The previous post on why the negative exponential model still works made the argument that average densities in rings around the CBD would only be modestly affected by the presence of outlying employment centers. Another approach to thinking about these issues focuses on accessibility to employment throughout the urban area.
Accessibility to employment varies, of course, across an urban area and can be determined for every location in the area. It is a measure of how many jobs are located close to a given location. A measure can be a simple as the number of jobs within some distances to a weighted sum of distances to all jobs in the urban area, with the weight given the jobs decreasing with distance. (Some form of the latter is much better.)
It has been shown that accessibility to employment is a better predictor of densities in census tracts than distance to the center. Accessibility is also more closely related to housing prices than distance, as it affects land rents (which is the way in which densities are affected).
Now turning to the question of why the negative exponential model still works for urban areas with increasingly more employment outside the CBD. For most plausible distributions of employment in an urban area, accessibility to employment will still decline with distance from the center. In fact, it is easy to show that accessibility will decline in that way even if employment were uniformly distributed across the area. Consider a circular urban area with a radius of 10 miles in which employment is evenly distributed and no employment is located outside. At the center of the urban area, the most remote job is 10 miles away. At a point on the edge of the area, the most remote job is 20 miles away. Consider the simple measure of the number of jobs within 5 miles of a location. For locations out to 5 miles from the center, the number will remain constant. Moving farther out, the number will decline steadily as one moves toward the periphery, as an increasing portion of the 5-mile circle around the location falls outside the urban area, the area with no jobs. More complex measures of accessibility will also decline with distance from the center, in a more uniform manner starting at the center.
For an area with multiple employment centers, employment accessibility will be varying with distances to those centers as well as to the center of the entire urban area. Accessibility will not be closely related to distance from the CBD. But the average employment accessibilities for concentric rings around the center will continue to decrease in a fairly steady fashion with distance from the CBD.
This provides a basis for observing one other difference in the results from tract- versus ring-based estimates of the negative exponential model. The model is estimated using distance to the center as the independent variable in a regression to predict density (using the log of density to form a linear expression). But what if accessibility to employment is the correct predictor of density, with distance being used only as a proxy? Then the tract-based model, with accessibility more weakly related to distance, will have greater error in the independent variable. (The error is not in the measurement of distance, of course, but in the use of distance to approximate accessibility.) And error in an independent variable in a regression will generally have the effect of attenuating the estimate of the regression coefficient, in this case, the density gradient.
Comparing the results of estimating the negative exponential model using both tract and ring density data shows this to be the case. For the earlier years, 1950 to 1970, the mean estimates of the density gradients were quite similar when using the tract and ring data. But after 1970, the mean estimates for the gradients become lower for the tract estimates as compared with the ring estimates. So the attenuation appears to exist in later decades. This is consistent with distance from the center becoming an increasingly poorer predictor of density at the tract level over time.
More detail on this ring-based analysis of the negative exponential decline of density is in the paper “The Monocentric Model with Polycentric Employment: Ring versus Tract Estimates of the Negative Exponential Decline of Density,” which can be downloaded here.