaller incompatible networks.7 If M does not invest immediately, M and N will be identical
when the uncertainty is resolved. In this case, they may choose to invest but are unable to coordinate
on a single standard and two incompatible network standards will emerge.
The optimal licensing fee is obtained as M’s choice that maximizes its profits while
recognizing N’s ability to invest in a new network standard. We find that the intensity of the network
effect has opposite impacts on M’s profit maximization condition and N’s acceptance condition.
Therefore, depending on the level of network intensity, the optimal licensing fee may be determined
by one or the other of the above conditions. In other words, M may choose to charge a licensing fee
that is lower than the highest level that the entrant would accept. We also find that the optimal
licensing fee is monotonically increasing in the level of uncertainty because it has a similar impact on
both M and N’s incentives.
We find that early investment always leads to a single standard. Unlike in the continued
monopoly case, the investment threshold now is monotonically decreasing in both the intensity of the
network effect and the level of uncertainty.
The rest of the paper is organized as follows: In Section 2 we model the network effects
through its impact on the demand function. Section 3 solves the monopolist’s investment problem
and Section 4 focuses on the investment decisions under imperfect competition. In Section 5, we
discuss implications of our results and provide some concluding remarks.
2
The Economic Impact of Network Goods
Networks conjure an image of a myriad of connections that provide physical links that
facilitate the movement of goods, people, or information between dispersed locations. Highway and
rail systems facilitate the smooth and rapid movement of goods and people. Telephone networks
7
A similar situation results when M commits an early investment but does not offer a licensing opportunity to N.
4
enable people throughout the world to talk to each other. Most readers would recollect the days
when only a handful of scientists had access to e-mail. Anyone outside this community would not
find many friends or colleagues belonging to the network of e-mail users and would not find e-mail
to be of much value. However, as the community of email users widened to reach a larger
population, individual users found e-mail to be increasingly valuable. It is the widespread adoption
that brought about value of the network. As each new user joined the network, all existing users
benefited from the ability to connect to the new user. This surge in value is referred to as a network
effect.
Network effects can also arise without physical connections between the users. Common
standards establishing logical connections between users bring about a similar effect. For example, a
particular word processor or spreadsheet program “connects” a network of users who can collaborate
and share documents. As in the case of physically connected networks, as the community of users
adhering to the common standard grows, so does the value to each user.
In yet other instances, networks are formed around systems of complementary goods or
services. The proliferation of video game titles around a particular game console creates such a
complementary system, where a network of users become “connected” and benefit from the
proliferation of variety over time. The value of an operating system depends on the variety of
software developed for it. Such complementary systems of a hardware/software paradigm prevail
not only in technology industries, but also apply to markets such as cars/gas stations, credit
cards/merchants, and durable equipment/repair services. The common theme between all of the
above networks is that the value to each user grows with the number of users. Users of
complementary networks also reap network benefits but they arise through a very different
mechanism. As more consumers adopt a complimentary network system, it leads to increased
incentive to innovate, thus resulting in a proliferation of complementary components. Consumers
benefit from this proliferation of variety. For instance, as more users adopted VHS players, more
5
movies became available in the VHS format. Just as they derived utility from increased connectivity,
consumers also derive utility from the increased variety.
Understanding the impact of network effects is vital to producers of a network good. An
obvious economic effect of networks arises from the high fixed costs and very low (near zero)
variable costs of production. As a result, producers experience increasing returns to scale for
network goods and has led to the emergence of regulated natural monopolies like
telecommunications, electric power, railroads, and water. The tremendous consumer benefits of
networks, however, can play an even more vital role when committing investments in networks.
Increased consumer utility, however, does not necessarily translate into higher prices for
network goods or higher profits to the network investors. Network builders must cope with the
troubling feature of adoption externalities that can prevent the capture of the value added from
network effects. When a consumer joins the network, each existing consumer also stands to gain
increased utility. The producer is not always able to reflect this effect in the price to pre-existing
consumers. Therefore, it is vitally important to find mechanisms to convey the equilibrium size of
the network to its potential users.
The focus of this paper is on such strategic aspects of investments in networks. We postulate
situations where the investment acts to internalize the network effect. Specifically, we modify the
inverse demand function such that the price of the good is affected by the network effect. This
allows us to study the impact of investment timing and network effects isolated from all the other
network ramifications. Let the inverse demand function for a network good be given by
P(q, θ ) = θ + v(q ) − q
where q is the total demand for the good. We assume that consumers are homogeneous in their
valuation of the network effect while the maximum autarky value a consumer may derive from the
product is θ. Alternatively, θ can be interpreted as the maximum quantity demand in the absence of
network effects. In either case, θ captures the uncertainty surrounding the size of the future market.
6
The presence of network effects increases the value to consumers and their willingness to pay. We
represent this network effect by v(q), which will be an increasing function of q. i.e., v ' (q ) > 0 .
There is a growing literature from which we can draw inferences about the form of the v(q)
function. Perhaps the most widely known form of the network value proposition is reflected in
Metcalfe’s Law, which states that the total value of a network increases in proportion to the square of
the number of users in the network. With homogenous consumers, the Metcalfe’s Law translates into
a linear v(q) function,8
v (q ) = βq
The parameter β can be interpreted as the “intensity” of the network effect. At the one extreme for
normal goods where consumers realize their entire value from the standalone use, β =0. As β
increases more and more value comes from network effects. For instance, telecommunications
networks will be associated with high β. However, in order to maintain bounded solutions β < 1.
3
Strategic Investment in Network by a Monopoly
In our first model, we assume that a single firm, M, has a monopoly over both the network
investment opportunity and the product market. At t=0, M has the opportunity to make an
irreversible investment I which enables the production of a network good at some future date (t=1).
We can think of this investment as R&D or some other fixed “entry fee” that allows the firm to
produce at time 1, when the market opens.9
While such an investment can impact future production costs (e.g., learning), in this model
we isolate its sole impact to be the credible communication of the size of the future network to
a licensing opportunity to N.
More generally, we know that network benefits tend to level off after it reaches sufficiently large size. In fact, in
some cases very large networks may even become cumbersome to navigate and create congestion, so that user
benefits may decline beyond a certain scale. These effects can be modeled by a more general function of the form,
v (q ) = βq α .
8
9
Since we deal only with a fixed amount of investment, and not one that varies with the amount of production, I is
unlikely to be incurred in building production capacity.
7
potential consumers.10 Specifically, by committing an investment at time 0 monopolist can credibly
announce the output level to potential consumers. The consumers will take into account the resulting
equilibrium network size and the ensuing network benefits when making their purchase decision.
The firm will, thus, choose equilibrium output level such that its time-1 profits are maximized with
the recognition that consumers will accept this quantity. In the presence of network effects, the
resulting higher quantity (larger network size) will lead consumers to pay a higher price, and thereby,
internalize the network effect. The investment becomes the mechanism through which the
monopolist can internalize the network externality.
If the investment is not committed at time 0, the Monopolist can still invest I at time 1 and
produce the network good. However, consumers then must form their expectations on the size of the
network exogenously.11 The output choice is now determined as a fulfilled expectations equilibrium
(FEE) and results in a smaller network than before. In turn, the equilibrium price of the network
good will be lower than when investment is committed at time 0.
The monopolist must incur some cost in producing the network good at time 1. We assume
that the unit cost remains constant regardless of the time of the investment. The unit cost is a
combination of all production costs and may be a function of the output, which we denote by k(q).12
For a network good, k(q) is likely to be decreasing in q, leading to increasing return to scale on the
supply side. However, in order to isolate the network effects arising from the demand side, we
assume the variable cost of production to be constant. i.e., k(q)=k.
The profit function for the monopoly at time 1 is given by π = q[θ + v(q ) − q − k ] , where θ is
fully revealed at time 1. At time 0, θ is uncertain and distributed on (0,∞), with expected value
E0 (θ ) = θ 0 > 0 . By convincing users through committing an immediate, irreversible investment I,
10
Katz and Shapiro (1985) consider a similar situation where investment can be used to credibly communicate
output levels in a multi-firm setting.
11
For example, based on predictions made by government agencies or market research firms.
12
In the Kulatilaka and Perotti (1998) model they allow for the early commitment of investment to lower k(q).
8
the Monopolist can optimize the output level by treating q as endogenous. The output level is chosen
by M, such that q MI = Arg max[q(θ + v(q) − q − k)]. The resulting equilibrium production
q
level (q MI ) , price ( PMI ) , and profits (π MI ) are given in column1 of Table 1.
If the monopoly does not invest at time 0, then it no longer has an ability to credibly convince
consumers about its future output level. Instead, consumers form expectations of the network size,
qe, and commit to pay a price based on this level of production. Therefore, M must take qe as given
in making the investment decision. This fulfilled expectations equilibrium is achieved when profit
[
]
maximizing output q MNI = q e . In other words, q MNI =q e= Arg max q(θ + v(q e ) − q − k) . The
q
resulting equilibrium values are given in column 2 of Table 1. Finally for comparison purposes, we
report the equilibrium quantities, prices, and profits earned by a monopolist in the absence of
network effects in column 3 of Table 1.
Table 1
The Impact of Network Effects on Production Decisions
No Network Effect
Network Effects
Quantity
q MI
If M invests at time 0
(1)
1
=
(θ − k )
2(1− β )
1
(θ + k )
2
Price
PMI =
Profit
π MI =
1
2
(θ − k )
4(1− β )
If M doesn’t Invest at time 0
(2)
1
(θ − k )
q MNI =
2−β
PMNI =
π MNI =

