Association between Populetion Density and The Market Areas of Towns

Transcription

1 M. J. Webber Association between Populetion Density and The Market Areas of Towns Abstract This paper examines the relationship between population density and town spacing in four kinds of environment. These are: (1) linear curves of demand, transport and production costs and an even population distribution; (2) nonlinear transport cost curves over space; (3) variable population density within hinterlands; and (4) variation of transport costs with population density. Previously, it has been shown that the criterion of free entry does not uniquely determine town spacing. The implications of four stronger criteria are therefore examined in this paper. These location criteria are: (a) the number of towns is maximized; (b) the number of towns in minimized, subject to all consumers being served; (c) the towns are all owned by one profit-maximizing monopolist; and (d) the average of some consumer utility function is maximized. In cases (a), (c), and (d) spacing decreases with density; in (b) spacing is an increasing function of density. Actual data are presented on the spacing of towns in Iowa and shopping centers in Chicago which indicate that spacing and population density are not associated. These results are consistent with the notion that entry is free but are not consistent with the stronger constraints employed in this paper. Consider a dispersed population which engages in nonservice activities and a set of nodes which service that population. The dispersed population lives outside the nodes, which may be towns servicing a rural population or shopping centers retailing goods to a population within a city. The provision of goods to people who live within the nodes is disregarded. M. J. Webber is assistant professor of geography at McMaster University.

2 110 / Geographical Analysis The population of the region has homogeneous tastes. Income and the state of technology are constant over the region. Information about demand and costs is assumed perfect. Consumers purchase a good from that node which offers it at the lowest delivered price. Intuition indicates that the density of towns in any part of the region may depend upon (1) the level of transport costs, (2) the density of population, and (3) economies of scale in the provision of nodal services. This paper examines the relationship between population density and the hinterland areas of towns and shopping centers within such a region. Several models are examined and their predictions compared with some data for Iowa and Chicago. A node is defined as a spatially associated congregation of sellers who retail at least a minimum bundle of goods (of course, they may sell more than that particular bundle). The particular minimum bundle which is set determines the values of the parameters of the models, but does not affect the methods of analysis. The nodes are assumed to have circular market areas; this is perhaps the least realistic assumption which could be made about market shapes, but it is made because of its analytical simplicity and because, as Mills and Lav [ 151 have shown, the assumed shape of market areas does not affect the main conclusions of the analysis. In particular, this assumption about the shape of market areas does not imply that competitive forces are too weak to cause complete packing. In another paper [22] it is shown that the twin assumptions that nodes enter freely and that they set profit maximizing prices do not uniquely identify the prices and the spacing of these nodes. Some stronger rule is necessary before both spacing and price may be determined. The implications of four such criteria are examined in this paper, namely: (A) that the number of nodes is maximized, (B) that the number of nodes is at a minimum, (C) that nodes operate pricing and spacing policies as if they were all owned by a single monopolistic firm, and (D) that node prices and spacing are such as to maximize the average of some consumer utility function. The rules are subject to the constraints that node profits be greater than or equal to zero (normal profits being included in costs) and that, given their area and shape, hinterlands are maximally packed. Prices are set so as to satisfy these rules. No attempt is made to determine how the policies of individual firms could generate such prices or market areas. The criteria are applied to four different environmental conditions, in models I through IV. MODEL I Model I is the simplest model presented in the paper. It rests upon a linear demand curve, constant transport rates, an even population distribution in the market, and a linear production cost curve. The quantity (4) of the bundle of goods sold by the node to each person in the market area is, as a function of price ( p),

3 M. J. Webber / 111 which is a linear demand curve. The price at the node per unit of the bundle of goods (the mill price ) is y. The transport rate per unit of goods per unit of distance is A. Consequently, the delivered price of a unit of the bundle of goods to a consumer who lives d units of distance from the town is p = y + Ad, (2) at which price the quantity sold to that consumer is Long [ 131 has shown that such a demand curve over space can only be valid in special circumstances; it is retained here because of its tractability. Population is evenly distributed over the node s hinterland at a density of p persons per unit area. Therefore at a distance of d units from the town there live 2ndp people, and total sales to that number of people are Consequently, within a distance of D units of the node, the total quantity of goods sold is Q(D, r) = D 0 2nd~[a - P(r + A41dd The cost of producing and selling this quantity of the bundle of goods is Therefore the total profit to the node from selling at price y over market area of radius D is

