A Model to Relate the Size of the Central Business District to the Population of a City

Transcription

1 Ronald Briggs" A Model to Relate the Size of the Central Business District to the Population of a City Abstract A set of dynamic models of CBD growth are developed. They hypothesize that growth rates depend upon the population growth rate together with the increasing unwillingness of people to travel to the city center as city size increases. A prediction of the model is that absolute decreases in CBD size can occur associated simply with population growth rather than technological change as is usually postulated. The models are tested using retail sales data for the United States. Good predictions are general1 y found, especially when the unique CBD of New York City is removed from the data set. However, the model which has the greatest theoretical validity relative to central place theory does not have the best fit to the data. Implications of this are discussed, along with those of utilizing transformations in regression models. The decline, either in relative or in absolute terms, of the Central Business District (CBD) as the major focus of economic activities within *The author would like to thank Dr. Emilio Casetti for his valuable comments and assistance. This study was completed while the author was partially supported by a from the U.S. Department of Transportation (DOT-OS-30093). The ideas expresseet entirely those of the author and should not be construed to represent the policies of the Department. Ronald Briggs is assistant professor in the Department of Geography and research associate at the Population Research Center, the University of Texas- Austin.

2 266 / Geographical Analysis the modern American city has been a subject of discussion for several years. Despite the voluminous body of literature upon this topic there have been few attempts to model, in quantitative terms, changes through time in the level of economic activities in the CBD. This paper outlines the beginnings of such a model. Discussion will concentrate upon the retailing function of the CBD as it relates to the population of the city, but a generalization of the model to include the office function will be suggested. A review of the existing literature relating to the CBD indicates both a need for, and an absence of, quantitative, predictive, process-oriented models concerning its growth and development over time. Studies which have been undertaken fall into three main groups. The first comprises discussions of general factors behind changes in the CBD through time. Particular stress has been placed upon factors contributing to the supposed suburbanization of activities traditionally performed in the CBD, a process which most authors attribute to technological change in the transportation industry, particularly the development of motor vehicles. Such studies include those by Horwood and Boyce [9], Vernon [22], Hoover and Vernon [8], Vance [21], Meyer, Kain, and Wohl [11, chap. 21, and Kain [lo]. While these studies have contributed greatly to our understanding of processes behind present changes in the CBD, their lack of a strong analytic framework precludes their immediate use for quantitative, predictive purposes. They also stress the overriding role of technological change in the recent decline in importance of the CBD. While the importance of this factor in recent history cannot be denied, its continuance into the future at the same rate cannot necessarily be assumed. Thus, technological change cannot necessarily form the basis of a predictive model. This paper will suggest that population change should provide this basis, and that population growth, in and of itself, can lead to an absolute decline in the CBD. A second set of existing studies concentrates upon the pattern of land use within the CBD and its change through time. Such studies had their origin in the classic works by Murphy and Vance [IZ, 13, 14, 151. These, as with the many which have followed, have primarily been descriptive case studies of one or a small set of cities, although Garner [ 71 does attempt to explain observed land use patterns in terms of Von Thunen s land use theory. From the point of view of developing a model of overall CBD change through time, these studies are not overly useful since their concern is with changes in spatial distribution patterns within the CBD as a consequence of an exogenously determined overall change in the role of the CBD. For instance, Scott s model of changing activity locations [ 18, p. 351 requires that at each time period the total magnitude of change in the system [the CBD] must be specified from outside the model. This suggests a need for models to determine this total change, yet these have not been extensively developed, the few existing to date forming the third major thrust of CBD research. Recent examples of case studies include Bowden [2], Weaver [23], and Davies [6].

