## January 15, 2011

### A Fermi Problem in Western Pennsylvania

Attention conservation notice: A 2000-word attempt to reduce decades of painstaking empirical work and careful theorizing in economic geography to a back-of-the-envelope calculation; includes a long quotation from a 19th century textbook of political economy. An outtake from a post that turned into a paper-in-progress, posted now because I'm stuck on a proof in another paper, and don't want to work on writing the next problem set for 402.

Physicists are fond of a kind of rough estimation exercise they call "Fermi problems", since our folklore attributes them to the great Enrico Fermi. A classic instance is the one I first encountered, as a physics undergrad at Berkeley: how many piano tuners are there in the East Bay? Well, there are about a million people living around the eastern shore of the San Francisco Bay, i.e., on the order of 106. How many people are there per piano? 10 per piano seems high, but 10,000 per piano seems low, say 103 per piano. How often a piano needs to be tuned? Clearly not every day, or even every week, but also not once a decade, so something like once a year. Thus the East Bay needs about 103 piano-tunings per year. How quickly can a piano be tuned? Probably in less than a week but more than an hour, so something like a day, or about 10-2 years. So there should be about 10 piano tuners in the East Bay. The professor, having elicited these numbers, then told us to "look it up in the phone book"; having pulled the same stunt myself since then, I can tell you that any number between 5 and 50 will be declared "the right order of magnitude".

Suppose we were interested not in greater San Francisco but Stewart Township, Pennsylvania, the site of Fallingwater: how many piano tuners does it have? Stewart Township has a population 7*102, and our reasoning above says it's got something like one piano, and so demands one day of piano-tuning per year. What does the piano tuner do the rest of the time? They could be an ordinary citizen, who only becomes a piano tuner once a year when it's called for. Or it could be that Stewart Township shares a specialist piano-tuner (or three) with the 6*105 other people of the Laurel Highlands. Since tuning a piano is a reasonably demanding skill, it's much more likely that it's done by a specialist.

What goes for piano tuners goes for other specialists. Most people need their skills rarely, or need only a small fractional share of their output, or need it only indirectly. (You want to hear piano music, so the pianist needs to find a tuner.) Small settlements cannot keep them occupied full time. But there are fixed costs to specialist services --- tools, of course, but more essentially the time and effort needed to acquire, maintain and develop the specialist's skills. It is more efficient for one specialist to serve many people, thereby spreading the fixed costs over many customers, which rules out the part-time amateur in each village. (More exactly, since the local amateurs lack the skills to do the job well, they can only compete with the specialists by being much cheaper, or if customers can't tell the difference.) This will tend to divide up a dispersed population into regions served by one or another specialist; increasingly specialized skills will require increasing large population bases.

It is not required by this argument that the specialists be located near each other; but it tends to happen. After all, they need each others' services, and being located near each other reduces transport costs for them, and there will often be economies of scope in setting up specialists near each other. (If everyone needs to take or make freight deliveries, they can share one set of loading docks, etc.) If demand is high enough to support multiple specialists, there can be "agglomeration economies": they can begin to benefit from each other by sharing information and knowledge, creating a local market for their specialist suppliers, etc. There is a famous passage from Alfred Marshall (in 1890) which is traditionally trotted out on these occasions, and far be it from me to break with tradition:

When an industry has thus chosen a locality for itself, it is likely to stay there long: so great are the advantages which people following the same skilled trade get from near neighbourhood to one another. The mysteries of the trade become no mysteries; but are as it were in the air, and children learn many of them unconsciously. Good work is rightly appreciated, inventions and improvements in machinery, in processes and the general organization of the business have their merits promptly discussed: if one man starts a new idea, it is taken up by others and combined with suggestions of their own; and thus it becomes the source of further new ideas. And presently subsidiary trades grow up in the neighbourhood, supplying it with implements and materials, organizing its traffic, and in many ways conducing to the economy of its material.

Again, the economic use of expensive machinery can sometimes be attained in a very high degree in a district in which there is a large aggregate production of the same kind, even though no individual capital employed in the trade be very large. For subsidiary industries devoting themselves each to one small branch of the process of production, and working it for a great many of their neighbours, are able to keep in constant use machinery of the most highly specialized character, and to make it pay its expenses, though its original cost may have been high, and its rate of depreciation very rapid.

