While I'm in this nerdy mood, let me point to this paper by Jaison Abel, Ishita Dey and Todd Gabe:
We estimate a model of urban productivity in which the agglomeration effect of density is enhanced by a metropolitan area’s stock of human capital. Using new measures of output per worker for U.S. metropolitan areas along with two measures of density that account for different aspects of the spatial distribution of population, we find that a doubling of density increases productivity by 10 to 20 percent. Consistent with theories of learning and knowledge spillovers in cities, we demonstrate that the elasticity of average labor productivity with respect to density increases with human capital. Metropolitan areas with a human capital stock that is one standard deviation below the mean level realize around half of the average productivity gain, while doubling density in metropolitan areas with a human capital stock that is one standard deviation above the mean level yields productivity benefits that are about 1.5 times larger than average.
This is a little technical, so let me translate: Cities tend to become more productive as they grow denser. On average, a city's workforce becomes 10% more productive when the city doubles in density. But that average obscures the importance of skills. Less skilled cities benefit a lot less than skilled cities from densification. In fact, skilled cities, on average, enjoy three times the productivity gain from denser growth than less skilled cities. This is yet more evidence of the increasing returns and agglomeration benefits from density.
This is a nice complement to the Glaeser and Resseger paper. Glaeser and Resseger found that workers in skilled cities become more productive as the city grows, while workers in unskilled cities do not. Abel et al find that workers in skilled cities become more productive as the city grows denser; workers in unskilled cities, less so.
The authors also test their conclusions using a variant of weighted density. They find even greater productivity gains (20% on average) when a city doubles its weighted density.
The authors use a coarse form of weighted density. They weight urbanized area density by county subdivisions. But how you chop up a city matters when calculating weighted density. In order to calculate weighted density, you first divide the city into a bunch of smaller regions. You then assign each region's density a weight equal to its share of the populations. In general, weighted density increases as you chop up the city into smaller regions.
I used census tracts for my weighted densities. There are many more census tracts than county subdivisions. I thus got a lot more stratification than they did -- e.g., their top weighted density was 19,000 ppsm, while mine was 33,000. How you divide a city for calculating weighted density is somewhat arbitrary, but I think using census tracts makes more sense than county subdivisions, which are more or less arbitrary. I suspect the authors would have found even greater returns to density had they weighted density by census tracts rather than by county subdivisions.
H/t Richard Florida.