Why does the negative exponential model still work?

It has long been observed that urban population and housing unit densities tend to decline as a negative exponential function of distance from the center of an urban area. And that is predicted by economists’ monocentric model, which assumes the concentration of employment at the center of the urban area and people’s desire for accessibility to that employment.

But the idea of a monocentric city is increasingly inappropriate with the continued growth of employment and employment centers outside of the CBD. As stated in the previous post, my research has shown that the negative exponential model has been doing less well in recent decades in predicting densities at the census tract level.

Yet many researchers continue to use the negative exponential model to describe the patterns of urban areas. And the estimates of the model parameters, the density gradient and the central density, continue to be reasonable, showing consistent trends over time. This, in spite of the fact that the fit of the model for some urban areas was spectacularly poor, with R2 values as low as 0.01 in the past 4 decades. Five areas had R2 values below 0.05 in 2010, with a total of 11 (a quarter) below 0.1.

So how to explain this apparent inconsistency? Imagine a city with density declining negative exponentially with distance from the CBD and all employment initially located there. An outlying employment center is developed. Presumably people will also value accessibility to that location, producing a density peak around that center as well, with densities declining with distance from the outlying center. These densities will be higher than if the outlying center did not exist, so the fit of the tract density data to the negative exponential model will be poorer.

Now consider a ring around the CBD that encompasses the outlying center. Densities in that ring will be higher near that center. But they will decline as you move around the ring away from that center. The density in most of the ring will be that determined by the distance from the CBD as if the new subcenter did not exist. The average density in the ring will be somewhat higher than if the outlying center did not exist, but not that much higher.

Real urban areas will have multiple outlying employment centers. It is most likely that they will be at varying distances and in different directions from the CBD. They will produce more local peaks in densities and increasingly poorer fit of the negative exponential model. But the densities of concentric rings around the center will only be increased modestly. The pattern of densities for the rings will continue to reflect a negative exponential decline in density with distance from the center, with perhaps some decrease in the density gradient due to the somewhat higher outlying ring densities.

This can be examined empirically. The performance of the negative exponential model in predicting tract densities can be compared with estimates made using the tract data aggregated into concentric rings. As stated in the previous post, the performance of the negative exponential model in predicting densities for the tract data declined significantly after 1970, with mean R2 values dropping from well over 0.3 to 0.19 in 2010. The R2 values obtained when estimating using the ring densities are much higher, 0.69 to 0.82, as would be expected with such aggregation. And they do not show a regular pattern of decline over the period from 1950 to 2010. Ring densities continue to show clear decline as a negative exponential function of distance from the center. Which is why the negative exponential function still works.

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.

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