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An agent-based model of residential patterns and social structure in urban areas

Un modèle multi-agents de localisation résidentielle et structure sociale en zone urbaine
Rémi Lemoy, Charles Raux et Pablo Jensen

Résumés

Le modèle standard de l’Economie Urbaine (Alonso, Muth, Mills) décrit analytiquement un équilibre statique de la localisation des ménages. Nous présentons un modèle multi-agents qui atteint cet équilibre d’une manière dynamique. Il nous permet de simuler l’évolution d’une ville en combinant des agents hétérogènes (selon les revenus et les préférences pour les aménités), un coût temporel de transport et diverses localisations d’aménités. Nous explorons les conditions permettant d’aboutir à des patterns résidentiels et des structures sociales de la ville non triviaux.

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Introduction

1Agent-based models have been widely used to simulate traffic at a microscopic level. The goal of the work presented here is to use this tool in Urban Economics to deal with the research question of the location of households with respect to their income.

2There are in the literature numerous analytical works on the Urban Economics model of Alonso, Muth, Mills (which is described for instance in Fujita (1989)), studying the factors explaining the location choices of households (Bruckner et al. (1999), Gofette-Nagot et al. (2000)). Exogenous amenities have been added to the model (Wu and Plantinga (2003)) to determine their influence.

3This work uses a simulation tool which is more and more widely used in social sciences: agent-based programming. This tool allows us to study economic agents living in a predefined simulation space. These agents interact in a simple way, and from their interactions, collective behaviours emerge, which are difficult to predict in an intuitive way or to compute analytically. Agent-based models have three main components: agents, an environment and rules of behaviour. The agent has internal states, some fixed and others that can change, as their preferences, and rules of behaviour. The environment is defined as a two-dimensional space supporting resources and can also be a communication network. It is a device that is separated from the agents and that interacts with them. Rules of behaviour determine the interactions between agents, between agents and the environment and within the environment. For this work we use the multi-agent programming language NetLogo and the integrated modeling environment that bears the same name.

4The study presented here is based on a model which has been widely studied, the standard Urban Economics model developed by Alonso, Muth, Mills to study households’ residential locations. Agents compete for land use in a city, they pay a transport cost to access their jobs and have income constraints. This model is exposed in section 1. It has been the object of numerous works, theoretical and also empirical ones, so that this model is suitable to be used for the conception of an agent-based model. The first stage of this work is to reproduce the classical results of the Urban Economics model. To build an agent-based model, the analytical model has to be discretized: there is a finite number of agents who interact individually. This is different from the analytical model, which is continuous, but it can be argued that social systems are indeed built on individual interactions of a finite number of agents. A comparison between the results of the simulations and those of the continuous analytical model is presented in section 1.

5The use of agent-based models allows us to easily handle agents’ states, rules of behaviour and the environment. Sets of agents such as neighborhoods can be used, so that it is easy to introduce neighborhood effects. Individual and collective behaviours can be monitored in a simple way. In addition, the simulations and the model are dynamical, which is not the case for the standard urban economics model and for most analytical economic models, which are equilibrium models. This time dependence allows us to see the equilibrium emerge from the interactions between agents, which is described in section 3, or to study non-equilibrium dynamics. The emergence of an equilibrium can have no relationship with the historical evolution of a real city, but the interactions between agents can also be used to study specifically features of cities which are linked to their historical evolution. As the results of the agent-based model are validated by comparison with the analytical model, the model can be made more complex by adding different ingredients, the first of which is heterogeneity through income groups. This allows us to explore phenomena which are difficult or impossible to treat analytically. This work is presented in section 4 with the introduction of transport time costs and amenities. The introduction of transport time cost can change the location pattern of income groups under a certain condition. Amenities for which income groups have the same preference have a complex influence on the city, but do not inverse the location pattern of income groups. Hence differentiated preferences for the amenities are introduced, leading to interesting two-dimensional effects. In section 5, several perspectives of this work, for instance the historical evolution of endogenous amenities, are presented.

