FACULTEIT)ECONOMISCHE)EN)SOCIALE)
WETENSCHAPPEN)&)SOLVAY)BUSINESS)SCHOOL)
!
!
!
!
!
ES$Working!Paper!no.!6!
!
!
!
Testing!Metcalfe’s!law:!pitfalls!and!possibilities!
Leo!Van!Hove!
!
!
!
!
!
!
!
!
!
Date:!August!1st,!2016!
!
!
!
!
Vrije!Universiteit!Brussel!$!Pleinlaan!2,[email protected]!
©!Vrije!Universiteit!Brussel!
!
!
!
!
!
!
!
!
!
!
!
!
This!text!may!be!downloaded!for!personal!research!purposes!only.!Any!additional!reproduction!for!
other! purposes,! whether! in! hard! copy! or! electronically,! requires! the! consent! of! the! author(s),!
editor(s).! If! cited! or! quoted,! reference! should! be! made! to! the! full! name! of! the! author(s),! editor(s),!
title,!the!working!paper!or!other!series,!the!year!and!the!publisher.!
!
!
Printed!in!Belgium!
Vrije!Universiteit!Brussel!
Faculty!of!Economics,!Social!Sciences!and!Solvay!Business!School!
B$1050!Brussel!!
Belgium!
www.vub.ac.be!
!
Testing Metcalfe’s law: pitfalls and possibilities
A small but burgeoning literature tries to assess whether Metcalfe’s law provides a
realistic yardstick for the value of specific types of networks. This paper uncovers a
number of flaws in the extant tests. Specifically, I point out that a proper test of
Metcalfe’s law – or of any of the competing ‘laws’ – requires a correct identification of
the type(s) of network effects involved, as well as of the relevant market(s). I also argue
that a multi-market setting typically calls for the scaling of network sizes and that one
may need to control for changes in network quality over time. Finally, one should not
mix up indicators at the individual and aggregate level. Armed with these insights, I reexamine Madureira et al. (2013)’s results. Unlike Madureira et al., I find that
Metcalfe’s law fits the data better than Briscoe’s law.
Words: 137
Keywords: Metcalfe’s law, network value, network effects, digital information networks, Internet
JEL codes: B5; D2; D8; L9; O3
1. Introduction: a world of networks
With the ever-increasing digitisation of production and consumption processes, a growing number
of goods and services in a growing number of industries have become, or are becoming, so-called
network goods; that is, goods that derive at least part of their utility from their connection to a
network of sorts. Social networks of all types – including those used in academia – spring to mind,
but physical goods are not immune to the trend either. Take the Dutch startup 3D Hubs. In spring
2013, 3D Hubs launched an online platform that connects 3D printer owners who have idle capacity
-1-
with ‘makers’ who would like to have something printed in 3D. In summer 2016, the 3D Hubs
“community” comprised some 30,500 printers in over 150 countries 1.
Given their increasing prevalence, a solid understanding of the economic value of ‘networks’,
‘communities’ or ‘platforms’ such as those just mentioned is becoming more and more imperative
for researchers, practitioners and policymakers alike. Leaving aside possible congestion, there is
little doubt that the value of a network increases as it adds members. The question is by how much.
A popular heuristic is Metcalfe’s law, which states that the value of a telecommunications network
is proportional to the square of the number of users. However, as Robert Metcalfe himself points
out in a 2013 article, until recently, “nobody (including me) has ever made the case for or against
Metcalfe’s law with real data” (Metcalfe, 2013, p. 26).
On the occasion of the 40th birthday of Ethernet, Metcalfe therefore revisited his law by
(successfully) fitting it to Facebook’s annual revenues over the period 2004-2013. Almost
simultaneously, Madureira et al. (2013) came up with an altogether different test of Metcalfe’s law
– as well as an alternative that they call Briscoe’s law. Specifically, Madureira et al. study the use
that European enterprises and individuals make of Digital Information Networks (DINs) such as the
Internet. Overall, Madureira et al. find that their models fit the data well and that, depending on the
type of Internet usage, “the value created [… is] linearly or quadratically dependent on the size of
the DIN infrastructure” (o.c., p. 254). This would validate, respectively, Briscoe’s and Metcalfe’s
law. More recently, Zhang et al. (2015) repeat Metcalfe’s test in a more systematic way using data
for both Facebook and (Chinese equivalent) Tencent. Van Hove (2016), in turn, extends Zhang et
al.’s approach by explicitly controlling for changes in network quality over time. Both Zhang et al.
and Van Hove find that Metcalfe’s law outperforms the competing laws.
1
Source: 3D Hubs, 3D Printing Trends, June 2016 <https://www.3dhubs.com/trends> (last accessed 29.06.16).
-2-
Unfortunately, all of the above efforts can be criticised in one or more respects. This paper uses the
extant literature, and especially the Madureira et al. article, as a springboard to point out a number
of pitfalls for researchers who may wish to examine the value of networks. Specifically, I point out
that a bona fide test of Metcalfe’s law – or any of the competing ‘laws’ – requires a correct
identification of the type(s) of network effects involved and of the boundaries of the market. I also
argue that a multi-market setting typically, but not always, calls for the scaling of network sizes,
and that in some cases, one may need to control for intertemporal changes in the nature of the
network or for inter-country differences. Finally, one should be sure to take the same point of view
– either that of individual users or that of network owners – for all indicators involved.
The paper proceeds as follows. Section 2 introduces Metcalfe’s law and its alternatives, briefly
touches upon Briscoe et al.’s (2006) critique of Metcalfe’s law, and summarises the existing tests –
with a focus on Madureira et al.’s efforts. In Section 3, I raise five questions concerning these tests.
Based on the answers (not all of which are conclusive), I then amend Madureira et al.’s approach in
Section 4 and re-examine a selection of their results. Section 5 concludes.
2. Metcalfe’s law
As mentioned in the Introduction, the most popular rule of thumb for the value of a network is
Metcalfe’s ‘law’, which focuses on the number of possible connections between members. In a
network with n members, each member can make n-1 connections 2. Metcalfe’s law assumes that all
these connections are equally valuable. Hence, for any given member, the utility of belonging to the
2
It is implicitly assumed here that the network is either the only one in the market or that there is complete
incompatibility between networks, so that interconnections with other networks can be disregarded.
-3-
network would be proportional to n-1. The aggregate value of the network – that is, number of
members times individual utility – would then be proportional to n(n–1), or roughly to n2.
Metcalfe’s law has been the subject of substantial criticism. This is how Metcalfe (2013, p. 26)
himself summarises it in his recent article:
Critics of my law have argued that N2 greatly overestimates the network effect. Not only is the law wrong, they
say, it is dangerous, because quantifying the network effect is central to society’s major infrastructural
investment decisions. Such decisions—thanks in part to Metcalfe’s law—went famously awry during the
dot-com bubble in the late 1990s. Today, detractors say, my law is causing stock markets to grossly overvalue
Internet-based companies such as Google, LinkedIn, Facebook, Twitter, and Snapchat.
The critique that has resonated most is the one put forward by Briscoe et al. (2006). Briscoe et al.
proffer two arguments, one of a theoretical and one of a semi-empirical nature. The theoretical
argument is that Metcalfe’s law clashes with the observed behaviour of networking companies,
because in the real world, only companies of roughly equal size are eager to interconnect without
side payments, whereas “if Metcalfe’s Law were true, then two networks ought to interconnect
regardless of their relative sizes” (o.c.). However, Van Hove (2014) shows that this inference is
flawed and that even under Metcalfe’s law, larger networks may have sound economic reasons to
resist interoperability. The essence of Van Hove’s refutation is that because Briscoe et al. reason in
aggregate and static terms, they overlook the fact that interconnection alters the utility which
individual users derive from the respective networks and that interconnection thus transforms the
networks’ competitive positions. Indeed, by agreeing upon interoperability, bigger networks turn
the smaller ones, ceteris paribus, into formidable competitors. Van Hove concludes that “to the best
-4-
of [his] knowledge, there is no valid theoretical reason to reject Metcalfe’s law” 3 (o.c., p. 7) and
that determining which growth ‘law’ is more realistic “is ultimately an empirical question” (o.c., p.
1).
This brings us to Briscoe et al.’s (2006) semi-empirical argument against Metcalfe’s law, namely
that it is wildly overoptimistic. [I call the argument semi-empirical because Briscoe et al. do not
underpin their claim with data from real networks.] In particular, Briscoe et al. take issue with
Metcalfe’s assumption that all connections are equally important 4 and consider Zipf’s law, which
assumes decreasing marginal utility, to be more realistic. Briscoe et al. therefore propose that the
value of a network of size n grows in proportion to nlog(n) – or nln(n) in both Madureira et al.’s
(2013) and Metcalfe’s (2013) notations. Madureira et al. call this “Briscoe’s law” (and do not so
much see it as an alternative to Metcalfe’s law, but rather as an extension of it that would hold for
large networks). Metcalfe (2013, p. 29) calls it “Odlyzko’s law” and points out the following:
3
A new (semi-)theoretical argument has now emerged that merits further scrutiny. Schneider (2014) constructs a
dynamic model in which two price-setting firms in an emerging network industry try to turn their own technology into
the standard. The model also provides a quantitative method for the valuation of the firms, and Schneider shows that
firm value depends strongly on the choice of the consumer utility function. In particular, Schneider finds that a linear
network strength function – think Metcalfe’s law (even though Schneider does not use this term) – “leads to price
explosions in the model and therefore unrealistically high firm values” (o.c., p. 76). Interestingly, a logarithmic
specification – in line with ‘Briscoe’s law’, as introduced later in this section – produces a more realistic and robust
model. But Schneider ends his article with a caveat: “for any real-world valuation of companies, the task of analysing
and comparing different network strength functions should be carried out in more detail in future research” (o.c., p. 85).
4
One possible justification is that while users will not, in practice, typically interact with everyone else in the network,
they nevertheless value the possibility of being able to communicate with the n-1 other members when needed.