1
1 
k
θ + 1 −
2−β
 2−β 
1
(2 − β )
2
(θ − k )2
(3)
qM =
1
(θ − k )
2
PM =
1
(θ − k )
2
πM =
1
2
(θ − k )
4
Note: There is no production when θ ≤ k .
We make several important observations. First, independent of when I is committed, the
presence of a network effect produces a larger network with higher prices and yield higher profits to
M (compare columns 1 and 2 with 3.). Second, investing at time 0 prior to the resolution of
9
uncertainty, results in higher quantities, prices, and profits when compared to when investment is
postponed. (Compare column 1 with 2.) In other words, q MI > q MN > q M , PMI > PMN > PM , and
π MI > π MN > π M .13 In all cases, the firm will not produce if the realization of θ is lower than the
production cost, k. As we expect, the intensity of the network effect monotonically enhances the
effects due to early investment.
Although the immediate commitment of investment leads to higher profits, the investment
requires an irreversible expenditure of I. If θ turns out to be less than k, the firm would regret
having made the investment. Therefore, the investment decision would be based on a comparison of
the ex-ante expected profits under the two investment scenarios. The expected value when M makes
the investment at t=0, VMI , is obtained by taking the expectations over the region of positive profits
and netting out the investment.
+
 1
(θ − k )2  − I
V = E0 
 4(1 − β )

I
M
Similarly, if no investment is made at time 0, the investment and production decisions at t=1 are
made if and only if π MNI > 0 . Hence, the firm value is given by
V
NI
M
 1

(θ − k )2 − I 
= E0 
2
 (2 − β )

+
The investment decision of the Monopolist is based on a comparison of the value functions
VMI and VMNI . Since these require taking expectations, they depend on the distribution of θ.
Proposition 1: For general but well-behaved distributions of θ, there is a unique threshold level of
expected demand ( θ 0* ), which must be exceeded in order for the Monopolist to commit I at time 0.
13
These results hold true for more general network value functions including the case where networks effects are
tempered by congestion.
10
Furthermore, θ 0* is monotonically decreasing in the intensity of the network effect, β. The behavior
of θ 0* to changes in σ is ambiguous.
Proof: See Appendix 1.
In order to gain more insight into the characteristics of the investment threshold we assume
θ 
 1

that θ is log-normal distributed: ln  ~ N− σ 2 , σ 2  , such that E (θ ) = θ 0 , the expected value of

 2
θ 0 
θ. 14 The value functions can then be expressed as,
VMI =
NI
M
V



1  2
k
k
k 
2
2
θ 0 exp(σ )Φ 32 σ − σ1 ln  − 2kθ 0Φ 12 σ − σ1 ln  + k Φ − 12 σ − σ1 ln  − I
4 (1− β ) 
θ0 
θ0 
θ 0 



2
 
 2


d1 
d1   k 
d1 
2
3
1
1
1
1
1


Φ
−
=
θ
exp
σ
Φ
σ
−
ln
θ
Φ
σ
−
ln
−
I
σ
−
ln
+
−
2k








(
)


