Central Place Theory (Anderson 2012)

22 Central place theory In the last chapter, we saw evidence of hierarchical structure in city s\ ize dis- tributions. As geographers, we are not much satisfied with size distri\ butions; our main interest is in spatial distributions. We will see, however, that th\ e idea of a hierarchical size distr...

22 Central place theory In the last chapter, we saw evidence of hierarchical structure in city s\ ize dis- tributions. As geographers, we are not much satisfied with size distri\ butions; our main interest is in spatial distributions. We will see, however, that th\ e idea of a hierarchical size distribution is a useful point of departure in develop\ ing models of the spatial distribution of cities.The topic of this chapter is central place theory, which is a framework for explaining a spatial-hierarchical ordering of cities. This is one of the\ “classic” topics of research in economic geography, but in recent years it has fallen som\ ewhat out of fashion. In fact, many recent texts don’t even cover it, or if the\ y do they focus on its outcomes – especially the peculiar hexagonal pattern of urban \ market areas that it predicts – rather than its underlying mechanisms. The main re\ ason that central place theory has become passé is that its predicted patterns don’t fit the urban systems of the twenty-first century very well. But it still yields some \ powerful insights when its underlying logic is understood. As in the case of location “theory,” central place “theory” \ would be more accurately defined as the central place model. (However, we will stick \ with convention and use the term “theory,” albeit loosely.) Recall tha\ t models are simplified representations of reality that are constructed to explore \ and illustrate mechanisms that give rise to observed patterns. Models cannot reproduce \ the real world precisely, but they provide possible explanations as to what under\ lies the things we see in reality. The spatial patterns of urban systems are driv\ en by an array of different and sometimes contradictory mechanisms. The most we can hop\ e to get from any model is a better understanding of some of the most importa\ nt of those mechanisms. Judged by that standard, central place theory still provides\ one of the most powerful models in economic geography. Perhaps the most important thing to understand about central place theor\ y is that it is meant to explain the spatial pattern of a particular class of\ cities known as central places. Roughly speaking, traditional classifications of citi\ es define three functional types: transportation cities, production cities and cen\ tral places. (In later chapters, we will propose some new types, including global ci\ ties and information cities.) Transportation cities are generally points of tran\ s-shipment where different forms of transportation come together and the economy is\ dominated by industries such as trade, shipping, warehousing and distrib\ ution. Production cities are manufacturing centers where a variety of inputs ar\ e assembled and transformed into goods of higher value. Central places are located i\ n pre- dominantly agricultural regions where they provide goods and services to\ a dis- persed population. In the early stages of their development, transportat\ ion and production cities are located to take advantage of spatial variation in \ the natural landscape. Transportation cities are generally located on rivers or at n\ atural harbors, or better yet where a river and a natural harbor come together.\ Production cities must locate where materials and energy resources are easily broug\ ht together. Central places, by contrast, are located in a relatively undifferentiate\ d landscape with an evenly distributed population of potential customers. As such, t\ heir locations are not dictated by natural features. One might think, therefo\ re, that they are “footloose” and that their location pattern could be more\ or less random. But because they compete for a finite market, their locations relative to one another affect their economic success. Of course, few cities fit into any of these ideal types – nearly ever\ y city has elements of all three. But, since the locational forces acting on the th\ ree types differ, a model that addresses the locational pattern of just one type will be m\ uch easier to develop. This, in essence, is the modeling strategy of central place \ theory. Christaller’s central place theory 1 Central place theory was developed in the 1930s by the German geographer\ Walter Christaller. While most people think of central place theory as a highly\ theoretical exercise, Christaller’s objectives were practical: he wanted to expla\ in the spatial pattern of urban places in the agricultural region of southern Germany. \ In studying the region, he observed two spatial characteristics. First, the city siz\ e distribution was clearly hierarchical with few cities at the highest order and increa\ sing numbers of cities at lower orders. (In central place theory, it is conventional\ to refer to levels of the hierarchy as “orders.”) Second, he observed that wi\ thin any order of cities the spatial pattern is dispersed. In other words, the highest-ord\ er cities are separated by long distances and the cities in each successively lower or\ der are