Measuring "Sprawl:" Alternative Measures of Urban Form in U. S. Metropolitan Areas
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Alternative Measures of Urban Form in U.S. Metropolitan Areas By Stephen Malpezzi and Wen-Kai Guo Revised, January 15, 2001 The Center for Urban Land Economics Research The University of Wisconsin 975 University Avenue Madison, WI 53706-1323 smalpezzi@bus.wisc.edu wgou@bus.wisc.edu http://wiscinfo.doit.wisc.edu/realestate Stephen Malpezzi is Associate Professor and Wangard Faculty Scholar in the Department of Real Estate and Urban Land Economics, and an associate member of the Department of Urban and Regional Planning, of the University of Wisconsin-Madison. Wen-Kai (Kevin) Guo is a Ph.D. candidate in Food Science at the University of Wisconsin-Madison. Comments on this and closely related work have been provided by Alain Bertaud, Michael Carliner, Mark Eppli, Richard Green, James Shilling, Kerry Vandell, Anthony Yezer and participants at the Homer Hoyt Institute/Weimer School's January 1999 and January 2000 sessions as well as the June 1999 Midyear meeting of the American Real Estate and Urban Economics Association. Comments and criticisms of this paper are welcome. The research we describe was supported by the University of Wisconsin's Graduate School, the Wangard Faculty Scholarship, and the UW Center for Urban Land Economics Research. Opinions in this paper are those of the authors, and do not reflect the views of any of the above individuals, or of any institution. Introduction Economists and other social scientists have tried to study urban form more or less rigorously for the better part of two centuries. The earliest commonly cited work is that by German scholars such as von Thunen (1826) and somewhat later work by Lösch (1944). In this century pioneering English-speaking scholars include Clark (1951), Hoyt (1939, 1966), and Burgess (1925). The 60s and 70s saw a further flowering with work such as Alonso (1964), Mills (1972), Muth (1969), and Wheaton (1977) among many others. Song (1996) provides a particularly nice discussion of some alternative measures. Excellent reviews and extensions of this large literature can be found in Anas, Arnott and Small (1998), Fujita (1989), and Turnbull (1995). The economics of location has been a fertile academic field for some forty years, and is enjoying resurgence due partly to the high profile of recent work on the economics of location by generalist economists such as Krugman (1991). But this academic resurgence is nothing compared to the eruption of interest in urban form by a wider range of political and social commenters. One watershed was certainly journalist Joel Garreau’s excellent popularization Edge City (1991), which broadened the audience for discussion of urban form. But the real explosion of interest in urban form has been due to the growing concern and exploding reference to urban sprawl. What environmental activists and now many others call sprawl is certainly not new to urban economists. The phenomenon of rapid growth on the periphery of the city is something that has been a core feature of most of the literature mentioned above, and the much larger literature that lies behind it. To give just one example, Edwin Mills’ classic 1972 book studies the decentralization of urban population in a large sample of U.S. cities going back to the latter part of the 19th century. Sam Bass Warner’s Streetcar Suburbs, coming out of a different scholarly tradition, is another well-known early examination of decentralization. Perhaps the first dividing line between urban economists and many other urban observers is the use of the word sprawl, with its pejorative connotations. While a number of authors have used the term sprawl in the academic literature (see references below), most urban economists have preferred less value-laden terms, such as urban decentralization (Mills 1999) or accessibility (Song 1996). But economists have lost the lexicographic battle. To give just one example, we did a simple Internet search on the term "urban decentralization" using Infoseek.com. The search engine returned 27 hits. We repeated an identical search using the term "urban sprawl." The engine returned 5,946 hits. With all the extraordinary attention paid to sprawl, it is quite interesting that only recently have some of those involved in the policy discussion attempted to define it. Consider the following quotation: ...sprawl in all its forms is seldom satisfactorily defined. Urban sprawl is often discussed without an associated definition at all. ... Some writers make no attempt at all at definition, while others engage in little more than emotional rhetoric, as in "the great urban explosion (which) has scattered pieces of debris over the countryside for miles around the crumbling centre ... a destruction of the qualities of the city" Despite its contemporary relevance, the quotation is from a paper by Robert Harvey and W.A.V. Clark, written 35 years ago.1 Our view is that discussions about sprawl, whether academic or policy oriented, are greatly hampered by loose definition and inadequate measurement. Our intention in this paper is to contribute toward improving the measurement of sprawl. Using a consistent data source, we compute some familiar measures such as average population density of the metropolitan area, and the familiar negative exponential density gradient. We also compute some less often used measures, such as those based on gravity models, and some which are fairly new, including measures based on order statistics, measures of fit of various models, and measures that incorporate the notion of autocorrelation. We also investigate the use of data reduction techniques to collapse some of these disparate measures into a univariate index. We also evaluate how well each one of our measures incorporates the information contained in the others, i.e., to get some sense of "which measure