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.

Transportation and “catalog” retailing

The last post discussed the role of urban transportation improvements leading to the development of the department store in the late nineteenth century. And of course the role of the automobile in shaping retail developments in the twentieth century is obvious. This got me thinking about the role of transportation (and communication) improvements in the evolution of retailing where the customer orders goods from a remote vendor and those goods are delivered to the customer.

We tend to think of modern developments such as e-commerce as novel developments. However, I’m going to start again in the late nineteenth century. But first, a brief excursion into the pros and cons of this type of purchase from the standpoint of the consumer. The major advantage is the selection of goods available, the ability to purchase things that are not available in local retail establishments, along with the convenience of being able to purchase the merchandise without having to travel to a store. The major cons are the inability to physically view the items to be purchased and the delay associated with the need for delivery, the lack of the instant gratification associated with physical purchase. Both introduce some uncertainties into the transaction. I am not mentioning price. The vendor saves money by not having brick-and-mortar stores, but this will be offset at least to some extent by the costs of shipping. This could go either way.

The mail-order catalog business emerged in the late nineteenth century with the major vendors being Sears, Roebuck and Montgomery Ward. The retailers made available a variety of merchandise to residents of rural areas that they otherwise would have been unable to acquire. The development of the railroads along with express freight services and parcel post to deliver the merchandise was undoubtedly a prerequisite. On the communications side, regular reliable mail service had been available for some time. But this mail-order business also required the development of printing technology that enabled production of the catalogs at a reasonable cost. (I don’t know just when this threshold may have been crossed, but I seriously doubt that Ben Franklin could have printed large numbers of Sears catalogs economically.) Just as transportation improvements enabled the rise of the general mail-order catalog, widespread use of the automobile made physical stores accessible to rural residents and led to its decline.

Another wave of remote shopping expanded in the second half of the twentieth century with the growth of specialized catalog shopping with telphone ordering, ranging from clothing (Lands End, L.L. Bean, etc.) to gourmet foods (Dean & Deluca). The attraction to the consumer was access to a wider selection and to specialized goods they could not purchase locally. Some improvements to delivery services helped. UPS did far better than the very long delivery times the post office provided, especially in the past. The role of improvements in communication should not be discounted. Toll-free 800 numbers made the calls free. I imagine customers would not have relished the idea of paying the expensive long-distance charges of the past to make purchases. Again, I don’t know at what point the costs of high-quality color printing for catalogs became reasonable, especially since they send out huge numbers. But it does seem that I saw a lot less color printing in the mid-twentieth century. These catalog retailers also innovated to minimize the risks associated with remote purchasing, offering no-questions-asked returns if something didn’t fit or even if you just didn’t like it.

We finally get to today’s e-commerce. It is noteworthy that Amazon started with books, which have two features favoring this model. First, for any given author and title, all books are the same. There is not the guessing that would be involved in choosing among several green sweaters. And second, with books, the breadth of selection is everything. No brick-and-mortar bookstore can possibly approach the inventory of an online retailer. Amazon and the other online retailers also adopted the policies of the catalog retailers (now, of course, also online) with easy returns and high levels of customer service. Zappos has no problem with your ordering multiple pairs of shoes in order to pick the one pair you want and send the others back.

Obviously the World Wide Web was the innovation on the communications side, making both obtaining informtion on available items and ordering quick and easy. And on the transportation side, the expansion of e-commerce is driving improvements in delivery services, with 2-day and even 1-day delivery becoming commonplace without excessive charges. This, of course, reduces the penalty of having to wait for delivery. Indeed, considering the likelihood of a lag between wanting to purchase an item and having the time to go out to a store, online ordering may be quicker.

Given the rapid developments in e-commerce and speedy delivery, we may be seeing only the first stages in the effects on physical retailers and therefore our urban areas.

Transportation and economies of scale in retailing

The automobile may be blamed for the evolution of big-box retailers, but the effect of improvements to intraurban transportation on retailing began much earlier. This can be seen clearly with the development of the department store in the latter part of the nineteenth century.

In the walking city of the early nineteenth century, most urban residents could only move around on foot. This necessarily limited the distances they could travel and the amounts of goods they could carry. Stores tended to be small and rather limited.

Transportation improvements–horsecars, cable cars, electric streetcars, and more–dramatically increased mobility in urban areas. Cities greatly expanded as residents took advantage of the greater ease of travel. Going to the developing central business districts several miles away became feasible.

A larger number of potential customers could travel to a store located in the downtown area, creating a greater market. This allowed the emergence of the modern department store carrying a far larger range of goods with greater selections. More volume provided greater economies of scale to the store in the sale of its merchandise. But these were also economies of scale from the perspective of their customers, who benefited from the convenience, wider selection, and lower prices.