4 112 / Geographical Analysis In model IA the number of firms is to be maximized; that is, a minimum of D is to be found, subject to P( D,y) 10. The anticipations and policies of individual firms are irrelevant to this analysis, as to all the models discussed in this paper; nodes are organized according to social criteriaas if by some all-powerful planner-not according to individualistic maximization. The value of y is chosen to minimize D, that is, to satisfy which holds if ap -= 0, ay (9) provided that second-order conditions are satisfied. The condition (9) is the same as that employed by Mills and Lav [I51 in their analysis of market areas under free entry. But (9) arises out of a social criterion (that the number of firms be maximized) and not from an analysis of firms' pricing policies. Therefore criticisms of Mills and Lav, that they assume that firms ignore the anticipated effects of price changes upon rivals' prices and upon market areas, are not criticisms of the derivation of (9); see also [ZZ]. From (9) it is found that a e ~ y=-+--- D. 2p 2 3 The second-order condition, that (10) minimizes D, is that which, substituting (9), is equivalent to a condition which is satisfied if the numerator and denominator are of different sign. The inequality in (11) is found to hold.

5 M. J. Webber / 113 Now substitute (10) into equation (7). The result is an expression for the node's profit as a function of D: 1 A P(D) = npd2[4p (a- Ptl)"-(a- 3 fm)d+- Py2 9 D2] - 6. (12) At D = 0, P(D) = -6. Consequently, the minimum value of D for which P(D) is nonnegative is the smallest positive root of the quartic P(D) = 0. If h = 0, implicit differentiation of the quartic yields whence market areas are found to be related to population density through a negative power function nd2 = ncpw2. But in general A > 0, and I am unable to solve this quartic analytically and have consequently analyzed the effects of its parameters numerically. In model IB the number of nodes is at a minimum-d is maximized. From equation (3) and the condition that sales to any person be nonnegative, it is readily seen that the maximum value that D can take for any given value of y is an equation which defines the range of the bundle of goods offered by the nodes. Substitute (13) in P(D, y): Equation (13) demonstrates that D is greater the lesser is y; hence, to maximize D is to minimize y, subject to P(y) L 0. But P(y) < 0 when y = 0, and therefore the minimum of y consistent with the condition that P(y) be nonnegative is the smallest positive root of (14). This quartic is also analyzed numerically. The monopolist of model IC does not wish to maximize either D or P( D, y), but to maximize II = n * P(D,y),

6 1 14 / Geographical Analysis where n is the number of nodes in the region. The value of n equals the area of that region (A) divided by the area assigned to each node. The area assigned to a node equals its market area together with the adjacent interstices, that is, equals the area of the circumscribed hexagon, namely, 6D2 tan (n/6). Let Z = tan (7r/6); then Equation (15) is to be maximized with respect to both D and y. Differentiating IT partially with respect to y, the price is found to be the same as that defined in equation (10). This price is substituted in (15), which is then differentiated with respect to D. Setting this derivative equal to zero yields the condition that PA A 6 o = - D - - (a - pe) + - D -~. 9 6 TP If (16) and the second-order conditions are satisfied, the monopolist has maximized his profits. Model ID investigates maximum utility solutions of the location problem. The utility (u) which an individual derives from his purchase of q units of good from a node in this system is u = u(q). It is required that utility increases as a function of the quantity bought, but at a decreasing rate: i.e., that du d2u --0,- dq2 '* dq One simple function which satisfies these requirements is u= q" =(a- Py-pAd)", O<n=l. (17) Therefore, the sum of the utilities of all the consumers who live within distance D of the node is D U(D, y) = 2np\ d(a - py - pad)"dd. (18) 0

7 M. J. Webber / 115 The sum of the utilities of all the people who live in the region is therefore - ATP - 3ZP2X2(n + l)(n + 2) D-'{(cx - PY)"+~ - (a - py - phd)"+l [a - py + (n + l)pad]}. (19) The quantity in (19) is to be maximized subject to the requirement that P(D,y) 2 0. Now au/ay < 0 for all economically meaningful values of the parameters. Therefore to maximize 6(D,y) is to minimize y: for a given value of D, y is minimized-that is, is set such that P(D,y) = 0. The solution of this condition yields ~ D2 p ] 1'2}. The value of y defined in (20) is now substituted in (19), which is then maximized with respect to D. This procedure does not correspond to any of the three procedures previously identified. Losch's criterion that the number of firms be maximized does not define a utility maximum (in our sense), nor does the utility maximizing price correspond to the price used by Mills and Lav in their analysis of welfare maxima under free entry. U(D, y) is maximized numerically. Before examining the solutions of these conditions, consider the minimum density of population necessary to support a node profitably. A given node with market area D maximizes P(D, y) if y is such that that is, if a 0 h y = D. 2p 2 3 The second-order condition for a maximum is met, since

8 116 / Geographical Analysis Profit for this node is maximized if D is set at its maximum value consistent with the condition that sales to persons at D be nonnegative: from (3), If (21) and (22) are substituted in P(D,y), it is found that P(D,y) 2 0 if and only if 256 P 3X '6 ' 9r(a - P6)4* (23) 1000 minimum number of nodes "T 100 0?! a U - 0 L a2 L E.- I I0 I K- Minimum density necessary lo support o node profitably PARAMETER VALUES a 20 0 B A 05 \ maximum number of nodes \ I I 10 Population density, P I FIG. la. Association between population density and the market areas of nodes, for four constraints. Transports costs are higher in Fig. lb than in Fig. la.

