by Gene Callahan
On his blog, Daniel Kuehn notes that “relations in economies are not stable.” In fact, we can go further: Relations in the social sciences are not stable. As an illustration, consider Zipf’s Law as applied to city size.
In 1700, London had about 575,000 people. According to Zipf’s Law, the next-sized city should have had about 280,000 or 290,000 thousand. What was the actual size of the second largest city? As far as I can determine, it was Norwich, with a population of about 30,000. (My source for the population figures is 1688: The First Modern Revolution.) Zipf’s “Law” is off by a factor of about ten in this instance.
What I suspect is that there is some historical circumstance that leads to Zipf’s Law applying to city size in recent centuries, which was not present in 1700. As political scientist Terry Nardin put it: “Generalizations about how people usually behave are not scientific generalizations about a truly time-independent class of phenomena; they are more or less well-disguised descriptions of customs specific to a particular historical situation.”
ThinkMarkets 
If we think of why we have cities at all, one answer stands out above the rest: the division of labor. And Adam Smith pointed out that the division of labor is limited by the extent of the market. At more subsistence levels, the “extent of the market” – the demand for anything other than the bare essentials – probably didn’t sustain cities in the same way. In other words, once you get to a certain size distribution you might not be looking at a violation of Zipf’s law so much as a simple left-censored sort of truncation of the distribution you’re applying it to!
I wonder if any work has been done on more recent size distributions for societies closer to subsistence – it would be interesting to observe the emergence of the relation over time.
Back in college, I did a paper for a short course on the economy of Hong Kong, where I looked at the size distribution of cities in China and talked about its implications for the re-integration of Hong Kong into China.
I completely forget exactly what I ultimately concluded, but one thing I do remember discussing is the fact that the deliberate urbanization by Chinese central planners distorted the Zipf’s law relation in a way that you don’t see in urban networks that grow more organically. It may very well be that London has certain monopoly priveleges that are driving this too. This is very interesting stuff Gene, thanks!
I’m getting out of my depths here… but isn’t this sort of relation supposed to have scaling properties? Would it help to look at a sub-region of England, or perhaps Western Europe as a whole?
I think you’re right, generally.
I will note that if relations in economies were stable, we wouldn’t have economies. Unstable relationships are typical of scale free networks, while stable relationships are typical of hierarchical networks. The latter are found in organizations, like families and firms, while the former are found in self-organizing processes.
There are a variety of requirements to achieve the conditions of self-organization. Among them is density. If one does not find Zipf’s laws at work, one does not have self-organization going on. In the case given here, there may have been local self-organization, but not global self-organization – local free markets, but not global. Self-organization also requires modularity. If one has highly interacting cities, one has the conditions of modularity. Zif’s law begins to take effect. If one has a highly fragmented situation, the conditions are not met, and one wouldn’t expect to find Zipf’s law at work.
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“I’m getting out of my depths here… but isn’t this sort of relation supposed to have scaling properties? Would it help to look at a sub-region of England, or perhaps Western Europe as a whole?”
If we took Western Europe in 1700, then we get a second city that’s too big — Paris was only 5 or 10% behind London.
Gene, what if you just ignore the cities whose sizes don’t fit?
The “problem” – if that is what it is – is that humans have free will. The “laws” of social science are ex post facto observations, not invariant predictions. Humans are not billiard balls.
She – “Let us move unto Londonne where be more Opportunitie.”
He – “Would but that we could, but fain tis a corruption of Zipf’s Law. Thus we muste remayne here.”