0
0
2
2
2
2
σ
σ
σ
θ0 
θ 0   2 − β 
θ0 


 
(2 − β ) 
1
where d1 = k + (2 − β ) I .
Figure 1: VMI , VMNI against θ0, β=0.75 and 0.5
3.5
3
VMI (β = 0.5)
2.5
VMNI (β = 0.5)
2
1.5
1
0.5
θ0
0
-0.5
0
0.5
1
1.5
-1
2
2.5
3
θ0*
-1.5
14
This allows us to perform mean-preserving changes to the level of uncertainty by examining the sensitivity to σ.
11
Figure 1 plots VMI and VMNI functions against θ 0 . Several features about these plots are worth
noting. First, both functions are monotonically increasing in θ0. Second, since investing
immediately incurs the initial investment, VMI (0) < VMNI (0) . It is also easy to show that
lim (VMI (θ 0 ) − VMNI (0))> 0 . Hence, from the intermediate value theorem we know that there is a
θ 0 →∞
( )
( )
unique investment threshold, θ*0 at which VMI θ 0* = VMN θ 0* and the firm is indifferent between
investing at t=0 or postponing the decision to t=1.
We now focus on the impact of β on θ 0* . In cases where there is a greater network effect
(i.e., higher β), both VMI and VMNI functions have greater value. However, the impact of the network
effect is stronger on VMI than on VMNI . Consequently, the investment threshold falls with rising
network effect, β.
An important feature of our model is its ability to study the impact of uncertainty on the
investment threshold. Figure 2 plots the investment threshold against σ for three different values of
β. We note that for very high levels of β (β = .75), the threshold decreases monotonically with σ.
However, as the network effect intensity diminishes (β =.25), the threshold first rises with increasing
σ, but as s increases without bounds the threshold falls.
An interesting interpretation of the results comes from drawing an options analogy. Investing
immediately can be thought of as the acquisition of a growth option the value of which is given by
VMI . In contrast, postponing the investment decision until time 1 retains the value of the wait-toinvest option, represented by VMNI . When the network effect is strong (high β), increasing
uncertainty raises the value of the growth option from network effect more than the value of the waitto-invest option, lowering the investment threshold. When the network effect is weak, increasing
uncertainty makes the option of waiting more valuable at first, but eventually the growth option
12
dominates. Consequently, the investment threshold rises with increasing σ but eventually falls when
σ increases without bound.15
Figure 2: Investment Threshold θ 0* against σ for β=0.75, 0.5 and 0.25
4
θ0*
β = 0.25
3.5
3
2.5
β = 0.5
2
1.5
1
β = 0.75
0.5
σ
0
0
4
0.2
0.4
0.6
0.8
1
1.2
Strategic Investment under Potential Imperfect Competition
We now turn to the investment problem in the face of potential competition in the future.
Unlike in the previous model, now the source of advantage from early investment arises from
establishing a network standard. Licensing this standard, in effect, allows the firms to coordinate
around a single network standard.16
We retain the assumption that the early investment opportunity lies with a single firm. If M
invests at t=0, it establishes a standard that may offer to license the standard to another firm, N. Firm
N then has the choice of accepting M’s standard at a per-unit fee l or retains the option to develop a
new standard at time 1. Firm N makes the choice knowing that it would have an investment
opportunity at time 1. If N chooses not to adopt M’s standard, then it can invest in a new standard
15
The logic is similar to that when the early investment advantage comes from the cost side (see Kulatilaka and
Perotti 1998).
16
This model can be interpreted in settings other than technology standards. For instance, in the music industry
bands may choose to give free access to their music (initial investment) in an attempt to establish a community of
fans. The band can capture the value of the resulting network by selling complementary goods (T-shirts) to the fans.
13
after all uncertainty regarding θ is fully resolved. Investment by N at time 1 will also require
spending I to produce a network good that is a perfect substitute (in its autarky value) to M’s product.
The presence of network effects plays a vital role in M’s choice of the licensing fee, N’s
adoption decision, and hence, M’s investment timing decision. When N licenses M’s standard, M
and N together will create a larger market, and will be in position to charge a higher price for the
product because of higher network value. Thus, by investing early and setting the appropriate
licensing fee, M can establish its standard as the industry standard and collect a licensing fee from
the other firm in addition to producing a good that has a greater network value. Thus, early
investment is a mechanism that enables the monopolist to establish an industry-wide standard,
internalizing the network effects through pricing and licensing contract. If M does not invest, the
two firms will make investment decisions simultaneously after the uncertainty is resolved, and it is
impossible for them to coordinate, so if they both invest, they will develop two different standards
and produce incompatible network goods. Each firm’s customers form their own smaller network
and realize a lower network value than those in a larger common network.17
The sequence of events and decisions are depicted in Figure 3. At time 0, M decides whether
or not to invest immediately. If M invests and develops a standard, then M decides whether to offer
N to adopt this standard and sets a licensing fee l. New entrant N chooses between adopting M’s
standard and developing a new standard. Although there is a time sequence to the decisions up to
now, new information becomes available only at time 1 when θ is fully revealed.
Consumers have exogenous expectations of the quantities and network size(s). Unlike in the
earlier model, the time-0 investment decision does not affect consumer expectations. When the
market opens, M and N choose the optimal quantity of the network good. If M does not invest, then
M and N have identical opportunities at time 1 and simultaneously decide whether to invest in new
17
Customers of both firms would be willing to pay a higher price if the products are compatible as could interact
with users of the other firm’s product.
14
standards. When θ is large enough to trigger investment at time 1, both firms will invest and produce
incompatible products whose users belong to different networks.
Figure 3: Sequence of Events
t=0
t=1
Accept
M’s standard:
M, N produce
(q1+q2)
q1
q2
Offer l
Not
Accept
Invest
q1
q2
Not
Invest
q1
Invest
q1
q2
M, N’s standards
M, N produce
q1, q2 respectively
Not
Invest
q1
M’s Standard
M produce q1
No Offer
Not Invest
q1
q2
M, N’s standards
M, N produce
q1, q2 respectively
Not
Invest
q1
M’s standard
M produce q1
Invest
Not
Invest
M’s standard
M produce q1
Invest
Invest
M’s decision node
M, N’s standards
M, N produce
q1, q2 respectively
Invest
Not
Invest
q2
N’s standard
N produce q2
0 standard
0 production
N’s decision node
We solve M’s investment and licensing fee choice through backward induction. We first
solve for the optimal licensing contract between M and N assuming that M has invested at t=0, then
solve for M’s investment threshold at t=0. First, suppose M has built a standard at t=0 and then
15
offers a contract to N with a unit licensing fee of l. If N accepts the contract and adopts M’s
standard, we can solve M’s and N’s profit maximization problems to get the equilibrium quantities
and profits: q1* , q 2* , π 1* , π 2* . These are expressed as functions of θ and l, where subscripts 1 and 2
represent firms M and N respectively. Table 2a summarizes the equilibrium quantities and profits.
We allow for a general specification that also includes licensing fees that are negative (subsidies) or
zero (open standard). We later prove that the optimal licensing fee is positive and therefore, our
subsequent discussion is restricted to the case of positive licensing fees.
Table 2a: Equilibrium Quantities and Profits
When M invests at time 0 and N adopts M’s standard
L
M
θ
*
Quantity q1
l>0
l=0
l<0
N
Profit
π 1*
θ > k + (2 − β )l
θ − k + (1 − β )l
3 − 2β
k < θ ≤ k + (2 − β )l
θ − k
2 − β
θ −k 
 2 − β 