is best." We also estimate simple models of the determinants of each of the measure, using right-hand side variables suggested by the urban economics literature as well as some of the sociological explanations for decentralization.2 While we investigate a large number of measures, we certainly do not exhaust all possible measures. For example, our data tell us little directly about how much space within a tract is devoted to one land use or another, or how much space is privately owned vs. publicly owned. These are important components of some people’s views of sprawl. Our explanatory models are exploratory and simple. We mention here and discuss again below the fact that more complete models would deal with the potential endogeneity of some of our right hand side variables, as well as utilize a more complete vector of determinants. For example, for tractability this paper abstracts from the effect of housing prices as an equilibrating mechanism, in contradistinction to Malpezzi and Kung (1998), which argues that housing price gradients and location may be jointly determined. We also limit ourselves to data from the 1990 Census, that is, our measures are developed from a single cross section. Measures that focus on tract level changes are only one class of many possible extensions to our measurement effort here. Previous ResearchThe literature on urban form is huge. Our intention in this paper is to focus on measurement, so our review here is extremely selective. Readers interested in a broader review of the literature on sprawl are referred to Malpezzi (2000), Ewing (1997), and Gordon and Richardson (1997) among others. Those who wish a more detailed review of the academic approach to urban form should consult Anas, Arnott and Small (1998), as well as McDonald (1989), Wheaton (1979), and Ingram (1979). Surely the simplest measure of sprawl, and one used any number of times by urban economists and others, is the average density of the metropolitan area. Brueckner and Fansler (1983) and Peiser (1989) are among well-known papers by urban economists that use this measure. Far too many papers to cite focus on the negative exponential density gradient and its many derivatives and extensions. According to Greene and Barnbrock (1978), the first to use the negative exponential function was the German scholar Bleicher (1892).3 In many respects, the function was popularized by the work of Colin Clark in geography, and later by Edwin Mills, Richard Muth, and others in economics. Many authors have noted that the monocentric negative exponential is not always a terribly good fit for many metropolitan communities; see Richardson (1988), Kau and Lee (1976), and Kain and Apgar (1979) for examples. Here we note the following key points. (1) Without doubt the univariate negative exponential fits some cities reasonably well, and others quite badly. (2) Despite this, it is still often used partly because of the advantages of having a univariate index of decentralization or sprawl (see, for example, Jordan, Ross and Usowski 1998). (3) It is apparently neglected in the literature that the measure of fit of such a simple univariate model is in its own way a measure of sprawl, as we will discuss below. Most individual papers that measure decentralization, or 'sprawl,' focus on one measure or at most a few measures. There are exceptions, of course. Several papers have examined differences among functions theoretically and using simulation methods. Ingram (1971) examined average distances, negative exponential functions, a linear reciprocal function, among others, and Guy (1983) examined a number of accessibility measures in a broadly similar fashion. Broadly speaking, these papers clarify differences among candidate functions, and tell us which might best capture stylized facts of observed patterns, but do not offer empirical tests per se. Some papers have tested a limited number of specific hypotheses, e.g. whether a single parameter exponential function performs as well as some flexible form (e.g. Kau and Lee). In many respects Song (1996) is a paper that parallels this one, in that it uses actual data to test a wide range of alternative forms. Song estimates a variety of gravity, distance and exponential models using tract level data from Reno, Nevada. Best-fit criteria suggest that gravity measures and, especially, a negative exponential measure, perform much better than linear distance measures. As Song is careful to note, results from a single metro area are suggestive, but it remains to examine other forms, and especially to test forms across other metropolitan areas. Most analysts would admit the possibility of differing performance for a given estimator in, say, Los Angeles compared to Boston or Portland, for example. In our paper, we follow Song in examining a range of possible measures of urban form, but rather than focus on a single location, we examine a wide range of U.S. metropolitan areas. More recently, Galster et al. (2000) have independently undertaken an exercise in some respects similar to ours. Galster and colleagues estimate a series of measures of urban form for a dozen large MSAs (in contrast to our measures, for some 300). Later, we will briefly compare our results to Galster et al.'s, and to Song's. The Measurement of Urban Form In this section we discuss some measures of sprawl and related measures of urban form. First we introduce some notation. We use a capital P to indicate the population of a metropolitan area, and small p to indicate the population of a tract. Capital A denotes the area of the metropolitan area, and small a denotes the area of the tract. Capital D refers to the density of the metropolitan area, i.e. D=P/A, and small d, d=p/a, is the density of a tract. Distance from the city center is denoted by the letter u. Letters i and j index tracts within a metropolitan area, and k indexes metropolitan areas themselves. Generally we construct