POPULATION DENSITY IN A CENTRAL-PLACE SYSTEM*

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1 JOURNAL OF REGIONAL SCIENCE, VOL. 00, NO. 0, 2014, pp POPULATION DENSITY IN A CENTRAL-PLACE SYSTEM* Yorgos Y. Papageorgiou McMaster University, School of Geography and Earth Sciences, Hamilton, ON L8S 4L8, Canada. yorgos@mcmaster.ca ABSTRACT. The existing empirical literature about polycentric population density has focused on the urban scale, and the alternative models proposed in that context have been justified using heuristic arguments. This paper describes how polycentric density distributions can, in general, be endowed with a theoretical framework which differs from the existing literature with respect to the treatment of centers: instead of assuming that they represent places of work, it assumes they are places that provide goods and services to households. This imposes a hierarchical structure on the model, which allows replacing the set of distances to all centers (typically used in the existing literature as the same explanans irrespectively of location) with a smaller set of distances that corresponds to the number of levels in the hierarchy and varies with location. The central-place framework used also provides a direct link between a polycentric model and the Clark formula, in the sense that the latter can emerge through a smoothing procedure of the former. Finally, in the context of central places, the scope of related empirical investigations can be extended naturally from the urban to the regional scale. This is the scale of a simple test presented here, which has been specifically included to support the corresponding theoretical arguments about the structure of a polycentric density gradient. The paper concludes with some expected problems and advantages of applying these ideas to the urban scale. 1. INTRODUCTION The population density gradient, introduced by Bleicher (1892) and re-introduced by Clark (1951), asserts that the residential population density D at a particular location is a decreasing exponential function of the distance x between this location and a single city center, namely (1) D [x] = D [ 0 ] exp ( x), where is a parameter. In Equation (1), the Clark formula as it came to be known, has been tested successfully by many authors for different parts of the world and for the past 200 years. 1 It has been given a justification based on urban economic theory by Muth (1969) and, more generally, by Papageorgiou and Pines (1989). And even though the model has been typically applied to locations within the city, this restriction on D [x] is not evident. Bogue (1949), in fact, has maintained that Bleicher s exponential decline can extend up to 300 miles away from a large city center. The first empirical studies of a generalized Clark formula that admit a number of centers within the city, rather than a single one, were published by Griffith (1981a, 1981b). In its simplest form, Griffith s generalization was written as (2) D [x] = N A i exp ( a i x i), i=1 *The author would like to thank Alex Anas, Richard Arnott, Art Getis, Gianmarco Ottaviano, and three referees for their insightful comments. Received: December 2012; revised: November 2013; accepted: November For a review of early empirical tests of the Clark formula and its variations see Berry, Simmons, and Tennant (1963). DOI: /jors

2 2 JOURNAL OF REGIONAL SCIENCE, VOL. 00, NO. 0, 2014 where x determines location, N denotes the number of centers in the city and x i is the distance between location x and center i. IfN = 1, (2) reduces to the Clark formula with A 1 = D [ 0 ]. The same model was applied by Gordon, Richardson, and Wong (1986) and by Small and Song (1994) among others. A second alternative formulation in the published literature, namely, ( ) N (3) D [x] = Aexp a i x i includes Heikkila et al. (1989), as well as a variation by McDonald and Prather (1994) and by McMillen and McDonald (1998). 2 The passage from a monocentric to a polycentric city in this literature hinges on an implicit, common assumption that access to all centers in the city, more precisely to all employment centers, is valuable, hence that population density depends on distance to all these employment centers. A number of papers on the identification of such centers is discussed by Anas, Arnott, and Small (1998) in some detail, together with the studies mentioned in the previous paragraph, and related issues. Anas et al. also survey the large and growing theoretical literature on urban agglomeration and polycentricity. But no theory is offered to support either (2) or (3). Indeed, the only argument that aims to justify the choice of a polycentric density function is a pragmatic one presented in Heikkila et al. (1989). According to this argument, if the employment centers are perfect substitutes then the polycentric density function must be the upper envelope of the corresponding monocentric density functions because, for each urban location, only the closest center matters. 