θ ≤ k
0
0
θ > k
θ −k
3 − 2β
θ ≤ k
0
θ > k − (1 − β )l
θ − k + (1 − β )l
3 − 2β
max (0 , k + l ) < θ ≤ k − (1 − β )l
0
θ ≤ max (0 , k + l )
0
(θ
*
Quantity q 2
(
− k ) + (5 − 4 β )(θ − k )l − 5 − 5 β + β
2
(3 − 2 β )2
 θ − k

 3 − 2β
2
)l
2



θ − k − (2 − β )l
3 − 2β
 θ − k − (2 − β )l 


3 − 2β


0
0
0
0
θ −k
3 − 2β
2
 θ −k

 3 − 2β



2
2
0
0
θ − k − (2 − β )l
3 − 2β
 θ − k − (2 − β )l 


3 − 2β


θ − k − l 
 2 − β  l


θ −k −l
2−β
θ − k − l 


 2−β 
0
0
0
(
− k ) + (5 − 4 β )(θ − k )l − 5 − 5 β + β
2
(3 − 2 β )
2
2
)l
2
Note that when the realized demand θ is below the operating cost k, neither firm would
produce. In this case, M would regret its investment, which in effect, kills the option to postpone.
However, this potential loss is offset by several benefits. For higher levels of θ, M gains a strategic
advantage from its early investment. When demand falls in the range k < θ < k + (2 − β )l , N is
dissuaded from entering the market and M retains its monopoly. When θ is even higher
16
π 2*
2
0
(θ
Profit
2
2
( θ > k + (2 − β )l ), the licensing fees act so that M produces a higher quantity than N. In other words,
the choice of the licensing fee can influence the resulting market structures.
If N does not accept the fee, we solve for q1** , q 2** , π 1** , π 2** as functions of θ. The resulting
equilibrium quantities, prices, and profits are given in Table 2b.
Table 2b: Equilibrium Quantities and Profits when M invests at time 0 but
N develops new standard at time 1
M
θ
N
**
Quantity q1
Profit
π 1**
θ >k
θ −k
3− β
θ − k 
 3 − β 


θ ≤k
0
0
2
**
Quantity q 2
Profit
π 2**
θ −k
3− β
θ − k 
 3 − β 


0
0
2
As before neither invests or produces for θ ≤ k . For θ > k the two firms engage in symmetric
Cournot competition.
Optimal Licensing Fee
If N accepts the contract and adopts M’s standard, the expected profits for M and N are
E (π 1* (l )) and E (π *2 (l )) respectively (M’s investment is sunk cost). If N rejects M’s standard, M
+
+
( )
+
(
)
+
and N expect values E π 1** and E π **
2 − I . Thus, N chooses to accept the contract and license
(
)
+
(
)
+
M’s standard if and only if E π *2 (l ) ≥ E π **
2 − I . We assume that N accepts the standard when
indifferent.18
After committing the investment, M must decide whether to offer N a licensing contract, and
if so, the optimal licensing fee, l * . If M does not allow N to license the standard, M expects to earn
E (π 1** ) , which is the same payoff it gets if the offer l is made but declined by N. Thus, the best M
+
can do with a licensing contract is one that maximizes expected profit, and satisfies N’s acceptance
18
Since we take expectations over the positive range, the strategy for N to stay out of the market with a reservation
utility of zero is dominated and will not be chosen.
17
condition. Therefore, when offered the optimal licensing fee l* determined by the following
maximization problem, N accepts the licensing contract and adopts M’s standard.
Max
l ∈(−∞,+∞)
E (π 1* (l ))
+
(M’s profit maximization condition)
E (π *2 (l )) ≥ E(π **
2 − I)
+
s.t.
+
(N’s acceptance constraint)
Before solving for the optimal licensing fee l* we will closely examine each firm’s decision
criterion separately. We define l1* as the solution to the unconstrained optimization problem
(
)
*
Max E(π 1* ) and l 2* as the solution to E π *2 (l2* ) = E (π **
2 − I) . The optimal licensing fee l is
+
l
+
+
determined as l * = Min(l1* , l 2* ) .19 The specific values of l1* and l2* depend on the parameter β and the
distribution of θ. As before we assume that θ is log normal distributed in solving for l1* and l2* .
Given θ 0 is sufficiently large to justify M’s investment at t=0, Table 3a shows how l1*
changes with β and σ. Reading down the columns, we see that l1* decreases in β. In the presence of
a stronger network effect, a monopoly will charge a lower licensing fee to entice the entrant to adopt
its standard and contribute to the network size. The increased network size allows the monopoly to
profit more from its own production and gain more licensing fees. Reading across the rows, we see
that l1* monotonically increases in σ. As the future demand uncertainty increases, it will make M
less likely to retain the monopoly when the market opens and, thus, will charge a higher licensing
fee.
The simulation results also examine the behavior of l 2* under changes in β and σ. (See Table
3b.) The entrant N is willing to accept a higher licensing fee when the network effect is stronger,
because it becomes more profitable to produce a compatible good for a large network than to develop
a new standard and operate a smaller network. The impact of uncertainty on l 2* is not
19
See proof of Proposition 2.
18
Table 3: Optimal Licensing Fee
(a) l1*
β
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
1.363
1.158
1.109
1.096
1.098
1.103
1.106
1.098
1.069
0.2
Inf
1.793
1.433
1.285
1.210
1.166
1.136
1.109
1.071
0.3
0.4
0.5
σ
0.6
4.786
2.273
1.757
1.504
1.355
1.255
1.177
1.103
inf
4.370
2.770
2.092
1.726
1.495
1.328
1.189
11.369
5.013
3.237
2.396
1.911
1.590
1.346
Inf
10.398
5.559
3.613
2.616
2.014
1.596
0.7
0.8
0.9
1
27.034
10.567
5.910
3.825
2.694
1.981
64.850
22.232
10.469
5.967
3.801
2.569
inf
52.421
20.056
9.921
5.650
3.477
136.295
41.517
17.561
8.840
4.909
(b) l2*
σ
β
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
1.645
1.684
1.733
1.788
1.853
1.931
2.025
2.142
2.291
0.2
1.797
1.853
1.918
1.993
2.083
2.192
2.324
2.488
2.697
0.3
1.929
2.001
2.085
2.185
2.305
2.451
2.630
2.856
3.145
0.4
2.030
2.123
2.232
2.363
2.522
2.717
2.960
3.269
3.669
0.5
2.096
2.215
2.358
2.530
2.742
3.004
3.333
3.757
4.312
0.6
2.123
2.278
2.465
2.694
2.976
3.329
3.778
4.359
5.131
0.7
2.111
2.314
2.562
2.867
3.245
3.722
4.335
5.137
6.217
0.8
2.064
2.333
2.661
3.068
3.577
4.225
5.066
6.181
7.706
0.9
1.994
2.348
2.785
3.329
4.016
4.901
6.064
7.631
9.810
1
1.918
2.383
2.962
3.692
4.627
5.847
7.475
9.707
12.871
0.6
2.123
2.278
2.465
2.694
2.976
3.329
2.616
2.014
1.596
0.7
2.111
2.314
2.562
2.867
3.245
3.722
3.825
2.694
1.981
0.8
2.064
2.333
2.661
3.068
3.577
4.225
5.066
3.801
2.569
0.9
1.994
2.348
2.785
3.329
4.016
4.901
6.064
5.650
3.477
1
1.918
2.383
2.962
3.692
4.627
5.847
7.475
8.840
4.909
(c)  l *
σ
β
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
1.363
1.158
1.109
1.096
1.098
1.103
1.106
1.098
1.069
0.2
1.797
1.793
1.433
1.285
1.210
1.166
1.136
1.109
1.071
0.3
1.929
2.001
2.085
1.757
1.504
1.355
1.255
1.177
1.103
0.4
2.030
2.123
2.232
2.363
2.092
1.726
1.495
1.328
1.189
0.5
2.096
2.215
2.358
2.530
2.742
2.396
1.911
1.590
1.346
19
straightforward; it depends on the intensity of network effect. When β is low, l 2* first increases in σ
but then decreases; when β is high however, l 2* increases in σ. The entrant is trading-off two
alternatives: to invest in its own standard or to adopt M’s standard. When the network effect is weak,
as uncertainty increases, the benefits of developing its own standard first decrease and then increase
(because of the same reasons that determine the investment threshold in the monopoly case);
therefore, the licensing fee that makes it indifferent is just the opposite. With a strong network effect
however, even though the expected profits from developing its own standard increase with
uncertainty, increasing uncertainty has an even stronger impact on the expected profits from adopting
M’s standard, thus N is willing to accept even higher licensing fees.
The optimal licensing fee l * is determined by the minimum of l1* and l 2* . When the network
effect is small, l * is determined by N’s indifference condition. i.e., l * = l2* . But as the intensity of
the network effect increases, M’s profit maximization is achieved with a lower licensing fee and
l * = l1* . Table 3c presents the comparative statics of l * with respect to β and σ.
When the network effect is significant (high β) and uncertainty low (low σ), l * is determined
by l1* and N’s acceptance constraint is not binding, in other words, M charges a low fee even though
N would accept a higher one. As the network effect becomes less significant or as uncertainty
increases (moves toward lower β or higher σ), M wants to charge a higher licensing fee, and N’s
acceptance constraint starts binding. For example, when β=0.1, the optimal licensing fee is
determined by N for all σ’s except for σ=0.1. When β=0.5, the optimal licensing fee is determined
by M for σ≤0.4, and by N for higher σ; l * keeps increasing, but at a smaller slope when determined
by N. When β=0.9, the optimal licensing fee is determined solely by M for the range of σ of interest,
and is strictly increasing. When the uncertainty level is fixed and the intensity of network effect
changes, the optimal licensing fee increases in β when determined by N and decreases when by M.
20
For example, when σ=0.1, the optimal licensing fee is determined solely by M, thus decreases in β.
When σ=0.5, l * is decided by N for β≤0.5, where it keeps increasing and by M for higher β where it
declines accordingly. The pattern is the same for β=0.9.
This is particularly significant for low uncertainty in demand (low σ) and a strong network
effect (high β). When σ is low, it is highly unlikely for a large θ to occur, and a high licensing fee
means that the market condition for N to produce is highly unlikely to happen, and thus the total
expected licensing income with higher per-unit fee can be too small to compensate for the loss of
licensing income when N is not producing.
When β is high, due to the strong network effect, the advantage of being a monopoly
becomes less significant, because a firm can charge a higher price for a product with a larger
network, even if the larger network is a result of other firms producing compatible products. On the
other hand, the disadvantage of being a monopoly, i.e. the loss of licensing income from competitors
is more significant, because under higher β, N would produce more if it were. Therefore, M will
choose a lower licensing fee to take advantage of the network effect and avoid being a monopoly
when it does not payoff.
We now formalize these properties of the optimal licensing fee in Proposition 2.
Proposition 2:
θ 
 1