Coming to shop at the department stores via public transportation did have one limitation, however. Customers purchasing large numbers of items or very large items could find it difficult or impossible to carry their purchases back home with them. The stores recognized this problem and offered delivery of merchandise purchased in the various departments of the store.

This evolution depended solely on the transportation improvements made in the late nineteenth century. It had nothing to do with the automobile. Indeed, at least some department stores continued to assume that significant numbers of their customers would come to their downtown stores using public transportation at least into the 1950s. When growing up and shopping at the large downtown department stores in Milwaukee during that decade, the stores were continuing to offer their delivery services. Of course now, the assumption more often is that customers will be arriving by automobile and can take all but the largest items home themselves.

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.

Centralization in large urban areas

Many have examined the decentralization of population and housing units over time. A common approach has been to use the density gradient from the exponential model as a measure of centralization. I have estimated the parameters for the model for large urban areas since 1950. I wanted to consider how well the density gradient actually performed as a centralization measure (which will be the subject of the next post). But to do so, I needed a separate, good measure of the centralization of housing units.

I reviewed a variety of centralization measures in the literature and was not satisfied with any of them, so I developed my own. I wanted a measure that made maximum use of the data on the distribution of housing units by census tract. And I wanted the measure to be interpretable, to have meaning beyond a larger value indicating housing is more centralized. The measure involves calculating two values: One is the mean distance housing units in the urban area are from the center. The other is the mean distance they would be from the center if housing units were uniformly distributed in the area, densities everywhere the same, no centralization. The ratio of the actual to the uniform distance would, of course, be 1 if housing were uniformly distributed and would decline with decreasing mean distance to the center and greater centralization. The minimum value would be 0 if all housing were located at the center. I wanted a measure of centralization that would increase with greater centralization, so this ratio is subtracted from 1. This measure, which I am calling the centralization ratio, is the proportional reduction in mean distance housing units are located to the center compared with a uniform distribution. So a centralization ratio of 0.25, for example, would mean that the mean distance to the center is a quarter less than for an even distribution.

I calculated the centralization ratio for 59 large urban areas for each census year from 1950 to 2010. The widely expected decentralization did occur, on average, with the mean value dropping from 0.25 to 0.18 over this period. But decentralization was far from universal; 14 areas saw increases.

Levels of centralization varied greatly across the urban areas. The highest and lowest values in 2010, for example, were 0.46 and 0.08. New York, Chicago, and Philadelphia were the areas with the highest levels of centralization, not surprisingly. Tampa-St. Petersburg, El Paso, and Jacksonville were the lowest. Urban areas in the Northeast had the highest mean centralization in 2010, followed by those in the Midwest. Urban areas in the South had the lowest levels of centralization (and would have been even lower if Washington-Baltimore, more like other large urban areas in the Northeast Corridor, had been excluded). The very largest urban areas also tended to have higher levels of centralization.

More detail on this analysis using the centralization ratio is in the paper “The Degree of Centralization in Large Urban Areas in the U.S., 1950–2010,” which can be downloaded here.

One (or maybe one and one-half) cheers for Walmart

I don’t particularly like Walmart. I find their stores crowded and unpleasant and I don’t shop there. They do not treat their employees well, a true failing given that a successful competitor, Costco, is able to do much better.

Urbanists have criticized Walmart for their negative effects on downtown areas in small and medium-sized cities. A Walmart opens, drawing customers away from the established shops. The result can be a shuttered main street.

Walmart attracts those customers, of course, with its low prices, significantly less than those that were being charged by the downtown merchants. It is able to do this because of its tremendous purchasing power, economies of scale, and highly efficient inventory and distribution systems.

Opponents have tried, sometimes successfully, to block the development of new Walmarts. They obviously believe that the preservation of the current merchants and the downtown is worth continuing to pay the higher prices that those merchants charge. That is, if those opponents would ever choose to shop at Walmart in any event.

The opponents may argue that Walmart’s prices are not that much lower than their competitors in any area, which may now be true. On the other hand, the price differences may have been greater when compared to those merchants who failed because of the competition from Walmart.

But whatever the magnitude of the price difference, being able to purchase the goods you buy at lower prices effectively raises your real income. And the effect will be greater for those with lower incomes, who spend a higher proportion of their income on goods sold at places like Walmart. (That is why a general sales tax is regressive.)

For a family with a very modest income, saving even a few percent on the things they buy might make it possible for the family to go out every few months to a movie or for a dinner at Olive Garden or wherever. It could make the difference between having a number of special gifts for each child at Christmas as opposed to providing one “real” gift and wrapping up packages of socks or underwear to have more presents under the tree.

For a very poor family, the savings could make it possible to buy a pair of new shoes for each child at the start of the school year. Or it might even mean having enough food for the last few days of the month.

I am not sure that keeping out Walmart and paying higher prices to maintain existing merchants in the downtown will look like such a good tradeoff to these families.

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.