1. Model

1.1. Standard Urban Economics model

6This model has been developed by Alonso, Mills and Muth to study the location of agents representing households in a city where they compete for land. These agents have daily monetary and time costs to access their workplace, which is located in the central business district (CBD) of the city. Hence this model is called monocentric model. They can also have preferences for amenities in later versions of the model (Wu and Plantinga (2003)). In the simplest version of the model, agents are single workers, but they can represent households in other versions. They rent their housing, and the landowners rent to the highest bidder. Agents compete for land use with one another and with an agricultural land use, represented by an agricultural rent Ra, which is one of the parameters of the model.

7Agents have a utility U = αln z + βln s, where z is a composite good representing all consumer goods except housing, s is the surface of housing, α and β are parameters describing agents’ preferences for composite good and housing surface. These last parameters are chosen so that α + β =1, because it makes analytical calculations easier. This “log-linear” form of the utility is often used and we use it in the agent-based model, but analytical calculations are also often performed on a Cobb-Douglas function (the exponential of this log-linear function), which gives roughly the same results, or even without specifying the concrete form of the utility function. Agents also have a budget constraint Υ = z + tx + ps, where Y is their income, t the transport cost per unit distance, x their distance to the CBD and p the price per unit surface of their housing.

8The model which is reproduced here with an agent-based system is a closed city model: the population N of the city is chosen exogenously and remains constant. This model can be solved analytically for a two-dimensional space if Ra = 0 (see Fujita (1989)). For Ra > 0 it can be solved numerically for one income group. With a population divided into several income groups, a specific algorithm is needed for the resolution of the model, whose general form is described in Fujita (1989). We did not write such an algorithm, so that we will only be able to compare quantitatively the results of the agent-based model to the results of the standard model with one income group. But the simulations presented here will also consider the case of two income groups.

9The standard urban economics model and its results are presented in a more detailed way in numerous works, for instance Fujita (1989) or Fujita and Thisse (2003).

1.2. Agent-based model

10In the agent-based system, the space is a two-dimensional grid and each cell can be inhabited by one or several agents, or be used for agriculture. The CBD is represented by a point at the center of the simulation space.

11At the beginning, the city population N is fixed and agents are placed at random locations. They can be divided into two income groups or not. Prices are all equal to the agricultural rent p0 = Ra. Agents choose the surface of housing which provides them with the higher possible utility at a given location with price p0:

Image 10000000000000640000003CCD83DB5DA1A0DE40.png

12This determines the quantity of composite good they consume and also their utility. This expression for the optimal surface can be obtained by differentiating agents’ utility with respect to surface s at fixed location and price:

Image 10000000000000FA000000796A012F78AC36D811.png

13Hence the expression of the optimal surface, with α + β = 1.

1.2.1. Moves

14Agents can move with no cost. The iteration n of a simulation, changing the variables from their value at step n to their value at step n = 1, goes as follows. An agent which will be candidate to a move and a cell are chosen randomly. The price of this cell, located at a distance x of the center, is pn. The optimal surface that the agent can choose in the candidate cell is s = β(Υ – tx) / pn, which allows us to compute the utility that the agent would get thanks to the move and to compare it to its current utility.

15If the agent candidate can gain utility ΔU > 0 by moving in the candidate cell, he moves. In this purpose he raises the price of the candidate cell by proposing a bid

Image 10000000000000D20000003BFA0EB978A8F3582C.png

16where ε is a parameter that we introduce to control the magnitude of this bid. Prices evolve quicker if this parameter is high. socc is the surface occupied by other agents in the cell and stot the total surface of the cell. The factor socc / stot makes the bid higher if the cell is more occupied, that is to say, attractive. The first agent to move into an empty cell does not raise the price. The price is a price per unit surface and it is a variable linked to a cell. When an agent bids higher, the price is changed immediately for all agents in the cell. Their surface of housing is also changed according to s = β(Υ – tx) / pn+1 and their utility is computed again.