3 RonaldBriggs / 267 Studies within this third group have commonly utilized a regression framework. Dependent variables, primarily the absolute value, or percentage relative to the SMSA of CBD retail sales, have been regressed against such independent variables as SMSA population, CBD office sales, and shape of the SMSA [3, 171. Alternatively, measures of CBD change between two time periods have been regressed against measures of change in the magnitude and location of the SMSA population, amongst other variables [19]. In a slightly different framework, Thomas [20] has advocated the applicability of the law of allometric growth to the relationship between CBD size and urban population. Concerning factors influencing the development of the CBD, the major conclusion to be drawn from these studies is that population change is by far the most important. This empirical evidence suggests therefore, that population change should form the basis of any predictive model of CBD development. Although studies in this third group come closer than those in the first two groups to achieving quantitative predictions of changes in the CBD, the models utilized are not expressly concerned with the process and dynamics of such change. While the regression or correlation models may approximate the available data, and usefully indicate relationships between certain variables, they cannot be accepted as models of a dynamic system without a closer examination of the actual processes involved. Since both the empirical results above and the theoretical arguments outlined below suggest that the population of the city is the primary determinant of changes in the CBD, attention is now directed toward the development of a dynamic model to relate these two variables. THE MODELS Central place theory suggests that the function of the CBD is to supply the surrounding population with goods and services. With increasing urbanization and the decreasing significance of agriculture, it is reasonable to assume that this population is simply the population of the associated urbanized area, for empirical purposes the SMSA. Given this, the single most important factor governing the size of the CBD should be the population size of the associated SMSA. Thus, any model of CBD size should begin by relating it to the population of the city. The basic aim of the model outlined in this paper is to define the form of the function relating these two variables, namely, CBD retail sales and the population of the city. An increase (or decrease) in the population of a city should result in a corresponding increase (or decrease) in retail sales in the CBD. At any one moment in time, the level of CBD retail sales is the sum of all previous changes in retail sales brought about by changes in population over all previous time periods. Letting ds/ dp denote change in CBD retail sales ds for a change in SMSA population dp, that is,

4 268 / Geographical Analysis the rate of change of sales with respect to population, CBD sales can be represented by :; S= -dp, where the integral merely represents summation over infinitely small changes in P. If factors controlling the rate of change (ds/dp) can be specified, then an expression can be derived relating retail sales S to the population of the city P. The following argues that this rate of change is itself a function of population size and the level of retail sales. Thus, ds - = f(s, P). dp The function f is derived in two steps. The first incorporates what may be termed the direct impact of population change on retail sales. Basically, it incorporates the impact on CBD sales of changes in the number of people in the city, implying the hypothesis that growth in retail sales is generated by population growth. The second step allows for indirect effects brought about by the impact of population change on other variables which influence the population s per capita purchases in the CBD. It basically involves a second hypothesis, namely, that as population grows, the increasing unwillingness of people to travel to the city center decreases CBD sales. At least three possible formulations can be used to incorporate the direct effect. One is to assume that a change in population will generate a constant positive change in sales: _- ds (3) dp- a. This implies a simple linear function relating sales and population: Model IA: S= ap+ c. (4) A second approach is to assume that an increment of population brings about a positive percentage increment in sales: 1 ds -ds=adp or -- dp - as. S If the constant a is greater than zero, then this derivative will be always

5 Ronald Briggs / 269 positive, implying P is related to S by a continually increasing function of the form: Model HA: S= cop. (6) A third alternative is the elasticity formulation whereby a percentage increase in population is related to a percentage increase in sales: 1 1 ds S -ds=a-dp, or -=a- S P dp P (7) This implies P is related to S by a function of the form: Model IIIA: s= CP". (8) Of these three formulations the most theoretically valid appears to be the second. Central place theory implies that a unit increase in the population of a city will not always generate the same increase in retail sales. Rather, the increase in sales is dependent upon the city's position in the hierarchy. A unit increase in the population of a high-order city will generate more sales than in a low-order city because of the greater range and variety of goods available in the former, and thus higher levels of purchasing by each individual. This implies a continually increasing function relating S to P. Only Model I1 meets this criteria for all valid values of the constant a. The direct effect of population growth implies at least some increase in sales, thus the constant term a should always be positive. In Model I such a value would imply a constant rate of increase in sales. In Model I11 values of a less than unity would generate increasing sales but at a decreasing rate. Neither of these implications is consistent with central place theory's implication of sales growth at a continually increasing rate. Thus, Model 11, which for all positive values of the constant has this characteristic, is the most theoretically valid. The above models will hold only if no factors other than the direct effect of population growth are relevant. However, population increases imply changes in other variables which lead to indirect effects on CBD growth. It is characteristic of city growth in the western world that, with increasing population, rather than there being an increase in population density, the areal extent of the city increases and the population density gradient from the city center decreases [I]. As a consequence, the total distance travelled by the population to reach the CBD increases. Furthermore, increases in the size of the city imply increasing congestion at the city center. These increases in both travel distance and congestion 2The elasticity formulation corresponds to the law of allometric growth suggested by Thomas [203 as applicable to CBD development.