Alternatively, one could see the proportionality factor in Metcalfe’s law – see equation (a) below – as capturing,
amongst other things, the average number of members that people do interact with.
-5-
Odlyzko’s law suffers from the same two main problems as my own. First, like N2, N x ln(N) goes to infinity
with N, and network values, like trees, do not grow to the sky. Odlyzko’s law has network value growing less
than Metcalfe’s law, but still without bound. Second, neither of us has attempted to validate our law with data
from real networks. Until now.
Indeed, as pointed out in the Introduction, recently Metcalfe himself and Madureira et al.,
independently from one another, came up with empirical tests of Metcalfe’s law. Metcalfe’s article
also triggered two follow-up papers, by Zhang et al. (2015) and Van Hove (2016).
Metcalfe’s own test is a simple one: he takes data on Facebook’s annual revenues as a surrogate for
the value of its network, plots data for the period 2004-2013 in a graph, programmes the Metcalfe’s
law function in Python with a slider attached to the proportionality factor, and, after adjusting the
slider, obtains “a pretty good visual fit” (2013, p. 30); see his Figure 4. Apart from being not
especially methodical (there is, for example, no attempt to fit Briscoe’s law), a problem with this
test is that, as Metcalfe acknowledges himself, “Facebook creates much more value than is captured
and monetized by Facebook selling ads” (ibid.) 5. It is no coincidence that Ross (2003), in an article
about technology ‘laws’ in IEEE Spectrum, argues that “Metcalfe's [law] can't be quantified,
because value – what economists call utility – can't be measured; you just know it when you see it”.
It is thus perhaps wise that, in their test, Madureira et al. (2013) make no attempt to directly
measure and explain the utility and value of DINs, but rather try to circumvent the problem by
focusing on the use that is made of these networks. Specifically, Madureira et al. put the Holonic
Framework (HF) developed by Madureira et al. (2011) to the test. This framework consists of
13 mechanisms – called capabilities – that enterprises and individuals alike can use to derive utility
5
This would not be a problem if Facebook’s 'monetisation ratio' were constant. But there is little doubt that Facebook
has, over time, become more astute at capturing the value it creates.
-6-
from accessing digital information. One such capability is ‘selectibility’, defined as the “capability
of a node/user in a network to scan or search for the unknown or generate courses of action that
improve on known alternatives” (Madureira et al., 2013, p. 248, Table 1). Madureira et al.
operationalise 9 of these capabilities by means of indicators taken from a Eurostat database on IT
usage in 33 European countries. Selectibility, for example, is proxied by the fraction of enterprises
using Internet search engines (o.c., p. 250, Table 2). Madureira et al. also posit that there is a causal
chain of the following type: “DINs → capabilities → economic value” (o.c., p. 249). As already
alluded to, they only operationalise the first step in this chain, but they assume that enterprises or
individuals use capabilities because “they have direct returns on value from that use” (o.c., p. 254).
Hence, Madureira et al. argue, the selected usage indicators yc (with c for capability) can be seen as
proxies for the “real economic value in €s” (ibidem).
In this setup, Madureira et al. then want to test whether and how usage of the capabilities – over
time and across countries – correlates with the size of the relevant DIN. However, the Eurostat
dataset used by Madureira et al. does not provide absolute numbers of enterprises and households
that have access to the Internet, only fractions. As a result, Madureira et al. cannot, as they would
wish, simply test Metcalfe’s law as
yc = kc,Mn2,
(a)
with kc,M = the “coupling strength between the size of the network and the value generated by
capability c” (o.c., 2013, p. 247) and with the subscript M referring to Metcalfe’s law. [Note that I
identify equations that do not appear in Madureira et al.’s article by means of the letters (a), (b),
etc.; the intended benefit being that the equations that are numbered correspond one-on-one with
equations in the original article.] Luckily, or so Madureira et al. argue, “thanks to the
proportionality of yc with [the size of the network squared], [network size] may also be expressed in
-7-
terms of the relative size of the network. Such a transformation only affects the value of the
proportionality constant kc,M” (o.c., p. 248). Madureira et al. therefore rewrite the above equation as
yc = kc,Mx2,
(1)
with x = the relative size of the relevant network (the maximum value of which is 1), and with kc,M
having a different value than in eq. (a).
Where the left-hand side of equation (1) is concerned, Madureira et al. try to match as many of the
HF capabilities as possible with indicators in their Eurostat database, and succeed in finding
suitable proxies – some better than others, as they admit themselves – for 9 out of the 13 (o.c., p.
249). Importantly, the data points that Madureira et al. collect in this way, for 33 European
countries over the period 2002-2009, are not solely of the country-year type. Madureira et al. also
add observations at the level of economic sectors as well as regions within a country 6. Depending
on the capability, this gives them between 191 and as many as 3,635 observations (o.c., p. 252,
Table 3). With these data, Madureira et al. then estimate, capability per capability, the
proportionality factor kc,M by minimising the sum of the squared residuals of model (1).
Turning to Briscoe’s law, here Madureira et al. argue that they “need to know the absolute size n of
the DIN, rather than its relative size x” (o.c., p. 248). This would be because, as I explain in
Appendix 2, “we cannot replace n with x in Briscoe’s law as we did in Metcalfe’s” (ibidem).
Madureira et al. therefore compute n from
x = n/I,
n = xI,
(3)
with I being “the potential maximum size of the DIN” (ibidem), and in this way obtain model (4):
yc = kc,BxIln(xI).
6
In Section 3 (under Question 2) I will argue against mixing different types of observations in this way.
-8-
(4)
Madureira et al. find that, overall, both model (1) and model (4) – which capture, respectively,
Metcalfe’s and Briscoe’s law – fit the data “quite well” (o.c., p. 254), with one exception. For
selectibility, model (1) fails. Instead, usage of this capability behaves linearly, with a slope close
to 1. Madureira et al. (o.c., p. 253) point out that for selectibility they do not have observations at
small network sizes, and argue that the observed linear behaviour is actually the upper part of a
quadratic curve. More generally, Briscoe’s law fits the strongly coupled capabilities – which
include ‘adoptability’ as well as ‘selectibility’– better than Metcalfe’s law, and the opposite is true
for the capabilities with a lower k. Madureira et al. (o.c., p. 254) conclude that “these results are in
concordance with observations about the validity interval of Metcalfe’s law (Briscoe et al., 2006)”.
Finally, there are two brand new leaves on the branch of the literature started by Metcalfe (2013)
himself. Zhang et al. (2015) administer three improvements to Metcalfe’s original test: they
examine two cases instead of one – not only Facebook but also China’s most popular social
network, Tencent; they compute fit parameters and do not simply rely on a visual fit; and, most
importantly, they compare the performance of four laws rather than limiting the analysis to just one
from the start. In particular, besides Metcalfe’s and Odlyzko’s laws, Zhang et al. also try their hand
at Sarnoff’s law (which holds that the value of a network increases linearly with the number of
users) and even Reed’s law (which asserts that it scales exponentially). Zhang et al. find that
Metcalfe’s law fits the data best and that the fit is even better for Tencent than for Facebook. Van
Hove (2016), in turn, points out that the value of a social network may, in part, also be driven by
increases in the variety and quality of the services offered. Van Hove therefore extends Zhang et
al.’s approach by explicitly controlling for such changes. He finds that this only strengthens Zhang
et al.’s conclusions: Metcalfe’s law now outperforms the other laws even more clearly.
-9-
3. Pitfalls and possibilities
As mentioned in the Introduction, none of the tests described in the previous section are perfect. In
this section, I use the existing literature as a springboard to highlight a number of pitfalls for
researchers who might wish to examine the economic value of networks. I do this by raising five
questions, which I have ranked from the more general to the more specific/technical. I do not claim
to have all the answers, but, at the very least, I hope I am asking the right questions. I scrutinize the
methodology of Madureira et al. (2013) in more detail than the rest because their tests are the most
comprehensive, but, in part precisely because of this, at the same time also the most open to
criticism.
My high-level goal in this section is thus to lay bare the assumptions underlying the key choices
made by the authors mentioned in Section 2. However, where relevant, I also point out whether
other authors who empirically examine network effects make similar assumptions or not. To be
clear: these authors do not really test any of the network value laws. They typically simply posit one
functional form or another and then estimate the strength of the network effects. Still, their
approaches and results can be informative.
Question 1 – Does Metcalfe’s law hold for any type of network?
The most fundamental question that is triggered by in particular Madureira et al.’s efforts is whether
it is realistic to expect Metcalfe’s law to hold as is for any type of network – or, more specifically,
for all the capabilities that Madureira et al. analyse. As was already hinted at earlier in this paper,
- 10 -
Metcalfe devised his law with communications networks in mind 7. Crucially, such networks are
two-way networks and the value creation is similar on both sides. A message from node A to
node B creates value for A in pretty much the same way as a message from B to A creates value
for B. Note in this respect that Metcalfe (2013) himself and the two follow-up papers test
Metcalfe’s law with data for networks that are – at least at first sight – all about interaction, namely
social networks.
Where Madureira et al. are concerned, several of the capabilities in their Table 1 (2013, p. 248)
would effectively seem to be two-way; ‘coordinatibility’ and ‘trustability’, for example. But
capabilities such as ‘selectibility’ and ‘adoptability’ would appear to be predominantly one-way,
particularly in view of the way in which they are operationalised. Take ‘adoptability’. This is
defined as the “capability of a node/user in a network to acquire novel knowledge from other
nodes/users and integrate it in its existing internal knowledge structures” (o.c., p. 248; Table 1). It is
proxied by the “fraction of individuals that have used [the] Internet for training and education”
(o.c., p. 250; Table 2). To the extent that the latter is not a P2P phenomenon – that is, it is not just
about using the Internet to look up peer-produced content that might be useful for one’s education 8
– there would appear to be two distinct sides to this mechanism. Let me refer to them as ‘trainees’
and ‘trainers’. Clearly, in such a (semi-9)formal setting, the value creation process is not the same
on both sides. More specifically, the network size that determines the utility of the mechanism is
7
See Metcalfe (2013, p. 27-28) for a vivid account of how he devised the ‘law’ to help the 3Com sales force convince
potential buyers of Ethernet technology that the value of LAN email would establish itself if only they added a
sufficient number of nodes.