a measure for each metropolitan area, and for notational simplicity we usually drop the subscript k. Average Density The most common measure is average density in the metropolitan area.: (1) Average MSA density; for each MSA, D=P/A. In our database of MSA results, described below, this variable is denoted MSADENS. While widely used, the limitations of the measure are obvious. Consider two different single-county MSAs of equal area and equal population. Suppose the first contains all of its population in a city covering, say, a fourth of the area of the county, the rest of which is rural and lightly settled. Suppose the other MSA has a uniform population distribution. Our measure, average density, is the same. But most observers would consider the second MSA as exhibiting more "sprawl" than the first. Of course there is no reason to limit ourselves to average densities. Other moments, and nonparametric measures can also be considered, as below. Alternative Density Moments In this paper we construct several new indicators of population density gradients, based on the densities of the Census tracts in each MSA. The starting point for each MSA is to compute these tract densities, and then to sort tracts by descending density. We then construct several indicators of "sprawl", one for each MSA: (2) Maximum tract density, DENMAX = max(di) (3) Minimum tract density, DENMIN = min(di) (4) DENMED: the density of the "median tract weighted by population," that is, median(di) when tracts are sorted by density, the tract containing the median person in the MSA. For example, suppose the population of the MSA is 100 people, in 7 tracts: Tract Tract N Density Population 1 10 30
2 9 30 3 8 10
4 7 10 5 6 10
6 5 5 7 4 5
(5), (6): DENQ1 and DENQ3, the corresponding 1st and 3rd quartiles of tract density, constructed as above. (7), (8): DENP10 and DENP90, the corresponding 10th and 90th percentiles of tract density, constructed as above. Measures of Dispersion in Tract Densities (9): DENCV, the coefficient of variation of tract densities. (10). DENGINI, a Gini coefficient measuring variation in tract density, constructed as follows. Sort tracts within MSA in descending density. Compute cumulative population in percent, CPP and cumulative area in percent, CAP, as you move down tracts. Compute difference between CPP and CAP for each tract. Then sum over all tracts within each MSA. (11). Theil's information measure, an alternative to the Gini coefficient:  where ai is the area of the ith tract, A is the MSA area, pi is the population of the ith tract, and P is the MSA population. Population Density Gradients The measure of city form that has been most often studied by urban economists is the population density gradient from a negative exponential function, often associated with the pioneering work of Alonso, Muth and Mills, but as noted earlier first popularized among urban scholars by the geographer Colin Clark. More specifically, the population density of a city is hypothesized to follow: 
where d is tract population density at distance u from the center of a city; d0 is the density at the center; e is the base of natural logarithms; gamma is "the gradient," or the rate at which density falls from the center. The final error term, ε, is included when the formulation is stochastic. Among the other attractive properties of this measure, density is characterized by two parameters, with a particular emphasis on γ, which simplifies second stage analysis. The function is easily estimable with OLS regression by taking logs: ln d(u) = ln d0 - γu + ε which can then be readily estimated with, say, density data from Census tracts, once distance of each tract from the central business district (CBD) is measured. Thus, we construct measures (12) our gradient, gamma (denoted KMB1_1 in the database), and (13) density at the center, d0 (denoted INTB_1). The exponential density function is sufficiently important to warrant brief discussion. This particular form has the virtue of being derivable from a simple model of a city, albeit one with several restrictive assumptions, e.g. a monocentric city, constant returns Cobb-Douglas production functions for housing, consumers with identical tastes and incomes, and unit price elasticity of demand for housing. As is well known, the standard urban model of Alonso, Muth and Mills predicts that population density gradients will fall in absolute value as incomes rise, the city grows, and transport costs fall. Extensions to the model permit gradients to change with location-specific amenities as well (Follain and Malpezzi 1981). The negative exponential function often fits the data rather well, for such a simple function in a world of complex cities. Sometimes it does not fit well, as we will confirm. Many authors have experimented with more flexible forms, such as power terms in distance on the right hand side (of which more below). The world is divided up into two kinds of people: those who find the simple form informative and useful, despite its shortcomings (e.g. Muth 1985), and those who believe these shortcomings too serious to set aside (e.g. Richardson 1988).4 In fact, given the predicted flattening of population density gradients as cities grow and economies develop, it can be argued that the monocentric model on which it rests contains the seeds of its own destruction; and that a gradual deterioration of the fit of the model is itself consistent with the underlying model. Extensions of the Simple Exponential Gradient As already noted, the simplest, and most widely used model for estimation is: ln d = a + b ln u where d is the tract's density, and u is distance from the center. We are relying on this simple model for our second stage work, but we have also computed three additional models, with right hand side variables: (14) A quadratic model, i.e. with terms u and u2. (15) A cubic model, u, u2, and u3. (16) A fourth power model, u, u2, u3 and u4. In