Why does a city twice as big do more than twice as much?

Because, according to a line of research called urban scaling, cities follow scaling laws much as animals do, and for many things the law is not a straight line. Urban scaling is the study of how urban indicators change systematically with city size, treating cities as complex systems. Its central finding, from Luis Bettencourt, Geoffrey West and Jose Lobo, is that these indicators follow power laws in population: socio-economic quantities scale superlinearly, while measures of infrastructure, like the number of gas stations, scale sublinearly.

Start with animals. Max Kleiber found in the early 1930s that metabolic rate scales with the 3/4 power of body mass. A cat 100 times heavier than a mouse uses only about 32 times the energy. Bigger animals are more economical per gram. Physicists, including West, a theoretical physicist who went on to lead the Santa Fe Institute, asked whether cities behave the same way.

In part, they do. Infrastructure shows economies of scale: double a city and it needs less than double the gas stations. But the human side runs the other way. Larger cities tend to be more innovative and productive, and also face more crime and disease spread. One summary of West's work says that when a city's population doubles, salaries per capita generally increase by 15%. Urban economists see something similar in the urban wage premium: wages are larger in larger cities.

The proposed reason is that city life is an emergent phenomenon, arising from many people interacting in close proximity. More people, more connections, more output, and more friction.

The theory is contested. Critics argue other functions could fit the data equally well, and that the exponent depends on where you draw the city's boundary.