When the demand uncertainty is log-normal distributed, i.e., ln  ~ N− σ 2 , σ 2  , the
 2

θ 0 
optimal licensing fee l * is positive and
a)
If M’s expected profits for any arbitrary licensing fee is less than the expected profits
(
)
+
to a monopolist in the product market, i.e., E π 1* (l ) <
21
1
2
(2 − β )
∫ (θ − k ) f (θ )dθ ,
∞
k
2
then the optimal licensing fee is l * = l 2* , where l 2* is determined implicitly by
(
)
E π *2 (l2* ) = E (π **
2 − I) .
b)
+
+
( )
If there exists a finite lˆ > 0 such that E π 1*
(
+
l = lˆ
>
1
2
(2 − β )
∫ (θ − k ) f (θ )dθ , the
∞
2
k
)
optimal licensing fee l * = min l1* , l 2* , where l1* is determined implicitly by M’s
profit maximization condition,
l1* −
(5 − 4β )θ 0 Φ(σ2 + σ1 ln θ 0 − σ1 ln(k + (2 − β )l1* ))
=0.
2(β 2 − 5β + 5)Φ (− σ2 + σ1 ln θ 0 − σ1 ln (k + (2 − β )l1* ))
Proof: See Appendix 2.
Proposition 2 shows that the optimal licensing fee is only sometimes decided entirely by N’s
acceptance of the licensing contract. In particular, it may not be in the best interest of M to charge an
infinitely high licensing fee regardless of N’s reaction. Under certain conditions, M will charge a
finite licensing fee, not to deter N from developing a new standard that creates two smaller networks
and lowers M’s profits, but because M can earn higher profits in a single network.
The reason for this counter-intuitive result lies in the tradeoff between the two effects of the
licensing fee on M’s profit. A higher licensing fee increases the range of θ over which M retains a
monopoly (k, k + (2 − β )l ) and N’s entry is deterred. However, being a monopolist has a cost in
that M cannot collect licensing fee from N over a larger range of θ. The choice of the licensing fee
affects not only the range of monopoly but also the level of revenues to M. The licensing revenues to
M when N enters the market depends on the per-unit licensing fee paid by N and the level of N’s
production. With a higher l* M gets higher per-unit royalty from N but it also lowers the quantity
that N produces. Therefore, the net impact of increasing l* on the licensing revenue is not
necessarily positive. The combined effects of an increase in the licensing fee on M’s expected
profits can be ambiguous.
22
( ( )) .
We have shown that M’s expected profit from the optimal licensing contract is E π 1* l *
+
( ( )) ≥ E(π ) . The
M will choose to license his standard to N rather than not if and only if E π 1* l *
+
** +
1
( ( )) ≥ E(π ) is always satisfied, and thus as long as M
proof of Proposition 3 shows that E π 1* l *
+
** +
1
has invested at time 0, it will always license the standard to N, resulting in one, compatible network
in the market.
Proposition 3: If M invests at t=0, then M chooses to license its standard to N with a licensing fee of
l*, as specified in Proposition 2, and N accepts and adopts M’s standard.
Proof: See Appendix 3.
Proposition 3 illustrates the decision path along the game tree in Figure 3. We have shown
that when M commits the investment at time 0, it will always choose to offer to license its standard to
N. Previously we have shown that the licensing fee can be chosen to ensure that N adopts M’s
standard. Therefore, an optimally managed investment will result in a single network. In other
words, failure to set the correct licensing fee or the decision to not license the standard will result in
sub optimal value. We now turn to M’s investment decision at time 0, given that the subsequent
management of the investment will be optimal.
Investment Threshold
As with the earlier case of the pure monopolist, we can now examine M’s investment
decision at t=0 and the investment threshold θ 0* . We know from Proposition 3 that if M invests, M
will offer the optimal licensing fee l* to N and subsequently N will accept and adopt M’s standard.
When M makes its investment decision at t=0, its expected net payoff from investing is
E (π 1* (l * )) − I , which is a function of θ 0 = E0 (θ ) .
+
If M does not invest at time 0, then at time 1, M, together with N, will decide whether to
invest, and if so, what the optimal quantity he should produce. If M invests at t=1, the optimal
23
quantities and associated profits are q1*** , q 2*** , π 1*** , π 2*** as functions of θ (see Table 2c). M’s
( ) − I at t=1 and E(π
expected profit is E π 1***
+
***
1
− I) at t=0.
+
Table 2(c) Equilibrium Quantities and Profits
When M does not invest at time 0, M and N develop their own standards at time 1
M
θ
***
Quantity q1
θ >k
θ −k
3− β
θ ≤k
0
N
Profit
π 1***
θ − k

