Scaling and universality in urban economic diversification

Abstract

Understanding cities is central to addressing major global challenges from climate change to economic resilience. Although increasingly perceived as fundamental socio-economic units, the detailed fabric of urban economic activities is only recently accessible to comprehensive analyses with the availability of large datasets. Here, we study abundances of business categories across US metropolitan statistical areas, and provide a framework for measuring the intrinsic diversity of economic activities that transcends scales of the classification scheme. A universal structure common to all cities is revealed, manifesting self-similarity in internal economic structure as well as aggregated metrics (GDP, patents, crime). We present a simple mathematical derivation of the universality, and provide a model, together with its economic implications of open-ended diversity created by urbanization, for understanding the observed empirical distribution. Given the universal distribution, scaling analyses for individual business categories enable us to determine their relative abundances as a function of city size. These results shed light on the processes of economic differentiation with scale, suggesting a general structure for the growth of national economies as integrated urban systems.

Keywords: power laws; scaling laws; universality; urban indicators; urban scaling.

Figure 1.

(a) The total number of establishments N_f scales linearly with city size: N_f ∼ N^α, where α = 0.98 ± 0.02 with R² = 0.97. (b) The number of distinct business types D(N) normalized by its maximum value D_max at various levels of classification, r, based on the NAICS scheme, from the lowest resolution (three-digit) to the highest (six-digit) denoted by green circles, blue triangles, red diamonds and black squares, respectively (corresponding values of D_max are 317, 722 and 1160). All values are scaled by the corresponding size of the classification scheme at that resolution, D_max, such that all values fall in between 0 and 1. The black solid line and orange dashes are the predictions from equation (2.5), with and without ϕ.

Figure 2.

Rank-abundance of establishment types. (a) The number of establishments at rank x ranging from 1 to 90 in descending order of their frequencies (from common to rare) for New York City, Chicago, Phoenix and San Jose. Establishment types are colour coded by their classification at the two-digit level. (b) Universal rank-abundance shape of the establishment type by dividing N_x by the population size of city in semi-log for all ranges. All 366 metropolitan statistical areas are denoted by grey circles. Seven selected cities are denoted by various colours and shapes; New York city, Chicago, Phoenix, Detroit, San Jose, Champaign-Urbana and Danville are, respectively, marked by red squares, pink diamonds, orange triangles, yellow left triangles, green right triangles, sky blue pluses and blue crosses. The black dash line and the black solid line are fits to equation (2.3) without and with ϕ, respectively. The inset shows the first 200 types on a log–log plot showing an approximate Zipf-like power law behaviour.

Figure 3.

Multidimensional allometric scaling of industry types. (a) The number of lawyers' offices scales super-linearly with population size, N_lo ∼ N^1.17±0.04 with R² = 0.92. (b) The rank of lawyers' offices goes up with population size, expressing an increase in their relative abundance: x_lo ∼ N^−0.4±0.06 with R² = 0.32. (c) Histogram of scaling exponents γ for all establishment types at the two-digit level. While primary sectors disappear, managerial, professional, technical and scientific establishments increase in relative abundance, helping to explain the increased productivity of larger cities despite the slow addition of new business types. Each scaling analysis is shown in the electronic supplementary material.