## Thursday, June 23, 2011

### How cities differ from animals

kw: ideas, scaling, review reference

I am reading Where Good Ideas Come From: The Natural History of Innovation by Steven Johnson. It is just a bit bigger than the average book at some 75,000 words, but it is packed with ideas. I simply have to explore a few of them as I read.

A major point in the book's introduction is the concept of scaling. For example, we would expect larger mammals to require more food then smaller ones, just to keep a larger body warm. But how, exactly, does the rate an animal burns food increase with increased mass? Max Kleiber studied this in the 1930s, and by the mid 1940s had derived Kleiber's Law, that metabolic heat production increases as the 3/4 power of mass for mammals. This chart is from his 1947 report, "Body Size and Metabolic Rate", published in Physiological Reviews.

The 3/4 power is the fourth root of the cube, so we can apply it thus. One dot in the middle of the chart represents the data for a woman: Mass = 50kg, Met = 1,500 kcal/day. Down and to the left, we find a mouse: Mass = 20g = 0.02kg, Met = 3 kcal/day. Ratio of masses = 2500; ratio of metabolic rate = 500. 25003 = 15.625 billion; the fourth root of this = 354. This is pretty close; using logarithms to find the exact exponent we find it is 0.79. The red line on the graph follows an exponent of 0.75. (If I had Kleiber's original data for woman and mouse, perhaps the numbers would match more closely, and perhaps not.)

It has been said, particularly in literature, that a city is like a large animal, with its own metabolism. However, when cities have been studied, many important parameters are found to scale with an exponent greater than one. The easiest to study is total cash flow, the sum total of the income of a city's residents:

This chart, from this government report, uses the natural logarithm of population and total income, which makes it harder for most people to parse. I'll pick the upper-right point (probably NYC), and the one at lower left closest to the trend line. Big city: 18.8 million people, \$440 billion; small city: 57,000 people, \$800 million. Dollar ratios = 550; population ratio = 330. The trend line has an exponent of 1.2, and from these two points I find an exponent of 1.09. The difference per capita is stark: \$800M/57k = \$14,000; \$440B/18.8M = \$23,400, or 67% more. No wonder so many people flock to larger cities! (Except when recession hits…)

This is not just that the rich tend to live in larger cities (I live in a suburb of a city of 80,000 and there are millionaires aplenty). Rather, Johnson's point in his book is that larger aggregations of people yield disproportionately greater amounts of innovation, invention, and entrepreneurship. For example, it takes about a thousand patients to support a physician, and at least a few hundred to support a lawyer. In the smallest towns, you might have no more than one or two local physicians or lawyers, or none.

I recently visited three small towns in Missouri. Malta Bend, where my paternal ancestors lived, has fewer than 300 residents, and not a single retail or professional establishment. Even the single Methodist church is served by a circuit rider. The residents all go to nearby Marshall, population 11,000 to shop, see the doctor, and everything else. The circuit rider also lives in Marshall. A bit farther from Marshall there is Grand Pass, population 53. They also go to Marshall for everything except neighborly socializing.

A second item larger cities provide is a greater diversity of "stuff." An inventor needs "stuff". I remember growing up in Pasadena, California. There was a military surplus store, where all the radio hams and other homebrew hobbyists got "stuff". A town much smaller than Pasadena, population 147,000, but near gigantic Los Angeles, can't support a surplus outlet. And Los Angeles is one of only three huge metro areas I know of that support a walk-in parts store for antique cars. I used to shop there for parts for a 1948 truck I had in the 1970s. They had walls of parts for Model A and T Fords! The proprietor bragged to me that you could build an entire 1955 VW bug from parts he had on hand. I wonder if that is still true. To go a step further, my brother bought two junkyard Subarus, one wrecked in front, the other wrecked in back, and put together one working car, though he had to go to the same outlet for a few parts. "Stuff". He'd have got nowhere in Malta Bend, or even Marshall.

The prime difference between a city and an animal is that a city is a colony. I wonder if metabolic rates or other parameters for colonies, such as anthills, bee hives and termite mounds, follow a super-linear scaling law, one with an exponent greater than 1.0. For whatever reason, innovation in cities is a super-linear function. As the book proceeds, the author promises to ferret out other factors of similar power. If innovation is what you need, such factors are a must-know. Stay tuned.