Researchers have long noted that population and housing unit densities decline with distance from the center of a city. This has been observed in the past and the present, in urban areas around the world.

Urban economists have developed a model of urban settlement that explains this pattern as the result of people trading off accessibility to the center (minimizing transportation costs) and the desire for more space (which will be less expensive farther from the center). This is called the monocentric model, as the most basic form assumes that all employment is located at the center, to which everyone commutes. With some reasonable choices of functional forms and model parameters, the model predicts that density will decline as a negative exponential function of distance from the center, which is the pattern that has been observed.

The negative exponential model provides a simple way of describing the general distribution of people and housing in an urban area. The model includes two parameters, the density at the center of the city and the gradient, the rate at which density declines with distance. Numerous studies have looked at the changes in density gradients over time and have found that the gradients have declined, indicating a decentralization of population and housing units.

My urban patterns dataset provides the opportunity to estimate the parameters of the negative exponential model for housing unit densities over an extended period (from 1950 to 2010) for a sample of large urban areas (43 with a single center) defined consistently over the period. As have other studies, I found a consistent pattern of decline in the density gradient over the entire period. I also found a significant decrease on average in the central densities. But interestingly, about a quarter of the areas saw increases in their central densities from 1950 to 2010. These were generally areas that had experienced above average rates of growth. They were becoming much larger urban areas and were producing the higher densities near the center that are typically associated with larger urban areas.

The early studies of density gradients used data for small areas like census tracts (often just a sample) to estimate the model parameters. Needless to say, this was extremely tedious and time-consuming before the advent of machine-readable data and geographic information systems. Mills (in *Studies in the Structure of the Urban Economy*, 1972) devised an ingenous method of estimating the model using only the populations (or other quantities) for the central city and for the entire metropolitan area).

My dataset allowed me to examine one other change over time that many of the other studies of negative exponential density trends could not–how closely the pattern of densities in the census tracts conformed to the exponential pattern. Mills’ method, used by many succeeding researchers, simply assumed that density declined according to the negative exponential model. Using only 2 data points, they had no way of examining the extent to which the density pattern fit the negative exponential distribution (or even that it did). But with my data, using densities for the census tracts and regression to estimate the parameters of the model, I also obtained the measure of the fit of the model, *R ^{2}*, the extent to which the model was correctly predicting the observed densities.

Now we go back to the assumption of the monocentric model that employment was located at the center of the city. While not strictly true even in the past, in 1950 it was still the case that the dominant employment location for most urban areas was still the central business district. But since that time, urban areas have experienced employment growth outside of the CBD, with the emergence of outlying employment centers, some of which have become very large. In other words, urban areas are no longer monocentric. They have become polycentric cities. So what effect has that had on the performace of the negative expontial prediction of the monocentric model?

The *R ^{2}*–the fit of the negative exponential model–varied widely across the urban areas. However, the mean values across the 43 acres remained fairly steady from 1950 through 1970, 0.33 to 0.36. (There are good reasons it is not higher, not the least of which is the presences of nonresidential land uses not accounted for by the model.) But after 1970, the mean

*R*values dropped in every decade, to a low of 0.19 in 2010. Bottom line: Densities no longer conform as closely to the predicted pattern. The negative exponential model no longer works as well as it had in the past. This is certainly consistent with the transformation to more polycentric urban areas.

^{2}More detail on this analysis and the overall examination of the negative exponential decline of density is in the paper “The Negative Exponential Decline of Density in Large Urban Areas in the U.S., 1950–2010,” which can be downloaded here.