Tag Archives: negative exponential

The negative exponential model and the size of cities

Researchers have long noted the tendency for densities to decline as a negative exponential function of distance from the center. They have looked at declines in the density gradient over time as a measure of decentralization in urban areas. They have noted the relationships of the estimated parameters of the model–the density gradient and the density at the center–to a variety of characteristics of urban areas, including, naturally, the size of the area. The consistent finding has been that the gradients tend to be smaller for larger urban areas, while the central densities tend to be larger.

Consider the relationships among the three–the gradient, the central density, and the size of the urban area. If density declines with distance following the negative exponential model, these three values must necessarily be mathematically related. But what affects what? It seems reasonable to believe that the size of the urban area is primarily affected by factors other than the parameters of the negative exponential model.

But what about the model parameters? Housing is long lasting and once established, the patterns in developed areas can remain remarkably stable for many decades. The density of urban development was much higher before widespread use of the automobile. And it turns out that the central densities are very strongly related to the sizes of urban areas in 1910. So it may not be unreasonable to conclude that, at least to some extent the density gradient is determined by the central density and the size of the urban area.

Solving for the mathematical relationship between the gradient, central density, and size yields a somewhat complex expression. However, a simplified approximation can be used. This approximation has the density gradient being directly proportional to the square root of the central density and inversely proportional to the square root of the size of the urban area.

As described in an earlier post and in a paper, I had used my urban patterns data to estimate the parameters of the negative exponential model for large urban areas in the United States from 1950 to 2010. It was straightforward to test for the conformity with the expected relationships among the density gradient, central density, and the size of the urban area. The gradient was indeed approximately inversely proportional to the size of the area, as expected. And the gradient did increase with the central density, though the proportionality was closer to the density itself rather than the square root. It may be possible that this is the result of the fact that the census tract densities in my data (and used by most other researchers) are measures of gross density including nonresidential uses, streets, and vacant land and are therefore lower than the net residential densities within the residential areas alone.

More information on this analysis, including the mathematical derivation of the relationship among the 3 values, is in the paper “Negative Exponential Model Parameters and the Size of Large Urban Areas in the U.S., 1950–2010,” which can be downloaded here.

The negative exponential density gradient and decentralization

Many researchers have used the density gradient from the negative exponential model to study the decentralization of population and housing units in urban areas. The density gradient is the rate of decline of density with distance from the center of the city. A decrease or flattening of the density gradient has been considered to be evidence of the decentralization of population or housing. And the density gradient has been used as a measure of the amount of centralization in an urban area that could be used to compare levels of centralization with other urban areas.

I have estimated the density gradients for 43 large urban areas for each of the census years from 1950 to 2010. And I have developed a separate, “pure” measure of centralization of housing units which I described in the previous post. I am calling this measure the centralization ratio. So this gave me the means of actually looking at the extent to which the density gradient was a good measure of centralization and decentralization.

First, I looked at changes in the density gradient over time and compared it to changes in the centralization ratio. The relationship was reasonably strong. It is appropriate to use the change in the density gradient as a measure of decentralization.

Then I looked at the relationship between the magnitudes of the density gradient and the centralization ratio at single points in time. This time, virtually no relationship. The density gradient does not work as a measure of the level of centralization in an urban area that could be used to make comparisons with other urban areas.

What gives? Why such different findings? The key lay in the fact that the density gradient is strongly inversely related to the size of an urban area. Using the density gradient to predict the centralization ratio resulted in no relationship. But add number of housing units in the urban area to the model, controlling for the size of the area, and a strong relationship emerged. And this is why the change in the density gradient works as a measure of change in centralization over time. The size of the urban area is being subtracted out when you look at the change (with the exception of any change in size over the period).

Someone committed to the idea that the density gradient is a good measure of centralization might object that I have only shown that the centralization ratio and the density gradient are different, not that one is a better measure of centralization. I think I make a good case for the use of the centralization ratio. Also, in developing the measure, I calculated other measures of centralization for a sample of a dozen areas and they were all highly correlated. And an anecdotal point: The three urban areas in my study with the highest centralization ratios were New York, Chicago, and Philadelphia. And all three had density gradients that were below the mean for the 43 large urban areas I looked at.

Accessibility to employment will always decline with distance from the center

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.

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.

Density declines and the emergence of the polycentric city

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, R2, 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 R2–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 R2 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.

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.