8
Note that the Eurostat database that is exploited by Madureira et al. has a separate indicator for the “percentage of
individuals using the Internet for consultation with the purpose of learning”.
9
The Eurostat ICT database also has a separate indicator for the percentage of “individuals who have used Internet, in
the last 3 months, for doing an online course (of any subject)”.
- 11 -
not the same for trainees and trainers, with the utility of the network for trainees depending on the
number of trainers, and vice versa.
This brings to mind the distinction between direct and indirect network effects. Direct network
effects arise when direct contact between users is of value, as in social networks or in the telecom
industry. Indirect network effects are of the ‘hardware/software’ type, “where one user’s adoption
has no direct impact on the utility of other users, but may have lagged, indirect effects through the
provision of software” (Katz and Shapiro, 1994, p. 96) – or, to put it more generally, through an
increase in the variety of complementary products or the number of ‘complementors’. In the 3D
printing example used in the Introduction, the hub is only sufficiently useful for interested
consumers if there is a critical mass of participating printer owners in their city. Conversely, the
attraction of the hub for printer owners depends on the number of interested consumers in the
network.
Looking at Madureira et al.’s Table 2, they would appear to have modelled all nine capabilities in
terms of direct network effects, whereas for some there would seem to be two distinct sides to the
mechanism, implying that the relevant actors for the left-hand and right-hand side of their equations
are different. Let us again consider ‘adoptability’. As argued above, the relevant network size x
would not seem to be the fraction or absolute number of households with access to the Internet
(which is what Madureira et al. use), but rather, if such an indicator exists, the number of websites
with useful information. For ‘cooperatibility’, which is operationalised as the “fraction of
enterprises that have ordered products or services via the Internet” (o.c., p. 250; Table 2), the
market would also appear to be two-sided, with buyers on one side and sellers on the other. In other
words, here the relevant x would seem to be the number of enterprises that sell over the Internet,
- 12 -
rather than simply those that have access to the Internet 10. In short, Madureira et al. could have
made some of their tests more specific 11.
In practice, the situation can be even more complex because goods or services can be subject to
both direct and indirect network effects. Consider a mobile payments app that can be used for
person-to-person payments as well as for payments at the point-of-sale. The utility of such an app
for individuals depends both on the numbers of users and the number of merchants that accept it.
Such settings require a specific approach. Boudreau and Jeppesen (2015) examine the case of
on-line multi-player game platforms for which unpaid complementors develop so-called ‘mods’
(game modifications). In order to be able to estimate both ‘same-side’ and ‘cross-platform’ network
effects, Boudreau and Jeppesen estimate a linear system that consists of two equations (o.c., p.
1768). In the first equation, the number of new mods generated for a given platform in a given
month depends on usage of the platform by players (cross-platform effects) and on the size of the
complementor side (same-side effects). The second equation captures the positive feedback effect
whereby platform usage depends on the size of the complementor side (again cross-platform, but in
the other direction). And then Boudreau and Jeppesen do not even look into any same-side
10
Note that the Eurostat ICT database has information on the number of “enterprises which have received orders via
Internet over the last calendar year (excluding manually typed e-mails)”. Notation: e_isell.
11
Madureira et al. are not the only ones in the literature to struggle with identifying and, in particular, empirically
distinguishing the two types of network effects. McIntyre (2011), for example, examines the market for application
software in the US in 1986-1998 (in five segments: word processing, spreadsheets, desk-top publishing, personal
finance, and computer-aided design). Such software is subject to both direct and indirect network effects: users want to
be able to exchange files (direct effects), and at the same time a higher installed base provides incentives for creators of
complements (indirect effects). However, McIntyre only models direct network effects – although in the case of
personal finance software the latter would appear to be limited.
- 13 -
interactions on the user side, as they stress in an earlier version of their paper (Boudreau and
Jeppesen, 2012, p. 16).
To sum up, a first lesson would appear to be as follows. Metcalfe’s law was devised for
communications networks, where direct network effects reign supreme. Once indirect network
effects enter into play – in some cases on top of the direct effects – one needs to reassess what the
relevant network size is.
Question 2 – What are the boundaries of the market?
As explained in Section 2, when assembling their dataset, Madureira et al. do not limit themselves
to collecting yearly data for 33 countries. They also collect data points “for various economic
sectors and geographic regions” (o.c., p. 249) within these countries, and subsequently put all these
observations together in one sample.
This approach raises two (related) concerns. For one, is it appropriate to mix observations collected
at different levels? And also: do Madureira et al. consider the relevant level(s)? In both cases, the
key question is: what drives individual utility? Is it the number of Internet users within the sector,
regionally, and/or nationally? Perhaps the network effects are (also) situated at the European or
even the global level?12 While the answer may differ depending on the capability, the issue would
12
For example, where ‘selectibility’ is concerned, which is proxied by the fraction of enterprises using Internet search
engines, companies might benefit from finding information from all over the world – provided that they have a good
command of English and/or foreign languages. Conversely, for other types of networks, the relevant market might
actually be smaller than that of a sector or a country. Where 3D Hubs is concerned, for example, it is no coincidence
that its website contains maps that indicate the locations of the participating printers city per city. Indeed, the envisaged
market is distinctly local – with printers ultimately being a mere “bike ride” away.
- 14 -
seem to be of an either/or nature. Consider the capability ‘cooperatibility’. Suppose that EU
companies frequently engage in online sourcing outside their own country and sector, but never
outside the EU. If this is the case, then the EU level is the relevant level – a level that Madureira et
al. do not consider. It would also appear to be the only relevant level. Internet penetration at the
country level would not be a correct metric because there are foreign enterprises that create value.
The implications of an incorrect choice of dimension are obvious. Let us imagine a scenario in
which, within a given year, the number of, say, Portuguese enterprises with an Internet connection
is flat, but that the total number of connected enterprises in the EU increases dramatically. Under
my assumptions, this substantially increases the utility of the Internet as a sourcing channel for
Portuguese enterprises. But if one were to work with country-level observations (or lower) one
would not pick this up.
My answer to Question 2 is therefore as follows. First, one should not mix observations relating to
different levels, as this will result in an underestimation of the true strength of the network effect.
Second, given that selecting the relevant level is essentially an empirical issue, it is of vital
importance to acquire an accurate understanding of the ‘boundaries’ of the network/market, perhaps
based on survey evidence. In the absence of such direct indications, one could try to identify the
relevant installed base by testing at what level the fit of the regressions is strongest 13.
13
I owe this suggestion to an anonymous referee. If one pursues this line of reasoning, although Facebook is typically
seen as a global network, it would be interesting to replicate the Facebook tests of Metcalfe, Zhang et al., and Van Hove
with national data, so as to check whether Facebook is not in effect ‘multi-domestic’; that is, whether the Facebook
networks of most users are not predominantly national, so that the network effects mainly work at a national level.
There are indications that would appear to back this up. For one, while pioneering social networks Friendster and Orkut
were quickly eclipsed by Facebook on a worldwide scale, they initially remained popular in certain ‘pockets’ – in
Southeast Asia and Brazil, respectively (Sources: Waters, R., “Now you see it, now you don’t. Mobile life can be
- 15 -
Question 3 – Should one use absolute numbers or relative network sizes?
In other words: to scale or not to scale? Most authors are not faced with this question because they
examine a network that is deemed to be global (Metcalfe, 2013; Boudreau and Jeppesen, 2015;
Zhang et al., 2015 – for the Facebook case) or because they use data for a single country (Ohashi,
2003 14; Zhang et al., 2015 – for Tencent).
In the setting of Madureira et al., it all depends on the answer to Question 2 above. If the relevant
market is at the EU or global level (and if one thus uses, as advised, only EU-level or global
observations), there is obviously no problem. As long as one considers a single market, absolute
numbers and penetration rates will, by and large, dovetail 15. If, however, the markets are national
and if one wants to consider multiple countries, as is the case for Madureira et al., then the question
does arise.
This said, according to Madureira et al., the question is irrelevant in practice. As explained in
Section 2, in deriving their test of Metcalfe’s law, Madureira et al. claim that replacing n by x only
affects the proportionality factor, so that it does not matter whether one uses absolute (n) or relative
numbers (x). In Appendix 1, I show that within Madureira et al.’s setup and logic this is correct, but
short”. Financial Times, March 13, 2013; “The value of friendship”. The Economist, February 4, 2012, p. 19-20).
Second, there is anecdotal evidence of countries where the growth of Facebook within the country surpassed the overall
growth rate. In Portugal, for example, the use of Facebook among Portuguese internet users jumped from 10 to 63 per
cent in one year’s time, between July 2009 and July 2010 (Source: “Facebook on course to reach 1bn users”, Financial
Times, July 22, 2010). Note that Tencent is an altogether different case: because it is only available in Chinese, it is
essentially a national phenomenon – even though it is of use to Chinese abroad.
14
Ohashi (2003) studies the VCR market in the US.
15
That is, abstracting from population growth.
- 16 -
only because they assume that I, the potential maximum size of the DIN, is the same for all
observations. In a multi-market setting, this obviously clashes with reality. In the Appendix I show
that when I is not constant, one cannot simply replace n by x. Hence, testing Metcalfe’s law with
absolute or relative data does make a difference. The intuition is straightforward: scaling with a
constant, as Madureira et al. do, is not, in fact, scaling at all.