3 And if the employment centers are perfect complements then the polycentric density function must be described by (3) because, for each urban location, all N centers matter. Finally, according to Griffith (1981a), since population density results from the layering of influences generated by N centers, the polycentric density function can be described by (2). Alternative (3) was first proposed by Papageorgiou (1970), and its conceptual framework was published in Papageorgiou (1971) without its empirical supplement. Later on, this model has been theoretically derived in Papageorgiou and Pines (1999). And although formally identical to the model used by corresponding, subsequently published empirical studies, it differs fundamentally with respect to the treatment of centers. Whereas existing studies connect density at every location x on the LHS of (3) with distances to all N centers on the RHS, the original model in Papageorgiou (1970) imposed an additional structure on the N centers by assuming that they belong to a central-place hierarchy with n levels. Within a particular order centers are perfect substitutes so, for every location, only the closest center of any particular order is relevant. By contrast, across orders, centers are perfect complements. Consequently, for every location, only the closest centers that correspond to each hierarchical level are relevant. Thus employment centers in the existing empirical literature correspond to commercial and service centers in the original proposal; and instead of connecting every location x with distances to all N centers, the original proposal connected every location x with a subset of n distances between that location and the closest set of centers that provide goods and services of order 1,..., n. In particular, if at location x the closest center is of order i, thoseatx obtain all goods and services 1,..., i from that closest center. This additional structure introduces parsimony to i=1 2 In this variation, distance was replaced by its inverse to allow for potential development even in places at a sufficiently long distance from a single center, where such a distance would otherwise force predicted population densities to become sufficiently close to zero irrespectively of other proximate centers. 3 This polycentric density form has been proposed by White (1976).

3 PAPAGEORGIOU: POPULATION DENSITY IN A CENTRAL-PLACE SYSTEM 3 the model by replacing the same large set of explanatory variables used irrespectively of location with a smaller number of explanatory variables, specific to location, which agree with casual experience about how and why location matters. 4 Empirical studies in the literature on polycentricity have been confined to the city scale. An exception to this rule can be found in Papageorgiou (1970) which includes a yet unpublished, empirical application of (3) at the regional scale. Although dated, part of this application can be useful as a companion to the theoretical framework presented here. The simple test borrowed from the original application matches exactly the assumptions of the theory and can simply serve as an ex post motivation for it. These two components, together with the conceptual framework borrowed from Papageorgiou (1971), provide the foundations for this paper. Although already published, the conceptual framework has preceded the onset of the polycentric urban literature by a decade, thus remaining unnoticed. Also unnoticed remained the theoretical framework in Papageorgiou and Pines (1999), which was buried within a chapter of extensions to Alonso s (1964) standard monocentric city model with no adequate reference to the thriving empirical literature on polycentric cities. In isolation, each of the three components is lacking. In combination, they provide a consistent, self-contained narrative which can interest people working in polycentric density gradients because it develops a formal argument concerning the choice among empirical specifications of a generalized Clark formula; it includes a first example of its extension to the regional scale; it suggests the development of a potential centralplace framework parallel to the employment-centers framework already applied at the urban scale; and it implies the possibility of empirical work on a unified spatial framework that combines urban and regional central-place data at the same level of detail. 2. A GENERALIZED CLARK FORMULA Derivation 5 Consider a central-place system in Christaller s (1933) tradition, containing a number of centers that represent the only places which can provide goods and services to the population in the system. These centers form an n-order hierarchy, where n is fixed. The number of centers belonging to a particular order decreases as the order increases. The highest order n has a single center. Assume, as Christaller did, that centers of a particular order provide all goods and services also provided by lower-order centers, but no goods and services of higher order. Thus a center of order i also operates as a center of order 1,..., i 1. The goods and services of order i are represented by a single composite good of that order, any amount of which can be found at the same quality and fixed price in every center that provides goods and services of order i. All composite goods are perfect complements. A perfectly homogeneous population of individuals, who aim to maximize their utility level subject to their budget constraint, is distributed within that system. Since all individuals consume all types of composite good, they typically interact with more than one center in order to obtain the n composite goods. The interaction between any individual and the centers is determined by a fixed vector of frequencies m (m 1, m 2,..., m n), 4 A different kind of additional structure was introduced by Sivitanidou (1996), who replaced centerspecific distances for all locations with corresponding proximity-ordered distances specific to location. More recently, Baumont, Ertur, and Le Gallo (2004) use only the main center and the closest subcenter for each location in model (3). This specification underlies the contrast between the local influence of subcenters and the global influence of the main center. 5 This section follows Papageorgiou and Pines (1999, pp ).