9 M. J. Webber / 117 nodes cannot minimum number of nodes maximum profits PARAMETER VALUES AS IN la. except that A Z I 5 maaimum utlllty (n=l 0) \ maximum number of nodes I I 1 Population density, P FIG. lb. This inequality defines the minimum population density at which nodes can make a profit. If p is given the minimum value defined in (23), then the price and the spacing identified in (21) and (22) are found to satisfy the solution conditions of all the four location criteria: at this threshold density all four criteria predict the same price and nodal market area. The threshold density increases as the square of transport costs. Fig. 1 graphs the solutions of conditions (12), (14), (16), and (19) for varying population densities and for two levels of transport rates. The upper line on these graphs indicates the response of the maximum market area to changes in population density. At A = 0, the maximum market area encompasses the entire region, independently of p (notice that (23) is satisfied). But at positive costs of transport, the maximum market area increases with increases in population density; this result arises because higher densities permit firms to cover their costs with lower prices, which in turn enable the firms to serve larger market areas. The lowest line on the graphs relates the minimum market area

10 maaimum number of firms maximum profits 05 1 ~~ FIGURE 20 maximum utility (n=l) "1 PARAMETER VALUES AS IN la 1 I I I0 100 Population density, 0 FiGURE 2b l ,000 10, N= P k I 10 I00 Populotion density, P FIGURE 2c

11 0) N W (II a= 3 LL 0 lay b apou lad 0 0

12 120 / Geographical Analysis to density changes. As density of population increases so the minimum market area falls, approximately as,rr~2 = (a and b are parameters). As transport costs increase, the minimum market ares increases while the maximum market area is diminished. Now, it is shown in [22] that if the entry of nodes into the service industry is free, the spacing of those nodes is indeterminate, within the limits imposed by the minimum and maximum market areas. It follows from this and from the results in Fig. 1 that the assumption of free entry does not by itself predict that the spacing of nodes is related to population density: the market areas of freely entering nodes may increase, diminish, remain constant, or vary randomly (within limits) as population density increases. It follows also that the degree of variation in the spacing of nodes is less where population densities are lower and where transport costs are higher. It is possible, however, that the spacing of nodes is constrained by a criterion which is stronger than the assumption of free entry. Losch [I41 assumed that the number of nodes is maximized; the utility of consumers may be maximized; total node profits may be maximized. These three criteria all generate a negative association between population density and the spacing of nodes. If utility or profits are maximized, nodes have smaller market areas at higher levels of transport costs than at lower. Thus, an observation that nodal spacing is actually negatively associated with population density in a given region is evidence that some criterion stronger than free entry constrains the spacing of nodes in that region; and if market areas increase with increases in transport costs those market areas are at their minimum size whereas if hinterland areas fall as transport costs increase, nodes are organized to maximize either their total profits or consumer utility. The effects on market areas of variations in a, p, 8, and 6 may be investigated by taking implicit derivatives of (12), (14), (16), and (19). The threshold density, below which no node is profitable, rises as p and 6, but falls if a and 8 increase. If nodes are at their maximum distance apart, an increase in a causes an increase in market areas, while an increase in p, 8, or 6 causes a diminution of areas of hinterlands. The other three criteria predict exactly opposite effects of the parameter changes: increases in a are associated with reductions of market area, while p, 8, and 6 are positively related to hinterland areas. Thus, increases in node overhead and operating costs and in the slope of the demand curve cause increases in market areas if those areas are of less than maximum extent, but decrease market areas if those hinterlands are as large as possible; the y-intercept of the demand curve is positively associated with hinterland area when market areas are maximized, but otherwise to raise the demand curve is to decrease market areas. Fig. 2 demonstrates several additional attributes of the location patterns which are generated by the four criteria. In Fig. 2a mill prices are