Now that we know that the choice matters – at least in a multi-market setting – the next question is
obviously: what is best, absolute or relative? On closer scrutiny, this may well depend on the type
of network/capability. Where communications and social networks are concerned, it would seem
that users essentially value market penetration. Consider LinkedIn. The penetration rate of LinkedIn
determines the probability that a new professional contact is also on LinkedIn, and thus determines
whether or not the service is a sufficiently efficient tool for managing one’s contacts. Just as for,
say, e-mail, this would seem to point towards the use of scaled numbers, provided that the borders
of the networks of the bulk of the users coincide with the borders of their country or sector; that is,
provided that the setting is multi-market rather than European or global. Another useful way of
thinking about the issue is to think in terms of users’ maximum utility or maximum willingness to
pay; that is, their utility in a scenario where literally everyone in a country has e-mail or has a
LinkedIn account. Is this maximum utility, ceteris paribus, higher in larger countries than in smaller
ones? If the answer is no, scaling is imperative.
Following this line of thinking, when it comes to other networks/capabilities, absolute numbers
might well be more appropriate. Consider a wiki 16 and suppose that people only value information
in their own language and/or written by people in their country or sector. Suppose furthermore that,
16
Wikipedia – itself a wiki – defines a wiki as “an application, typically a web application, which allows collaborative
modification, extension, or deletion of its content and structure”.
- 17 -
across countries/sectors, individuals are equally likely to contribute to the wiki and that their entries
are of equal quality. Then the collective knowledge produced by the citizens of a more populous
country/sector will be larger, and thus more valuable. In such a setting – and provided that the
above assumptions apply – scaling would be incongruous.
Upon perusing papers that estimate network effects in a multi-market setting (but do not actually
test any of the ‘network value laws’), I found multiple papers that scale network sizes in line with
the above reasoning but I also came across one counterexample. By coincidence, all papers relate to
the telecom sector.
Allow me to give three distinct examples from the first group of papers. Liikanen et al. (2004) use
data on the mobile phone industry in 80 countries over 1992-1998 and find, besides positive withingeneration network effects, that the stock of first-generation (1G) adopters has a positive effect on
second-generation (2G) diffusion. Liikanen et al. measure the installed base of both analog and
digital phones as the number of subscribers divided by population. Suarez (2005) analyses the
technology choices made by mobile operators in 47 countries in North, South, and Central America
during the period 1992-2001. At the time, operators had a choice between three incompatible 2G
technologies (GSM, TDMA, or CDMA). For technical reasons, network effects played a role in this
decision. To give just one example, in the absence of multi-technology handsets, operators could
only provide their subscribers with a roaming feature (the use of a phone outside its primary
coverage area) for networks of other operators using the same technology (o.c., p. 713). To capture
this direct B2B network effect, Suarez constructs a variable called ‘strong-ties network’ by, first,
determining for each country its top three international calling partners, second, calculating, for
each of these three countries, the total number of subscribers using a particular technology divided
by the total number of active subscribers, and, third, computing a simple average of these scores.
- 18 -
What is crucial for our purposes is that Suarez does not use absolute numbers, but works with
(average) penetration rates 17. For their part, Karaçuka et al. (2013) focus on a single country and
analyse network effects on the mobile telecom market in Turkey. Their setting is nevertheless, at
least potentially, multi-market – because operators’ market shares vary substantially between
regions within Turkey. To test whether the network effects are, effectively, local, Karaçuka et al.
explain individual consumers’ choice of network in a conditional logit model comprising, amongst
other variables, the market shares of the network at the national level and at the level of the
province 18. In other words, these authors also make use of relative network sizes – in order to take
into account that the provinces differ in size.
The counterexample can be found in Fuentelsaz et al. (2015). Using a longitudinal panel spanning
the period from 1998 to 2008 and covering 65 mobile communications providers in 20 European
markets, Fuentelsaz et al. analyse how operators’ strategic actions (such as entry timing and
internationalisation) impact the value of their network. For the latter measure, Fuentelsaz et al. build
on Zipf’s law (rather than Metcalfe’s law), but amend it to take into account differences in network
17
Incidentally, Suarez also has a variable called ‘world network’. This is constructed as the total number of operators
worldwide using a particular technology divided by the total number of active operators in any given period (o.c.,
p. 714). However, this variable turns out not to have a significant effect on operators’ technology decisions, leading
Suarez to conclude that “when choosing which technology to purchase, cellular operators tend to pay more attention to
decisions made previously by other operators in a selected subset of countries with which they have strong ties than to
the overall situation in the world” (o.c., p. 716; my italics). There is a clear link here with our Question 2: in Suarez’s
setting the relevant market proves to be ‘supra-national’, rather than either national or global.
18
Again there is a link with our Question 2: Karaçuka et al. find that whereas national market share has no effect on
network choice, market share at the province level is a very significant determinant. Karaçuka et al. conclude from this
that the network effects are not nation-wide, but rather local. They put forward two reasons: “(a) consumers can usually
better observe the choices made in their local surrounding, and (b) most mobile calls are typically still made to
customers within the same local area” (2013, p. 343).
- 19 -
intensities as a result of providers charging more for off-net calls; see (o.c., p. 875-877) for details.
Oddly enough, whereas Fuentelsaz et al. do state that “in mobile communications, competition
takes place within national markets” (o.c., p. 874), they use absolute subscriber numbers for their
measure of network value. If, as argued above, subscribers value market penetration, such an
unscaled measure is biased against networks that are active in small markets.
Focusing again on the Madureira et al. article, allow me to point out that their answer to my
Question 3 is not very consistent, in that they work with scaled data when testing Metcalfe’s law,
but use absolute numbers when testing Briscoe’s law. This obviously ties in with their argument,
mentioned above, that the choice is irrelevant. Hence, when testing Metcalfe’s law, they work with
scaled data because the data happen to come in that format, or so it seems 19. However, as argued,
the choice does matter. In my (partial) re-examination of Madureira et al.’s regressions, in Section
4, I will therefore, in the absence of clear indications either way, attempt to use both scaled and
unscaled data.
However, let me also point out that Madureira et al.’s use of a scaled dependent variable is not
neutral either. Indeed, because yc is the fraction of enterprises or individuals who have a certain
capability (see their Table 2), the maximum value for yc, which is a proxy for the value created, is
the same in all countries. Hence, given my argument above about the link between the size of a
country and maximum utility, the use of a scaled variable would only seem justifiable if the size of
the network – the variable on the right-hand side – was also scaled. In my re-examination I will
therefore also experiment with a combination of unscaled dependent and independent variables,
19
Cf. “We use [relative sizes], because our data set from Eurostat provides direct numbers for the fraction of potential
members being connected to DINs, …” (o.c., p. 248).
- 20 -
even though such an unscaled dependent variable also has its drawbacks – as I will explain below,
in my answer to Question 5.
Question 4 – Are there disturbing factors that need to be controlled for?
An implicit assumption behind Metcalfe’s law is that, apart from the number of users, the network
itself does not change; that is, the technology remains the same and the operator continues to offer
the same service(s). If the network that is studied is, say, a network of fax machines, this
assumption can likely be ignored without creating major problems because intertemporal
improvements in quality, if any such improvements occur, are probably minor.
However, social networks for one are different. As I put it in Van Hove (2016): “the Facebook of
2014 is not the Facebook of 2004”. Facebook now offers a wider variety of services, which are
arguably of better quality. Hence, an increase in the value created by Facebook may not only be the
result of the company’s (spectacular) growth in terms of number of users. This is why in Van Hove
(2016) I explicitly incorporate a quality indicator into Zhang et al. (2015)’s tests for Facebook and
Tencent. In concrete terms, based on financial information taken from annual reports, I compute,
year per year, a ‘cost per (active) user’ or ‘cost per node’ (CPN). The key assumption – which can
be criticised; see Van Hove (2016) – is then that a constant CPN is an indication of constant quality,
or, in other words, that cost increases beyond linear growth are a sign of improvements in quality. I
find that the inclusion of such a quality variable as an additional driver of network value does not
invalidate Zhang et al.’s conclusions, on the contrary. I do point out, however, that future research
might want to look into interaction variables between quality and size.
- 21 -
Let me stress that I am not the first to control for quality in a paper that looks into network
externalities. For example, in studies on the mobile telecom sector, it is quite evident that one
should control for improvements in (local) coverage quality. In a paper on the US wireless industry,
Weiergräber (2014, p. 15) uses the average satisfaction level of an operator’s customers in a
specific region. In a study on Rwanda, Björkegren (2015) exploits direct information on the
location of cell towers. Finally, in a paper about on-line video games Liu et al. (2015) use two
proxies to capture game quality: ratings by professional critics and the percentage of players who
had spent more than 100 hours playing the game by the time that they provided their own ratings 20.
Question 5 – Which point of view? Network owner or individual users?
Apart from the methodological problems, this is in fact a non-issue. Even if I offer a number of
reflections in the Conclusion as to which approach would seem to be the more promising, in
principle one could just as well try to explain aggregate network value or individual utility.
However, Madureira et al. would seem to have a consistency issue, in that they do not take the same
point of view on both sides of their equations. On the left-hand side, Madureira et al. measure, as a
proxy for individual utility, certain usages that people/enterprises make of the Internet. In other
words, Madureira et al. take the point of view of the members of the network, the users. In all logic,
the right-hand side of their regressions should then also reflect this and should feature the size of
the network that matters for users; that is, xi rather than x i2 . Indeed, while under Metcalfe’s law
€
20
Note that Liu et al. use panel data and include the two variables primarily in order to control for differences in game
quality across games.
- 22 -
aggregate network value grows quadratically, individual utility grows linearly; see Figure 3a in
Choi and Lee (2012, p. 189) 21.
Unlike Madureira et al., Metcalfe (2013) and Fuentelsaz et al. (2015) take the point of view of
network owners 22 – respectively, Facebook and mobile operators. It is therefore no coincidence
that they have on the right-hand side of their regressions aggregate network value – respectively, n2
and nlog(n).