17An agent may want to move into a candidate cell that is already full. This case is treated in the next paragraph.

1.2.2. Displacement and time decreases

18In the situation where an agent wants to move into the candidate cell, proposing a higher bid, but this cell is already full, we need a mechanism to still allow prices to evolve, in order to have a fluid housing market and to reach the equilibrium. Then two options are available: either the price of the cell is raised for agents living in it, without the candidate moving in (and he needs to wait until he is offered an attractive cell where there is enough space available), or the agent moves into the cell and then some agents previously in the cell need to be relocated. We choose the second solution.

19Thus, one or more agents are randomly chosen within the candidate cell and are sent “to the hotel”, until there is enough space for the agent candidate to move in. These agents which have been displaced are then the next candidates for a move. And they remain priority candidates until they have all found a new location. While they are searching for another cell to live in (rather than living “at the hotel”), their utility decreases exponentially following Un+1 = Un –(Un – Umin) / Te, where Un is the initial utility, Un+1 the decreased utility, Umin the minimal utility in the city and Te a parameter governing the speed of this decrease. Thanks to this mechanism, agents at the hotel find a new cell to live in.

  • 1 With two income groups, it is difficult to determine whether a cell is full or not, as rich and poo (...)

20There is another comparable mechanism of decrease in this model: the time decrease of prices of vacant cells. With the bid mechanism presented before (agent and cell chosen at random), the price of a cell where several agents move in successively can increase and reach a value which makes the cell unattractive. In this case, the agents will progressively move to more attractive cells. We choose to decrease exponentially the price of cells which are not completely full1. The decrease formula is

Image 100000000000011300000035FE055E133B07A265.png

21where ΔU = (Umax – Umin) / Umax measures the homogeneity of the utility in the city, Tp is a parameter determining the speed of decrease of prices, sfree = stot – socc and stot are the non-occupied surface and the total surface of the cell. If no agent moves in, the price decreases according to this formula until it reaches the agricultural rent, where the decrease stops. The factor sfree / stot makes this decrease quicker as the cell is emptier and thus less attractive. The ΔU factor makes the decrease slower as the utility becomes more homogeneous, that is, when the system gets closer to the equilibrium. The different parameters of the model are listed in Table 1. Their default value has been chosen according to several criteria.

22The value of the parameters of the model itself: α, β, Y, t, N, Ra, stot, has been chosen mainly for technical reasons regarding the comparison between the analytical and the agent-based model, but naturally other values could have been chosen, without changing the qualitative behaviour of the model. The calibration of the model with a set of values of parameters that are in agreement with empirical studies is one of the perspectives of this work and will be discussed in section 5.

23The parameters ε, Te and Tp, which are specific to the agent-based model, have been chosen such that the competition between agents on the housing market is efficient and the system reaches the equilibrium in the whole city. The agent-based model has different behaviours and for instance does not reach an equilibrium (the utility of agents does not become completely homogeneous across the city) for certain values of these parameters, but the study of these different behaviours is beyond the scope of the present paper.

24Additional parametersγ, a0 and d are introduced to describe the amenities studied in paragraph 4.3.

Parameters

Description

Default value

α, β

Preferences for composite good and housing

0,9; 0,1

Y

Income

300

t

Transport cost (unit distance)

5

N

Population

700

Ra

Agricultural rent

10

stot

Surface of a cell

12

ε

Bidding parameter

20

Te

Time decrease of the utility of displaced agents

7000

Tp

Time decrease of the price of non-full cells

15

γ

Preference for the amenity

0,1

a0

Attractiveness of the amenity

4

d

Range of attraction of the amenity

2

Table 1: Parameters of the model

1.3. Socioeconomic outcome

25In order to explore the outcomes of the models developed in section 4, we study some variables extracted from these models. By comparison with the values of these variables in a reference simulation (a simulation of a monocentric city with two income groups with the same transport cost, without amenity), we can observe the socioeconomic effects of the modifications introduced in the standard model.