3− β



***
Quantity q 2
θ −k
3− β
2
Profit
θ − k

3− β
0
0



2
0
( ( )) − I = E(π
Proposition 4: M’s investment threshold θ 0* is the solution to: E π 1* l *
π 2***
+
***
1
− I) , where
+
under the lognormal distribution for θ
(
)
E π 1* (l * ) =
+


k
k
k 
1  2 σ 2 3
Φ 2 σ − σ1 ln  − 2kθ 0Φ 32 σ − σ1 ln  + k 2Φ 32 σ − σ1 ln 
2 θ 0 e
θ0 
θ0 
θ 0 



(2 − β ) 
 1
 2 σ2 3
1
−
−
θ e Φ ( 2 σ − σ1 ln d 2 )
2
2 0
(
)
(
)
2
−
β
3
−
2
β


 (5 − 4 β )l * − 2k
2k 
+
+
θ Φ ( 12 σ − σ1 ln d 2 )
2
2 0
(2 − β ) 
 (3 − 2 β )
 k 2 − 5 − 4 β kl * − 5 − 5β + β 2 l * 2

(
)
(
)
(
)
k2 

+
−
Φ(− 12 σ − σ1 lnd 2 )
2
2


3
−
2
β
2
−
β
(
)
(
)

[
]
where d 2 = k + (2 − β )l * θ 0 , and
[θ e
E (π − I) =
***
1
+
[
2 σ
0
2
]+ 
Φ(32 σ − σ1 lnd 3 ) − 2kθ 0Φ(12 σ − σ1 lnd 3 )
(3 − β )
2
]
where d 3 = k + (3 − β ) I θ 0 .
24
2