But for individual members, aggregate network value is not what matters. This can be illustrated as
follows. According to the Eurostat data used by Madureira et al., in 2009 Internet access among
households happened to be identical in Italy and Cyprus, with xi = 53%. If we are to suppose that
consumers value market penetration and that the relevant market is essentially national in nature,
then the upshot is that, in 2009, access to the Internet was equally valuable for citizens of both
countries. Using aggregate network value – individual utility times the number of members – would
give a completely different picture. To see this, we need data on ni; that is, on the number of
households in Italy and Cyprus. Such numbers are not readily available on a cross-country basis,
but estimates can be obtained by dividing total population by the average number of persons per
21
In their paper on massively multiplayer online role-playing games (MMOGs), Liu et al. find that the effect of the size
of the installed base on individual game ratings (their proxy for consumer utility) exhibits non-linear dynamics over the
product life cycle: the effect is non-significant in the first four weeks after game release, highly significant in weeks
5-17, and significant but less strong in the remaining weeks. While Liu et al. do not present their regressions as a test of
Metcalfe’s law, this result would in fact seem to disprove it. However, as Liu et al. themselves point out,, their data “are
not typical time series. Even though the ratings occur over time, they are posted by different users. […] In this sense,
the ratings data have a critical cross-sectional element”. The strength of their results also hinges on whether product
ratings are a good proxy for consumer utility; see Section 5.
22
The same is true for the follow-up papers on Metcalfe.
- 23 -
household 23. Based on such estimates, roughly 13.8 million Italian families had access to the
Internet in 2009 vs. 183,000 in (much smaller) Cyprus. Hence, the aggregate value of the Internet
would have been (proportional to) 13,836,472 * 0.53 = 7,333,330 in Italy and 183,640 * 0.53 =
97,329 in Cyprus. Clearly, the huge difference – a factor of roughly 75 – only reflects the difference
in market size. Aggregate network value is an adequate measure of the value to be reaped by, say,
Internet Service Providers, but it is not as if access to the Internet was 75 times more valuable for
the average Italian than it was for the average Cypriot.
Let me give another example. Suppose that I am correct in arguing, in my answer to Question 3
above, that the utility of having an e-mail account is determined by the penetration rate of the
technology. Suppose Internet penetration is 10% in Belgium and 50% in Germany. Then, if
Metcalfe’s law holds, individual Germans should find access to the Internet five times more useful
than individual Belgians. At the country level this should, ceteris paribus, result in a share of e-mail
adopters that is five times higher in Germany than in Belgium – in line with the linear relationship
that Metcalfe’s law presumes between network size and individual utility. A quadratic specification,
as in Madureira et al.’s equation (1), would dramatically alter the scales between the two countries.
All this explains why in my re-examination below, my preferred specifications will be models with
individual utility on the right-hand side.
4. Madureira et al.’s results: a re-appraisal
In this section I apply, step by step, several of the amendments suggested above – that is, as many
as possible – to (a selection of) Madureira et al.’s data and compare the results with their original
results.
23
Source: Eurostat, Population and Social Conditions, <http://ec.europa.eu/eurostat/web/main/home> (last accessed
28.10.2015). Respective Eurostat references: tps00001 and tsdpc510.
- 24 -
4.1. Data
As mentioned in Section 2, a peculiarity of Madureira et al.’s set-up is that they do not only collect
data points at the national level, but also for “various economic sectors and geographic regions”
(o.c., p. 249). Their article does not provide further details, but e-mail exchanges with lead author
Antonio Madureira made clear that “regions” does not refer to the observations at the supranational
level (for the EU28, etc.) that appear in the publicly-available Eurostat data, but rather to various
(and varying) breakdowns of the national figures.
Quite apart from the problem of identifying exactly which breakdowns were used for which
capability, I decided not to use sub-national figures and to limit myself to the data points at the
national level – mainly for two reasons. First, I think it is unwise to mix different-level
observations; see the discussion of Question 2. Second, if one accepts that a choice needs to be
made, the national level would seem to make the most sense. Indeed, given that Madureira et al.
look into the use that individuals and companies make of the Internet, it is difficult to argue that
markets would have sub-national delineations 24. Conversely, I would have liked to alternate
country- and European-level observations, but for the European aggregates there are simply not
enough observations available 25.
24
To give just one example: one of the breakdowns in the Eurostat database used by Madureira et al. (a breakdown that
may or may not feature in their dataset) splits up households depending on whether they live in densely-populated,
intermediate urbanized, or sparsely-populated areas. It is improbable that people living in a sparsely populated area in,
say, Italy, use the Internet to sell goods only to people living in other sparsely-populated areas in Italy.
25
Madureira et al.’s dataset spans just 8 years, and due to missing values I would, in practice, have 3 observations for
adoptability and a maximum of 7 for biddability. I have therefore only been able to use country-level observations. This
said, the two capabilities that I examine relate to households (see lower in the main text) rather than to enterprises. And
for households it is less evident that the ‘right’ level is the European level; see Section 4.2.
- 25 -
Given this preference for national-level data, I decided to limit myself to the two capabilities that
relate to individuals, namely biddability (operationalised as the fraction of individuals who have
used the Internet for selling goods) and adoptability (proxied by the fraction of individuals who
have used the Internet for training and education). The rationale behind this decision is that when it
comes to the capabilities that relate to enterprises, Madureira et al. not only include regional
observations but also data points at the level of sectors within a country. As a result, the differences
between their dataset and a dataset comprised solely of national-level data would become too big.
To illustrate this, note that Madureira et al. have 191 observations for the ‘biddability’ capability of
European households. Using the same database and concentrating on the same period (2002-2009),
I ended up with 178 observations at the national level. However, for the capabilities that relate to
enterprises, Madureira et al. have between 805 and no fewer than 3,635 observations; see their
Table 3.
4.2. Results
Tables 1a and 1b summarise the results of my re-examination of Madureira et al.’s original results.
As explained, I have limited myself to the two capabilities that relate to individuals. For each
capability and law, I first simply restate Madureira et al.’s results; for example, where adoptability
and Metcalfe’s law are concerned, see row (M.a) in Table 1a. [The M stands for Metcalfe, just as
the B stands for Briscoe.] In the second row, I have then re-estimated the models used by Madureira
et al. for the more limited samples that I have (which contain only country-level observations).
Where biddability is concerned, this straightforward replication is reassuring: both the coupling
strengths and the R2s are similar or very similar (and the RSDs are actually lower) 26. For
26
Note that my estimates of the coupling strength fall within the 95% confidence interval of Madureira et al.’s estimates
and are thus not statistically different.
- 26 -
adoptability, the differences are larger – especially where Briscoe’s law is concerned – but then the
difference in the number of observations is substantially larger, too. The third row again presents
estimations of Madureira et al.’s models, but for an even smaller sample – in order to have the best
possible point of comparison for the tests that follow.
From the fourth row onwards, I then progressively apply the amendments suggested in Section 3.
But before we address the results, let me first recapitulate the amendments suggested in Section 3,
and explain whether and how I have applied them.
In my answer to Question 1, I argued that for some of the capabilities the network effect would
seem to be of the indirect type, which would call for a more specific measure of the network size on
the right-hand side (rather than just the total number of households/enterprises that have access to
the Internet). However, for the two capabilities that I examine, I could not conceive of a way to do
this. For ‘adoptability’, a more specific indicator does not, as far as I know, exist. For ‘biddability’,
there may well be two sides to the market – namely buyers and sellers – but the problem is that
many of the individuals who sell via the Internet will probably also be buyers.
- 27 -
Table 1a
Estimates of coupling strengths: comparison of tests, adoptability
Test
Source
Explanation
n
(changes compared to
Madureira et al.)
Metcalfe’s law
(M.a)
y c,i
(M.b)
Madureira
et al.
(2013)
this paper
(M.c)
this paper
= kc,M x i2
€
(M.d)
= kc,M n i x i
y c,i
(M.e)
y c,i = k c,M n i
(M.f)
yabsc,i = k c,M n i
€
2
2
(M.g)
= kc,M x i
y c,i
€
Eq. (1)’
kc
RSDkc
R2
(%)
220
0.68 ± 0.05
7
0.85
€
97
0.79 ± 0.03
4
0.86
87
0.79 ± 0.04
5
0.84
3.3E-14
± 0.5E-14
7.1E-6
± 1.8E-6
4.9E-8
± 0.4E-8
0.55 ± 0.02
16
0.31
25
0.16
7
0.69
3
0.91
1.1E+5
± 0.2E+5
2.5E-6
± 0.4E-6
1.31 ± 0.04
15
0.35
15
0.35
3
0.92
this paper
Q3 – RHS variable scaled
differently; Eq. (A2.c)
87
this paper
Q3 – RHS variable unscaled;
Eq. (a)
87
this paper
Q3 – RHS variable unscaled +
LHS variable unscaled
87
this paper
Q5 – individual utility on RHS
87
(M.h)
€
yabsc,i = k c,M x i
this paper
Q5+Q3 – individual utility on
RHS + LHS variable unscaled
87
(M.i)
y c,i = k c,M n i
this paper
Q5+Q3 – individual utility on
RHS + RHS variable unscaled
87
yabsc,i = k c,M n i
this paper
Q5+Q3 – individual utility on
RHS + RHS variable unscaled
+ LHS variable unscaled
87
Eq. (4)’
220
19.7E-12
± 0.2E-12
1
0.99
(B.b)
Madureira
et al.
