From an excellent New York Times Magazine article on urban mathematics:
West and Bettencourt refer to this phenomenon as “superlinear scaling,” which is a fancy way of describing the increased output of people living in big cities. When a superlinear equation is graphed, it looks like the start of a roller coaster, climbing into the sky. The steep slope emerges from the positive feedback loop of urban life — a growing city makes everyone in that city more productive, which encourages more people to move to the city, and so on. According to West, these superlinear patterns demonstrate why cities are one of the single most important inventions in human history. They are the idea, he says, that enabled our economic potential and unleashed our ingenuity.
And earlier in the piece:
In recent decades, though, many of the fastest-growing cities in America, like Phoenix and Riverside, Calif., have given us a very different urban model. These places have traded away public spaces for affordable single-family homes, attracting working-class families who want their own white picket fences. West and Bettencourt point out, however, that cheap suburban comforts are associated with poor performance on a variety of urban metrics. Phoenix, for instance, has been characterized by below-average levels of income and innovation (as measured by the production of patents) for the last 40 years. “When you look at some of these fast-growing cities, they look like tumors on the landscape,” West says, with typical bombast. “They have these extreme levels of growth, but it’s not sustainable growth.” According to the physicists, the trade-off is inevitable.
I’ve come to think that places like Phoenix are both over-deplored by their detractors and over-hyped by their “counterintuitive” proponents. Fast-growing sunbelt cities are generally growing fast for the very understandable reason that local policymakers want the cities to grow rapidly. The population of the United States of America is growing, and the rural/metro balance continues its long-term shift in favor of metropolitan areas. The people have to go somewhere. So naturally they wind up going where it’s legal to build homes for them. If high-productivity cities insist on preventing increased density in their most desirable neighborhoods, then people will naturally end up in Phoenix or the exurban fringe. This is bad for the environment and bad for the economy and entirely avoidable, but only if the incumbent residents of high-productivity metro areas want to avoid it.