4 4 JOURNAL OF REGIONAL SCIENCE, VOL. 00, NO. 0, 2014 where m i represents an individual s number of trips per unit of time to obtain goods and services of order i. 6 Since higher-order goods and services are more specialized, frequencies of interaction decrease as the order increases: m 1 > m 2 >... > m n. Since individuals consume all types of composite good, an individual s location is defined by a fixed vector of distances x (x 1, x 2,..., x n), wherex i is the distance between location x and the closest center that provides the composite good of order i. And since higher-order centers provide all lower-order goods, the distance vector must obey x 1 x 2... x n. Finally, if no trips to other destinations are recognized, an individual located at x faces a total transportation cost n (4) T [x] = tm i x i, i=1 where t is the fixed transportation rate. 7 An individual located at x has a fixed income Y. This is spent to obtain amounts Z 1, Z 2,..., Z n of the n composite goods and H [x] of housing. The individual faces a fixed price vector P (P 1, P 2,..., P n) for the composite goods and a price R[x] for housing. Given that individuals enjoy a fixed level U of utility in equilibrium, their minimum expenditure function is given by E [ P,R[x], U ] n [ ] [ ] = P i z i P,R[x], U + R[x] h P,R[x], U, i=1 where z i and h denote compensated demands. And since R[x] is a parameter in E [ P,R[x], U ], the envelope theorem can be applied to obtain the derivative property of the expenditure function, namely E (5) R[x] = h[ P,R[x], U ]. Now assume that the expenditure function yields a unitary price elasticity of the compensated demand for housing. This and (5) imply 2 E E/ R[x] = h 2 R[x] R[x] R[x] h [ ] P,R[x], U = 1. R[x] The solution to the above second-order differential equation is given by E [ P,R[x], U ] [ ] [ ] = C 1 P,U ln R[x] + C2 P,U where C 1 and C 2 represent some functions. Applying once again the derivative property of the expenditure function yields h [ P,R[x], U ] = C [ ] 1 P,U (6). R[x] Next, since income and expenditure must be balanced, we have E [ P,R[x], U ] = Y T [x] which, upon differentiation and subsequent use of (4) and (5), gives the wellknown complementarity principle of (Muth, 1969, p. 21) (7) h [ P,R[x], U ] R = tm i x i 6 We do not allow either multipurpose shopping or an influence of distance to the frequency of interaction. 7 If the closest center to a location is of order j then x i = x j for i = 1,..., j.

5 PAPAGEORGIOU: POPULATION DENSITY IN A CENTRAL-PLACE SYSTEM 5 generalized in the case of a central place hierarchy. Replacing (6) into (7) yields R[x] = tm i [ ] R[x] i R[x] x i C 1 P,U with solution R[x] = R [ 0 ] ( ) n exp i x i ; and applying (6) once again, we arrive at the generalized Clark formula (3) written as D [x] = D [ 0 ] ( ) n (8) exp i x i with D [ 0 ] R [ 0 ] /C 1 [ P,U ]. Notice that the identical structure of the exponential component in the polycentric rent and density models follows from imposing a unitary price elasticity of the compensated demand for housing. Finally, notice that since m 1 > m 2 >... > m n, we have 1 > 2 >... > n. 8 Spatial Morphology Roughly speaking, the spatial morphology of the derived polycentric rent and density models can be summarized as follows: (i) Rents and densities attain a global maximum at the location of the highest-order center. This happens because they have negative first partial derivatives with respect to distance and because x = 0 holds only at the location of the highest-order center. (ii) Rents and densities at the location of ith-order centers decrease as those centers become more distant from higher-order centers. This happens because rents and densities at zero distance from an ith-order center correspond to locations such that x j = 0for j i and x j > 0for j > i. It is also known that (iii ) local maxima correspond to the locations of centers; (iv) there may be centers that do not correspond to local maxima; and (v) maxima correspond only to the locations of centers. 9 Alternative Interpretations The generalized Clark formula (3), as derived at the beginning of this section, can be interpreted in the context of a city where various order centers that belong to a centralplace hierarchy are distributed within a relatively small area. This interpretation corresponds to the scale of the existing empirical literature on the polycentric city surveyed by Anas et al. (1998). An alternative interpretation of this model describes a region that contains a system of towns and cities in it, aggregated to single points which are endowed with the hierarchical properties already explained. This is the object of the original test in Papageorgiou (1970), part of which is presented in the following section. Finally, the model may be thought of as a detailed description of a region if the previous two interpretations are combined, so that the production and market activities taking place within settlements are no longer aggregated into single points. This interpretation can be applied to an empirical research program of the future. In summary, the location of an nth-order center may denote either the CBD of a city, or the location of the highest-order city in a region, or the CBD of the highest-order city in a region. i=1 i=1 8 This constraint on the empirical parameters i applies only if individuals make single-purpose trips, that is, they only purchase a single type of composite good per trip. If multipurpose trips are allowed such regularity need not apply. 9 For a detailed discussion and related proofs see Papageorgiou (1971).