(2013)
this paper
97
3
0.91
(B.c)
this paper
9.9E-12
± 0.3E-12
9.7E-12
± 0.3E-12
2.5E-6
± 0.3E-6
3
0.91
15
0.35
16
0.33
3
0.91
€
(M.j)
€
€
Briscoe’s law
(B.a)
€
y c,i
= kc,B x i I ln(x i I )
€
(B.d)
ln(n i )
y c,i
= kc,B n i
(B.e)
y c,i
= kc,B n i ln( n i )
(B.f)
yabsc,i
€
(B.g)
€
y c,i
ln(n i )
y c,i
(B.j)
yabsc,i
€
87
this paper
Q3 – RHS variable unscaled
87
this paper
Q3 – RHS variable unscaled +
LHS variable unscaled
Q5 – individual utility on RHS
87
1.4E-7
± 0.2E-7
0.08 ± 0.00
87
32.1 ± 1.4
4
0.86
this paper
ln( n i )
this paper
87
6.6E+5
± 1.0+5
4
0.33
ln( I i )
ln( n i )
Q5+Q3 – individual utility on
RHS + LHS variable unscaled
this paper
87
2.13 ± 0.09
4
0.86
= kc,B ln( n i )
this paper
Q5+Q3 – individual utility on
RHS + RHS variable unscaled
Q5+Q3 – individual utility on
RHS + RHS variable unscaled
+ LHS variable unscaled
87
4.8E+5
± 0.6E+5
13
0.40
yabsc,i = k c,B
(B.i)
Q3 – RHS variable scaled
differently
ln(I i )
(B.h)
€
this paper
ln(I i )
= kc,B n i ln(n i )
= kc,B
87
= kc,B
€
€
Notes:
n = number of observations; RSD = relative standard deviation. Q3 and Q4 refer, respectively, to Question 3
€and 4 in Section 3.
- 28 -
Table 1b
Estimates of coupling strengths: comparison of tests, biddability
Test
Source
Explanation
n
(changes compared to
Madureira et al.)
Metcalfe’s law
(M.a)
y c,i
(M.b)
Madureira
et al.
(2013)
this paper
(M.c)
this paper
= kc,M x i2
€
(M.d)
= kc,M n i x i
y c,i
(M.e)
y c,i = k c,M n i
(M.f)
yabsc,i = k c,M n i
€
2
2
(M.g)
= kc,M x i
y c,i
€
Eq. (1)’
kc
RSDkc
R2
(%)
191
0.19 ± 0.03
16
0.86
€
178
0.21 ± 0.01
3
0.84
124
0.23 ± 0.01
4
0.86
1.2E-6
± 0.1E-6
2.8E-14
± 0.4E-14
2.1E-8
± 0.0E-8
0.14 ± 0.01
10
0.46
14
0.29
2
0.94
5
0.76
3.3E+4
± 0.5E+4
7.9E-7
± 0.8E-7
0.46 ± 0.01
15
0.27
10
0.43
3
0.91
this paper
Q3 – RHS variable scaled
differently; Eq. (A2.c)
124
this paper
Q3 – RHS variable unscaled;
Eq. (a)
124
this paper
Q3 – RHS variable unscaled +
LHS variable unscaled
124
this paper
Q5 – individual utility on RHS
124
(M.h)
€
yabsc,i = k c,M x i
this paper
Q5+Q3 – individual utility on
RHS + LHS variable unscaled
124
(M.i)
y c,i = k c,M n i
this paper
Q5+Q3 – individual utility on
RHS + RHS variable unscaled
124
yabsc,i = k c,M n i
this paper
Q5+Q3 – individual utility on
RHS + RHS variable unscaled
+ LHS variable unscaled
124
Eq. (4)’
191
2.6E-12
± 0.2E-12
12
0.83
(B.b)
Madureira
et al.
(2013)
this paper
178
4
0.76
(B.c)
this paper
2.4E-12
± 0.1E-12
2.5E-12
± 0.1E-12
8.1E-7
± 0.8E-7
5
0.76
10
0.43
11
0.42
3
0.93
€
(M.j)
€
€
Briscoe’s law
(B.a)
€
y c,i
= kc,B x i I ln(x i I )
€
(B.d)
ln(n i )
y c,i
= kc,B n i
(B.e)
y c,i
= kc,B n i ln( n i )
(B.f)
yabsc,i
€
(B.g)
€
y c,i
ln(n i )
y c,i
(B.j)
yabsc,i
€
124
this paper
Q3 – RHS variable unscaled
124
this paper
Q3 – RHS variable unscaled +
LHS variable unscaled
Q5 – individual utility on RHS
124
4.6E-8
± 0.4E-8
0.03 ± 0.00
124
7.0 ± 0.6
8
0.54
this paper
ln( n i )
this paper
124
1.7E+6
± 0.3E+6
19
0.19
ln( I i )
ln( n i )
Q5+Q3 – individual utility on
RHS + LHS variable unscaled
this paper
124
0.36 ± 0.04
12
0.63
= kc,B ln( n i )
this paper
Q5+Q3 – individual utility on
RHS + RHS variable unscaled
Q5+Q3 – individual utility on
RHS + RHS variable unscaled
+ LHS variable unscaled
124
1.2E+5
± 0.2E+5
16
0.24
yabsc,i = k c,B
(B.i)
Q3 – RHS variable scaled
differently
ln(I i )
(B.h)
€
this paper
ln(I i )
= kc,B n i ln(n i )
= kc,B
124
= kc,B
€
€
Notes:
n = number of observations; RSD = relative standard deviation. Q3 and Q4 refer, respectively, to Question 3
€and 4 in Section 3.
- 29 -
Question 2 related to the geographical dimension. Where biddability is concerned, it can be noted
that, although cross-border online shopping is on the increase in the EU, according to the European
Commission “the Internet is used to make purchases mainly from sellers or providers based in the
respondent’s own country” 27. In 2012, only 15 per cent of EU consumers had purchased once or
more times from an online seller/provider in another EU country in the previous 12 months 28.
Gomez-Herrera et al. (2014) use data from an online consumer survey in the EU27 to investigate
whether geographical distance still matters for e-commerce in physical goods. They find that
distance-related trade costs are greatly reduced, but that socio-cultural variables such as language
increase in importance. Gomez-Herrera et al. conclude that, on balance, “there are no indications
that home bias is less significant online than offline” (o.c., p. 84). For adoptability, the home bias
could be smaller because we are talking about pure information products, but here too the linguistic
and cultural fragmentation of the EU market might play a role.
Turning to the scaling issue of Question 3, in order to be able to scale, I needed data on the number
of households per country. Such data are not readily available, but, as explained, estimates can be
obtained by dividing total population by the average number of persons per household 29.
27
Source: European Commission, “Consumer attitudes towards cross-border trade and consumer protection”, Flash
Eurobarometer, Nr. 358, June 2013, p. 16.
28
There are, however, countries where more consumers have purchased from a site located in another EU country than
from a site in their own. This is, not surprisingly, especially the case in smaller countries – such as Malta (42 per cent
cross-border vs. 11 per cent domestic), Luxembourg (41 vs. 14), and Cyprus (31 vs. 5) – but it is also true for a country
like Ireland (48 vs. 40) (Source: European Commission, “Consumer attitudes towards cross-border trade and consumer
protection”, Flash Eurobarometer, Nr. 358, June 2013, p. 17). This indicates that the geographic dimension of the
relevant market can actually differ from one country to the next.
29
Note that, to the best of my knowledge, these data are only available at the national level. This constitutes a third,
pragmatic reason to only use national-level data. Note also that data on the number of active enterprises (not used in my
regressions, for reasons explained above) can be found in the Eurostat Structural Business Statistics. See
- 30 -
Unfortunately, for the latter statistic there are several missing values, resulting in a substantial loss
of observations. As can be seen in Table 1b, for biddability the number drops from 178 in row (b)
to 124 in row (d). In order to make sure that differences in the results are not caused by differences
in sample size, I perform all regressions with overlapping sets; see row (c). In specification (d) I
first scale the network size variable on the right-hand side in a different way compared to Madureira
et al. Indeed, in Appendix 2 I argue that Madureira et al. make a mistake when formulating
Metcalfe’s and Briscoe’s laws in relative terms. In specification (f), I use an unscaled network size
variable but combine this with an unscaled usage indicator on the left-hand side.
To continue, given the wide range of countries covered and the many service facets involved, I did
not see a straightforward way to take up the suggestion proffered in Question 4 to control for
improvements in quality over time – even though there is little doubt that, for example, between
2002 and 2009, average internet connection speed increased substantially in many a country.
Finally, in line with my answer to Question 5, in specifications (g) to (j) I have replaced aggregate
network value by individual utility. As announced, I try both scaled and unscaled dependent and
independent variables but my preferred specifications – indicated by boxes in column three – are
obviously those where both are either scaled or unscaled.
A first result that catches the eye in Tables 1a and 1b relates precisely to the remark just made:
models with inconsistent scaling – a scaled dependent variable combined with an unscaled
independent variable or vice versa – almost invariably give poor results, which is reassuring. In 10
of the 12 cases – see specifications (c), (h), and (i) – the R2s lie in the range 0.16-0.43 and the only
<http://ec.europa.eu/eurostat/web/structural-business-statistics> for data on 2008-2012. Data for earlier years can be
found at <http://appsso.eurostat.ec.europa.eu/nui/show.do?dataset=bd_9a_l_form> (last accessed 22.10.15).
- 31 -
real exception is the 0.86 for specification (B.i). I discard all these specifications and do not discuss
them any further.
Where the choice between aggregate network value and individual utility – the object of discussion
in Question 5 – is concerned, the results would seem to favour the latter. To see this, one needs to
compare specifications (d) and (g), and (f) and (j). In 5 of the 8 cases, the specification with
individual utility on the right-hand side yields (substantially) better results, in once case – (M.f) vs.
(M.j) in Table 1b – one could call it a draw (with R2s of, respectively, 0.94 and 0.91), and in 2 cases
the specification with aggregate network value performs best – because specification (j) performs
badly for Briscoe’s law.
This brings us to the central question in Madureira et al.’s paper: which ‘law’ performs best? To
answer this question we have to perform pairwise (M.z)-(B.z) comparisons. In the Tables I have
visualised the results of these comparisons by putting the highest R2 in bold each time. At first
glance, the answer would seem to depend on the capability being examined: for adoptability (in
Table 1a) Briscoe’s law has the highest number of bold R2s; for biddability (in Table 1b) it is the
opposite. However, if one discards the inconsistent specifications (c), (h), and (i) as well as
Madureira et al.’s specifications (a), (b), and (c) – which, as explained, I believe to be misspecified
– and, in particular, if one concentrates on my preferred specifications (g) and (j), the picture
changes completely. Indeed, if one looks at the boxed R2s, it becomes apparent that Metcalfe’s law
consistently outperforms Briscoe’s law, and in 3 out of the 4 cases, to a substantial degree.