6 270 / Geographical Analysis would decrease the attractiveness for the individual of the CBD as a retail source. The second hypothesis, therefore, is that as city size increases people are less willing to travel to the city center and retail sales decrease. This hypothesis may be incorporated into the models by using Casetti s Expansion7 method [5]. The constant term a can be made a function of population and assumed to be a measure of the willingness of the population to travel to the CBD as the size of the city increases. In the absence of knowledge concerning the likely form of the function, a simple linear relationship is assumed: a= a, - a,p. (9) By substituting (9) successively into (3), (9, and (7), we obtain three differential equations defining the function f, relating the rate of change of CBD sales to population: ds - = (a; - aip) dp ds - = (ay - al P)S dp Two derivations follow from these equations. First, it can be noted that where.. a p=1, a2 This implies the existence of some critical population size (9) below which population increase will generate increases in CBD sales and above which it will generate decreases in sales. This is a most important conclusion in the light of most presently existing explanations for observed decreases in CBD sales. These generally posit technological change in transportation as the responsible factor. However, the present model suggests that population growth alone can generate decreases in CBD sales without the occurrence of any technological change. A second derivation follows from integrating both sides of equations (lo),(ll), and (12), and yields three models relating CBD sales to the population of the city:

7 Ronald Briggs / 271 Model IB: Model IIB: Model IIIB: EMPIRICAL EVALUATION Data and Techniques To test the models, to assess the adequacy of the underlyinghypothesis and to estimate values for the critical population level P, multiple regression techniques,were applied to both the A and B groups of models, using logarithmic transformations where necessary. The actual models fitted are shown in Table 1. The A group allows testing of the hypothesis that growth in retail sales is generated by population growth. The B group considers in addition, the second hypothesis, namely, that CBD sales decline as population growth generates increasing unwillingness to travel to the city center. Since these models differ from those in the first group only by the addition of a second term, if this adds significantly to the explained variation it can be concluded that the second hypothesis is valid. In choosing a data set a major problem arose in that, while the models are basically dynamic, they do not incorporate technological change. For a dynamic.mode1 longitudinal data is generally the most appropriate, however, in the last decades marked technological change has occurred. Since one of the most interesting implications of the present model is that a decline in CBD sales can be expected from population growth alone without any technological change, cross-sectional data was used in an attempt to control for technological change. This, of course, leaves TABLE 1. Scnm.ui~ OF MODELS A Model I (Linear) OriginalS=aP+c Model I1 S = ce" Regression formulation S= ap+ c Ins = c + ap Model 111 (Elasticity) s = CP" 1nS = c + alnp Regression formulation S = a, + a,p - a,pe/2 Ins = a, + a,p - a,p2/2 Ins = a, + a,lnp - a,p