our database of results, these coefficients are represented by variables KMBa_b, where a represents the order of the model, and b represents which term. For example, KMB3_1 is the coefficient of linear distance in the cubic model. The intercepts from these models are denoted INTB_a, where a is again the order of the model. Measuring Discontiguity A simple and natural measure of discontiguity is: (17) The R2 statistic from the univariate density gradient regressions, denoted RSQ_1. Consider the two panels of Figure 1. Panel A shows a very highly stylized city with a given density gradient, as does Panel B. In Panel A, we have drawn a pattern consistent with very contiguous density patterns as one moves from the center of the city outwards. The second panel shows a city with the same density gradient, but a much more discontiguous pattern. The R-squared from the density gradient regressions is a natural measure of this discontinuity. However it should be noted that a low R-squared is a sufficient but not necessary condition for such discontinuity. To see this, consider a city where the density gradient is very contiguous from tract to tract, but assume that the gradient varies by direction as well as distance from the CBD. As example would be a metro area in which the gradient declines very rapidly with distance in one direction, but very slowly in another. Suppose this difference is very systematic, and density changes slowly as one rotates from left to right around the central point of the city. Such a city would not be truly discontiguous by most people’s thinking, but would have a low R-squared for a simple two-parameter density regression of the usual kind, where it is maintained that density varies with distance but not direction. Of course it would be possible to estimate distance density gradients that vary by direction as well as distance (see Follain and Gross 1983), but undertaking such an exercise would require resources beyond our present ones. (18) The difference in R2 statistic from the univariate density gradient regressions, and the R2 statistic from the fourth power density gradient regressions, DRSQ1_4. Another variation on the preceding theme; if the univariate model is a good one, then adding successive power terms adds little to the explanatory power of the regression. If, on the other hand, the improvement in fit is large, the simple model is less satisfactory. Measures of Spatial Autocorrelation The r-squared measure from the negative exponential regression will capture some of this, but we can construct examples where, for example, spatial auto correlation is high but the R-squared is low. Consider a case where spatial auto correlation is high and positive (i.e., very little sprawl on this element) but where density varies tremendously by radial location (direction north or south, for example). Since our simple negative exponential model imposes that density is a function of distance but not direction, we would find a low R-squared even though the spatial autocorrelation in such a city could be high. A more generalized measure of such autocorrelation is therefore desirable. (19) Moran’s I (denoted MORAN_I). One commonly used measure is Moran’s I, which is effectively a correlation coefficient, constructed using a weighting matrix where weights depend upon location. More specifically, the formula as usually written is: where n is the number of tracts, and C is an n by n matrix that incorporates the information of which tracts are contiguous. Specifically, each row represents a tract, and contiguous tracts have ones entered in their corresponding columns. Other elements of C are zero.  One practical difficulty in computing Moran’s I for a large number of places (tracts and MSAs) is to develop an algorithm to compute the matrix C. In this version of our paper we use an algorithm from Isaaks and Srivastava (1990) which uses a quadratic approximation. That is, when distances are small, the elements of C are approximately 1, but as they get larger given the elements are quadratic in distance, they rapidly approach 0. Other measures of spatial autocorrelation are possible, of course. Moran's I is isotropic, i.e., it depends on distance from the tract in question, but not direction. To the extent that direction as well as distance does matter, an anisotropic measure that accounts for direction would be a natural extension. Dubin, Pace and Thibodeau (1998) and Gillen, Thibodeau and Wachter (1999) discuss the use of such anisotropic indexes for single metropolitan areas. Computational difficulties have so far kept us from producing such an index for our full set of metropolitan areas, but given resources, this would be an appropriate extension for future work. Compactness In Bertaud and Malpezzi (1999), Alain Bertaud developed a compactness index, rho, which is the ratio between the average distance per person to the CBD, and the average distance to the center of gravity of a cylindrical city whose circular base would be equal to the built-up area, and whose height will be the average population density: 
where rho is the index, d is the distance of the ith tract from the CBD, weighted by the tract's share of the city's population, w; and C is the similar, hypothetical calculation for a cylindrical city of equivalent population and built up area. A city of area X for which the average distance per person to the CBD is equal to the average distance to the central axis of a cylinder which base is equal to X would have a compactness index of 1. In this paper we use the simpler weighted average of distances from one set of points in a metropolitan area to another to compute two distance measures. Distances are corrected for the earth's curvature. (20) DCENTAVG is the weighted average distance to the center, where tract populations are the weights. (21) DCENTMED is the weighted median distance to the center, where tract populations are the weights.