It can also be noted that, even though my samples are smaller, my preferred specifications for
Metcalfe’s law tend to have lower RSDs and/or higher R2s than Madureira et al.’s specification (a):
for adoptability I have an R2 of 0.91-0.92 vs. 0.85 for Madureira et al., for biddability the
- 32 -
corresponding values are 0.76-0.91 and 0.86. This could be seen as providing indirect support for
my criticisms. Finally, if we continue to concentrate on Metcalfe’s law, as far as biddability is
concerned, my unscaled specification (M.j) outperforms the scaled variant (M.g). For adoptability
there is hardly any difference.
5. Concluding remarks
This paper has scrutinised the empirical literature on Metcalfe’s law and its alternatives, the goal
being to lay bare the key choices that need to be made when testing these ‘laws’. I present the
lessons learned in the form of answers to five questions, and apply some of the suggested
amendments to Madureira et al. (2013)’s validation efforts.
After making these modifications, I find markedly different results. Madureira et al. conclude that,
overall, their results are “qualitatively the same irrespectively of using Metcalfe’s law or Briscoe’s
adaptation of it” (o.c., p. 55). For adoptability, Briscoe’s law even shows a better fit. Conversely,
for the two capabilities that I consider – one of which is adoptability – I find that Metcalfe’s law
performs better. This finding comes with four caveats. For one, I have only been able to run tests
for two of the nine capabilities considered by Madureira et al. Secondly, it remains uncertain at
what level – regional, national, supra-national, … – the tests should be conducted. Thirdly, my tests
inherit Madureira et al.’s implicit (and unproven) assumption that, for a given capability, the
proportionality factor between network value and network size is identical across countries 30. And,
finally, I take Madureira et al.’s (2011) Holonic Framework as a given.
30
In their case, depending on the capability, across countries as well as across (sectors and) regions within a country.
Note that, in their study of the diffusion of mobile telecommunications in 29 OECD countries and Taiwan, Jang et al.
(2005) find differences in the magnitude of the network externality coefficient. But then, the technology was introduced
- 33 -
More generally, the paper shows, if anything, that examining network value is far more complex
than a simple expression such as Metcalfe’s law conveys, especially in a multi-country and/or
multi-sector setting. The main takeaway would appear to be that future efforts would benefit greatly
from additional ‘preparatory research’; that is, research that would help towards making wellfounded choices concerning, for example, the geographical boundaries of the market that is
examined. Surveys into whether users value market penetration or absolute numbers would also be
welcome. So would research that looks into people’s willingness to pay – why not by means of
choice experiments, as in Sobolewski and Czajkowski (2012). In short, I concur with Karaçuka et
al. when they write that “[i]n the analysis of network effects on consumer choice both industry-level
and firm-level studies utilize what has been called ‘macro empiricism’ […], inferring individuals’
preferences from the observation of aggregate market behaviour” (2013, p. 335-336).
The quote from Karaçuka et al. is also a nice stepping stone to the highest-level question that the
present paper will probably trigger among readers: Which of the two branches of the literature that
have been identified is the most promising? Should one try to proxy and explain aggregate network
value – as in Metcalfe (2013), Zhang et al. (2015), and Van Hove (2016) – or should one instead try
to test Metcalfe’s law at the root and focus on (individual) consumer utility – as in Madureira et al.
(2013)? In fact, both types of indicators have their problems.
Using the revenues of the operator to proxy aggregate network value is particularly problematical
when the monetisation is indirect; for example, via advertising – as in the case of Facebook. Also,
as argued in Van Hove (2016), if the company behind the network offers a variety of services, it
at different times in different nations. As Jang et al. point out themselves (o.c., p. 138-139), countries that introduced it
late in the process experienced shorter penetration periods, because in the meantime communication costs had fallen
dramatically, so that it was easier to achieve critical mass.
- 34 -
may prove necessary to filter out revenues that are not driven by network externalities (and thus do
not obey Metcalfe’s law). Tencent, for example, derives substantial revenues from e-commerce
transactions (o.c.). The available data may not always allow us to filter out such revenues
sufficiently accurately.
Data on usage would seem intrinsically better suited, but can also have their downsides. By simply
tallying the number of users, as Madureira et al. do, one does not in fact capture their full
willingness to pay; one measures the number of economic agents whose willingness to pay exceeds
the cost involved 31. There may, however, be ways to do better. One possibility would be to exploit
on-line product ratings, as Liu et al. (2015, p. 680) do in their paper on on-line video games: “we
propose online product ratings as an emerging data source to directly study network externalities.
The literature shows that these ratings are a useful measure of consumption experience and
satisfaction, which directly relate to consumption utility that underlies network externalities […].
As various Internet mediums make online product ratings widely available, they provide an exciting
opportunity for research on network externalities”. However, given that not all users post ratings,
their representativeness could be called into question 32. Also, as Liu et al. themselves point out, “a
larger installed base might affect user ratings through a social proof effect (i.e., there are a lot of
people playing, so it must be good)” (o.c., p. 689).
31
To be entirely correct: Suarez (2005), in his multinomial logit model, measures not only whether mobile operators
deemed a given technology sufficiently useful, but also whether they saw it as more useful than the two other options.
Also, revenue-based indicators will typically not capture consumers’ full WTP either; unless, that is, under a pay-asyou-go revenue model (or first-degree price discrimination).
32
One of Liu et al.’s criteria for inclusion of a game in their dataset is a minimum of 100 ratings. Their final dataset
consists of 1,695 individual user ratings posted on third-part website Gemspot.com (over a period of 49 months).
- 35 -
Brighter prospects would therefore seem to lie in the use of (individual-level) big data of another
nature. Björkegren (2015) has comprehensive transaction data from Rwanda’s dominant mobile
phone operator covering a period of 4.5 years and no fewer than 5.3 billion transaction records in
total. Crucially, the billing structure at the time of study is “empirically convenient” (o.c., p. 15):
99% of the accounts are prepaid, so that “the person placing a call pays for it on the margin, by the
second” (o.c., p. 3). In other words, the calling decision reveals that the caller is willing to pay, at
least for the cost of the call, allowing Björkegren to estimate the utility of adopting a phone based
on its eventual usage (in terms of call duration) 33. This would seem to be an ideal setting in which
to test Metcalfe’s law, particularly since, as pointed out in Section 4, Björkegren’s data also allows
him to control for coverage. But, all in all, other lines of research deserve encouragement too.
In a world of networks, there is obviously value in examining network value from every possible
angle.
Acknowledgements
I am indebted to Antonio Madureira for providing me with details about his dataset and to two
anonymous referees, the editor, an associate editor, and my colleagues Marc Jegers and Benny Geys
for comments on earlier versions of this paper. Special thanks go to co-editor Tobias Kretschmer
for pushing me to take my paper the extra mile.
33
Björkegren’s approach is similar to that of Ryan and Tucker (2012), who study the adoption of a desk-top based,
internal videocalling technology in a large multinational investment bank. Ryan and Tucker have detailed data on all
2,169 potential adopters in the firm, from the time of installation up to the network’s steady state more than three years
later, and end up using data on 463,806 calls. Because the bank covered all monetary costs of adoption and usage, the
authors infer the utility of the videoconferencing system for employees from the number of calls made.
- 36 -
References
Björkegren, D. (2015). The adoption of network goods: evidence from the spread of mobile phones in Rwanda. Mimeo,
Brown University, November 23 <http://dan.bjorkegren.com/files/danbjork_rollout.pdf>.
Boudreau, K. J., and Jeppesen, L. B. (2012). Competing with a crowd: informally organized individuals as platform
complementors. Mimeo <http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1812084>.
Boudreau, K. J, and Jeppesen, L. B. (2015). Unpaid crowd complementors: the platform network effect mirage.
Strategic Management Journal, 36(12): 1761-1777 (DOI: http://dx.doi.org/10.1002/smj.2324).
Briscoe, B., Odlyzko, A., and Tilly, B. (2006). Metcalfe’s law is wrong. IEEE Spectrum, 43(7):34-39
<http://spectrum.ieee.org/jul06/4109>.
Choi, H. and Lee, B. (2012). Examining network externalities and network structure for new product introduction.
Information Technology and Management, 13(3):183-199 (DOI: http://dx.doi.org/10.1007/s10799-012-0125-x).
Fuentelsaz, L., Garrido, E., and Maicas, J. P. (2015). A strategic approach to network value in network industries.
Journal of Management, 41(3):864-892 (DOI: http://dx.doi.org/10.1177/0149206312448399).
Gomez-Herrera, E., Martens, B., and Turlea, G. (2014). The drivers and impediments for cross-border e-commerce in
the EU. Information Economics and Policy, 28:83-96 (DOI: http://dx.doi.org/10.1016/j.infoecopol.2014.05.002).
Jang, S.-L., Dai, S.-C., and Sung, S. (2005). The pattern and externality effect of diffusion of mobile
telecommunications: the case of the OECD and Taiwan. Information Economics and Policy, 17(2):133-148 (DOI:
http://dx.doi.org/10.1016/j.infoecopol.2004.05.001).
Karaçuka, M., Çatik A. N., and Haucap, J. (2013). Consumer choice and local network effects in mobile
telecommunications in Turkey. Telecommunications Policy, 37(4–5):334–344 (DOI:
http://dx.doi.org/10.1016/j.telpol.2012.10.005).
Katz, M. L., and Shapiro, C. (1994). Systems competition and network effects. Journal of Economic Perspectives,
8(2):93–115 (DOI: http://dx.doi.org/10.1257/jep.8.2.93).
