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Theoretical Geolocation Models
5 Theoretical Geolocation Models
In order to account for precise locations, geographers mainly employ empirical models. They use statistical models to deal with surveys made by businessmen. The content of these surveys is organized in relation to four types of factors supposed to analyze the locations. Surveyors identify first of all the advantages of certain territories, whether natural resources necessary for production, such as the presence of an ore, or good transport infrastructures, especially the presence of an international airport or highway access. The tax system, which plays a certain role on a national level among districts and is even more significant on an international scale, for example, among European countries, is equally important. Another set of factors includes externalities. The so-called static externalities depend mostly on the size of the market. In order to sell their products or services, businesses are advised to be located as close as possible to a large market, like a megalopolis. The presence of prestigious high schools and universities, as well as a good cultural reputation, represent other appealing factors. The so-called dynamic externalities correspond instead to a potential of information exchange with similar or complementary firms. Another group of factors is represented by how trained the workforce is and, even more significantly, by wage levels. These are the reasons behind the phenomenon of outsourcing toward Asia, Eastern European and North African countries. Finally, we should also consider some factors that, on the contrary, ensure the spatial dispersion of businesses rather than their concentration. However, the location of a business largely depends on its functional features, besides these external factors. Businesses with low competition, unaffected by labor costs and heavily dependent on intermediate goods tend to be located near large labor and consumer markets. Luxury goods industries are a good example.
130
Geographical Models with Mathematica
Conversely, a business that employs unskilled labor relies on a natural resource, is able to handle low transport costs and is not afraid to move away from a large urban center, as the locations of steel industries can prove. All these studies are similar from a methodological point of view. The only variable is the kind of firms considered. For example, just recently public authorities have been increasing the number of inquiries related to the location of businesses that had moved offshore and are now relocating to their home country. However, the results of these inquiries are nearly always processed statistically with the data analyses described in Chapter 2. On the contrary, spatial analysis tools, which will be presented in the following chapters and are more suitable for this type of issue, are not used often enough. For a long time, computer programs did not implement theoretical models. This is certainly no longer true. Most classic theoretical models, studied by the first spatial economists, are now available in Wolfram Demonstrations Project programs, sometimes under a different name. Thus, we have not reprogrammed them in this work. In this chapter, we will only show and comment on some graphic results. 5.1. Von Thünen and d’Alonso’s monocentric and polycentric models D’Alonso’s model, derived from the von Thünen model and applied to a city rather than to the countryside, has been programmed in Mathematica by R.J. Brown [BRO 05a, BRO 05b]. Its graphic result shows the structure of a concentric system within a monocentric city. The author identifies five kinds of land uses. At the center, in the Central Business District, commercial activities and banking are predominant. Then, after a belt of small industries, we can find a residential pocket, encircled in turn by industries before the beginning of a rural section. We would only have to change the names and some of the parameters, and then add a sixth concentric area, in order to adapt this program to von Thünen’s initial model and, therefore, analyze the distribution of farmland around a market. Figure 5.1 shows the initial situation. If we click on the “Show table” box, the program can even show us the area and income corresponding to each of the five activities. With the instruction Manipulate[], geographers can modify several parameters. They can change the initial conditions by dragging one or more sliders. For example, they can make the residential sector or the first industrial belt more or less significant. According to these inputs, they obtain a new diagram of the urban structure. This dynamic contribution makes it much easier to understand these
Theoretical Geolocation G Mod dels
131
geographhical phenomeena. Besides this t representaation of the ciity, the prograam yields the areass and income for f each belt.
