Modeling urban housing market dynamics: Can the socio-spatial segregation preserve some social diversity?

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Abstract

Addressing issues of social diversity, we introduce a model of housing transactions between agents who are heterogeneous in their willingness to pay. A key assumption is that agents' preferences for a location depend on both an intrinsic attractiveness and on the social characteristics of the neighborhood. The stationary space distribution of income is analytically and numerically characterized. The main results are that socio-spatial segregation occurs if – and only if – the social influence is strong enough, but even so, some social diversity is preserved at most locations. Comparison with data on the Paris housing market shows that the results reproduce general trends of price distribution and spatial income segregation.

Introduction

People's choice of residential location and the way they are distributed across cities matter, from both social and economic points of view. This paper seeks to explain, from the dynamics of price formation in an urban housing market, how individuals who are heterogeneous in their willingness to pay are distributed over a city. It shows that, under certain conditions on social interactions, housing price formation can entail income segregation, even if a space of social diversity remains.

A large literature is concerned with evaluating the extent and impact of housing price discrimination in education, housing and the labor market. While Brueckner et al. (1999) explain that the relative location of different income social groups depends on the spatial pattern of amenities in a city, Gobillon et al. (2007) highlight the adverse labor-market outcomes due to spatial mismatch, with low-skilled inhabitants of the inner-city suffering a greater distance to jobs and consequently a higher level of unemployment. Understanding the formation of prices in the real estate market and the way that prices are distributed over space is clearly an important issue. Following the path opened by Rosen (1974), most studies have focused on explaining prices through hedonic estimations, showing how the price per square meter can be influenced both by variables intrinsic to the apartment or house and by extrinsic variables concerning the surrounding area and its amenities. The role of these extrinsic variables has been particularly explored. Baltagi and Bresson (2011) underline how much the location influences the price of a dwelling. Ioannides and Zabel (2003), Figlio and Lucas (2004), Bono et al. (2007) or Seo and Simons (2009) emphasize the importance of the quality and density of the neighborhood, the reputation of nearby schools and the level of security. Following Tiebout (1956), these authors point out that the decision of where to live is based on families' preferences for the quality of public services and amenities, particularly education. Thus, prices on the real estate market vary with the quality of a bundle of public services which are capitalized into housing prices.

One question underlined by the studies cited above is that of people's preferences. As shown by Kirman et al. (2007), agents associate payoffs to links with their neighbors. The importance of neighbors is also emphasized by Fagiolo (2005). But do people prefer to live with people who are richer than they are, or poorer? In other words, what determines the social component of the attractiveness of a location? The literature on well-being tends to argue that people feel better when those around them are poorer—for a detailed survey see Luttmer (2005). Clark and Oswald (1994) show empirically that unemployed people are less unhappy when they live with other unemployed people. Goyal and Ghiglino (2010) explore the role of shifting social interactions: they use examples to illustrate how poorer individuals lose while richer ones gain as we move from an economically segregated society towards an integrated society. In line with the literature on hedonic prices which highlights the importance of the environment, we explore here the consequences of the alternative hypothesis that individuals prefer to live with people who are at least as rich as themselves. Indeed, some preliminary studies suggest that the interplay between social preferences and the preference for a high quality of local amenities has more striking effects when the social preference is to live with richer neighbors.

When the prices depend not only on the intrinsic characteristics of the goods but also on the level of surrounding amenities, as well as on some social preferences, the space is differentiated and a social mismatch may result. Different measures of segregation have been proposed. Alesina and Zhuravskayay (2011) measure segregation of different ethnic, religious and linguistic groups within the same country from quite an exhaustive data set covering several different countries. Cutler et al. (1999) show the influence of legal barriers on social segregation, while Jenks and Meyer (1990) or Cutler et al. (2008) evaluate the effect of segregation on the socioeconomic performance of minorities. Echenique and Fryer (2007) develop a spectral index of segregation related to the characteristics of social interactions. This index is defined at the individual level and is higher when the considered individual interacts with segregated individuals. Ballester and Vorsatz (2010) propose a random-walk-based segregation measure. For the present work, interesting results are obtained by considering simple measures of multigroup segregation, which compare, for a population in a given local neighborhood, the observed distribution of a given feature (here income) with the uniform distribution. A basic segregation index in the line of Reardon and Firebaugh (2002) and Alesina and Zhuravskayay (2011) is proposed and compared to the information-theoretic measure of Theil (1967), which was introduced into economics for measuring income heterogeneity.

The pioneering modeling work of Schelling (1971) describes residential segregation as emerging from social preferences alone. Since then, extensions have been proposed by several authors in order to integrate a housing market. In Bernard and Willer (2007), the price of a house depends on an intrinsic component of the location, randomly allocated, and on the composition of the neighborhood. In Fossett and Senft (2003), the price only depends on an intrinsic component of the location uniformly distributed over the city. However, the level of quality of locations does not directly impact the choice made by individuals. In Zhang (2004), the price varies according to the density of occupation of the neighborhood, but does not take into account any measure of house quality. In Bernard and Willer (2007), the choice of a location depends on the mean status of the neighborhood, the status being a wealth-related quantity randomly allocated to the individuals. The present work goes further, dealing with the individual attractiveness of a location and its influence on prices: in our model, the individuals choose a location according to its attractiveness, which is a dynamic quantity depending on both the intrinsic characteristic of the location and the (time-dependent) social composition (measured by the levels of income) of the neighborhood. The prices then depend on the attractiveness through the market dynamics. The framework introduced here can easily be adapted to make the attractiveness reflect different characteristics of the locations and different types of social preferences. It also has the advantage of allowing for detailed mathematical analysis.