13 M.J. Webber / 121 TABLE 1 ASSOCIATION BETWEEN POPULATION DENSITY AND LOCATION PARAMETERS FOR FOUR CONSTRAINTS NODES AND FOR FREE ENTRY Constraint Association between p and D Y sd2p Q Maximum market area positive negative positive positive Minimum market area negative positive negative negative Free entry (indeterminate) Maximum U negative positive positive positive Maximum P negative negative positive positive graphed as a function of p: prices decrease as density increases if nodes are organized to maximize consumer utility or to have maximum market areas; otherwise prices increase with density-i.e., nodes which maximize profits or have minimal hinterlands absorb some of the increases in transport costs which are caused by low population density. All criteria predict that the quantity sold per head of the market population rises as population density rises. Naturally, of all criteria, the utility maximizing solution predicts the greatest level of sales per consumer. Fig. 2c reveals that hinterland populations increase with increases in density, except when nodes have minimum market areas. Total sales made by each node are positively associated with density, again unless markets are of minimum area (Fig. 2d). Finally, sales per unit area of the hinterland rise with population density. Thus, if the location pattern is constrained only by the ability of nodes to enter the service industry freely, increases in population density generally raise sales per head and per unit area of the hinterland (though variations about this relationship are possible); however, prices, hinterland populations and total sales per node need not be related at all to population density. Table 1 summarizes some of the results contained in Figs. 1 and 2. In principle at least, these results uniquely associate empirical observations with one of the four constraints (or with free entry). The constraint that market areas are as large as possible is the only one which predicts that areas of hinterlands increase with greater population densities. Only if the number of nodes is maximized do sales per node and hinterland populations fall with increases in density. The profit maximizing constraint is the sole criterion which predicts that both market area and price fall as population density rises. It is only if nodes are organized to maximize the sum of consumer utilities that price, sales per node, and hinterland population all increase at greater densities. The absence of these relationships indicates either that the model is inapplicable to the region observed or else that the sole constraint on the location pattern derives from the free entry of nodes. MODEL I1 Several of the assumptions of model I are unrealistic. The effect of these assumptions is examined in models 11, 111, and IV. In model

14 122 / Geographical Analysis 11, the assumption that transport costs are a linear function of distance is relaxed. The remaining assumptions of model I are retained. Hoover [ 9, p. 211 shows typical cost curves for transporting freight. Such increasing, but convex up, curves may be approximated by a rate made up of an overhead cost plus a variable cost which is a linear function of distance. Consequently the delivered price per unit of a bundle of goods to a consumer d units of distance away from the node is where y is the overhead transport cost. Clearly the effect of the overhead transport cost is exactly the same as that of a tax on the quantity of goods sold. The quantity sold to that consumer is whence the total profit to the node from sellingat price y over a hinterland of radius Dis P(D,y) = (7-8)2mD2p - - D] - 8. (26) 3 The constant term a in the spatial demand curve (3) is replaced in (25) by the new term (a - Py ), and the existence of overhead costs of transport is apparently equivalent to a reduction in a. Therefore, the association between population density and the spacing of nodes is unaffected in form by such transport overhead costs. As these overhead costs increase, the threshold density rises and markets decrease in area if they are of maximum area (otherwise market areas increase as overhead costs of transport increase). Consequently, the main conclusions of the analysis are unchanged by the introduction of a more complicated transport cost function. MODEL I11 The third model examines the impact of population density variations within a hinterland upon the area of that market. The analysis is again conducted in terms of an isolated node, though this is a much less satisfactory procedure when density varies than when density is constant: it is imperfect because of boundary interactions between two hinterlands of varying density. Two cases are considered-in the first, density varies with distance from the node, while the second case is characterized by density variations around the node. Only the criterion that the number

15 M.J. Webber / 123 of nodes be maximized is examined in detail, for it can be shown that the effects of population density variations within hinterlands are similar for all constraints. At distance d, the quantity of the bundle sold per person is given by equation (3): Q= - P(y + Ad). The density of population varies with distance from the node such that p=++ed. (27) Consequently the total quantity sold at distance d is whence the total quantity sold up to distance D is and the profit made on selling this quantity is Following the procedure outlined in equations (8) and (9) for identifying the price which minimizes D, the value of y is found to be Y= 6+a + 4 (a~ - +PA)D - 3@AD P + SPED 2 Equations (29) and (31) may be substituted in (30) to yield the node s profit at its hinterland minimizing price. The result is a seventh degree polynomial in D, which may be solved numerically. It is found that for a hinterland of given mean population density, market areas are smaller when E is negative than when it is positive. Other relationships retain their original form. The important result is that the mean population density over the market area is not the sole control of hinterland size, for the area of hinterlands also depends on the form of the density variations. Thus, if density is not constant, to compare mean population density with the area of hinterlands is to introduce error.