Fig gure 5.1. The von Thünen–A Alonso model (source: [BRO O 05a, BRO 0 05b], W Wolfram Demo onstrations Pro oject). For a color c version of the figure, se ee w www.iste.co.u uk/dauphine/m mathematica.ziip
The author of the previous model, R.J. Brow wn [BRO 05aa, BRO 05b],, has also D ns Project proggram that developeed another equually interestinng Wolfram Demonstration simulates the formatioon of multiplee centers, the appearance of o edge cities, around a metropolis. As in thee previous case, this smalll program alllows us to display the w this tim me, however, is polycentricc. The urban fabric is dynamiccs of a city which represennted in three dimensions. d U Users of this program can create one to tthree new edge citiies. Moreoverr, this program m allows us to vary the distaance between the main center annd these secoondary centerrs, to locate the t latter by modifying a direction parameteer, and even too link them within w a conurb bation that woould appear if a circular transportt corridor, likee a railway neetwork or a road system sim milar to the rinng road in Paris, were w built. In spite of its siimplicity, thiss little prograam is quite innstructive. Figure 5.2 5 illustratess a stage of this appearaance of edge towns arounnd a city represennted by an urbaan peak. As with w the previoous program, if i we drag cerrtain sliders, a conurbation can form or break apart according to the valuues considered d.
132
Ge eographical Mod dels with Mathematica
Figure 5.2. Formation of a polycentrric conurbation n (source: 0 BRO 05b]], Wolfram De emonstrations Project) [BRO 05a,
5.2. Ste einer’s mode el generalize es Weber’s The problem put forward by Fermat in th he 17th Centuury underlies Weber’s W point minimizes the sum of disttances from the t three verttices of a model. Which triangle with all anglees less than 1220°? This pro oblem was firsst solved by T Torricelli, metric solution: the straight lines betweenn this point, M M, and the who provvided a geom three verrtices A, B annd C form anngles of 120°. In a standardd case, illustraated by a mechaniical model, a system is in a state of eq quilibrium whhen potential energy is minimal. The solutioon is describeed by the so o-called Variggnon triangle.. Finally, e ans have put foorward algebraic and numerric solutions. several econometricia Mostt importantly,, Fermat’s iniitial problem led to the deevelopment oof several generalizzations that aimed a to takee actual consttraints into account. The first one, conceiveed by Steinerr, consists in merely consiidering not onnly three, butt a larger number of points. Foor geographerrs, this amou unts to determ mining a locaation that ferent transferr costs linkeed to the includes several asseembly sourcees, with diffe productss that have to be incorporatted. This situaation is more similar to ourr present-
Theoretical Geolocation Models
133
day industrial economy, especially in relation to automotive industries or tablet manufacturers, when assembly plants are supplied by several sources. What is more interesting to geographers is that the distance considered is often weighted. At the moment, it represents a price-distance that incorporates the weight of the materials and their gangue. Thus, Weber’s model, described in Chapter 2, becomes a specific case of the generalized Fermat’s model. In order to show the extrapolation to different products, we use the program written by J. Rangel-Mondragon [RAN 12], which is still available in the Wolfram Demonstrations Project. Figure 5.3 has been obtained with this program. Besides Fermat’s optimal point, the program displays the centroid. With this dynamic program, readers can, on the one hand, choose the number of points – from 3 to 10 – and, on the other hand, displace each point as they wish. Therefore, they can even match the location of the points to a specific case, for example, an assembly plant supplied by several shops with known spatial coordinates. Thus, this program is not merely didactic. It can be adapted quite easily to test Weber’s theory on any region of the world. We only have to replace the monochromatic background with the image of a map and then displace the points on this more realistic background.