8 272 / Geographical Analysis the results open to criticism on the grounds of the longitudinal/cross-sectional fallacy, unless ergodicity is assumed. The data were obtained from the 1963 and 1967 Censuses ofretaizing for CBD sales and SMSA p~pulation.~ Only those cities for which CBD sales were available for both time periods were utilized. A problem was encountered as to how to cope with those SMSAs having more than one CBD and an arbitrary decision was made to omit them unless the sales of the larger CBD were more than twice those of the smaller, in which case the larger CBD alone was used. The result was a set of 109 cities for the United States. A further problem arose as to how to incorporate the two major metropolitan areas of the United States-New York and Chicago-which have been defined as Standard Consolidated Areas (SCA) by the Bureau of the Census. To better represent their size, the population of the SCA rather than the SMSA was used. In addition, the models were fitted, not only to the complete set of 109 cities for the U.S., but also to two subsets which excluded New York in the first case and both New York and Chicago in the second. Results for Hypotheses The first hypotheses concerning the direct impact of population growth on sales is clearly upheld. Over all data sets, the A group of models are significant at better than the.001 level, with percentage of the explained variation (R2) ranging from 46 percent to 87 percent (Tables 2,3, and 4): Similarly the first term in the B group of models is significant in all cases. Results for the second hypothesis, concerning the negative impact of increased travel distance and congestion on retail sales, are less conclusive. When the models are fitted to all cities, only Model I1 has the negative second term required to support the hypothesis (Table 2). However, when New York City is excluded, both Models I and I1 have negative second terms which are statistically significant, although this is not the case for Model 111 (Tables 3 and 4). It can be argued that the unique position of New York City as the primary commerce and trade center for the United States results in an atypical CBD structure. Its sales are 3.5 times those of the next largest metropolitan area, Chicago, despite having only twice the population. When it is removed from the regression analysis, there is a marked change in the R2 values (Tables 2 and 3), which is not the case when Chicago is removed (Tables 3 and 4). If this uniqueness argument is accepted and reliance is placed on the results for the data sets which exclude New York City, then evidence is stronger for accepting the hypothesis based on the fits for Models I and I1 than for rejecting it on the basis of Model 111. Thus it can be concluded that, as a consequence of increased travel distance and congestion, population growth alone can generate absolute decreases in CBD sales. 3The PO ulation fi es are for 1960 and that significance levels are irrelevant since a true random sample is not involved. 41t coul B be argue fi"

9 Model I Model I1 (Linear) Model 111 (Elasticity) 1963 S = P ins = (10-7)~ (27.09) (1 1.18) A (P < 1nS = lnP (17.53) Re =.873 Re =.539 Re =.74 B S = WP + l.l84(1o-*)p2/2 Ins = (10-7)P (10-14)P4/2 1nS = hP (10-*)P (6.23) (7.48) (10.47) (5.90) (9.71) (2.49) (p <.001) (p <.001) (P <.mu (P <.Ow (p <.001) (p <.01) Re =.917 Re =.653 Re = A S = P (26.56) (P <.m1) 1nS = (10-7)P (10.98) Re =.868 Re =.530 Re =.664 1nS = nP (14.54) (P < B S = ~ (10-8)pe/2 Ins = (lOW7)P (10-14)P2/2 hs = hP + l.025(10-7)p (5.62) (9.87) (9.94) (5.44) (7.43) (3.05) (p <.001) (p <.001) (P <.mu (P <.001) (P <,ow Re =.931 Re =.632 Re =.691

10 1963 A B 1967 A B TABLE 3 COEFFICIENTS, t VALUES, AND SIGNIFICANCE LEVELS of Models Fitted to All Cities except New York S.C.A. Model I (Linear) Model I1 (Elasticity) Model 111 S = P (14.92) IR2 =.677 ZnS = (10-7)P (10.58) (P < I R2 =.514 1nS = nP (15.88) R2 =.704 S = P (10-8)P2/2 1nS = (10-6)P (10-'3)P2/2 (10.10) (4.58) (11.60) (7.43) (P <.00u 1nS = nP P (8.23) (0.39) (p <.001) (N.S.) R2 =.731 R2 =.681 R2 =.704 S = P 1nS = (10-7)P (15.84) (10.30) (P <.oou ZnS = InP (13.08) IR2 =.703 R2=.500 R2 =.617 S = P (10-8)P2/2 (9.24) (3.43) (p <.001) (p <.005) 1nS = (10-7)P (1O-l3)P2/2 (10.16) (6.22) (P < (P < 1nS = ZnP (10-')P (5.88) (1.39) (p <.001) (N.S.) IR2 =.733 I R2 =.635 R2 =.624