26We study mainly some variables of the models: the utility of the individuals, which is associated to their welfare and gives an economic outcome of the models, the cumulated distances of agents’ trips to work, which give the environmental outcome of the models, and the social inequalities, which are given by observing the difference in the utility of individuals of different income groups.

2. Comparison with the analytical results

27As we will see, the simulations allow us to reach the equilibrium of the standard Urban Economics model of Alonso, Muth, Mills. This equilibrium corresponds to a situation where no agent can raise his utility by moving, and therefore no agent has an incentive to do so. In each income group, individuals have an identical utility across the city. The cells which are occupied are those closest to the city center (CBD), the prices at the border of the city are equal to the agricultural rent and prices increase when the distance to the center decreases. The surfaces of housing increase when the distance to the center increases.

28The results of the simulations do not match exactly the analytical results because of the effects of the discretization (which leads for instance to a border of the city which is not exactly at the same distance from the center all around the city). The discrete character of the simulation appears in particular on the density curve, which is like a step function in the simulations and a continuous function in the analytical results.

29The results of the model can thus be compared to the analytical results when they exist: presents such a comparison for a city with only one income group. It shows the density, rent and surface curves as functions of the distance to the center for the analytical and the agent-based model. Because of the discretization, cells are not completely full in the agent-based model and the density is in general lower than the analytical density. The city is slightly more spread, rents are slightly higher, surfaces slightly lower and the equilibrium utility is slightly lower.

Image 100000000000035C000006B98AB700E3F0BAA448.png

Figure 1: Comparison between the results of the agent-based model and the analytical results: density, rents and surface as functions of the distance to the center.

30It would be interesting to compare the results of the simulations with the results of discrete models of Urban Economics, which the simulations should reproduce exactly. However there are few or no analytical results concerning discrete models of circular cities. Analytical works deal mainly with continuous models or discrete models of linear (one-dimensional) cities.

3. Emergence of a city

31We now describe how a city emerges from the interactions between individuals during a simulation. Initially, all agents are located randomly and all rents are equal to the agricultural rent. Then agents move and bid higher, and the rent curve evolves from a flat rent to the equilibrium rent, and the density curve evolves towards the equilibrium density. Figure 2 shows how the shape of the city evolves during a simulation. In this simulation two income groups are present, “poor” agents in red and “rich” agents in blue. Paragraph 4.1. describes the equilibrium configuration in a detailed way.

Image 10000000000005140000011A5866724BC003DC1E.png

Figure 2: Evolution of the shape of the city during a simulation. The CBD is a green point. Cells whose background is grey indicate that both poor and rich agents live there: at the equilibrium, the city is completely segregated and there are no more such cells. n indicates the mean number of moves per agent since the beginning of the simulation.

32At the beginning of the simulation ((a) and (b)), few people are displaced (as described in paragraph 1.2.2), agents gather at the city center without competing much for land, because many cells close to the center are still not full. But when all agents are concentrated around the center (from (d) on), most moves result in one or several agents being displaced, for few cells have a sufficient free surface to allow an agent to move in with an interesting utility. This feature of the model arises because the vacancy rate of the standard Urban Economics model we reproduce is zero.

33The variable which shows the proximity to the equilibrium is the homogeneity of the utility of agents. To describe this homogeneity, we use the variable ΔU =(Umax – Umin) / Umax, which gives the relative inhomogeneity of the utility. With two income groups, we use the maximum of this variable in both income groups. Initially, this variable has a quite high value as all agents are located randomly, and it decreases during the simulation. We choose to stop the simulation when the relative variations of utility in the income groups are smaller than 10–6, which means that this variable has decreased by approximately five orders of magnitude.

4. Results of the simulations

4.1. Two income groups: model 1

34With two income groups, the utility of “rich” agents is higher than that of “poor” agents at the equilibrium, as they do not have the same income, but utility is homogeneous across the city within each income group. On Figure 3 we give the shape of the city with two income groups, and the rent as a function of the distance to the center.