Figure 5.3. Weber-Steiner’s model (source: J. Rangel-Mondragon [RAN 12], Wolfram Demonstrations Project)
134
Geographical Models with Mathematica
In spite of the informative qualities of these exercises, Weber’s model is too rigid if applied to the actual world. In fact, its results can lead us to set up a manufacturing plant in a totally deserted place. In the real world, businessmen prefer setting up their activity in a pre-existing city, even if still taking Weber’s hypotheses into account. This choice allows them to benefit from the so-called external economies (the presence of several amenities, housing, commerce, personal and business services, etc.). An optimal location is determined in a space of points, which represent several appealing places. Only some very specific industries, often linked to mining or causing pollution, can ignore discontinuities in residential spaces and be set up on sites that lack equipment and workforce. 5.3. Central place models in the making As far as we know, the central place models conceived by Christaller or Loesch have not been elaborated in terms of computer programming. Consequently, geographers cannot carry out instructive simulation experiments, or display certain dynamics and compare them with a basemap that outlines a geographical feature. However, it is possible to carry out experiments that compare a hexagonal model to a “field feature” shown on a map. The solution consists in superposing a network of hexagons onto the map in an image format. After importing the map image, we only have to process it with the simple program presented below, which is drawn from the help section describing the instruction RegularPolygon[]: b = {{3/2, Sqrt[2]/2}, {3/2, -Sqrt[3]/2}}; pts = Tuples[Range[0, 5], 2].b; tiles = Table[RegularPolygon[p, {1, 0}, 6], {p, pts}]; Graphics[{White, EdgeForm[Black], tiles, Red, Point[pts]}]
However, this approach only takes into account a level of hexagonal frames. Another solution consists of creating a figure made up of nested hexagons and superposing it onto a map. Moreover, in order to avoid making mistakes, we should take care to make the two graphs even. On the contrary, when using the 10th version of Mathematica, it is relatively easy to carry out several grid transformations and, therefore, to bring out certain sectors like in Loesch’s theory. Finally, we could consider the possibility of calculating the differences between the model and the field conditions. More recently, in the last versions of Mathematica, new kinds of topological and geometric processing should make it easier to elaborate these models. Another solution consists in calculating the R statistics described in depth in Chapter 7. This statistical indicator allows us to distinguish between concentrated (R = 0), random (R = 1) and regular (R = 2.15) distributions of cities. However, this regularity is not necessarily hexagonal. It can correspond to a square or triangular mesh.
Theoretical Geolocation Models
135
5.4. Conclusion If theoretical location models are very useful from a didactic point of view, the empirical ones turn out to be more effective when we want to understand the very large variety of the locations of present-day activities. Consequently, statistical models are favored in most studies. Besides, geolocation is developing thanks to the diffusion of new technologies. It is becoming widespread thanks to GPSs, tablets and mobile phones. These technologies allow us to answer simple questions like where are the Italian restaurants located within 2 km of my phone? However, as things stand now, these tools cannot process relatively complex locations and are even more clueless when it comes to analyzing them.
In order to account for precise locations, geographers mainly employ empirical models. They use statistical models to deal with surveys made by businessmen. The content of these surveys is organized in relation to four types of factors supposed to analyze the locations. Surveyors identify first of all the advantages of certain territories, whether natural resources necessary for production, such as the presence of an ore, or good transport infrastructures, especially the presence of an international airport or highway access. The tax system, which plays a certain role on a national level among districts and is even more significant on an international scale, for example, among European countries, is equally important. Another set of factors includes externalities. The so-called static externalities depend mostly on the size of the market. In order to sell their products or services, businesses are advised to be located as close as possible to a large market, like a megalopolis. The presence of prestigious high schools and universities, as well as a good cultural reputation, represent other appealing factors. The so-called dynamic externalities correspond instead to a potential of information exchange with similar or complementary firms. Another group of factors is represented by how trained the workforce is and, even more significantly, by wage levels. These are the reasons behind the phenomenon of outsourcing toward Asia, Eastern European and North African countries. Finally, we should also consider some factors that, on the contrary, ensure the spatial dispersion of businesses rather than their concentration. However, the location of a business largely depends on its functional features, besides these external factors. Businesses with low competition, unaffected by labor costs and heavily dependent on intermediate goods tend to be located near large labor and consumer markets. Luxury goods industries are a good example.