16 124 / Geographical Analusis But density may also vary around the node. To examine the effects of such density variations, suppose that there exist two homogeneous regions, one of density p and the other of density E. The boundary between the two regions is a straight line and on this line is located the node. In the region which has density p, sales are made up to distance D, while the market area extends to distance E where the population density is E. Total sales within the market are therefore Q(D,E,y) = IT [ (a ipe) (pd2 + re2) --(pb 3 The problem is thus to minimize the market area 1 PA + re3). (32) A = n(d2 + E2)/2, (33) subject to P(D,E,y) 1 0. The first problem is to find the price that minimizes A, and the first-order condition for this is that (34) which is met, for D and E positive and different if and only if -- ad --- ae a? a? - 0. both of which equal zero if ap - = 0. a? (35) Condition (35) is thus the first-order condition that must be imposed on y for the price to minimize the area of markets. This price is Only if D = E does this price equal that given by model I; the greater the difference between D and E, the greater the difference in the market area minimizing price between model I and that given by (36).

17 M. J. Webber / 125 The profit made by the node at this area minimizing price is found by substituting (32) and (36) in P(D,E,y). It is P(D,E) = r (pd2 + re2) 1. - (a- Pe)A PA2 (pd3 + re3)' (pd3 + re3) + - (37) 6 18 (pd2f re2) If D = E, the minimum market area-the smallest positive root of (37)-is the same as the minimum market area of a node with the same mean population density: (p + 4/2. Otherwise the minimum market area is not the same as that predicted by model I for the same average density. The main practical importance of model I11 lies in its implications for testing. If population density varies within the hinterland of a node, that market has an area different from that which it would have were the entire hinterland of the same average population density. And the difference in area is greater the greater are the differences in density within the market. Consequently if data are collected on mean population densities within nodal hinterlands and related to data on the market areas of those nodes, we should not expect a perfect relationship between the two. The errors thus introduced are greater the more hinterlands differ in the degree to which density varies within them. MODEL IV The third unrealistic assumption, particularly in regions where population density varies widely, is that the level of transport costs is independent of population density. Variations in A and p may be due to two causes. First, at relatively low densities, A may vary inversely with p because of the association of both with (say) the physical environment: for example, mountainous terrain may be associated with low densities and high transport costs whereas flatter land may support a greater population density and lower per unit transport costs. On the other hand, transport costs also vary with the density of use of the transport media and therefore with the density of population. Pelensky [ 171 has data which indicate that the cost of travel by car for one person in Sydney is about 16 cents/mile (occupant's time being valued at $2 per hour) whereas on the open road it is about 8 cents/mile; see also Neutze [IS, p Only the second kind of association is analyzed here, and for simplicity it is assumed that A = E + Jlp.

18 126 / Geographical Analysis Therefore the quantity sold at distance d is whence the total quantity sold within distance D is Profit on this quantity is defined as in model I: The methods of solution are the same as those employed in the previous models. If the location pattern is constrained so that the number of firms is maximized, the hinterland minimizing price is which for given levels of transport rate is identical with the price determined in model I. The profit made at this price is found by substituting (40) and (39) in P(D,y). It is P(D)=-S+21Tp{ai:e --D} 3 D2 P -- ( + *p)d3 6 This equation is set equal to zero and solved numerically for 0, to find its minimum positive root. Once more, market area and population density are negatively related: minimum market areas are smaller in regions where E and I/J are greater than where these parameters are smaller. Similar results are found for models in which consumer utility or total nodal profits are maximized. The fact that transport rates vary with population density has a more complicated effect upon the location of nodes when the market areas of these nodes are maximized. The condition that hinterlands are of maximum area is that the expression in (14) be equal to zero:

19 M.J. Webber / 127 Now it is apparent that an increase in the numerical value of the right-hand term causes P(y) to fall and therefore raises the minimum value of y at which P(y) = 0. The parameters being positive, an increase in p causes an increase in the numerical value of this term if and only if For given values of E and JI, this condition is less likely to be met at low than at high densities; in particular, the condition may be met when population density is high but not when density is low. It follows that, as population density rises, prices may at first fall and then begin to rise, and in consequence-as (13) makes clear-market areas will at first increase and then diminish. The simple positive association between population density and maximum market areas is destroyed if transport costs vary with density, particularly in regions where density variations are wide. EVALUATING THE MODELS This theoretical analysis indicates which constraints are necessary to maintain the usual belief that the spacing of nodes is negatively associated with population density in the hinterlands of those nodes. If the number of nodes is maximized or if nodes are organized to maximize consumer utility or total profit, the spacing of nodes is a negative function of density in each of the models examined. If hinterland areas are maximized, those areas increase as density rises, unless transport costs vary with population density, when it is possible that market areas are diminished as density rises. If population density varies within hinterlands, the association between mean density and spacing is subject to error. If the location pattern is constrained merely by free entry, there generally need be no relationship between density and spacing (unless transport costs vary with density in such a manner that maximum market areas fall with increases in density, when free entry may generate a loose negative relationship between spacing and density). It is now necessary to compare these conclusions with some data for Iowa and Chicago, in an attempt to determine which constraints operate in reality. Such comparison is simplified by an additional assumption: that all services of a higher order (that is, with a smaller number of retail outlets) than the minimum bundle of goods are provided only in nodes which also sell that minimum bundle. This assumption has previously been made by Christaller [8] and by Berry and Garrison [3]. Berry and Garrison [4] and Berry and Tennant [5] contain data which show it to be a reasonable assumption. It follows that the models define the relationship between population density in a hinterland and the market areas of nodes which supply goods and services of an order greater than some minimum specified level.