(22) The linear gravity function, GRAVLIN, is is the weighted average distance from the center of each tract to every other tract, in turn in fact the same as above:   (23) The exponential gravity function, GRAVEXP, can be written: Combining Measures Given the multidimensionality of sprawl, sprawl is a natural candidate for data reduction techniques. We used the well-known method of principal components. Principal components analysis derives a vector p = Za', where Z is a matrix whose columns consist of n observations on K variables and a' is a K-element vector of eignenvalues. We choose the a's so that the variance of p is maximized subject to the normalization condition that a1 + a2 + ... + aK = 1. Given inevitable collinearity, we choose 12 of our 22 spatial measures as elements of Z: INTB_1, the intercept from the simple univariate exponential model; KMB1_1, the coefficient of distance in the simple exponential model; RSQ_1, the R-squared from the simple exponential model; DRSQ1_4, the improvement in R-squared from the univariate exponential model to the fourth power model; DENCV, the coefficient of variation of tract densities; DENMED, the density of the median tract, when tracts are ordered by density; DENP90, the density of the 90th percentile tract, ordered by density; DENGINI, the Gini coefficient of tract densities; MSADENS, the average density of the entire MSA; DCENTMED, the weighted median distance to the center of the MSA; GRAVLIN, the linear gravity measure; GRAVEXP, the exponential gravity measure. We extract three principal component measures from this data, and label them PC1, PC2 and PC3. Other Possible Measures We have seed that in addition to the traditional gradient measure, many measures of urban form have been put forward and studied. The simplest, of course, is the average density of the city or metropolitan area. We have proposed a fairly large set of other measures to include here, but we have not exhausted the possibilities. Others include measures such as functions based on densities other than the negative exponential, such as the normal density (Ingram 1971; Pirie 1979; Allen et al. 1993). Many additional measures could be developed using techniques developed by urban geographers and others for the analysis of data exhibiting spatial autocorrelation. Moran's I, which we have computed, is one such measure but there are others. Anselin and Florax (1995), Pace and Gillen (forthcoming) and Pace, Barry and Sirmans (1998) describe these techniques in greater detail. A few papers have examined land use conditions on the fringe as opposed to the metropolitan area as a whole. (See Brown, Phillips and Roberts (1981)). Also, we note that the American Housing Survey has data on land area for single-family houses. To our knowledge, no one has used this data in the analysis of sprawl. For example, median lot size of single-family homes built in the last five years would be one possible indicator that could be constructed. We did undertake some preliminary work with this data. Problems arose from the fact that such data are only reported for single-family units. But the biggest problem with this potential measure is that preliminary analysis of AHS data tells us that their are many missing observations, and further (and more worryingly) missing plot area is correlated with other housing characteristics and income, suggesting potential biases in measures created from this dataset. Further work along these lines remains for future research. Directory: documents documents -> Concept stage documents -> Concept stage documents -> The great global switch-off documents -> Nonprofit grant application documents -> The Truth about the Rwandan Genocide documents -> Annual InStructional unit plan documents -> Chaos equipment included Download 0.75 Mb. Share with your friends:
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