Growth, innovation, scaling, and the pace of life in cities

Abstract

Humanity has just crossed a major landmark in its history with the majority of people now living in cities. Cities have long been known to be society's predominant engine of innovation and wealth creation, yet they are also its main source of crime, pollution, and disease. The inexorable trend toward urbanization worldwide presents an urgent challenge for developing a predictive, quantitative theory of urban organization and sustainable development. Here we present empirical evidence indicating that the processes relating urbanization to economic development and knowledge creation are very general, being shared by all cities belonging to the same urban system and sustained across different nations and times. Many diverse properties of cities from patent production and personal income to electrical cable length are shown to be power law functions of population size with scaling exponents, beta, that fall into distinct universality classes. Quantities reflecting wealth creation and innovation have beta approximately 1.2 >1 (increasing returns), whereas those accounting for infrastructure display beta approximately 0.8 <1 (economies of scale). We predict that the pace of social life in the city increases with population size, in quantitative agreement with data, and we discuss how cities are similar to, and differ from, biological organisms, for which beta<1. Finally, we explore possible consequences of these scaling relations by deriving growth equations, which quantify the dramatic difference between growth fueled by innovation versus that driven by economies of scale. This difference suggests that, as population grows, major innovation cycles must be generated at a continually accelerating rate to sustain growth and avoid stagnation or collapse.

Fig. 1.

Examples of scaling relationships. (a) Total wages per MSA in 2004 for the U.S. (blue points) vs. metropolitan population. (b) Supercreative employment per MSA in 2003, for the U.S. (blue points) vs. metropolitan population. Best-fit scaling relations are shown as solid lines.

Fig. 2.

The pace of urban life increases with city size in contrast to the pace of biological life, which decreases with organism size. (a) Scaling of walking speed vs. population for cities around the world. (b) Heart rate vs. the size (mass) of organisms.

Fig. 3.

Regimes of urban growth. Plots of size N vs. time t. (a) Growth driven by sublinear scaling eventually converges to the carrying capacity N_∞. (b) Growth driven by linear scaling is exponential. (c) Growth driven by superlinear scaling diverges within a finite time t_c (dashed vertical line). (d) Collapse characterizes superlinear dynamics when resources are scarce.

Fig. 4.

Successive cycles of superlinear innovation reset the singularity and postpone instability and subsequent collapse. (a) Schematic representation: vertical dashed lines indicate the sequence of potential singularities. Eq. 4, with N ≈10⁶, predicts t_c in decades. (b) The relative population growth rate of New York City over time reveals periods of accelerated (superexponential) growth. Successive shorter periods of super exponential growth appear, separated by brief periods of deceleration. (Inset) t_c for each of these periods vs. population at the onset of the cycle. Observations are well fit by Eq. 4, with β = 1.09 (green line).