Image 10000000000003E8000001A631335BC6EFEBCF75.png

Figure 3 Left panel: City with two income groups, rich in blue and poor in red. The CBD is a green dot. Right panel: equilibrium rent as a function of the distance to the center.

35The values of the parameters are those used in Table 1. The city population is composed of two groups of 700 individuals each: “poor” agents (in red) with income Υp = 300 and “rich” agents (in blue) with income Υr = 300 × 1,6 = 480. As in the analytical equilibrium, rich agents are located at the periphery of the city, where they pay lower prices and have higher housing surfaces, but also with higher transport costs.

36On Figure 4 are shown the density and the housing surface as functions of the distance to the center.

Image 10000000000003BE000005B6B69DF5FF38C37A16.png

Figure 4: Population density (number of agents per cell) and housing surface as functions of the distance to the center.

37The agent-based model with two income groups has the same behaviour as the analytical model for the city shape, density curve (discretized), rent and surface curves. But as said previously we did not build an algorithm for the resolution of the analytical model in this case, so that we can not compare quantitatively the results.

4.2. Value of time: model 2

38The equilibrium of the standard Urban Economics model we just presented gives a configuration where rich households live in the periphery of the city and poorer households near the center, which is in agreement with empirical results in most North American cities, but many European cities have an inverse configuration (see Bruckner et al. (1999)). One feature of the Urban Economics model that could account for this difference is the introduction of a difference in transport time cost. As indicated by empirical observations, rich people have a higher value of time than poorer people, and thus a higher global transport cost per unit distance.

39This can be introduced in the model by adding a transport time cost (per unit distance) ct / v, where ct is the value of time and v the transport speed. The global transport cost can therefore be written

Image 10000000000000AF00000039E59C1C6769165AF8.png

40where x is the distance to the center and t the monetary cost.

Image 1000000000000404000001EC3DA8F6B92C6417D5.png

Figure 5: Shape of the city and equilibrium rent with a transport time cost much higher for rich agents than for poor agents.

  • 2 This result is valid with the log-linear utility function used here, but also with a Cobb-Douglas f (...)

41Analytical treatment and agent-based simulations agree on the results of such a change in the model: to have rich agents located in the center and poor ones at the periphery, the ratio of the global transport costs per unit distance of rich and poor agents must be superior to the ratio of their incomes2:

42Tr / Tp Yr / Yp (1)

43with Tr and Tp the global transport costs per unit distance of rich and poor agents respectively. This situation is represented on Figure 5, where the monetary cost of transport has a value t = 2 for both income groups, and transport time cost has a value

Image 100000000000005B0000001EBB95256CF33B13AD.pngfor poor agents and Image 10000000000000F70000001EB931B239C8537EAB.pngfor rich agents. Then we have Tr / Tp = 1,7 > Yr / Yp = 1,6.

44The condition (1) for the inversion of income groups can also be written as Tr / Yr > Tp / Yp. This means that the income group which will be located near the center of the city is the one for which the unit global transport cost represents a higher share of the income.

45Empirically, it can be observed that the value of time does rise with income (de Palma and Fontan (2001)), but not quickly enough to make the condition 1 true if the monetary cost t and speed v are kept identical for both income groups. In this model, the value of time alone can not account for the difference of location of households depending on their income in European and North American cities.

46The introduction of congestion within this model decreases transport speed and increases as a consequence the time share of global transport cost, which tends to make the monetary cost negligible. However, in this case, the income elasticity of the value of time needs to be greater than one to have condition (1) true, whereas it is usually supposed in Urban Economics studies that this elasticity is smaller than one.

47Numerous analytical works deal with the question of the value of time and its influence on the location of households: the importance of this factor is still debated in the literature (see Gofette-Nagot et al. (2000), Bruckner et al. (1999), LeRoy and Sonstelie (1983), Wheaton (1977)).

4.3. Introduction of amenities: model 3

48We would like now to study the influence of amenities on the location of agents. These amenities can be of different types: environmental amenities (woods or forest for example), historical amenities related to the historical evolution of the city (monuments, parks) or amenities linked to modern entertainment places (for instance cinemas, swimming pools or tennis courts). Several types can naturally be found at the same place.