20 128 / Geographical Analysis Iowa Data Assume first that the notion of a node may be adequately translated into the empirically defined term town. Assume also that the population of towns is a function of the number of different services which they provide. This assumption has received empirical support. For example, Berry [2, pp ] shows that in three regions of the United States (SW Iowa, SW South Dakota, and NE South Dakota) the population of centers is dependent upon the number of different kinds of business that they provide to their market areas; the correlation is Similar findings have been reported by Stafford [ 191 for southern Illinois, King [Ill for Canterbury, and Scott [I81 for Tasmania. Consequently, a town may be defined as an urban area containing more than some specified number of people rather than as providing a minimum range of services. However, since the correlations between population and number of functions, though high, are not perfect, this definition has introduced an additional error term into the predicted relationship, As the population necessary for a place to be defined as a town varies, so the parameters of the relationship between population density and town spacing may vary. As the minimum population is increased, we should expect the market areas of towns to become larger. But once the population has been fixed (for example, at 2,500 people), the models predict that the size of a town has no influence upon the size of its market area for the minimum bundle of goods; Losch [ 14, p makes this clear. But both Long [I31 and Webber and Symanski [23] show that more sophisticated demand curves and economies of transport for shopping cause market areas to vary with the size of node. Yet in fact there exists no significant relationship between the size of sixty-two Iowa towns of population greater than 2,500 in 1960 and distances to nearest neighbors irrespective of size. (The method whereby these sixty-two towns were chosen is explained below.) The models have been predicated upon the assumption that towns serve a rural population with goods. No account has been taken of a town s need to serve itself. The association between population and number of functions holds true only where towns function primarily as service centers, while rural population density is irrelevant to spacing if towns do not operate primarily as service centers, as House [lo] has shown. Consequently the towns in the sample should act mainly as service centers for the surrounding rural population. This is the reason why Iowa has been chosen for this analysis. As noted elsewhere [21], Iowa towns do function largely as service centers: most of the manufacturing activity is directed towards primary food processing and other farm-linked activities. During 1950, in only three SMSAs was more than 10 percent of the workforce engaged in manufacturing durable goods [S, pp. LI-LII]. Distortion of the urban network by agglomeration economies and nonservice functions is thus minimized. Variables other than market area and population density have not been measured. While farm incomes do vary slightly between different

21 M.J. Webber / 129 regions of Iowa [ 6, pp , it is unlikely that rural nonfarm incomes vary much over the state. The influence of other socioeconomic characteristics on the parameters of the demand curve a and p cannot be estimated. Transport costs A and the parameters of the cost curve 6 and 8 are unlikely to vary in a state as homogeneous as Iowa and settled in so short a period of time. It is therefore assumed that variables other than population density remain constant over the state. Data to test the models have been drawn from the U.S. Bureau of the Census [20]. This source provides population estimates for all incorporated places in Iowa in Two definitions of towns have been employed. In the first a town has been defined as any set of contiguous incorporated places, bearing the same name, which contains a total population greater than 2,500. In the second the minimum population level is set at 5,000. The problem of aggregating incorporated places which do not bear the same name has in general been avoided, except that satellite and suburban towns of the central cities of SMSAs have been included in those central cities. Thus, at the 2,500 population level, Des Moines gains three satellites, Waterloo gains two, and Davenport, Burlington, and Cedar Rapids gain one satellite each. At the 5,000 population level, Des Moines gains two satellites, while Cedar Rapids, Burlington, and Waterloo each gain one. The rural population of a county is then defined as that portion of the county s population which is resident outside such towns. The hinterland of a town is defined as that area which lies closer to that town than to any other town. The areas of these Thiessen polygons are the towns market areas. No allowance is made in drawing these polygons for differences in the sizes of towns, because we are concerned with the provision by each town of the same minimum set of services and because consumers are assumed to patronize the nearest source of such services. All towns whose hinterland touches the boundary of the state are excluded from the analysis, for their market areas may be affected by the location of towns in neighboring states. There remain sixty-two towns of population greater than 2,500 and twenty-seven of population greater than 5,000. The rural population density in each town s hinterland has been estimated from data on the rural population density of counties. The rural population density of the jth town s hinterland (p,) is defined as where p{ is the area of county i within the hinterland of town j and pf is the population density of county i. This estimate of the rural population density in a town s hinterland is relatively crude, for it makes no allowance for spatial variations within a county in rural population

