I just read a terrific little book by Tom Standage, The Victorian Internet, a history of the rise of the telegraph. The story is, just as he claims, a great metaphor for the rise of the Internet; indeed, it is a stunningly close parallel in many respects. Standage uses the story in part to caution against the naive view that the Internet will eliminate nationalism, foster world peace, or promote a new golden age of culture. But the story also offers a more mundane cautionary lesson, suggesting that we should be skeptical about some of the enthusiastic claims about the rules under which the "new economy" works.

One of the key propositions of the "new economy" view is the idea that networks are inherently a source of very strong increasing returns; enthusiasts like Kevin Kelly like to invoke "Metcalfe's Law", which says that the usefulness of a network is proportional to the square of the number of people it connects, because that is the number of possible directions of communication. If you think that something like Metcalfe's Law actually applies, it has dramatic implications for economic dynamics. It suggests, for example, that networks are hard to get started, even when the technology is there, because it isn't worth investing in a connection unless enough other people are already connected. But once a network passes the tipping point at which connecting starts to happen, it should experience explosive growth, because each successive connection will be more valuable than the one before.

A telegraph example actually demonstrates the force of this argument quite nicely. Imagine a nation consisting of a number of equal-sized cities, each with 100 people. (Numbers are chosen for expository convenience, not realism!) If someone builds a telegraph line connecting two of the cities, it will make 10,000 two-way communications possible (from each of the inhabitants of one city to each of the inhabitants of the other). But when a third city is added to the network, this adds 20,000 possible communications, because the new city can communicate with the 200 people already in the network. And the fourth connection adds 30,000 possible links.

It's easy to see that in this case investors might be doubtful about the potential business on a single telegraph line; only once several lines had already been built would the economics of building still more become favorable, and then they would become ever more favorable.

The history of the telegraph, however, doesn't actually look that way. There was explosive growth, all right: the U.S. telegraphic network expanded 600-fold between 1846 and 1852. But the pause between when the technology was ready and the commercial applications began was negligible: as soon as an experimental line between Baltimore and Washington was up and running, investors were up and running too.

Why didn't investors hesitate? For one obvious reason: cities were not all the same size, and they could start by building lines connecting the biggest cities. A line between New York and Philadelphia already connected a large number of potential customers. And conversely, later lines did not necessarily add more potential communications than the existing ones: they connected to a bigger existing base, but they ran to smaller cities. In short, the inequality of city sizes meant that the network was not all that subject to increasing returns after all.

Of course, this all depends on the distribution of city sizes; but we know something about that. Somewhat mysteriously (see my book The Self-Organizing Economy) the size distribution of cities in the United States has long been quite well described by the "rank-size rule": the second city has half the population of the first, the third 1/3 the population, and so on. So imagine a country whose biggest city has 120 people, the next 60, the third 40, and so on. And now ask how many possible communications are added when each city enters the network, assuming - as is reasonable - that cities enter in size order.

Well, the connection between the two biggest cities will create 7200 (120 × 60) possible communications. Adding the third city to the network will add another 7200 (180 × 40). Then the network starts to run into diminishing returns: the next connection adds 6600 possible communications, the one after that 6000, and so on. The size of the base to link to keeps getting bigger, but the size of the next city keeps getting smaller, and the latter effect dominates.

The point is not that networks necessarily face diminishing rather than increasing returns; rather it is that increasing returns are by no means guaranteed. Against Metcalfe's Law must be set DeLong's Law (after Berkeley's Brad DeLong, who has made this point several times): in building a network, you tend to do the most valuable connections first. Is the net effect increasing or diminishing returns? It can go either way.

Increasing returns are, of course, more fun to think about - but that is itself a reason for caution. "New economy" types have a tendency to tell great stories, both about the economy and about themselves. Alas, the fact that a story is entertaining doesn't mean that it is true.