11 TABLE 4 COEFFICIENTS, t VALUES, AND SIGNIFICANCE LEVELS OF MODELS FITED TO ALL CITIES EXCEPT NEW YORK AND CHICAGO S.C.A.s Model I Kinear) Model I1 Model I11 (Elasticity) S = P (12.11) ZnS = (10-7)P (9.84 (P < ZnS = ZnP (14.91) (P.001) Re =.583 Re =.520 Re =.679 ~~~ ~~ S = P (10-8)Pe/2 (11.98) (6.83) t Re =.712 ZnS = (10-6)P - 3.O77(10-l3)P2/2 (11.45) (7.30) (P c.ow Re = 656 1nS = ZnP (10-9)P (7.76) (0.02) (p <.001) (N.S.) Re =.679 S = P (12.40) (P < R2 =.594 ZnS = (10-7)P (9.60) Re =.468 ZnS = nP (12.21) R2 =.587 S = P (10-8)P2/2 (12.05) (6.69) (p <.001) (p <.001) InS = (10-6)P (10-13)P2/2 (10.51) (6.51) ZnS = ZnP (10-8)P (5.57) (0.90) (p <.001) (N.S.) Re =.716 R2 =.622 Re =.590

12 276 / Geographical Analysis Model I Model I Model I All Cities - 11,305,213 12,591,404 - Exdudin New York 8ity 6,392,887 8,473,520 4,533,277 5,128,624 - Exdudin New York 8ity and Chicago 4,302,353 4,871,645 3,844,653 4,360, ,620,462 - Critical Population Values Table 5 presents the predicted critical population values (fi) beyond which absolute sales declines will occur. It is apparent that these values are very sensitive to both the particular set of cities used in the analysis and the particular model fitted. Furthermore, since they are estimated from cross-sectional data, but using a basically process-oriented model, great reliance should not be placed upon their exact values for planning or policy decisions. If accurate estimates are necessary for this purpose they must be obtained from longitudinal data with a more sophisticated model which incorporates the technological changes inherent in any time-series data. The Models The simple models tested here explain at least 60 percent of the variation in CBD retail sales. However, there are marked variations in their predictive abilities above this level as measured by the coefficient of determination (R2). Model I is generally superior to the other two, with Model I11 performing slightly better than Model 11. These differences are much reduced when New York City is removed from the data set. These results give rise to an important paradoxical situation. Model 11, which has the strongest theoretical base since it is consistent with central place theory, has the poorest empirical fit. Model 111, which Thomas [20] has argued for on theoretical grounds because of its conformation with the law of allometric growth, is still markedly inferior to Model I. Furthermore, the values of the parameters estimated for Model I11 are inconsistent with central place theory. Consistency requires a and a, to be greater than unity for Models IIIA and IIIB respectively, which is not the case (Tables 2, 3, and 4). On the other hand, Model I, which has the least theoretical validity, has the best empirical fit. The results suggest a need to develop a theoretical base to account for the particularly good fit of the linear model (Model I). They also suggest that central place theory may not be the most appropriate

