Scaling and Hierarchy in Urban Economies

In several recent publications, Bettencourt, West and collaborators claim that properties of cities such as gross economic production, personal income, numbers of patents filed, number of crimes committed, etc., show super-linear power-scaling with total population, while measures of resource use show sub-linear power-law scaling. Re-analysis of the gross economic production and personal income for cities in the United States, however, shows that the data cannot distinguish between power laws and other functional forms, including logarithmic growth, and that size predicts relatively little of the variation between cities. The striking appearance of scaling in previous work is largely artifact of using extensive quantities (city-wide totals) rather than intensive ones (per-capita rates). The remaining dependence of productivity on city size is explained by concentration of specialist service industries, with high value-added per worker, in larger cities, in accordance with the long-standing economic notion of the “hierarchy of central places”.

💡 Research Summary

The paper conducts a thorough re‑examination of the “super‑linear scaling” hypothesis that has been popularized by Bettencourt, West and collaborators, who claim that many urban indicators—including total economic output (gross metropolitan product, GMP) and total personal income—scale with city population N as a power law Y ∝ N^b with an exponent b > 1. Using data for 366 U.S. metropolitan statistical areas (MSAs) from 2006, the author first reproduces the original finding: a log‑log ordinary least‑squares regression of total GMP on population yields b ≈ 1.12, R² ≈ 0.96 and a root‑mean‑square (RMS) error of 0.23 in log‑space.

The key contribution, however, is to shift the focus from extensive (aggregate) quantities to intensive (per‑capita) ones. By defining per‑capita output y = Y/N and fitting several alternative functional forms—(i) a logarithmic model y = r ln N/k, (ii) a logistic model, (iii) a non‑parametric smoothing spline s(ln N), and (iv) a naïve constant‑mean model—the author shows that all models explain roughly the same amount of variation. RMS errors for the four models range from 0.225 to 0.234 (≈ ± 26 % prediction error), while the constant‑mean model has an RMS of 0.27 (± 30 %). Corresponding R² values lie between 0.23 and 0.29, far lower than the 0.96 obtained for the aggregate regression. This demonstrates that population size accounts for only about a quarter of the variance in per‑capita productivity; most of the variation is unrelated to city size.

A mathematical argument clarifies why the aggregate regression appears so strong even when per‑capita output is independent of N. If y and N are independent, ln Y = ln y + ln N is the sum of two independent random variables; the variance of ln Y is then Var