6 6 JOURNAL OF REGIONAL SCIENCE, VOL. 00, NO. 0, A REGIONAL POPULATION DENSITY TEST 10 The purpose of this test was to examine the validity of the generalized Clark formula (8) in a regional context. It used data from the northeastern part of Ohio which covered 19 counties and, during the 1960s, it included 113 urban places. 11 This region contains one metropolis (Cleveland), a few cities (Akron, Canton, Youngstown, etc.), a number of towns and many smaller settlements. Consequently, a four-order hierarchy was imposed with Cleveland as the single, fourth-order center. The problem was to determine the position of every other place in a three-level system. Toward this end, it was assumed that settlements of the same order form homogeneous groups with respect to population and number of establishments providing various types of goods and services. Thus the population of an urban place, together with the corresponding number of establishments in five census categories, were used in order to obtain this classification. 12 The data were orthogonalized using principal components analysis, with the first two components accounting for over 95 percent of the original data variation. Their scores were used as inputs to a discriminatory analysis for three groups. This procedure led to a hierarchical structure with 7 third-order, 28 second order, and 77 first-order centers, which agrees with the expectation that lower-order central places exist in greater numbers. Population density data presented here were obtained for 50 central places divided into three subregions adjacent to Cleveland, which were determined by the corresponding different data sources available (see Figure 1 and the appendix). The first (subregion I, Cuyahoga county) represents the system of centers inside and around the Cleveland metropolitan area. The second (subregion II, Lorain county), and the third (subregion III, Medina, Summit and Portage counties) contain centers around Lorain and Akron, respectively. An OLS estimation placed gross population density data against a set of straightline distances to the closest ith-order center, i = 1,...4, measured for convenience in inches from a large-scale map. And because each density observation was allocated to the corresponding center, the distance to the lowest-order center was always zero. Thus the model to be tested was written as 4 (9) ln D [x] = a 0 a i x i. Notice that this model matches exactly the assumptions of Section 2. For every order i, density at any location is connected with the distance to the closest center that provides goods and services of that order because centers of the same order are perfect substitutes. And travel patterns generated from any location connect density there with distance to all types of center because centers of a different order are perfect complements. Moreover, for any location, if the closest center is of order i > 1, density there is connected with the same distance traveled for goods and services of order j i because higher-order centers contain all lower-order functions. Finally, the assumption of identical individuals in Section 2.1 is consistent with the use of distance variables alone. Also notice that the inclusion of this test is not meant to examine the assumptions of the theory because it does not allow for the consideration of reasonable alternatives to i=2 10 This and the subsequent section are based on Papageorgiou (1970, chapter 3). 11 These data can be made available if requested. 12 Five groups of retail establishments were used, the selection being based both on the homogeneity of the group and on its relative order as indicated by the corresponding total number of establishments in Ohio. Two groups with high frequency of occurrence (relatively low order), two with intermediate and one with low frequency of occurrence (relatively high order) were used.