13 Ronald Briggs / 277 conceptual basis for the study of CBD growth. This theory is basically static and does not explicitly direct itself toward growth processes. When concepts of the theory are utilized in a growth framework, as was done in this study, the resulting model provides a relatively poor fit to the data. Thus, the need for a theoretical base specifically incorporating growth processes is suggested. CONCLUSIONS AND GENERALIZATIONS It cannot be denied that the models presented here are simplistic ones which need much refinement. However, they represent one of the few attempts to develop a model of the CBD which is explicitly dynamic and process oriented. Only by utilizing such an approach can we hope to understand changes through time in systems. Furthermore, in terms of substantive results it has been suggested and partially verified that declines in CBD retail sales are not necessarily dependent upon technological change in the system, but can follow merely from population growth within the system. Refinements need to go in three directions: the first, toward a more accurate prediction of retail sales; the second, toward a prediction of the other major activity in the CBD-the office function; the third, toward the incorporation of technological change. As a predictor of retail sales, the weakest part of the model is the assumption that the unwillingness of people to travel to the CBD is a direct linear function of population growth. The need is for a more exact specification of how increases in population generate increases in travel distance and in congestion. This should be derived as a function of population distribution and density and thus may account for differences in retail sales between cities having similar population levels. The relationship between travel distance and congestion, and the unwillingness to go to the CBD must then be determined. This would appear to be the appropriate place in which to introduce technological change in the transportation industry, perhaps the single most important of the technological changes since it directly impacts upon the relative facility with which the CBD can be reached. The influence on retail sales of changes in office employment within the CBD is another critical variable which must be incorporated into the model. Although this could be exogenously specified, it is suggested that the two hypotheses made in the retail model have counterparts in relation to office employment; thus, the model presented here would also seem capable of predicting this other activity. The hypothesis can be made that the rate of growth in office employment is proportional to the level of office employment. If the CBD is a major employment source, then increases in the labor force (i.e., in population) should generate increases in employment in the CBD. The primary advantage of the CBD as a location for offices results from the functional linkages existing between different units [ 161. As the size of the CBD increases

14 270 / Geographical Analysis the possibilities for linkages also increase. As a consequence, growth in office employment is itself a function of level of employment. Therefore, office employment is a continually increasing function of population, It can also be hypothesized that, as congestion and length of the journey to work increase with increasing population, people will be less willing to undertake a CBD journey to work. It should be noted however, that the relative influence of these factors may not be the same for both the retailing and office functions. Incorporation of some or all of these factors into the basic model formulated in the present paper would go far toward providing a dynamic, predictive model for CBD growth. Besides implications for the understanding of CBD growth there are also some important theoretical and methodological issues raised. First, the linearized models tested by regression analysis in this paper frequently appear in other regression studies. Their form usually follows from transforming a variable in order to better fit the data or meet the assumptions of regression. The derivations hefe demonstrate that, if such transformations are utilized, they can have important implications for underlying growth processes or theoretical constructs. Also, the dangers of obtaining estimates from a process-oriented model applied to cross-sectional data is illustrated by the critical population values, the magnitude of which does not seem consistent with current trends. Finally, results of this study showed that the model with the greatest theoretical validity had the poorest empirical fit. Whether these results should be interpreted as suggesting inadequacies in central place theory, particularly when its basically static concepts are applied in a dynamic framework, or as a consequence of the failure to incorporate as yet the other critical variables discussed above, is an important question for future research. On a more basic level, it demonstrates that apparent theoretical validity does not guarantee a superior empirical fit. This suggests the importance of testing not just the model which apparently has the greatest theoretical validity, as has been recently done both by Thomas [ 201 and Briggs [ 41 with conflicting results, but of comparing its successes with those of others with less apparent theoretical strength. By so doing, more valid theory may be approached. LITERATURE CITED 1. BERRY, B. J. L. J. W. SIMMONS and R. J. TENNANT. Urban Population Densities: Structure and Change. Geographicaf Reoiew, 53 (1963), BOWDEN, M. J. Downtown thru Time: Delimitation, Expansion and Internal Growth. Economic Geography, 47 (1971), BOYCE, R. and W. A. V. CWRK. Selected Spatial Variables and Central Business District Sales. Papers and Proceedings, Regional Science Association, 11 (1963) BRIGGS, R. A Model to Relate the Size of the Central Business District to the Population of a City. Paper presented at the Fall Meetings, Southwestern Division, Association of American Geographers, Sept CASETTI E. Generating Models by the E ansion Method: Applications to Geographic Research. Geographical Analysis, 4 (19%), DAVIES, R. L. The Retail Pattern of the Central Area in Coventry. The Retail Structure

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