spread out as much as possible. Since there are more cities at the lower\ orders, the average distance between cities becomes progressively lower for lower or\ ders of cities. He also observed something that is not explicitly spatial in nat\ ure, but which is a critical characteristic of the urban hierarchy: the number of goods\ and services on offer is higher at higher levels of the hierarchy. Christaller develo\ ped his model to describe economic mechanisms that would give rise to a spatial-hierar\ chical distribution with these three characteristics. As we will see, central place theory produces some complex spatial patte\ rns. But to get an understanding of its underlying mechanisms it’s best to sta\ rt with the simplest situation imaginable. To that end, make the following two simpl\ ifying assumptions: 1 The entire population of consumers is equally distributed in a bounded l\ inear market at a density of n per kilometer. Central place theory289 2 There is only one undifferentiated good (or service) that may be offer\ ed atcentral places. We have used the first assumption before in the case of strategic loca\ tion in chapter 16. For the moment, we adopt it because it makes graphical representation \ of the model simpler. We will eventually generalize the model to two-dimensiona\ l space. The second assumption means that any central place will be limited to of\ fering just one good or service and that, if there is more than one central place, e\ ach will offer exactly the same thing. (To be general, we will refer to any good or se\ rvice offered at a central place as a central function.) In a real agricultural landscape, the most typical central function might be a general store, but, since such an es\ tablishment offers a variety of goods and services, it is simpler to think of the ce\ ntral function as offering a single good (for example, an ice-cream stand) or a singl\ e service (for example, a barber shop). We now need a couple of assumptions about cons\ umers. 3 Consumers have downward sloping demand functions such that above some price they purchase nothing from the central function (see Figure 22.1). 4 The effective price for a consumer is the sum of a fixed purchase pric\ e P cand a travel cost that is proportional to their distance from the central pl\ ace. To make assumption 4 more explicit, the effective price of the central f\ unction to a consumer located at distance d ifrom a central place is P i
P ctd i where tis a constant transportation cost per unit of distance. The fact that P cis constant means that firms at central places do not compete with one an\ other on price – an important limitation of central place theory to which we w\ ill return later. Assumptions 3 and 4 together imply that we can define a spatial demand\ curve around the location of a central place as shown in Figure 22.2. The height of this graph shows the amount that someone located at any distance from the cen\ tral place will consumer per unit of time (say a year). The slope of the curve de\ pends both on the slope of the demand function (Figure 22.1) and on the transportation rate t. The distance defines the point in the linear market where demand goes \ to zero. This implies that P 0
P ctd R and therefore d R
P 0 P c t Consumers located at a distance greater than d Rconsume nothing from the central place. 290 Systems of cities We can use this diagram to determine the total sales of the central func\ tion. Recalling that n is the density of consumers, total sales are equal to the area under the spatial demand curve times the density times the average revenue: 2 SR
nd RQcPc From this we can observe two things. First, sales are increasing in the \ density nof consumers. Second, since d Ris decreasing in tand S Ris increasing in d R, a decrease in the transportation rate will bring about an increase in sales. Central place theory 291 Price P 0 PC QC Quantity threshold Quantity Q C dR dR Distance Spatial demand curve d T dT range Figure 22.1 Demand curve Figure 22.2Spatial demand curve, range and threshold A second critical distance d Tis called the threshold distance. Because of fixed cost, a firm that wants to provide the central function must have a ce\ rtain minimum sales level S Tin order to break even. The threshold distance defines the length of the linear market necessary to provide this minimum sales level. In the \ figure, the threshold distance is smaller than the range, which means that the firm \ earns revenues that are greater than the break-even level. If d T> d Rthen the central function is not economically viable because the length of market that it\ must command to break even is greater than the maximum distance that people w\ ould travel to consume it. Figure 22.3a shows what happens if there are two firms at different locations offering the central function. If they locate well apart as shown, they \ will not affect each other’s sales because no part of the line (and therefore no con\ sumers) will be within the range of both. In fact, a number of consumers will be within \ the range of neither, so the market is not completely covered. In Figure 22.3b, we see what happens if a third firm enters the market. In this case the other two sh\ ift their locations to become evenly spaced at a distance 2d R, so once again their ranges do not cross. This