49We introduce in the agents’ utility function a term accounting for the presence of amenities:

50U = αln z + βln s + γln(1 + a(r)),

51where a(r) is the amenity function and γ a parameter giving agents’ preference for amenities. a(r) is a decreasing function of distance r to the amenity, which takes the form a(r) = a0 exp(–r/b), with the values of parameters given in Table 1.

4.3.1. Distance to the center

52We study the influence of the distance d between the amenity and the CBD. Results are given on Figure 6.

Image 10000000000005EA00000382BA192B31EBC85F80.png

Figure 6: Amenity on the west side of the center at a varying distance d. Darker colors indicate higher prices in each income group. Color scales vary from one subfigure to the other.

53Depending on the distance d, the influence of the amenity on the shape of the city and on rents is more or less marked. It can be seen that for an amenity at a distance d=18 in this model (Figure 6(f)), some rich agents form an edge city around the amenity, close to the border of the main city. Such a configuration corresponds to a leapfrog development, and the presence of an attractive amenity far away from the center can thus be considered as a possible cause of such a phenomenon.

Table 2 presents the value of variables allowing us to characterize the equilibrium of each simulation. The variables given in the first columns are the utility of rich agents Ur, of poor agents Up and their difference. Due to the logarithmic form of the utility function, its variations are very small, so that we present the variations of the exponential of the utility instead. This exponential corresponds to a Cobb-Douglas function exp U = zα sβ, which is also often used in Urban Economics studies and gives the same results as the log-linear utility. The other variables are the mean commuting distance of rich Image 100000000000002B0000001948A09A10747D8919.pngand poor Image 100000000000002C00000019D58F89902EFE6EE6.pngagents, the total commuting distance Ddot, the total rent paid in the city Rtot, the mean price Pmean, the total surface of the city Stot and the mean density ρmoy.

Image 10000000000004B000000297BBAACF934B957FF0.png

Table 2: Comparison of the different models with amenities.

54The economic outcome of the introduction of amenities is naturally positive, because a new source of utility is added. Agents’ utility increases when the distance between the amenity and the center decreases (as more and more agents feel the effect of the amenity). From a social point of view, an amenity close to the center reduces the gap in utility between rich and poor agents, as poor agents mostly benefit from it. An amenity far away from the center does just the contrary.

55The city becomes denser if the amenity is introduced within the city or close to it, and is spread if the amenity is far away. The environmental outcome is a consequence of this result: commuting distances are reduced in the first case and increased in the second. Amenities close to the center influence more poor agents’ commuting distances, and those far away influence mainly rich agents’. From looking at the results of an amenity at distance d=10 from the center, it can be understood that the influence of an amenity is not obvious. In this configuration the city is made both denser because the mean density increases (the total surface of the city decreases) and more spread, for commuting distances increase. The reason is that the effect of the amenity has two components: it concentrates agents around the amenity, making the city denser, and it distorts the shape of the city, increasing commuting distances in this case.

56According to this model, a city planner wanting to make a city denser and to reduce commuting distances should introduce new amenities as close to the center as possible. It can also be observed that an amenity close to the center results in an increase in rents, which can be a problem in big cities where they are already very high, whereas an amenity far away from the center makes the rents lower.

57The parameters a0 and b, which describe the amenity, can naturally change its influence on the city, but changing them does not result in surprising outcomes: increasing them increases simply the influence of the amenity on the city. Leapfrog development occurs only for an attractive amenity which is at a certain distance from the city, weakly attractive amenities do not lead to this phenomenon.

4.3.2. Differentiated preferences for the amenity

58We also want to study the influence of individual preferences of agents on their location. Indeed, the model with amenity presents always the same configuration with poor agents close to the center and rich agents at the periphery, following the “North American” configuration of cities. But European cities tend to present the inverse configuration and we would like to see if a difference in agents’ preferences for amenities could result in such a configuration.

