Stephen Laniel’s Unspecified Bunker

This is the most I’ve enjoyed an economics book since Sam Bowles’s Microeconomics. Like Bowles’s book, it offers crystal-clear writing attached to non-scary mathematics. It’s less wide-ranging than Bowles’s book, but the amount of amazement per page is comparable. In the case of The Spatial Economy, the amazement comes from how much mileage they get from very simple models.

They’re trying to answer some questions that lay dormant in economics for many years, namely: why do cities form where they do? How important are first-mover advantages for cities? (I.e., what would have to happen for New York to no longer be top dog?) How do national boundaries affect the distribution of industries within a given nation? How does trade liberalization (for instance, NAFTA) change that distribution? How can the same methods used to study cities help us understand international trade?

One way to start modeling cities is to assume a built-in tension in their size: as a city grows more dense, it draws more businesses to it, but also pushes some out into the suburbs. It draws them because they want to be where the customers are (”forward linkage”) and also want to be where their suppliers are (”backward linkage”). It pushes them away because of land rents: the denser the city becomes, the higher the rent.

One of the most interesting parts of the book is that they don’t actually model land rents at all. In essence, all of their models derive from varying assumptions about transportation costs, forward linkages, and backward linkages. From these meagre beginnings, they get quite stunning results. One particularly charming chapter evolves their model through time as the population increases and companies move out into the hinterlands; cities merge and shift and grow, all from a quite simple dynamic, with a few assumptions about costs and market structure.

At every step, they go through basically the same motions. First, derive the wage at any point in the city or its suburbs, using some fairly unrealistic assumptions about zero-profit businesses. Do this for a second, nearby city as well. Deflate the wages in both cities by something akin to the Consumer Price Index; this gives you the real wage in both places. Then divide one by the other to get the ratio of real wages; if this number is not equal to 1, then employees have an incentive to move from one city to the other. Evolve the model through time until the real wage ratio equals 1 everywhere.

Normally this wage-ratio formula is rather complicated and analytically intractable. They simplify things a great deal by coming at the problem from another direction: assume that all cities are identical at the start — in particular, that they all have identical populations. Now take the derivative of the wage ratio in the neighborhood of this “symmetric” configuration. (It took me a while to figure out that “symmetric” meant “equal everywhere”; it also took me a while to realize that when they say “linearize,” they mean “take the derivative.” Both usages make sense; I’m just slow.) If a small movement of employees from one city to another leads to a cascade, wherein a flood of people follow in pursuit of higher wages, then the symmetric equilibrium is not stable; the authors call this the “break point.” Symmetry breaking leads to the formation of a new city. Then the question becomes: is this new equilibrium with unequally sized cities sustainable, or will it also fall apart when a few employees pick up and leave? If this “asymmetric equilibrium” is stable for certain values of the parameters, then the authors label this bundle of parameters the “sustain point.”

The math is not hard, though the sheer volume of symbols is imposing. There’s some calculus — a derivative here, an integral there — but it’s mostly just a lot of algebra, and a lot of tricks for simplifying complicated expressions. The authors are good writers, so they refuse to use equations where words will say the same thing.

They’re also straightforward about the problems their models face. One of these problems is particularly intriguing to me, namely the well-known fact that the distribution of city sizes approximately follows a power law. That is, the second-largest city tends to be 1/2 the size of the largest city, the third-largest tends to be 1/3 the size, and so on. This discovery is most famously associated with Herb Simon’s paper “On A Class Of Skew Distribution Functions”. Krugman, at least, hates Simon’s presentation, as he’s said elsewhere, so The Spatial Economy recapitulates Simon’s proof in a much simpler way. The trouble for Fujita, Krugman, and Venables is that their models don’t lead to anything close to a power law in city sizes. The authors are perfectly straightforward about this rather large hole in their results.

You have to think about methodology when you’re reading a book like this. In particular: if the assumptions are extremely simplistic and wildly unrealistic, then how valuable are the conclusions that spring from those assumptions? The answer, I think, is: very valuable indeed, because the unrealistic bits don’t alter the main thrust of the argument. The stability of equilibria probably wouldn’t be changed if we got rid of the zero-profit condition, for instance. Likewise, if the wage ratio had to get really large before employees would move (call this “wage inertia”), that probably would only slow down the speed at which new cities grow; it likely wouldn’t affect the existence or location of equilibria themselves.

For a book with a few hundred equations, The Spatial Economy is remarkably readable. Anyone who’s interested in economics, urban growth, international economics, or simple evolutionary-game-theory models will find this book indispensible and charming.