The functional form of network effects

Abstract

Metcalfe’s Law states that aggregate network value is proportional to the square of network size, and that implies the individual user’s utility is a linear function of network size. This paper explores the functional form of network effects in a simple model of a telephone network, and finds that it can be linear, but only under strong conditions. The paper establishes weaker conditions under which a representative subscriber’s utility is a linear function of network size. When these linearity conditions are not satisfied, we show that under some assumptions, the value to a typical subscriber will be an s-shaped function of network size.

Introduction

The aim of this paper is to explore the functional form of the relationship between utility and the size of a network. The concept of network effect (or network externality) is well established in the economics literature. It describes the benefit that accrues to the user of a product or service because he or she is one of many who use it. Following seminal contributions by Rohlfs (1974), David (1985), Farrell and Saloner (1985), Katz and Shapiro (1985) and Arthur (1989), the concept has found widespread application—see David and Greenstein (1990) for an excellent early survey, a special issue of the Journal of Industrial Economics (Gilbert, 1992), a special issue of the International Journal of Industrial Organization (Economides and Encaoua, 1996), and the Internet Site for the Economics of Networks managed by Economides (1998).

How does the value of a network relate to its size? In trying to answer this question, we need to be clear about two points. First, what sort of network we are talking about? And second, are we talking of aggregate value (across all members of the network) or the value enjoyed by an individual user of the network?

Taking the first question, it is conventional in the literature to distinguish three different types of networks, where the relationship between aggregate value and size of network takes very different forms. The first is the broadcast network, connecting one broadcaster to many in the audience. This is the least remarkable form of network, and indeed some would argue it is not really a network at all. To a first approximation, the aggregate value of this network to the broadcaster is given by $\text{Sarnoff’s Law}$, which states that the value of the network is proportional to the size of the audience (Reed, 1999).

The second type of network is the two-way communications network, such as the telephone, e-mail or fax. Here, to a first approximation, the aggregate value of the network is given by $\text{Metcalfe’s Law}$, which states that the aggregate value of the network is proportional to the square of the number of users (Gilder, 1993). The rationale for Metcalfe’s Law is simple enough. When there are n users in the network, the number of pairwise conversations is _nC₂=n(n−1)/2. If each of these conversations is of equal value, and of value to caller and receiver alike, then the total value is proportional to n(n−1), or for large n, proportional to n².

The third type of network is sometimes called a Group-Forming Network (Reed, 1999). Here, the network is used to form communities such as online ‘chat rooms’, which take many forms and sizes. Roughly speaking, the value of such a network is given by $\text{Reed’s Law}$, which states that the aggregate value of the network is proportional to the number of non-trivial groups that can be constructed from n users. When there are n users, we can construct a total of 2ⁿ−n−1 non-trivial groups of sizes between two and n members.¹ If n is large, then aggregate value according to $\text{Reed’s Law}$ is proportional to 2ⁿ.

So far, we have just talked about the aggregate value of a network, to all users. But when we analyse individual decisions where network effects are important, the key relationship is between network size and individual utility (or value), rather than aggregate value. Following Sarnoff’s Law, if aggregate value is proportional to n, then individual utility in a broadcast network is a constant (independent of n)—except in as much as a popular broadcaster is better resourced, and therefore able to offer higher quality content. Turning to a two-way communications network, Metcalfe’s Law assumes that each individual values each potential conversation equally, and hence individual utility is proportional to n—or strictly speaking, proportional to n−1. This is a strong assumption and we shall return to it below, to see what happens when we relax it. Finally, in a group-forming network, $\text{Reed’s Law}$ assumes that each user attaches equal value to each potential group (s)he could belong to, and hence individual utility is proportional to 2ⁿ⁻¹. This follows because we can form 2ⁿ⁻¹ possible groups from n users all of which include a particular user (say 1).² This is an even stronger assumption.

These three ‘laws’ are useful as they give us an order-of-magnitude idea of how network value relates to network size. However, several commentators have noted that Metcalfe’s Law is misleading, because it is actually quite unlikely that users attach the same value to communication with each other network user. In particular, the first members of a network may add the most value, while later entrants add less in value.³ As a result, individual utility is not in reality proportional to n, but grows less than proportionately. More generally, the user will pay more attention to the composition of the network of other users than the size per se. In short, the value of a network depends in a more complex way on the set of users, and not just the number of users. If that critique applies to Metcalfe’s law, then it applies even more so to Reed’s Law, since it is most unlikely that the network user would attach equal value to each putative group.

The aim of this paper is to look in more detail at the functional form of the relationship between individual utility and the size of the network. In what follows, we shall focus only on the second type of network: the two-way communications network, such as telephone and e-mail. Section 2 sets out a model of a telephone network, and Section 3 shows that there are two sets of conditions under which the value of the network to an individual user is a linear function of network size (n), but both require quite strong assumptions. Section 4 then demonstrates some weaker conditions under which the utility enjoyed by an average (or representative) user is a linear function of network size. These are more palatable, but are unlikely to apply in all circumstances. What can we say when none of these conditions are satisfied? Section 5 shows that under some quite general conditions, the value to the typical user is an s-shaped function of network size (n).

Section 6 asks why the form of this relationship matters. In models with network effects, the character of the results obtained depends on the precise functional form of this relationship. Moreover, from a business viewpoint, some of the entrepreneurs who started up dotcoms might have been more cautious if they had realised that network effects show diminishing returns. Finally, when the functional form for pioneering network users is very different to that for late adopters, then it is not surprising to find that these two groups have very different attitudes about the growth of the network. These differences are especially pronounced when the network is subject to congestion costs. Section 7 concludes.