7 PAPAGEORGIOU: POPULATION DENSITY IN A CENTRAL-PLACE SYSTEM 7 FIGURE 1: Central Places Used for the Population Density Test. those assumptions. But it can provide a first glimpse of the possibility that this complex of assumptions can be useful for the description of spatial population distributions. The results are shown in Table 1. Each one of the four columns in this table corresponds to a tested combination of subregions. The subregions entering a combination are shaded in the diagram on top of the column. All estimates are significant at the level, have an adequate number of observations for inference, acceptable coefficients of determination and positive distance coefficients, thereby supporting the generalized population density model. Recall that the parameters i in (8), as explained at the end of derivation in Section 2, obey 1 > 2 >... > n under an assumption of single-purpose trips. This regularity appears in the last two columns. If it does not obtain, as in the first two columns, the model suggests a significant effect of multipurpose trips in these subregional combinations. For example, within the confines of Section 2, column 1 suggests that during the 1960s a large number of households around Cleveland often preferred to be serviced directly from the highest-order center in that region Such inferences implicitly assume that the only reason households interact with centers is in order to satisfy their needs for goods and services. This, however, is only half of the story. The other half, which has been the main preoccupation of the existing literature on polycentric cities, concerns trips to work. Combining those two kinds of trip confounds interpretation. For example in the case of column 1, since most subcenters are within 15 miles from Cleveland, interaction with that metropolis must include both trips for retailing and trips to work.

8 8 JOURNAL OF REGIONAL SCIENCE, VOL. 00, NO. 0, 2014 TABLE 1: Results of Alternative Population Density Tests 4. RELATIONSHIP BETWEEN THE CLARK FORMULA AND ITS GENERALIZATION The central-place framework used to derive (8) allows for the establishment of an explicit relationship between the monocentric and polycentric models. Population density decreases on average away from the main center, while the density of subcenters increases on average with the density of population. 14 Intuitively speaking, the latter happens because, as the population density increases, the market area necessary to support a center decreases. Therefore, within a central-place system, the spacing s of lower-order centers increases on average as their distance from higher-order centers increases: s i [x] = f i [x i+1,..., x n] for i = 1,..., n 1with f i > 0and j > i. x j Such a systematic deformation of Christaller s hexagonal patterns has been attributed to the existence of agglomeration economies in hierarchical systems of this sort. 15 Since the spacing of ith-order centers increases on average as their distance from higher-order centers increases, the average distances traveled by individuals to ith-order centers must be longer for more peripheral locations. This suggests that the average distance x traveled to an ith-order center from a location at x must depend on spacing as (10) x i [ si [x] ] = g i [x i+1,...x n] for i = 1,..., n 1with d x i d s i > 0, g i x j > 0and j > i. If g i is linear in its arguments, it becomes possible to link polycentric with corresponding monocentric density models using a series of n 1 steps. The first step consists of replacing D [x] with D [x 1, x 2,..., x n] D (1) [x 2,..., x n] using (10), which reduces the dimensionality of the density function s domain by one. Since g i is linear, D (1) retains the structure of (8). This implies that the spatial morphology of the generalized Clark formula D which was described in Section 2.2 applies to the new function D (1) as well, so that the locations of first-order centers can no longer correspond 14 See, for example, Stephan (1988) and Gusein-Zade (1993). 15 For an early reference see Isard (1956, pp ), especially the diagram on p. 272.

