Complexity science offers tools to better understand the processes underlying city development, which is particularly important for building cities that are environmentally, socially, and economically sustainable. Urban scaling laws are empirically observed power-law relationships between population (i.e., city size) and other characteristics of cities in the same country. The theoretical underpinnings of these relationships have been proposed in The Origins of Scaling in Cities model, which derives scaling exponents from fundamental concepts in economic geography and general first principles of physics. The model introduces a parameter called net urban benefit, defined as a city’s social output minus the dissipative costs of infrastructure, representing a new metric relevant to city sustainability. Here we present a new derivation for a functional form of the metric, as well as new derivations that reveal how cities behave when the metric is maximized. We show results for U.S. cities and discuss the implications for the sustainability of the U.S. urban system, including the feedback mechanisms that keep cities on their current trajectories toward maximizing their socioeconomic output while minimizing infrastructure costs.

复杂性科学为更好地理解城市发展过程提供了工具，这对于建设具有环境，社会和经济可持续性的城市尤为重要。城市规模定律是根据经验观察到的人口（即城市规模）与同一国家城市其他特征之间的幂律关系。这些关系的理论基础已在“城市规模化的起源”模型中提出，该模型从经济地理学的基本概念和物理学的一般性第一原理中得出规模化指数。该模型引入了一个称为净城市收益的参数定义为城市的社会产出减去基础设施的耗散成本，代表了与城市可持续性相关的新指标。在这里，我们为度量的功能形式提供了新的推导，以及揭示了最大化度量时城市表现的新推导。我们显示了美国城市的结果，并讨论了对美国城市体系可持续性的影响，包括使城市保持其当前轨迹，以最大化其社会经济产出，同时将基础设施成本降至最低的反馈机制。