59We first study what happens in the extreme case where poor agents have no preference for amenities γp = 0, and rich agents keep the default value of the preference parameter γr = 0,1.

60The results of a such model are given on Figure 7.

Image 100000000000061800000398C2D3B153413841C7.png

Figure 7: Amenity (yellow point) on the west of the center (green point) at a varying distance d, with a zero preference of poor agents for the amenity γp = 0. Darker colors indicate higher prices in each income group; color scales vary from one subfigure to the other.

61For a central amenity or an amenity which is very close to the center, there is indeed an inversion of the income groups: rich agents live in the center close to the amenity, then further away live poor agents, and then again rich agents who do not benefit much from the amenity.

62Such an amenity has a positive economic outcome for rich agents, and all the more when it is closer to the center. For poor agents, the economic outcome is negative when the amenity is close to the center and it becomes positive when the amenity is further away, because competition with rich agents is reduced. The environmental outcome is roughly the same as for an amenity for which rich and poor agents have the same preference.

63Then, as it seems unrealistic to say that poor agents have absolutely no preference for the amenity, we can ask if a differentiated preference between poor and rich agents is sufficient to have rich agents living in the center of the city. We use a strong central amenity for which poor agents have a preference γp = 0,1 (default value) and rich agents a higher preference γr = γp × fa. As shown on the first line of Figure 8, the location of rich and poor households depends strongly on the factor fa differentiating the preferences for the amenity, and a value fa = 1,4 is sufficient to have an inversion of the “American” configuration with the values of parameters used here.

Image 10000000000005DC00000453CFF16FBF8E43B0E4.png

Figure 8: Social structure of the city with an amenity (yellow point) for which rich agents have a higher preference than poor agents γr = γp × fa, for different values of the difference factor fa and of the distance d between the amenity and the center (green point), with b=4 (amenity with a wider range).

64The last two lines of Figure 8 show us that the previous configuration of rich agents in the center and poor agents at the periphery is very unstable with respect to the displacement of the amenity: even a small displacement can change the social structure of the city. This feature is specific to a two-dimensional model and shows the importance of two-dimensional simulations rather than linear (one-dimensional) cities.

65Comparison of Figure 6, Figure 7 and Figure 8 gives a good idea of the influence on the social structure of the city of differentiated preferences for the amenity between rich and poor agents.

5. Perspectives

5.1. Historical evolution with endogenous amenities

66This perspective has already been partially carried out, as shown on Figure 9. The aim is to explore the historical evolution of a “semi-open” city, where population grows in time with the arrival of new inhabitants. We introduce an endogenous amenity which is linked to the presence of rich households: the cells where rich households live become more and more attractive, whereas cells where poor agents live become unattractive. Rich agents have to be more sensitive to this endogenous amenity in order to obtain the “European city” situation presented on Figure 9. This can be linked to the differentiated preferences for exogenous amenities that we had to introduce previously to account for this structure.

Image 1000000000000744000001E48FD72A2BC70215E0.png

Figure 9: Historical evolution of a city with endogenous amenities linked to the presence of rich neighbourhoods.

67At the beginning of the simulation, the population is ten times lower than at the end, and no endogenous amenity is present, so that the city has a “North American” structure. The “European” structure emerges over time, rich households outbidding progressively poorer ones in the center of the city.

5.2. Calibration

68Another perspective of this work is the calibration of the agent-based model with values which are coherent with empirical values for the different parameters (income, transport cost, agricultural rent...), and which give coherent results (densities, surfaces, size of the city...). For the moment this calibration is impossible because of the absence of vertical housing in the model: all agents live on the ground, and the densities can not come close to real densities. The city size is also unrealistic as a result.

69In order to make such a calibration possible, a mechanism of building construction will have to be added, representing housing industry and determining the housing surface available in each cell. Such models have been introduced analytically by Muth (see Fujita (1989)).