9 PAPAGEORGIOU: POPULATION DENSITY IN A CENTRAL-PLACE SYSTEM 9 to local maxima. The same smoothing procedure is repeated to obtain D (2), D (3),and so on, where each step involves an averaging of x i values associated with a particular (x i+1,...x n). After completion of the n 1 steps a single global maximum is left and the resulting D (n 1) [x n], which corresponds exactly to (1), describes the general trend of the polycentric surface generated by D [x]. Using the data of Section 3 on a linear version of f i, a spacing test in Papageorgiou (1970) supports the assumptions f i / x j > 0for j > i and i = 1,..., 3. This, together with a linear version of x i obeying dx i /ds i > 0 for all three levels, lead to a linear g i with g i / x j > 0for j > i and i = 1,..., 3. Since the results of this spacing test were significant at the level we conclude that, in this instance, the Clark formula may indeed describe the general trend of its polycentric analogue, which provides a theoretical justification of some sort to the early Bogue (1949) claim that the monocentric density gradient can be extended to the regional scale as well. 5. CONCLUDING REMARKS The polycentric population density model (8) seems preferable not only because it stems from theory, but also because it can be simply estimated using linear regression rather than using nonlinear estimation frequently fraught with convergence problems as in the case of (2 ). Furthermore, the hierarchical structure imposed on (8) eliminates the well-known problem of great distance from a center, which has forced authors applying (3) in the context of employment centers to use somewhat arbitrarily the inverse of distance instead. It provides a natural link with the Clark formula. It is supported by casual experience, in the sense that the main factors people take into account when deciding about where to locate include, in addition to current employment, the availability of various amenities nearby rather than distance to all employment centers irrespectively of residential location. Finally, the hierarchical structure imposed on (8) adds parsimony by reducing the number of explanatory variables used in the existing literature. The identification and choice of centers at the regional level is trivial. By contrast, at the city level, this task is far from trivial, and it can be even more involved if one uses the approach taken here than the approach used in the existing literature. This happens because the backcloth of a central-place hierarchy, which underlies spatial population distributions, is more intricate than the single-tier pattern of employment centers. And the issue of how much detail should be allowed, that is, how far down the hierarchy of central places should one decide to go, remains an open question. This is precisely analogous to the general problem of choice of resolution, discussed by Anas et al. (1998), which also arises when employment centers are identified using the method proposed by Giuliano and Small (1991) and subsequently applied by many others. The data used in that case refer to gross density of employees per unit of land, and the choice of resolution involves a decision on minimum gross density for each in the cluster of contiguous zones to be combined which, in combination, must contain a minimum total number of employees. Variations of those two minima cause variations in resolution. 16 And if one uses threedimensional graphics to plot urban density across two-dimensional space, one is struck by how jagged the picture becomes at finer resolutions (Anas et al., 1998, p. 1431). However, taking into account the central-place approach, this is no longer surprising. For employment centers are also centers that provide goods and services to the city residents 16 For example, McMillen and McDonald (1998) identify a single large subcenter near Chicago s O Hare airport, which can be dissolved into five smaller ones by doubling the minimum gross density requirement.

10 10 JOURNAL OF REGIONAL SCIENCE, VOL. 00, NO. 0, 2014 and, moving from a coarser to a finer degree of resolution, simply disaggregates combined hierarchical levels to reveal lower-order centers in the hierarchy. According to the spatial morphology of Section 2, along this movement, new peaks increasingly appear as the lowest level to be allowed in further declines. The opposite movement exactly corresponds to the smoothing procedure of Section 4 that leads from the polycentric population density model (8) to the Clark formula. And what appears as irregular, or even random, at fine degrees of resolution, can be seen to obey an elaborate order if observed through the viewing glass of central-place theory. The requirement that both the number and the location of centers must be known before the empirical application of a polycentric density model, has led some early attempts to draw upon information accumulated by planning authorities and other local organizations. 17 Subsequently, the need for common ground fostered the development of standardized methods, such as the method proposed by Giuliano and Small (1991), which allowed direct comparisons across empirical works. Yet a certain degree of arbitrariness, exemplified by the problem of choice of resolution, persisted across alternative procedures. More recently, methods to overcome this problem include nonparametric identification techniques used by Craig and Ng (2001) and by McMillen (2001), as well as exploratory spatial data analysis used by Baumont et al. (2004). All along this path, which has started from the ad hoc and aims toward a sense of increasing objectivity, one can still find here and there mutually inconsistent implications among different studies. 