k 
1
1
 − IΦ(− 2 σ − σ lnd 3 )
 3 − β 

Proof: See Appendix 4.
Figure 4 illustrates the investment threshold as a function of σ for three different levels of β.
Notice that the impact of the network effect on the investment threshold is the same as in the
previous model. Ceteris paribus, the investment threshold declines with rising network effect. This
implies that when a market exemplifies stronger network effects, a firm with the monopoly right to
invest in a standard will take the opportunity and commit the investment at lower levels of expected
future demand. Owning a standard will allow the firm to persuade future competitors to adopt its
standard and thereby collect royalty fees. The production from competitors helps enlarge the
network, which also translates into higher prices and profits for the investing firm.
Figure 4: M’s Investment Threshold with Imperfect Competition
θ0*
2
1.8
β = 0.1
1.6
1.4
β = 0.5
1.2
1
β = 0.9
0.8
0.6
0.4
0.2
σ
0
0
0.2
0.4
0.6
0.8
1
1.2
The impact of uncertainty on the investment threshold is now slightly different from that
when the firm remains the monopolist. Without potential competition, increasing uncertainty may
raise the investment threshold when the network effect is mild; this is because in these situations the
increased uncertainty adds more value to the waiting to invest option than to the “growth option”
25
resulting from the network effect. When the monopolist anticipates imperfect competition,
increasing uncertainty always lowers the investment threshold. This result arises from the ability to
use the licensing fee to induce future competitors to adopt the standard, thus, sharing the market and
the associated risks. Thus increasing uncertainty leads to lower investment threshold.
5
Concluding Remarks
In this paper, we study the strategic impact of investing in a network under different market
structures. We find that even a firm that is assured of being a continued monopoly has an incentive
to commit to network investments before uncertainty about the future demand is resolved. The key
feature of a network that drives our model is that each consumer experiences value from the presence
of other consumers of the network good (network effect), in addition to the autarky value of
consuming the standalone good. The benefit to early commitment comes from the credible
communication to the users about the future size of the network and thereby, internalizing the
adoption externality. We obtain the threshold level of expected demand above which the monopolist
will commit the investment by trading off this strategic benefit of investment against the value of
waiting to invest. Not surprisingly, the threshold level falls monotonically with increasing intensity
of network effect. This result stresses the importance of having an accurate estimate of the network
effect. Overestimating the network effect may make firms commit investment prematurely. This
effect is most pronounced in environments with high uncertainty.20
When we allow for potential competitors in the network, the strategic value of early
investment arises from the establishment of a network standard that can be licensed out. The choice
of the licensing fee plays a vital role in the adoption decision by the competitors. We show that there
is a unique level for the optimal fee at which the profits of the firm committing the investment to
establish a network standard are maximized and the competitors choose to adopt this single standard.
20
The rush to invest in dotcoms during the late 1990s and the investments in 3G spectrum by European mobile
operators may be explained by a very high perceived network effects. The underestimation of the investment
threshold is further exacerbated in highly uncertain environments.
26
Setting too high a licensing fee can result in incompatible networks and will be sub-optimal for the
network builders. The optimal licensing fees are also affected by the intensity of the network effect
and the level of uncertainty regarding future demand.
When the uncertainty is very low, the optimal licensing fee becomes very small. In the limit,
if there is no uncertainty, then open standards can lead to the largest networks and highest profits to
the network-building firms. The impact of the intensity of the network effect on the optimal
licensing fee leads to more interesting implications. For very low uncertainty, the licensing fee is
determined by the investing firm’s profit maximization and monotonically decreases with increasing
intensity of network effects. For higher levels of uncertainty, the licensing fee is determined by the
competitor’s adoption decision for low levels of network intensity. In such cases, as the network
increases the competitor is willing to accept a higher licensing fee. However, for further increases in
the network effect it becomes in the investing monopolist’s interest to limit the licensing fee. When
this condition is binding the optimal licensing fee falls with network intensity.
These results highlight the critical importance of setting the license fee in environments with
high uncertainty and high network effects. An investor may be tempted to charge a higher licensing
fee simply because competitors are showing willingness to adopt the standard. However, would be
in their best interest to charge a lower licensing fee and grow the network to realize larger network
profits. This is consistent with the view that opening the standards to competitors can have a
bandwagon effect.21
Once the optimal licensing fees are determined, we also solve for the expected demand
threshold at which the early investment should be committed. As with the earlier case of the
monopolist, the investment threshold is decreasing in network intensity, β. Now the investment
threshold also decreases monotonically with increasing uncertainty (σ). Unlike in the case of a
guaranteed monopolist, the investment threshold is not as sensitive to the intensity of the network
21
For instance Shapiro and Varian (1999) argue that “Openness is a more cautious strategy than control. The
underlying idea is to forsake control over the technology to get the bandwagon rolling.” (p199)
27
effect. This is because the optimal licensing fee takes into account the impact of the network
intensity.
References
Dixit, Avinash. 1980. The role of investment in entry deterrence. Economic Journal. Vol. 90, 95-106
Dixit, Avinash and Robert S. Pindyck. 1994. Investment Under Uncertainty, Princeton University
Press, Princeton, New Jersey.
Economides, Nicholas. 1996. The economics of networks. International Journal of Industrial
Organization, Vol. 14, No. 6, 673-699.
Gilder, George F. 2000. Telecosm: how infinite bandwidth will revolutionize our world, Free Press,
First edition, New York, New York.
Grenadier, Steven R. 1996. The Strategic Exercise of Options: Development Cascades and
Overbuilding in Real Estate Markets, The Journal of Finance, Vol. 51, No. 5 (Dec.) 1653-1679
Katz, Michael L., Carl Shapiro. 1994. Systems competition and network effects, The Journal of
Economic Perspectives, Vol. 8, No. 2, 93-115.
Katz, Michael L., Carl Shapiro. 1985. “Network externalities, competition, and compatibility”,
American Economic Review Vol. 75, No. 3 (Jun) 424-440
Kulatilaka, Nalin, Enrico Perotti. 1998. Strategic Growth Options,.Management Science, Vol. 44,
No. 8 (August), 1021-1031.
Kulatilaka, Nalin, Enrico Perotti. 2000. Time-to-Market Advantage as a Stackelberg Growth Option,
in E. Schwartz and L. Trigeorgis (eds) Innovation and Strategy: New Developments and Applications
in Real Options, Oxford University Press
Leibowitz, Stan J., Stephen E. Margolis. 1994. Network externality: an uncommon tragedy, Journal
of Economic Perspectives, Vol. 8, No. 2, 133-150.
McDonald, Robert L. and Daniel R. Siegel. 1986. The Value of Waiting to Invest, Quarterly
Journal of Economics, Vol. 101, No. 4, 707-728
Shapiro, Carl and Hal Varian. 1999. Information Rules: A Strategic Guide to the Network
Economy, Harvard Business School Press, Boston, Massachusetts
Spence, A. Michael. 1984. Cost Reduction, Competition, and Industry Performance,
Econometrica, Vol. 52, No. 1 (Jan.), 101-122
Standage, Tom. 1998. The Victorian Internet: The remarkable story of the telegraph and the
nineteenth century's on-line pioneers, Walker & Co, New York, New York.
28
Appendices
Appendix 1: The proof of Proposition 1 (omitted)
Appendix 2: The proof of Proposition 2.
We find the optimal licensing fee l* by solving:
Max E (π 1* (l ))
+
l∈( −∞ , +∞ )
(A1)
E (π 2* (l )) ≥ E (π 2** − I )
+
s.t.
+
(N’s acceptance constraint)
We know that π 1* and π 2* are defined differently for positive and negative l. So we first
prove that the solution to (A1) is positive.
Lemma 1:
(a) The licensing fee at which N’s acceptance constraint is binding, denoted by l 2* , is
positive;
(b) l 2* is determined implicitly by:
E (π 2* (l 2* )) = E (π 2** − I ) , or
+
+
{
}
2
1
2 σ2
Φ(32 σ − σ1 lnd 2 ) − 2[k + (2 − β )l 2* ]Φ(12 σ − σ1 lnd 2 ) + [k + (2 − β )l2* ] Φ(− 12 σ − σ1 lnd 2 )
2 θ0e
(3 − 2β )
=
1
(3 − β )
2
{θ e
2 σ
0
2
1
1
2
 k  2 
1
1

+ 
 − IΦ(− 2 σ − σ lnd 3 )
 3 − β 

}
Φ( σ − σ lnd 3 ) − 2kθ 0Φ( σ − σ lnd 3 )
3
2
1
where
[
]
θ 0 = E0 (θ ) , d 2 = [k + (2 − β )l 2* ] θ 0 , d 3 = k + (3 − β ) I θ 0 .
(c) For any l < l 2* , N’s acceptance constraint is satisfied with “>” holding.
Proof: From Table 2, we have:
29
2
 k +(β −1)l  θ − k − l  2
∞
 θ − k − (2 − β )l 
 f (θ )dθ ,l < 0

 f (θ )dθ + ∫

∫
k + ( β −1)l 
3 − 2β

 max(0,k +l )  2 − β 

=
2
 θ − k − (2 − β )l 
 ∞
 f (θ )dθ l ≥ 0
∫k +(2− β )l 
3 − 2β



( )
E π 2*
+
( )
It can be easily proved that E π 2*
+
is a continuous (but not differentiable at 0), globally
decreasing function of l.
We have:
( )
Eπ
(
Eπ
* +
2
**
2
l =0
−I
=∫
∞
k
2
 θ −k 

 f (θ )dθ
 3 − 2β 
) =∫
+
∞
k + ( 3− β )
( )
Thus, E π 2*
 θ − k  2

 − I  f (θ )dθ

I
 3 − β 

(
+
)
+
> E π 2** − I .
l =0
( )
We also have E π 2*
+
l →∞
= 0.
( )
Therefore, there is one and only one positive l such that E π 2*
(
+
)
+
= E π 2** − I . We denote
this licensing fee by l 2* . Solving the integrals will lead to the equation in (b).
( )
Because E π
( )
∂E π 2*
∂l
* +
2
( )
∂E π 2*
is a decreasing function, in particular,
∂l
+
( )
< 0 , for any l < l , E π 2*
+
(
> E π 2** − I
)
+
+
< 0 and
l >0
holds.
l <0
Q.E.D.
( )
Lemma 2: E π 1*
+
> E (π 1* )
+
l >0
> E (π 1* )
+
l =0
Proof: From Table 2, we have:
30
l <0
( )
E π 1*
+
=
l >0
1
(2 − β )2
∫
k + ( 2 − β )l
k
(θ − k )2 f (θ )dθ