22 130 / Geographical Analysis density; but there is no reason to suppose that the errors introduced by the estimator (43) are systematic. Fig. 3 graphs the relationship between towns hinterland areas and their hinterland population densities, for towns of population greater than 2,500. Population density varies between about fifteen persons per square mile and fifty persons per square mile and hinterland area between about 200 square miles and 1,100 square miles. The data give evidence of only a very slight negative association between density and hinterland area. The ordinary least-squares regression equation is 7~ D2 = p, (44) with r = The association is not significant at p = Similar results were obtained after an analysis of data for The coefficient of correlation between market area and hinterland population density for towns of more than 5,000 people is r = While hinterland populations increase as density increases, the association is again not significant. Chicago Data The data upon the distribution of shopping centers in Chicago contrast with those on town patterns in Iowa. Only nucleated centers have been analyzed, and thus other portions of the service system, including ribbons and specialized functional areas, are omitted. Data are available by shopping centers, and the intermediary variable of population size is not now required as a defining variable. Thirdly, in view of the known interpenetration of market areas of urban shopping centers and of the ready availability of data on consumer shopping behavior in Chicago, the market areas of shopping centers have been defined in terms of consumer patronage rather than of simple geometry. And fourthly, since incomes vary widely over space, mean family income is included in the analysis as a control variable. The data are taken from Berry and Tennant [ 51. Berry and Tennant have classified centers into three groups: (a) those providing shopping goods (including department stores, shoe, toy, and music shops), (b) community centers, which contain clothing stores, bakeries, dairies, and post offices in addition to the stores contained in (c) other neighborhood centers with peak land values in excess of $700 per front foot (supermarkets, drugstores, barber shops, and restaurants). The CBD is excluded. For the regional (shopping goods) centers, data are published on the shopping goods trade area-its area, population density, and income density. Similar data are provided for the convenience goods trade area of all centers. Trade areas are determined from shopping trips sampled in the Chicago Area Transportation Study and exclude areas which are not used for residences. These trade areas are mapped

23 M.J. Webber / 131 t 1000 a a m a a a m a a a a a a I I I POPULATION PER SQUARE MU IN MARKET AREA FIG. 3. Association between hinterland areas of Iowa towns and their hinterland population densities, SOURCE: Computations upon data in [ZO]. in Berry [I, pp. 64, 681. Population and income densities are derived from data on census tracts for the year 1960; other data relate to the year The results of the various analyses are reported in Table 2. Although correlation coefficients are higher than those obtained for Iowa, at most only one third of intercenter variation in trade area is explained by population density and income. Population density is generally negatively associated with trade area, although the association is not significant overall and is positive for some classes of centers. There is some tendency for the population served to increase as density rises. The association between income and trade area is even weaker. Variance ratios are marginally significant in only two of the analyses. These results reinforce those obtained for Iowa, despite substantial differences in the type of area and method af analysis. DISCUSSION In both studies the data reveal only a very weak association between population density and the market area of nodes. Although any expected

24 132 / Geographical Analysis TABLE 2 ASSOCIATION BETWEEN POPIJUTION DENSITY, INCOME PER HEAD AND MARKET AREA OF SHOPPING CENTERS, CHICAGO 1961 ANALYSIS(') All A B C All High Statistic Centers Centers Centers Centers Income Centers All Low Income Centers SOURCE: Computations on data in [J]. NOTES: (1) All variables were transformed by taking natural logarithms. In the regression equation, Y=o+ b,x,+ b,x,+ e, Y is In (trade area is square miles), X is In (population density per square mile in trade area), X, is In (mean income per head in trade area), a, b, and b, are regression coefficients, and e is an error term. (2) These are simple correlation coefficients between the indicated variables. (3) R is the multiple correlation coefficient. (4) S( Y)e is the standard error of estimate of Y. (5) SE(b,) and SE(b,) are the standard errors of b, and b, respectively. (6) F is Snedecor's variance ratio. (7) These are partial correlation coefficients between the indicated variables, holding constant the variable after the stop. association is subject to error because of density variations within hinterlands, such error can hardly account for the fact that the relationship is almost undetectable in such disparate circumstances. Similar results have been reported by previous workers. King [ 121, in an analysis of the spacing of 200 settlements in the United States, discovered only a weak association between population density and the distance of a town from its nearest neighbor; the correlation between the density of the rural farm population and the distance of a town from its nearest neighbor of similar size was only r = Brush and Bracey [ 71 found that town spacing was similar in Wisconsin and southern England, despite different population densities. (Note that this last result is complicated by the difference between the two areas in transport costs.)