18 A reason for this difficulty can of course be attributed to model-specification differences which, together with a lack of theoretical support, renders unclear the relative empirical merits of each work. But a deeper problem along this path stems from an underlying, strongly inductive bias which at its extreme, ultra-empiricism, aims to let data speak alone in order to reveal the true, hidden order. And this prevents the use of existing knowledge. A lot, however, is already known about household needs, and about how they affect household location preferences, spatial interaction and the like. A decision to forget all these, thus forcing oneself behind a veil of ignorance, prevents the use of existing explanatory power. By contrast, use of this power, that is, use of significant knowledge already available, may lead to a better understanding. The aggregate data employed in Section 3 for the classification of urban settlements are inadequate for a classification at the city level. There, as a first approximation, one could perhaps repeat the method on employment density data already applied in the case of employment centers, at different degrees of resolution, until that total number of centers is obtained which allows for a sufficiently large number of smallest centers. These then can be allocated to the various levels of the central-place hierarchy using the method of Section 3. A more elaborate classification approach along these lines would employ the same data on employment density, together with data on the number of retail-establishment types per zone or, even better, with corresponding disaggregate data on business volume. Once the identification problem has been resolved by choosing the 17 Such a classification example for metropolitan Toronto has been provided by Simmons (1964) who identified a three-order hierarchy with 1 third-order, 7 second-order, and 38 first-order centers. The Simmons classification has been employed by Papageorgiou and Brummell (1975) to estimate the population density surface for metropolitan Toronto using an expanded version of (8). This, and a corresponding estimation at the regional scale of southern Ontario, have been obscured because they represented intermediate steps within a study aiming to draw inferences about spatial consumer behavior. 18 For instance, using the multiplicative form [model (3)], Heikkila et al. (1989) found that the coefficient of the distance from the CBD of Los Angeles was positive and insignificant. This finding is at odds with the findings of almost every other study (Alperovich and Deutsch, 1996, p. 175).

11 PAPAGEORGIOU: POPULATION DENSITY IN A CENTRAL-PLACE SYSTEM 11 proper level of resolution, the classification of centers into a hierarchy is feasible in any case. 19 APPENDIX: CENTRAL PLACES USED FOR THE POPULATION DENSITY TEST Subregion I: Cuyahoga County 10. Beachwood 29. E. Cleveland 64. Moreland 87. Seven Hills 11. Bedford 33. Euclid Hills 88. Shaker Hts 12. Bedford Hts 36. Fairview Pk 68. N. Olmstead 92. S. Euclid 13. Broadview 38. Garfield Hts 71. Oakwood 94. Strongville Hts 41. Highland Hts 76. Parma 98. University 14. Brooklyn 47. Lakewood 77. Parma Hts Hts 15. Brook Pk 55. Maple Hts 78. Pepper Pike 102. Warrensville 23. Cleveland 57. Mayfield Hts 81. Richmond Hts 24. Cleveland 61. Middleburg Hts 105. Westlake Hts Hts 83. Rocky River Subregion II: Lorain County 3. Amherst 8. Avon Lake 50. Lorain 89. Sheffield Lake 7. Avon 34. Elyria 69. N. Ridgeville 99. Vermilion Subregion III: Medina, Summit, Portage Counties 6. Aurora 28. Cuyahoga 63. Mogadore 97. Twinsburg 9. Barberton Falls 80. Ravenna 100. Wadsworth 16. Brunswick 58. Medina 96. Tallmadge REFERENCES Alonso, William Location and Land Use. Cambridge: Harvard University Press. Alperovich, Gershon and Joseph Deutsch Urban Structure with Two Coexisting and Almost Completely Segregated Populations: The Case of East and West Jerusalem, Regional Science and Urban Economics, 26, Anas, Alex, Richard Arnott, and Kenneth A. Small Urban Spatial Structure, Journal of Economic Literature, 36, Baumont, Catherine, Cem Ertur, and Julia Le Gallo Spatial Analysis of Employment and Population Density: The Case of the Agglomeration of Dijon 1999, Geographical Analysis, 36, Berry, Brian L. J., James W. Simmons, and Robert J. Tennant Urban Population Densities: Structure and Change, Geographical Review, 53, Bleicher, H Statistiche Beschreibung der Stadt Frankfurt am Main und Ihrer Bevölkening. Frankfurt am Main. 19 One referee suggested that the specification of a central-place hierarchy at the urban level might be estimated recursively, taking into account the decline of population density with distance from the CBD. First, estimate order n 1 centers using Giuliano and Small (1991), but with cut-offs declining with distance from the CBD. Second, estimate order n 2 centers using cut-offs which decline with distance from the CBD and from the closest center of order n 1; and so on.

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