130
Geographical Models with Mathematica
Conversely, a business that employs unskilled labor relies on a natural resource, is able to handle low transport costs and is not afraid to move away from a large urban center, as the locations of steel industries can prove. All these studies are similar from a methodological point of view. The only variable is the kind of firms considered. For example, just recently public authorities have been increasing the number of inquiries related to the location of businesses that had moved offshore and are now relocating to their home country. However, the results of these inquiries are nearly always processed statistically with the data analyses described in Chapter 2. On the contrary, spatial analysis tools, which will be presented in the following chapters and are more suitable for this type of issue, are not used often enough. For a long time, computer programs did not implement theoretical models. This is certainly no longer true. Most classic theoretical models, studied by the first spatial economists, are now available in Wolfram Demonstrations Project programs, sometimes under a different name. Thus, we have not reprogrammed them in this work. In this chapter, we will only show and comment on some graphic results. 5.1. Von Thünen and d’Alonso’s monocentric and polycentric models D’Alonso’s model, derived from the von Thünen model and applied to a city rather than to the countryside, has been programmed in Mathematica by R.J. Brown [BRO 05a, BRO 05b]. Its graphic result shows the structure of a concentric system within a monocentric city. The author identifies five kinds of land uses. At the center, in the Central Business District, commercial activities and banking are predominant. Then, after a belt of small industries, we can find a residential pocket, encircled in turn by industries before the beginning of a rural section. We would only have to change the names and some of the parameters, and then add a sixth concentric area, in order to adapt this program to von Thünen’s initial model and, therefore, analyze the distribution of farmland around a market. Figure 5.1 shows the initial situation. If we click on the “Show table” box, the program can even show us the area and income corresponding to each of the five activities. With the instruction Manipulate[], geographers can modify several parameters. They can change the initial conditions by dragging one or more sliders. For example, they can make the residential sector or the first industrial belt more or less significant. According to these inputs, they obtain a new diagram of the urban structure. This dynamic contribution makes it much easier to understand these
Theoretical Geolocation G Mod dels
131
geographhical phenomeena. Besides this t representaation of the ciity, the prograam yields the areass and income for f each belt.
Fig gure 5.1. The von Thünen–A Alonso model (source: [BRO O 05a, BRO 0 05b], W Wolfram Demo onstrations Pro oject). For a color c version of the figure, se ee w www.iste.co.u uk/dauphine/m mathematica.ziip
The author of the previous model, R.J. Brow wn [BRO 05aa, BRO 05b],, has also D ns Project proggram that developeed another equually interestinng Wolfram Demonstration simulates the formatioon of multiplee centers, the appearance of o edge cities, around a metropolis. As in thee previous case, this smalll program alllows us to display the w this tim me, however, is polycentricc. The urban fabric is dynamiccs of a city which represennted in three dimensions. d U Users of this program can create one to tthree new edge citiies. Moreoverr, this program m allows us to vary the distaance between the main center annd these secoondary centerrs, to locate the t latter by modifying a direction parameteer, and even too link them within w a conurb bation that woould appear if a circular transportt corridor, likee a railway neetwork or a road system sim milar to the rinng road in Paris, were w built. In spite of its siimplicity, thiss little prograam is quite innstructive. Figure 5.2 5 illustratess a stage of this appearaance of edge towns arounnd a city represennted by an urbaan peak. As with w the previoous program, if i we drag cerrtain sliders, a conurbation can form or break apart according to the valuues considered d.
132
Ge eographical Mod dels with Mathematica
Figure 5.2. Formation of a polycentrric conurbation n (source: 0 BRO 05b]], Wolfram De emonstrations Project) [BRO 05a,
5.2. Ste einer’s mode el generalize es Weber’s The problem put forward by Fermat in th he 17th Centuury underlies Weber’s W point minimizes the sum of disttances from the t three verttices of a model. Which triangle with all anglees less than 1220°? This pro oblem was firsst solved by T Torricelli, metric solution: the straight lines betweenn this point, M M, and the who provvided a geom three verrtices A, B annd C form anngles of 120°. In a standardd case, illustraated by a mechaniical model, a system is in a state of eq quilibrium whhen potential energy is minimal. The solutioon is describeed by the so o-called Variggnon triangle.. Finally, e ans have put foorward algebraic and numerric solutions. several econometricia Mostt importantly,, Fermat’s iniitial problem led to the deevelopment oof several generalizzations that aimed a to takee actual consttraints into account. The first one, conceiveed by Steinerr, consists in merely consiidering not onnly three, butt a larger number of points. Foor geographerrs, this amou unts to determ mining a locaation that ferent transferr costs linkeed to the includes several asseembly sourcees, with diffe productss that have to be incorporatted. This situaation is more similar to ourr present-
Theoretical Geolocation Models
133
day industrial economy, especially in relation to automotive industries or tablet manufacturers, when assembly plants are supplied by several sources. What is more interesting to geographers is that the distance considered is often weighted. At the moment, it represents a price-distance that incorporates the weight of the materials and their gangue. Thus, Weber’s model, described in Chapter 2, becomes a specific case of the generalized Fermat’s model. In order to show the extrapolation to different products, we use the program written by J. Rangel-Mondragon [RAN 12], which is still available in the Wolfram Demonstrations Project. Figure 5.3 has been obtained with this program. Besides Fermat’s optimal point, the program displays the centroid. With this dynamic program, readers can, on the one hand, choose the number of points – from 3 to 10 – and, on the other hand, displace each point as they wish. Therefore, they can even match the location of the points to a specific case, for example, an assembly plant supplied by several shops with known spatial coordinates. Thus, this program is not merely didactic. It can be adapted quite easily to test Weber’s theory on any region of the world. We only have to replace the monochromatic background with the image of a map and then displace the points on this more realistic background.