Again, in all but the earliest stages of economic development a localized industry gains a great advantage from the fact that it offers a constant market for skill. Employers are apt to resort to any place where they are likely to find a good choice of workers with the special skill which they require; while men seeking employment naturally go to places where there are many employers who need such skill as theirs and where therefore it is likely to find a good market. The owner of an isolated factory, even if he has access to a plentiful supply of general labour, is often put to great shifts for want of some special skilled labour; and a skilled workman, when thrown out of employment in it, has no easy refuge. Social forces here co-operate with economic: there are often strong friendships between employers and employed: but neither side likes to feel that in case of any disagreeable incident happening between them, they must go on rubbing against one another: both sides like to be able easily to break off old associations should they become irksome. These difficulties are still a great obstacle to the success of any business in which special skill is needed, but which is not in the neighbourhood of others like it: they are however being diminished by the railway, the printing-press and the telegraph.

What we have argued ourselves into, on the basis of little more than a realization that comparatively high fixed costs matter, is to think that there should be spatial clumps of economic activity, where we find a lot of specialists, and that these clumps should come in grades, with more clumps containing less-specialized enterprises with less-increasing returns, and fewer clumps containing the more-specialized, more-increasing-returns enterprises. We call the clumps "towns" and "cities". (And indeed, if I can trust my searching, the nearest piano tuner to Fallingwater is located in the town of Connellsville, population 9*103.) The gradations of the clumps form the "hierarchy of urban places", an idea which has been familiar to economic geographers since at least the work of Christaller and Lösch in the 1930s. It implies that there isn't just quantitatively more economic activity in a bigger settlement, but generally different kinds of activity. Stewart Township is not a scaled-down version of Connellsville, which is not a scaled-down Pittsburgh, which is not a scaled-down Chicago or New York.

Moreover, the argument is more generally than just specialized services. It turns on having low marginal costs (a day of a heart surgeon's time to do an operation) compared to high fixed costs (ten years of training to become a heart surgeon). But the fixed costs don't have to be time, and similar logic will work for just about any industry with increasing returns, if transport costs are not prohibitive. So as we move up the hierarchy of urban places, we should find not only more, and more specialized, service providers, but also more industries with increasing returns, and, you should forgive the expression, increasingly increasing returns at that. One way industries come to have increasing returns is by being relatively capital- (as opposed to labor-) intensive, which will tend to increase the output per worker.

All of the above applies with great force to creating and disseminating new abstract, formalized, discursive knowledge. It is highly specialized, the fixed costs of entering are very high, economies of scope are important, the effects of agglomeration are important, and the cost of transporting the finished product is zero. All else being equal, we should expected knowledge production to be concentrated towards the top of the urban hierarchy.

All of this is, as I said, very standard stuff in economic geography and urban and regional economics. I learned much of it at (pretty literally) my father's knee, and it was old when he learned it from his teachers. (There is even a version of it in ibn Khaldun's Muqaddimah, from 1377: see ch. 5, sec. 15--22 [pp. 314--318 of the Rosenthal/Dawood translation] on the crafts, and again ch. 6, sec. 7--8 [pp. 340--343] on the sciences.) Of course the version I gave above was a bit of a cheat, in at least two ways. First, it was a story about how a certain outcome would be efficient, but that efficiency rested on a lot of unspoken or hinted-at premises about the relative sizes of different sorts of costs and values. (How many camel caravans are there in the East Bay?) Second, even granting the efficiency, would it really be brought about by the acts of interacting decision-makers, in the absence of a super-detailed coordinating plan?

Both of these questions, but especially the latter, have been the focus of a lot of very interesting work in economics over the last few decades. (Filial piety requires me to recommend this paper as an overview, but it's good, so that's easy to do.) One of those involved in this has been none other than Paul Krugman, who was one of the people who realized that new techniques for modeling imperfect competition with increasing returns could be used to attack the origin of cities and of industrial clusters. One of the things he also realized is that the problem of where the specialists should locate themselves is one of symmetry breaking, just like many kinds of pattern formation from physics — and named it as such, in a lovely little book from 1996, The Self-Organizing Economy. A later book, Fujita, Venables and Krugman's The Spatial Economy elaborated on that analysis, showing how the mixing increasing returns with the logic of comparative advantage leads naturally to spatial patterns of what can only be called combined and uneven development, again through symmetry breaking. (The nucleation of a high-productivity center not only inhibits the growth of other centers near it, it de-industrializes its periphery.) In my humble supremely arrogant opinion, this one of the few places where interesting ideas from physics have been productively used in the social sciences.

Update, next day: typo fixed, thanks to Cris Moore.

Posted at January 15, 2011 17:45 | permanent link