1
1
(
(
θ − k + (1 − β )l )2 +
θ − k − (2 − β )l )l  f (θ )dθ

2
k + ( 2 − β )l
(3 − 2β )

 (3 − 2 β )
+∫
E (π 1* )
+
l −0
=
∞
1
∞
(θ − k ) f (θ )dθ
(3 − 2 β ) ∫
2
2
k
And
( )
E π 1*
l <0
=
k + ( β −1)l
1
(θ − k − l )lf (θ )dθ
(2 − β ) ∫max(0,k +l )


1
1
(
(
θ − k + (1 − β )l )2 +
θ − k − (2 − β )l )l  f (θ )dθ

2
k + ( β −1)l
(3 − 2β )
 (3 − 2 β )

+∫
∞
( )
It can be easily proved that E π 1*
+
> E (π 1* )
+
l >0
> E (π 1* )
+
l =0
l <0
.
Q.E.D.
Based on Lemma 1 and 2, we can focus on finding the optimal licensing fee l* in the positive
range. Solving (A1) is equivalent to solving
( )
Max
E π 1*
s.t.
E π 2*
l >0
+
(A2)
( )
+
(
≥ E π 2** − I
)
+
(N’s acceptance constraint)
Lemma 3:
a)
The unconstrained maximization problem Max
l >0
( )
E π 1*
+
has an interior solution if and only
if there exists a finite lˆ > 0 such that
( )
E π 1*
+
l =lˆ
>
1
(2 − β )2
∞
∫ (θ − k ) f (θ )dθ
2
k
(b) When the above condition is satisfied, the interior solution denoted by l1* , is determined
by the following implicit equation:
31
*
1
l
(5 − 4β )θ 0 Φ(σ2 + σ1 ln θ1 − σ1 ln (k + (2 − β )l1* ))
−
=0
2(β 2 − 5β + 5)Φ (− σ2 + σ1 ln θ1 − σ1 ln (k + (2 − β )l1* ))
(A3)
where Φ is the c.d.f. of standard normal distribution.
Proof:
( )
(a) Since E π 1*
+
l →∞
=
1
(2 − β )2
∞
∫ (θ − k ) f (θ )dθ , an interior solution must satisfy the
2
k
above boundary condition.
(b) When the condition in (a) is satisfied, an interior solution exists. It must satisfy
( )
∂E π 1*
∂l
+
= 0 , and
l =l1*
( )
∂ 2 E π 1*
∂l 2
+
( )
< 0 . Taking the derivative of E π 1*
+
l =l1*
l >0
with respect to l, and
making it equal to zero, we get (A3).
Q.E.D.
Proof of Proposition 2:
Lemmas 1 and 2 show that l* is positive.
( )
+
E π 1*
Lemma 3 shows that when the unconstrained maximization problem Max
l >0
has a
finite solution (denoted by l1* ), it is determined implicitly by
l1* −
(5 − 4β )θ 0 Φ(σ2 + σ1 ln θ 0 − σ1 ln(k + (2 − β )l1* ))
=0.
2(β 2 − 5β + 5)Φ (− σ2 + σ1 ln θ 0 − σ1 ln (k + (2 − β )l1* ))
The optimal licensing fee l* must satisfy N’s acceptance constraint. When l1* exists, if
l1* ≤ l 2* , by Lemma 1, l1* satisfies the constraint, and therefore l * = l1* . If l1* > l 2* , l1* does not satisfy
(
)
( )
the constraint; for all the l’s that satisfy the acceptance constraint i.e. l ∈ − ∞, l 2* , E1 π 1*
( )
maximized at the l 2* , because it can be proved that E π 1*
(
)
+
(
is
is increasing in l for l ∈ 0, l1* . To sum
up, l * = min l1* , l 2* , where l1* is determined by (A3) and l 2* is as specified in Lemma 1.
32
)
+
When the condition specified in Lemma 3(a) is violated, it means that for any l ∈ (− ∞,+∞ ) ,
(
)
E π 1* (l ) <
+
1
(2 − β )2
∞
∫ (θ − k ) f (θ )dθ , i.e. M would set the licensing fee infinitely high if he could
2
k
force N to accept it. In this case, the solution to (A1) will be decided solely by N’s acceptance
constraint binding. Therefore, l * = l 2* .
Q.E.D.
Appendix 3: The proof of Proposition 3.
The acceptance constraint in the maximization problem (1) guarantees that N accepts the
( ( )) ≥ E (π ) .
optimal licensing fee l*. Thus to prove Proposition 3, we only need E1 π 1* l *
( )
By Lemma 2, E π 1*
+
l >0
( )
+
> E π 1*
l =0
( )
. Since l*>0, E π 1*
+
l =l
*
+
1
( )
> E π 1*
+
l =0
** +
1
.
We have
( )
E π 1*
+
l =0
( )
E π 1**
+
=
=
1
(3 − 2β )2
1
(3 − β )2
( )
Obviously, E π 1*
∞
∞
∫ (θ − k ) f (θ )dθ ,
2
k
∫ (θ − k ) f (θ )dθ
2
k
+
( )
( ( ))
+
l =0
≥ E π 1** . Thus, E π 1* l *
+
( )
+
≥ E π 1** .
Q.E.D.
Appendix 4: The proof of Proposition 4.
By Proposition 3, we know that if M invests at time 0, then M will license his standard to N
with a licensing fee of l*, and N accepts and adopts M’s standard. Thus the expected payoff for
( ( )) , which is a function of θ . Since M has to make the investment upfront, the
+
investment is E π 1* l *
0
( ( ))
net payoff is E π 1* l *
+
−I .
33
If M does not invest at time 0, then at time 1 M decides simultaneously with N whether to
(
invest and produce the network goods. Thus the expected payoff for not investing is E π 1*** − I
)
+
(no information on θ is revealed between time 0 and 1).
( ( ))
Therefore, the investment threshold θ 0* is the solution to: E π 1* l *
( ( ))
solving the integrals will result in the analytical expressions of E π 1* l *
+
+
(
)
+
− I = E π 1*** − I , and
(
and E π 1*** − I
)
+
presented in Proposition 4.
Q.E.D.
34