5.3. Polycentric city

70The monocentric city is a powerful but very rough model and many attempts have been made in the literature to develop analytical models of polycentric cities. The aim of these works is to study not only residential locations of households, but also location of activities (see for instance Fujita and Ogawa 1982). This can also be done with the agent-based model presented here. We already studied the influence on the city of several exogenous business districts (but focused here on the influence of amenities), and a perspective of this work is to introduce endogenous centers. To this end, a new type of agents must be created to represent firms, and these new agents should compete with other agents for land. They also make location choices, but trying to maximize a profit function whereas households maximize their utility function. Such analytical models (of one-dimensional cities) are described in Fujita and Thisse (2003).

Conclusion

71From a methodological point of view, this work shows the interest of agent-based systems in the social sciences studies of collective phenomena. With the example of the standard Urban Economics model, we use this simulation tool to reproduce the results of an equilibrium model. To this end, we introduce an interaction between agents which leads the system towards the equilibrium. This can be seen as an improvement of the equilibrium model because simple interaction mechanisms can be studied in this way. And as this simulation model is dynamical, it can be an interesting tool to study the dynamics of urban location as a perspective of this work. In addition, these agent-based simulations allow us to modelize phenomena which are difficult to deal with analytically, such as the introduction of amenities in a circular closed city with two income groups. The effects are multiple: on the social structure and shape of the city, rents, surfaces of housing, density and commuting distances, so that the overall effect is difficult to predict. The use of an agent-based system is convenient, because it allows us to handle agents easily and to have an access to individual or global variables characterizing the state of the system.

72Thanks to this model we explore research questions such as the influence of a value of time on the location of households depending on their income. We show that the Urban Economics model using a log-linear utility function cannot explain the location of households in “European” cities (where richer households live near the center and poorer at the periphery), even by introducing a value of time which increases with income, if monetary transport cost and speed are kept identical for both income groups. Indeed, the value of time would have to increase much more rapidly with income than empirical works suggest, to produce an inversion of the “North American” city configuration.

73Then with the introduction of exogenous amenities, we see that a central amenity for which both income groups have the same preference keeps the “American” configuration. The influence of the amenity on the city shape and on global variables depends strongly on the distance between the amenity and the center of the city. It has a twofold effect, concentrating agents around the amenity (making the city denser), and distorting the shape of the city (which can increase commuting distances). An attractive amenity far away from the center can cause leapfrog development.

74Finally, we show that a central amenity for which richer households have stronger preferences than poorer households can inverse the social structure of the city, with richer households being located in the center like in “European”-type cities. But for a two-dimensional city this situation is unstable with respect to a small displacement of the amenity: moving it slightly away from the CBD breaks the circular symmetry of the city.

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Bibliographie

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Notes

1 With two income groups, it is difficult to determine whether a cell is full or not, as rich and poor agents have different housing surfaces at the same location, because of their difference in income. We choose to let the price decrease if the mean surface of housing smean of agents there is smaller than the free surface of the cell sfree.

2 This result is valid with the log-linear utility function used here, but also with a Cobb-Douglas function.

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Référence électronique

Rémi Lemoy, Charles Raux et Pablo Jensen, « An agent-based model of residential patterns and social structure in urban areas », Cybergeo: European Journal of Geography [En ligne], Systèmes, Modélisation, Géostatistiques, document 512, mis en ligne le 06 décembre 2010, consulté le 27 juin 2022. URL : http://journals.openedition.org/cybergeo/23381 ; DOI : https://doi.org/10.4000/cybergeo.23381

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Auteurs

Rémi Lemoy

Transport Economics Laboratory (LET), Complex Systems Institute Rhônes-Alpes (IXXI), University of Lyon, CNRS, France, remilemoy@gmail.com

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Charles Raux

Transport Economics Laboratory (LET), University of Lyon, CNRS, France, charles.raux@let.ish-lyon.cnrs.fr

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Pablo Jensen

ENS Lyon Physics Laboratory, Complex Systems Institute Rhônes-Alpes (IXXI), University of Lyon, CNRS, France, pablo.jensen@ens-lyon.fr

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