means that each of the three firms would have sales S R– in other words, their sales are unaffected by the presence of the other firms. \ But things get more interesting if a fourth firm enters the market. Assuming once aga\ in that the firms space themselves equally, the minimum feasible spacing is 2d Rwhere each firm is able to get its threshold sales level S T. If they were spaced closer, none of the firms would be economically viable. Thus, the threshold sales level \ and its related threshold distance determine the number of firms that can “\ fit” in this market. In our simple example, there is only one central function, so the locati\ on of each firm offering that function represents the establishment of a cen\ tral place. In other words, the pattern of firm locations determines the pattern and de\ nsity of urban locations in this very simple system of cities. The density of cit\ ies is determined by the threshold, which is essentially a reflection of scal\ e economies. The pattern of cities is dispersed as a result of an implied assumption \ that cities space themselves evenly so that a maximum number can fit in the market\ . This has been a point of much criticism of central place theory, since the me\ chanisms underlying this dispersed pattern are not defined. 3In fact, the Hotelling model of chapter 16 suggests that competing firms left to their own devices may cluster r\ ather than disperse. Because there is a limited range, however, a clustered pa\ ttern would miss potential sales to much of the dispersed population. Thus, the rang\ e provides at least a rationale, if not a precise mechanism, for the dispersed patt\ ern. We can extend the model to two-dimensional space by changing assumption \ 1 as follows: 1a The entire population of consumers is equally distributed in an undiffer\ entiated plane at a density of n per square kilometer. (Note that the density nhas a different meaning now because it is defined per square kilometer instead of per meter.) Figure 22.4 shows the two-dimensional versions of the range and the threshold distance as the radii of concentric circl\ es around the 292 Systems of cities central place. Here the “hats” over the distance variables dˆ Rand dˆ Tare there to indicate that the distance variables are defined in two-dimensional sp\ ace rather than one-dimensional space as before. The range distance dˆ Rstill has the same basic meaning as the maximum distance that anyone would travel to consume the \ central function. The threshold distance d ˆ T, however, has a slightly different meaning from before. In the two-dimensional case, it is the radius of a circle that e\ ncloses a population large enough to provide the threshold level of sales S T. Thus, with a given population density and demand curve, the threshold is defined in\ terms of a minimum market area that the central place must command. If we arrange central places spaced at 2 dˆ R(analogous to Figure 23.3b) we get the pattern in Figure 22.5. In this case the market is not completely covered by the Central place theory 293 2dR 2dR 2dT 2dT 2dT Figure 22.3aTwo firms with uncovered market Figure 22.3bThree firms, fully covered market Figure 22.3cFour firms, minimum spacing circles defined by the range, so some people in the plane will have ze\ ro demand. Extending the results of our one-dimensional model, we should be able to\ space the central places at 2dˆ Tas shown in Figure 22.6a. Now all the parts of the plane are within the range distance of at least one central place. But the cir\ cles defined by the range distance overlap. So how do we define a distinct market a\ rea around each central place? This is where the hexagons come in. The hexagonal market areas diagrams produced by central place theory are\ perhaps the most iconic image associated with economic geography. Sadly,\ if you ask a sample of students who have completed an introductory course in ec\ onomic geography to define central place theory, a fair number will probably \ tell you that 5 294 Systems of cities dˆR dˆT 2d ˆR Figure 22.4 The range and threshold in two dimensions Figure 22.5 Central places spaced at the range distance it’s the theory that market areas are shaped like hexagons. In fact, \ the hexagon is just a geometric convenience with no intrinsic meaning in the theory. Ce\ rtain geometric shapes – including triangles, squares and hexagons – are\ space filling, which means that they can be arranged to cover an entire space with no g\ aps or overlaps. The figures above indicate that circles are not space-filling \ shapes; if they do not overlap they leave gaps (Figure 22.5) and if they do not leave gaps they overlap (Figure 22.6a).Figure 22.6b shows that hexagons can provide a space-filling approximation to circles. Note that a straight line between the central place in the midd\ le of the figure and any other central place is exactly bisected by one side of the hexag\ on at a distance d ˆ T. This pattern can be extended to define exclusive market areas – e\ ach sufficiently large to generate the threshold sales level S T– around evenly spaced central places as shown in Figure 22.7. The market areas make it possible to know where a person at any point in the plane will go to consume the central \ function. Since central functions are undifferentiated, one always consumes from t\ he closest center. A person located within a given market area will always consume \ from the central place at its center. Central place hierarchy So far, the idea of an urban hierarchy has been absent from our model de\ velopment. Christaller’s theory explains the hierarchy as arising out of variabi\ lity of the thresholds for different central functions on offer. To explain this, fi\ rst alter assumption 2 as follows: 2a There are two undifferentiated goods (or services) with different thre\ sholds that may be offered at central places. Central place theory 295 2dˆT Figure 22.6a Central places spaced at the threshold distance Figure 22.6b The hexagonal market area For example, suppose the central function with the lower threshold is ba\ rber shops. The function with the higher threshold may be doctors. Since a doctor mu\ st make a higher investment in education and equipment than a barber, she needs \ a larger market area to make her practice economically viable.We have demonstrated that the spacing, and therefore the number, of cent\ ral places depends on the required threshold distance dˆ T, which in turn depends on the required threshold sales S T. Let’s say that Figure 22.7 represents the spacing of the central places defined by the central function with the lower thre\ shold. What can we say about the locations of the central function with the higher t\ hreshold? Clearly, with a higher threshold they must be spaced further apart, and \ therefore there must be fewer of them. But how will the patterns of the two types \ of central places relate to one another? Define the central function with the lower threshold as the first-order \ central function and all the places at which it is offered as first-order cent\ ral places. Now define the function with the higher threshold as the second-order cent\ ral function. If a pattern of first-order central places already existed and the sec\ ond-order function were an innovation just getting established, it is reasonable to assume \ that anyone looking for a location for the second-order function would start by choo\ sing among the existing first-order central places. Such places would already be \ familiar to the population of consumers and would have some elementary infrastructure th\ at would not be found at other points in the plane. In this case a second-order p\ lace would be defined as a place offering both the second-order service and the first-order service. In fact, this is a fundamental assumption of Christaller’s c\ entral place theory: 1 If a higher-order service is offered at a central place, all lower-order\ services will also be offered at that central place. 296 Systems of cities Figure 22.7 Region of hexagonal market areas The number of first-order centers that become second-order centers wil\ l depend on the relative size of the ranges of the two central functions. Defin\ e kas the ratio of the second-order market areas to the first-order market areas. For \ example, if k
3and the market area for the first-order central places is 10 square ki\ lometers, the market area for the second-order central places must be 30 square ki\ lometers. Figure 22.8 illustrates the spatial pattern of central places and market areas for two orders of central function and k
3. It is easy to verify that the market areas for the second-order central \ places are three times the size of those for the first-order central places. Each\ second-order market area includes one first-order market area at its center and one\ -third of each of the first-order market areas surrounding it. This also indicates th\ at there are three times as many first-order centers as second order-centers. We can extend this system to include a third-order central place functio\ n. Once again, we will use the k
3rule and assume that the market area for each third- order central place will be three times the market area for the second-o\ rder central places and therefore nine times as big as the market areas for the fir\ st-order central places. For example, assume once again that the first-order function i\ s barber shops and the second-order function is doctors. We now add ballroom dance inst\ ructor as the third-order function. This example may not make sense to you at fi\ rst. After all, we said earlier that doctors are a higher-order function than barbe\ r shops because of their greater investment in education and equipment. Certainly, a bal\ lroom dance instructor does not need as much education as a doctor and his equipment\ is not as expensive. So why does he provide a higher-order service? Consider th\ at nearly all men (so half the population) use barber shops and just about every\ one uses the services of the doctor, although not as often. A very small proportion o\ f the population, however, will use the services of a ballroom dance instructo\ r. (In economic terms, he faces a demand function that is lower than those for \ barbers and doctors.) It is therefore necessary to serve a market with a large \ population in Central place theory 297 Figure 22.8Two-level central place system (k