Liikanen, J., Stoneman, P., and Toivanen, O. (2004). Intergenerational effects in the diffusion of new technology: the
case of mobile phones. International Journal of Industrial Organization, 22(8-9):1137–1154 (DOI:
http://dx.doi.org/10.1016/j.ijindorg.2004.05.006).
Liu, Y., Mai, E., and Yang, J. (2015). Network externalities in online video games: "an empirical analysis utilizing
online product ratings, Marketing Letters, 26(4): 679–690 (DOI: DOI 10.1007/s11002-015-9390-x).
Madureira, A., Baken, N., and Bouwman, H. (2011). Value of digital information networks: a holonic framework.
Netnomics, 12(1):1–30 (DOI: http://dx.doi.org/10.1007/s11066-011-9057-6).
Madureira, A., den Hartog, F., Bouwman, H., and Baken, N. (2013). Empirical validation of Metcalfe’s law: How
Internet usage patterns have changed over time. Information Economics and Policy, 25(4):246-256 (DOI:
http://dx.doi.org/10.1016/j.infoecopol.2013.07.002).
McIntyre, D. P. (2011). In a network industry, does product quality matter? Journal of Product Innovation
Management, 28(1): 99-108 (DOI: http://dx.doi.org/ 0.1111/j.1540-5885.2010.00783.x).
Metcalfe, B. (2013). Metcalfe's law after 40 years of Ethernet. Computer, 46(12):26-31 (DOI:
http://dx.doi.org/10.1109/MC.2013.374).
Ohashi, H. (2003). The role of network effects in the U.S. VCR market, 1978-1986. Journal of Economics &
Management Strategy, 12(4):447–496 (DOI: http://dx.doi.org/ 10.1111/j.1430-9134.2003.00447.x).
Ross, P. E. (2003). 5 commandments: technology laws and rules of thumb. IEEE Spectrum, 40(12):30-35 (DOI:
http://dx.doi.org/10.1109/MSPEC.2003.1249976).
- 37 -
Ryan, S. P., and Tucker, C. (2012). Heterogeneity and the dynamics technology adoption. Quantitative Marketing and
Economics, 10(1): 63–109 (DOI: http://dx.doi.org/10.1007/s11129-011-9109-0).
Schneider, L. (2014). Firm value in emerging network industries. Information Economics and Policy, 26:75-87 (DOI:
http://dx.doi.org/10.1016/j.infoecopol.2014.01.001).
Sobolewski, M., and Czajkowski, M. (2012). Network effects and preference heterogeneity in the case of mobile
telecommunications markets. Telecommunications Policy, 36(3):197–211 (DOI:
http://dx.doi.org/10.1016/j.telpol.2011.12.005).
Suarez, F. (2005). Network effects revisited: the role of strong ties in technology selection. Academy of Management
Journal, 48(4):710-720 (DOI: http://dx.doi.org/10.5465/AMJ.2005.17843947).
Van Hove, L. (2014). Metcalfe's law: not so wrong after all. Netnomics, 15(1):1-8 (DOI:
http://dx.doi.org/10.1007/s11066-014-9084-1).
Van Hove, L. (2016). Metcalfe’s law and network quality: and extension of Zhang et al., Journal of Computer Science
and Technology, 31(1):117-123 (DOI: http://dx.doi.org/10.1007/s11390-016-1615-9).
Weiergräber, S. (2014). Network effects and switching costs in the US wireless industry, Discussion Paper No. 512,
Governance and the Efficiency of Economic Systems (GESY), University of Mannheim.
Zhang X., Liu J., and Xu Z. (2015). Tencent and Facebook data validate Metcalfe’s law. Journal of Computer Science
and Technology, 30(2):246–251 (DOI: http://dx.doi.org/10.1007/s11390-015-1518-1).
- 38 -
Appendix 1 – Absolute or relative does make a difference
In Section 3, I explain that Madureira et al. claim that replacing n by x in Metcalfe’s law only
affects the proportionality factor and I argue that this claim is incorrect. That is, in their setup it is
correct, but only because they make an unrealistic assumption. If we start from equation (1) – to
which I have now, for reasons that will become clear below, added subscripts i (which refer to
observations on, say, the country level) – and if we replace x by n/I – see eq. (3) – we successively
obtain:
y c,i = k c,M x i2
€
n
y c,i = k c,M ( i ) 2
I
y c,i =
€
(1)’
k c,M 2
n .
I2 i
(A1.a)
A comparison with eq. (a) in the main text shows that indeed only the proportionality factor has
€
changed. However, this is only because
Madureira et al. assume, for reasons that I set out in
Appendix 2, that I, the potential maximum size of the DIN, is the same for all observations 34. This
obviously clashes with both reality and their data. In the Eurostat data that Madureira et al. use for
xi – the fraction of enterprises/households that have access to the Internet, both at the national level
and at lower/higher levels (o.c., p. 249) – I is clearly not constant. Where the country-level
observations are concerned, Ii varies from one country to the next (and for that matter, although to a
lesser extent, from one year to the next). And when the observation relates to the Portuguese
construction sector in the year 2004 – to use the same example as Madureira et al. – the benchmark
is the total number of enterprises in that sector, in that country and in that year.
34
They do so when deriving their test of Briscoe’s law – see equations (3) and (4) in Section 2 – but the implications
extend to their test of Metcalfe’s law.
- 39 -
In short, contrary to what Madureira et al. assume, in their dataset the potential maximum size of
the DIN is not constant. In such a setting – that is, when xi = ni/Ii instead of ni/I – one cannot simply
replace n by x, as the impact is no longer limited to the proportionality factor. This can be
demonstrated as follows:
y c,i = k c,M x i2
€
(1)’
n
y c,i = k c,M ( i ) 2
Ii
y c,i =
€
k c,M 2
n .
Ii2 i
(A1.b)
A comparison with eq. (a) highlights that, in the (realistic) case of a non-constant I, testing
€
Metcalfe’s law with relative or absolute
data does make a difference. Q.E.D.
Appendix 2 – Formulating Metcalfe’s and Briscoe’s laws in relative terms
When elaborating on Question 3 – Should one use absolute numbers or relative network sizes? – in
the main text, I point out that Madureira et al.’s answer to this question is inconsistent, in that they
test Metcalfe’s law with relative and Briscoe’s law with absolute numbers. In this appendix, I show
(1) that just as for Metcalfe’s law in Appendix 1, working with a constant I in Briscoe’s law is
simply not realistic, (2) that, contrary to what Madureira et al. think, Briscoe’s law can be
formulated in relative terms, and (3) that Madureira et al. make a mistake when transposing
Metcalfe’s law from absolute to relative terms.
As pointed out in Section 2, Madureira et al. are convinced that in order to test Briscoe’s law they
need absolute network sizes n. Unfortunately, the Eurostat dataset only gives them relative numbers
x. To get around this problem, Madureira et al. “estimate n by multiplying x with the number of
Internet Protocol (IP) addresses advertised in 2010 (I ≈ 2.2 * 109 […])”; that is, “the largest
- 40 -
realistically achievable network size” (o.c., p. 250). However, just as for Metcalfe’s law, here too a
constant I just does not make sense. For country-level observations, the maximum network size in,
say, Portugal, is not the size of the global World Wide Web, but the total number of Portuguese
enterprises or households, depending on which capability is analysed. Hence, the absolute numbers
that Madureira et al. compute with a constant I are wide of the mark. The operation also completely
alters the relative levels between countries. To give a salient example, let us compare Portugal with
tiny Luxembourg, say, in 2005. In that year, Internet penetration among enterprises was 92 per cent
in Luxembourg and 86 per cent in Portugal 35. Thus, in Madureira et al.’s calculations the number of
Luxembourg enterprises with Internet access is xLU * I = 0.92 * 2.2 * 109 = 2.0 * 109. This is not
only a totally unrealistic estimate for a country the size of Luxembourg, but in addition also some
7% more than in Portugal. In reality, according to Eurostat data, in 2005, there were 23,194 active
enterprises in Luxembourg vs. 874,163 in Portugal 36. This gives, respectively, 21,338 and 751,780
enterprises with Internet access; that is, 35 times fewer in Luxembourg than in Portugal.
My second point is that Madureira et al. are mistaken in their belief that they have no choice but to
use absolute numbers; cf. “[i]n model (4), we must use absolute values due to the ln function
present” (o.c., p. 253; my italics). Given their equation (4), this is correct. However, in the relative
form of Briscoe’s law, ln(ni) should not be scaled as ln(ni/I) – as is implied by Madureira et al.’s
equation - but rather as ln(ni)/ln(I). Indeed, if the utility that users derive from a network of size ni is
ln(ni), then the maximum utility – the utility they would derive from a network that is adopted by all
I members of the population – is ln(I). With this type of scaling, it is perfectly possible to use
relative network sizes.
35
Source: Eurostat.
36
Source: Eurostat, Structural Business Statistics, <http://ec.europa.eu/eurostat/web/structural-business-statistics> (last
accessed 21.08.15).
- 41 -
Finally, let me point out that Madureira et al. make a mistake when transposing Metcalfe’s law
from absolute to relative terms. As Madureira et al. themselves point out (and as is also explained in
Section 2), the first n in n2 and in nln(n) “comes from the fact that there are n members” (o.c.,
p. 248). Hence, when formulating Metcalfe’s law in relative terms, there is no need to scale this n.
Only the second term, which represents the individual utility, should be scaled. Computing the
aggregate value of the network then only requires a simple multiplication with the number of
members, just like in the absolute version of the law. Algebraically, the correct relative version of
Metcalfe’s law – at least in Madureira et al.’s framework 37 – is thus not
y c,i = k c,M x i2
y c,i = k c,M
€
(1)’
ni ni
,
Ii Ii
(A2.a)
ni
Ii
(A2.b)
but rather
€
y c,i = k c,M n i
y c,i = k c,M n i x i .
€
€
37
In the main text I argue, in response to Question 5, that the x2 in equation (1)’ should in fact be x.
- 42 -
(A2.c)