25 M.J. Webber / 133 Significantly, in all these studies the population density variations within the regions investigated are not very great. In Iowa mean density ranges from 15 to 50 persons per square mile; in Chicago the absolute range is much greater, but even so 85% of the market areas have densities between 10,OOO and 50,000 persons per square mile of residential land; Brush and Bracey compared two regions of densities 30 and 182 persons per square mile; in King s study, rural farm density varies between 10 and 50 persons per square mile. The highest mean density in a hinterland is never much more than four or five times the lowest. By contrast, Chicago densities are commonly one thousand times as great as those in Iowa (though densities are measured differently in the two cases) and at this scale there are clear differences in the market areas of equivalent nodes. These results are consistent with the constraint that entry is free. When density variations are limited, density has an insignificant negative effect on spacing and positive effect on population served. On the larger scale, when transport costs vary significantly with density, density is negatively associated with spacing. These results are not consistent with any of the social constraints upon location patterns which have been investigated in this paper; they may be accounted for by the less restrictive constraint of free entry. But these results should not be regarded as establishing that the appropriate constraint upon location is free entry. In the first place, the models may be subject to fundamental error. There is another potential difficulty associated with this conclusion: that repeated tests, at different population-density and transport-rate levels, may also reveal a weak negative relationship between density and spacing when density varies within relatively narrow limits. Should that happen, a model predicated upon free entry of nodes on a homogeneous plain would not be an acceptable model of the location pattern. Some additional constraint upon entry would be necessary-for example, that markets are bounded. LITERATURE CITED 1. BERRY, B. J. L. Commercial Structure and Commercial Blight. Chicago: University of Chicago, Geo raphy of Market Centres and Retail Distribution. Englewood Cliffs: Prentice HaB, BERRY, B. J. L. and W. L. GARRISON. Recent Developments of Central Place Theory, Regional Science Association, Papers and Proceedings, 4 (1958), The Functional Bases of the Central Place Hierarchy. Economic Geography, 34 (1958), BERRY, B. J. L. and R. J. TENNANT. Chicago Commercial Reference Handbook. Chicago: University of Chicago, BOGUE, D.. and C. L. BEALE. Economic Areus of the United States. New York: Free Press of G r encoe, BRUSH, J. E. and H. E. BRACEY. Rural Service Centers in Southwestern Wisconsin and Southern England. Geographical Reoiew, 45 (1955), CHRISTALLER, W. Central Places in Southern Gemny. Englewood Cliffs: Prentice Hall, 1966.

26 134 / Geographical Analysis 9. HOOVER, E. M. The Location of Economic Actioity. New York: McGraw-Hill, HOIJSE,~. W. Medium Sized Towns in the Urban Pattern of Two Industrial Societies: Englan and Wales-USA. Planning Outlook, 3 (1953), KING, L. X The Functional Role of Small Towns in Canterbury. 3rd New Zealand 12. Geograp La1 Conference, Proceedings. (1961), A Multivariate Analysis of the Spacin of Urban Settlements in the United States. Annals of the Association of American Eeographers, 51 (1961), LONG, W. H. Demand in Space: Some Neglected Aspects. Regional Science Association, PaDers, 27 ( LOSCH. A. The Econom-ics of Location. New Haven: Yale Universitv.., MILLS, E. S., and M. R. LAV. A Model of Market Areas with Free Entry. Journal of Political Economy, 72 (1964), NEUTZE, G. M. Economic Policy and the Size ofcities. Canberra: Australian National University, PELENSKY, E. Cost of Urban Car Traoel. Australian Road Research Board, Special Report No. 5, Sydney, SCOTT, P. The Hierarchy of Central Places in Tasmania. The Australian Geographer, 9 (1964), STAFFORD, H. A. The Functional Bases of Small Towns. Economic Geography, 39 (1963), UNITED STATES BUREAU OF THE CENSUS. US. Census of the Population: 1960, Vol. I: Number of Inhabitants. Washington: G.P.O., WEBBER, M. J. Population Growth and Town Location in an Agricultural Economy: Iowa, Geographical Analysis, 4 (1972), ~. Free Entry and the Locational Equlibrium. Association of American Geographers, Annals, 64 (1974), forthcoming. 23. WEBBER, M. J. and R. SYMANSKI. Periodic Markets: A Classical Location Analysis. Economic Geography, 49 (1973),