3) order to have enough customers to make a ballroom dance studio viable. T\ he point of this example is that a higher-order central function is simply one th\ at requires a large market area, which may be the case for a variety of reasons.Figure 22.9 shows the central place system for three orders. Note that the relationship between the third-order centers and the second-order center\ s is exactly the same as the relationship between the second-order centers and the th\ ird-order centers. We could easily expand this to include a fourth, fifth or eve\ n higher order of center. The pattern will become more complex, but the relationship be\ tween orders will remain the same as long as k
3. This is an extremely stylized pattern generated by a model with strong a\ ssump- tions, but it offers an explanation for the three empirical regularities\ that Christaller observed in southern Germany: the hierarchical structure of city sizes, \ the dispersed patterns of cities within a particular order, and the provision of all l\ ower-order functions at higher-order central places (although, to be honest, the t\ hird is an assumption rather than a result of the model). The significance of k The choice of k
3for the development of the central place systems illustrated in Figures 22.8 and 22.9 was not arbitrary. A ratio of three between central places of 298 Systems of cities Figure 22.9 Three-level central place system ( k
3) different orders provides a neat, repeatable pattern of nesting of marke\ t areas. By contrast, there is no way to draw the spatial pattern of a k
2system. You can draw a hexagonal market area that is around a particular central place t\ hat is twice as large as that of the next lowest order, but it does not yield a patte\ rn such that, for example, all the second-order centers fall at the locations of fir\ st-order centers. It is, however, possible to devise central place systems for a number of\ other values of kincluding 4, 7, 9 and 12. 4We will not go into these in detail – if you are interested you can consult an advanced book on central place such as\ Berry (1967). But it is worth noting that systems based on values of kother than 3 produce systems that are different from the k
3system in interesting ways. The relative placement of first- and second-order central places for k
4and k
7are shown in Figure 22.10. Recall that, in the k
3system, first-order central places are located at the vertices of the hexagonal market area of a second-order central place. In the k
4system, the first-order places are located on the sides of the octagon. This arr\ angement actually makes more sense from a transportation perspective because it m\ eans that each first-order place is located on a straight line connecting two seco\ nd-order places, so a much more efficient road network could be set up to conne\ ct all central places. For this reason, the k
4 is known as the transport principle . In the k
7system, six first-order centers are located inside the triangular mark\ et area of the second-order center. In contrast to the k
3 and k
4systems, where the first-order places are located equidistant to three and two second-order\ places respectively, all first-order places are clearly in the “orbit” \ of one and only one second-order place. This makes sense in an administrative system where t\ he first- order places are under the jurisdiction of the second-order places. Thus\ , k
7is known as the administrative principle. Contemporary relevance of central place theory For a number of reasons, central place theory does not lend itself easil\ y to appli- cation in real urban systems. For one thing, its assumption of an undiff\ erentiated plane seldom applies in practice because variations in the natural lands\ cape and the structure of transportation networks tend to disconnect transportati\ on costs from Central place theory 299 k = 4 k = 7 Figure 22.10 The transportation and administrative principles straight-line distance. Also, it applies only to central places, whereas\ most urban systems are hybrids of central place, transportation, production and oth\ er functional types. It was designed to address a settlement pattern where most people\ live in rural areas and cities exist to serve them. In fact, the great majority \ of people in the twenty-first century live in cities. Finally, it assumes that cent\ ral functions are undifferentiated, so distance is the main concern in consumers’ spati\ al choices. As pointed out in earlier chapters, the modern economy is increasingly c\ haracterized by differentiated goods and services.Despite these problems, there have been a number of empirical studies ap\ plying principles from central place theory to real urban systems. Generally, t\ hey focus on places that come close to meeting the assumptions of the model: agric\ ultural regions in relatively undifferentiated terrain. Of course, the most famo\ us of these is Christaller’s own application of his theory to the pattern of citi\ es and towns in southern Germany at the beginning of the twentieth century. Perhaps the \ best known of more recent applications is Berry’s (1967) study of southwestern\ Iowa, which used modern statistical methods to define urban hierarchies based on t\ he variety of goods and services offered and traced spatial patterns of consumer tr\ avel to make purchases of various types. Various principles from the theory were confi\ rmed, such as the tendency of people to travel long distances only for higher-\ order functions and to shop in the closest place where the needed goods were a\ vailable. It also observed trends that can be explained using the underlying logic\ of central place theory, such as the disappearance of some of the lowest-order cent\ ers due to declining rural population densities. 