Figure 5.3. Weber-Steiner’s model (source: J. Rangel-Mondragon [RAN 12], Wolfram Demonstrations Project)
134
Geographical Models with Mathematica
In spite of the informative qualities of these exercises, Weber’s model is too rigid if applied to the actual world. In fact, its results can lead us to set up a manufacturing plant in a totally deserted place. In the real world, businessmen prefer setting up their activity in a pre-existing city, even if still taking Weber’s hypotheses into account. This choice allows them to benefit from the so-called external economies (the presence of several amenities, housing, commerce, personal and business services, etc.). An optimal location is determined in a space of points, which represent several appealing places. Only some very specific industries, often linked to mining or causing pollution, can ignore discontinuities in residential spaces and be set up on sites that lack equipment and workforce. 5.3. Central place models in the making As far as we know, the central place models conceived by Christaller or Loesch have not been elaborated in terms of computer programming. Consequently, geographers cannot carry out instructive simulation experiments, or display certain dynamics and compare them with a basemap that outlines a geographical feature. However, it is possible to carry out experiments that compare a hexagonal model to a “field feature” shown on a map. The solution consists in superposing a network of hexagons onto the map in an image format. After importing the map image, we only have to process it with the simple program presented below, which is drawn from the help section describing the instruction RegularPolygon[]: b = {{3/2, Sqrt[2]/2}, {3/2, -Sqrt[3]/2}}; pts = Tuples[Range[0, 5], 2].b; tiles = Table[RegularPolygon[p, {1, 0}, 6], {p, pts}]; Graphics[{White, EdgeForm[Black], tiles, Red, Point[pts]}]
However, this approach only takes into account a level of hexagonal frames. Another solution consists of creating a figure made up of nested hexagons and superposing it onto a map. Moreover, in order to avoid making mistakes, we should take care to make the two graphs even. On the contrary, when using the 10th version of Mathematica, it is relatively easy to carry out several grid transformations and, therefore, to bring out certain sectors like in Loesch’s theory. Finally, we could consider the possibility of calculating the differences between the model and the field conditions. More recently, in the last versions of Mathematica, new kinds of topological and geometric processing should make it easier to elaborate these models. Another solution consists in calculating the R statistics described in depth in Chapter 7. This statistical indicator allows us to distinguish between concentrated (R = 0), random (R = 1) and regular (R = 2.15) distributions of cities. However, this regularity is not necessarily hexagonal. It can correspond to a square or triangular mesh.
Theoretical Geolocation Models
135
5.4. Conclusion If theoretical location models are very useful from a didactic point of view, the empirical ones turn out to be more effective when we want to understand the very large variety of the locations of present-day activities. Consequently, statistical models are favored in most studies. Besides, geolocation is developing thanks to the diffusion of new technologies. It is becoming widespread thanks to GPSs, tablets and mobile phones. These technologies allow us to answer simple questions like where are the Italian restaurants located within 2 km of my phone? However, as things stand now, these tools cannot process relatively complex locations and are even more clueless when it comes to analyzing them.