Category Archives: Density

Scattered, leapfrog development vs. low-density development

Two residential development patterns are most often associated with urban sprawl. Scattered or leapfrog development refers to the building of new residences, either separately or in a subdivision, at some distance from existing built-up areas. Low-density development refers to the construction of individual houses on larger lots. It is possible, of course, for scattered development to also be done on larger lots, though this is not the distinguishing feature of such leapfrog patterns.

In looking at densities in areas larger than the actual residential lots, scattered development will also be low-density because of the vacant land that has been skipped over. But I think most people would agree that scattered, leapfrog development and low-density development are two distinctive types of residential development and sprawl.

But how different are the two? Here is a thought experiment: Imagine an undeveloped area of land one mile square at the edge of an urban area. Now consider two ways in which this land might be developed. The first will be very low-density development. The area is divided into 64 10-acre lots and a house is built on each, completely developing the area. (For simplicity, we will be ignoring the need for land for roads to provide access.) Let’s assume that the owners only landscape small areas surrounding their residences, leaving the remainder of their lots undisturbed.

Now consider the second alternative of extreme scattered development. Sixty-four houses are constructed on 1-acre lots that are fairly evenly distributed across the mile-square area. In this case, only 10 percent of the land has been developed, but the developed area is 10 times as dense as in the previous case. Now suppose that these 64 houses are exactly the same as the very-low density houses and are located in exactly the same places. There would literally be no way to distinguish the scattered development from the very low-density development based on any physical characteristics of the developments. The only way to tell whether the development is very low-density or scattered is by looking at the land records.

Of course that difference in ownership matters–to a degree. The owners of the homes on the scattered 1-acre lots have no control over the undeveloped 90 percent of the land, which could be developed at any time. In the case of the very low-density development, each owner exercises control over 10 acres surrounding the residence by virtue of ownership. One might assume that these very large lots were acquired because the owners wanted the space and the control. (Though it is possible that land use regulations and/or choices made by the prior owner or developer of the area limited options available to the purchasers of these lots.) It is very likely that you are not going to see the purchasers of these large lots soon subdividing their land for higher-density development.

But the operative word here is “soon.” Over time, as demand increases and conditions change, further subdivision and development in the very low-density area becomes an increasing possibility. I currently live in an area that was developed from that late 1960s through the early 1980s with lots around a half acre. There are several lots in the neighborhood where the owners have built a second, substantial house on the rear portion of the lot (more than just an accessory unit or “granny flat.”)

So the very low-density developed area perhaps is not that completely different from the area with the scattered development.

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.

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.

Factors associated with urban density and change

The previous post addressed housing unit densities and their change for 59 large urban areas from 1950 to 2010. This post continues by addressing several factors associated with levels of density and density change in the urban areas.

I started by looking at what might be related to housing unit densities in 1950 and 2010. Three factors were considered: First, the size of the urban area. In larger urban areas I would expect greater competition for locations closer to the center, raising land prices and increasing densities. Second, the size of the urban area at an earlier period–how much housing had been developed earlier and was older. This idea I had here was that housing built long ago, especially before the widespread use of the automobile, would be more dense. And since housing is relatively permanent, that would contribute to higher densities later on. Finally, if physical barriers such as mountains or wetlands limited the expansion of the urban area, densities might be greater.

Looking at densities in 1950, I used the number of housing units in the urban area in 1950, the 1910 population of the urban area as a percentage of the 1950 population for age of the housing, and an indicator for the presence of nearby mountains or wetlands (a judgement call on my part). The size of the area and the presence of more older housing were both significant; physical barriers were not.

For 2010, I repeated this but had the issue of what to use for the age of housing–population in 1910 as a percentage of current population or population in 1970, forty years before, as was the case for 1950. I initially tried each one. The size of the urban area was again highly significant and now physical barriers were as well. (Being significant now as opposed to not being significant in 1950 may make sense as the urban areas have expanded and are now encountering the barriers.)

The curious results came with the age of housing, earlier population as a percentage of current population. Using 1910 population, there was no association. Apparently too many changes had occurred over the century for this to continue to affect densities. Using 1970 population as a percent of 2010 population produced a very weak but statistically significant effect. However, I could think of no good reason why 1970 should be important and why housing built before that time should have an effect. So I tried population in other years as a percent of current population. To my surprise, the strongest relationship was for the population in 2000. That’s hardly a measure of the presence of old housing. But it is a measure of the amount of new, recently built housing. The more recent, new development, the lower the density. My guess as to what is happening is this: With new development, the urban area expands with some of that development taking place on rural land at the periphery. But initial development in those areas is scattered and random, leaving, at least for a period of time, vacant land. These newly developed areas meet the minimum urban density threshold and become part of the urban area. The vacant land contributes to lower overall densities. With more recent development, larger areas of this new development are added to the urban area, resulting in lower densities.

Looking at the change in densities from 1950 to 2010, three factors were considered and all were clearly associated with density change. Greater densities in 1950 at the start were associated with larger declines in density. Given the magnitudes of some density declines over the period, this almost had to be the case, as those declines could only have been possible in areas that started out with higher densities. The growth of the area, the change in the number of housing units from 1950 to 2010, was positively related to density. Increasing demand and therefore greater increase in the size of an urban area should have this effect. And finally, the presence of mountains or wetlands as physical barriers to urban expansion was clearly associated with greater increases in densities.

These were very simple, basic models predicting density and density change with 3 variables. But it was surprising how well they performed in accounting for the variation in density and density change. Each of the models predicting density in either 1950 or 2010 had values between 0.6 and 0.7. And the model predicting density change had a whopping of 0.86.

More detail on this analysis and the overall examination of densities and change is in the paper “Density of Large Urban Areas in the U.S., 1950–2010,” which can be downloaded here.

Densities of large urban areas, 1950–2010

For my urban patterns research, I have developed the dataset with housing units by census tract for 59 large urban areas delineated for each census from 1950 to 2010. (More information is in the posts describing the data  and urban area definition.) For the first, most basic inquiry, I examined the overall densities of those areas and how these have changed over the period.

Given the tremendous growth of suburban areas, my expectation was that densities would be declining significantly over time. The maximum density did decrease greatly over the period, from nearly 3,000 housing units per square mile to just over 1,900 units. The mean density across the 59 urban areas likewise declined steadily, though far less dramatically.

Looking at the changes in densities over time for the individual areas, however, yielded a very different, more complex picture. I divided the areas into quartiles based on their change in density from 1950 to 2010. The first two quartiles did indeed see significant declines in density. For areas in the third quartile, however, housing unit densities were more or less stable over the period. These areas had relative small changes, ranging from a decline of 200 units per square mile to an increase of just over 100 units per square mile. But most surprising were the 15 urban areas in the top change quartile, which experienced increases in housing unit densities over the period up to a maximum jump of over a thousand units per square mile.

As a group, the areas in the top quartile with increasing densities had among the lowest densities in 1950, half the mean density of areas in the first quartile. By 2010, the density increases for this group gave them the highest mean density for all of the quartiles.

The locations of the urban areas with declining, stable, and increasing densities is very striking, as can be seen on this map:

Areas by density change quartile





Most of the areas that saw declines in densities were located in the Northeast and Midwest (and densities dropped for all of the areas in those regions). The areas where densities increased were primarily in the West and in Florida. And almost all of the areas with relatively stable densities were in the South.

More detail on this research is presented in the paper Density of Large Urban Areas in the U.S., 1950-2010.