5 Despite the strong assumptions of central place theory, recent authors s\ till find it useful in modified version for the analysis of modern retail system\ s (Dennis et al., 2002). Also, providing a set of rather stark results, central place t\ heory generates a number of testable hypotheses that can be addressed by statistical ana\ lysis (Mushinski and Weiler, 2002). Some of the most interesting applications of central place theory are by\ archeologists (see Box 22). Since they often turn up evidence of settlement patterns from societies for which there is no written history, central place theo\ ry provides a useful framework for understanding the relationships between settlemen\ ts of different sizes in societies with dispersed agricultural populations and\ small urban populations (Kosso and Kosso, 1995). 300 Systems of cities Box 22 Central place theory and ancient Mayan settlement patterns In 1973, archeologist Joyce Marcus published a paper with the title “\ Territorial Organization of the Lowland Classic Maya,” in which she drew strong parallels between the spatial patterns of excavated Mayan settlements an\ d the settlement pattern envisioned by central place theory. The classic M\ ayan civilization flourished in what is now the tropical rainforest of the \ Yucatan Central place theory301 Peninsula between 600 and 900 C.E. In the nineteenth and twentieth centu\ ries archeologists turned up numerous “lost cities” of the Mayas, with \ large stone structures, regular urban patterns and carved hieroglyphs. As more and m\ ore cities were uncovered, a regular pattern began to emerge. Marcus borrowe\ d some regularities from central place theory to explain the pattern. The Mayan view of the universe helped create a spatial structure within which a hierarchical urban pattern could exist. They believed that heave\ n and earth were divided into four segments, and so they established four primary ceremonial and administrative cities broadly spaced across their\ lowland realm, which at the time was deforested and planted in corn and \ other crops. Each primary center had an acropolis, large plazas and numerous monuments. A large set of secondary centers each had a pyramid and some monuments. A third order of centers, which controlled shifting agricultu\ ral hamlets, could also be identified from the archeological findings. P\ articularly noteworthy was the fact that the major sites (primary and secondary cen\ ters) were quite evenly spaced at an average distance of 10.33 kilometers with\ a standard deviation of only 1.9. Furthermore, the hierarchy of centers wa\ s structured according to a hexagonal lattice very similar to that of cent\ ral place theory. Figure B22.1 shows Marcus’s idealized representation of the spatial Northern capital Northerncapital Western capitalEastern capital 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 Shifting hamlets Figure B22.1 Idealized spatial structure of classic Mayan settlements Source: Marcus, 1973: Figure 8. In a sense, the utility of central place theory does not rest solely in \ its ability to reproduce, predict or explain contemporary urban patterns. It has provid\ ed many insights that are useful even in contexts that are very different from t\ he world of undifferentiated geography and goods. For example, Christaller’s work\ introduced the notion that a city’s place in the hierarchy may be defined by t\ he variety of goods and services that it provides rather than simply by its population. Also\ , by defining a fundamental logic for the development of a spatial urban hierarchy and\ a concise set of results, central place theory established a point of departure fo\ r the study of systems of cities. When the observed reality doesn’t fit the result\ s of the model, the logic must be adjusted to generate new results that can be tested ag\ ainst reality. The result of this process has been the development of new models of urb\ an systems, some of which will be introduced in the next chapter. 302 Systems of cities system of cities. Here the points marked with a star represent the four \ primary centers, while those marked 2 and 3 represent secondary and tertiary cen\ ters respectively. A similar lattice would have surrounded each of the four p\ rimary centers, although it is only shown for one in the diagram. We should take care not to draw too strong an analogy from the ancient Mayan lowlands to the central places of southern Germany. After all, the\ relationships among centers were more ceremonial than economic, although\ a large agricultural surplus must have flowed up the hierarchy from the hamlets to the urban nobility. But the interesting thing is that socioec\ onomic processes appear to have given rise to a regularly spaced and hierarchic\ al settlement pattern. It is tempting to think that the pattern simply refl\ ects a grand design imposed by a powerful centralized authority, but Marcus dou\ bts whether the Maya either planned or recognized the pattern. Rather, it em\ erged out of dependency relationships between settlements with different funct\ ions and different levels of importance.