The present paper focuses on spatial income segregation, leaving aside all other features (such as ethnic characteristics) that may also cause segregation. The model proposed here takes some inspiration from Short et al. (2008) and Berestycki and Nadal (2010), who model the evolution of the spatial distribution of crime in a city, attributing to each location an attractiveness for illegal activity. The originality here is that each agent attributes to each location a specific level of attractiveness. This attractiveness results from a combination of an intrinsic or objective part, and a subjective part. The endogenous (subjective) attractiveness results from the individuals' intrinsic social preferences (for neighbors with similar or higher incomes). The main assumptions are: (i) people make decisions according to both their willingness to pay (WTP) and their individual evaluation of the level of attractiveness of the different locations, (ii) buyers, who are heterogeneous in their WTP, base their search for housing on the level of attractiveness of the location of the dwelling, (iii) agents are both buyers and sellers, and (iv) the intrinsic attractiveness is spatially heterogeneous.

The results are demonstrated through mathematical analysis and then empirically confirmed in the simulations and in the empirical analysis using data on the Paris housing market. The analysis of the stationary regime reached by the market dynamics is first performed for an arbitrary, spatially heterogeneous, intrinsic attractiveness. For illustrative purposes and to compare with the empirical data, the analysis is then conducted in more detail for specific cases. First, we consider the case of a monocentric city defined by an intrinsic attractiveness which decreases with the distance from the geographical center. This is a simple case motivated by the classical Von Thünen model and Alonso's study of land use (see Alonso, 1964), explaining how, generally, prices are higher in the center of a city or a county. The fact that prices diffuse from the center to the periphery (with higher prices in the center) has more recently been highlighted in a regional context by Clapp et al. (1995), Meen (1996), Berg (2002), Oikarinen (2004) or Holly et al. (forthcoming). The monocentric case also provides a first order approximation to the description of the Paris housing market. Second, we consider a more complex structure of the intrinsic attractiveness, allowing for a better comparison with empirical data on transaction prices in Paris.

This paper exhibits three important results. The first concerns the mathematical analysis of the model. This analysis is first presented for an arbitrary intrinsic attractiveness. This provides general results. The results are then specified for the particular case of a monocentric city. For the empirical validation, we consider a more complex structure of the intrinsic attractiveness which matches the Paris data quite well. Within this theoretical framework, we provide an analytic description of the spatial distributions of transaction prices and incomes in the city. We prove that, when the market is not saturated, social segregation occurs if and only if the social influence is strong enough.

In this case, and this is the second important result, the general analytical solution underlines the emergence of critical endogenous thresholds in income and intrinsic attractiveness. The income threshold induces a segregation between those with an income above the threshold (whom we shall call the rich agents) and the others. These rich agents can freely choose their location, and they are the only ones to have access to locations above the threshold in intrinsic attractiveness. For agents below the income threshold, the housing possibilities are restricted to a subset of locations. Applied to a monocentric city, this gives that: (i) people with a higher income live closer to the city center while the poorer people live in the periphery, (ii) the rich agents are the only ones to live within a certain endogenous critical distance from the center, but there is no segregation amongst the rich agents within this central domain, (iii) agents below the income threshold have access to locations at distances greater than endogenous thresholds (the poorer the agent the greater the critical distance), and (iv) at any location outside the center, except at the very periphery where only the poorest live, there is always some social mixing. And this is the third important result: if the first of these results, (i), is expected and in line with most of the literature, the preservation of social diversity, on the one side amongst the richest agents at the center, and on the other side in a large domain between the center and the periphery, is particularly original and proved robust for a wide range of parameter values.

The paper is organized as follows. Section 2 presents the assumptions and the dynamics of the agent-based model. Section 3 reviews the key parameters of the model. Section 4 deals with the mathematical analysis of the equilibrium states. Section 5 describes the conditions under which segregation occurs. Section 6 presents numerical simulations for a simple case, that of a monocentric city. Section 7 considers a case allowing for a better comparison between simulated and empirical data. The conclusion follows.

Section snippets

The agent-based model: assumptions and dynamics

An agent-based model of residential location is proposed. In this model, agents make their decision according to both their income (characterized by an idiosyncratic willingness to pay) and their individual evaluation of the level of attractiveness of the different locations. The assumptions of the model are presented below.

Model parameters and key features

Let us now summarize the set of key parameters that characterize the model. The model behavior depends on the following set of control parameters: (1) the incoming rate, that is the number Γ of new agents entering the economy per unit of time, (2) the rate α at which agents leave the city; the parameters in the attractiveness dynamics, that is, (3) the time scale 1/ω of relaxation towards the intrinsic value, (4) the weight of the social influence, ϵ, and (5) the parameter ξ which fixes the

Continuous time dynamics

For the mathematical analysis, the evolution of the above agent-based model is further formalized through partial differential equations, taking the continuous time limit (that is δt0).

In this continuous limit, the updating rule of the attractiveness Ak(X,t) of a location X seen by a k-agent, Eq. (3), givestAk(X,t)=ω(A0(X)Ak(X,t))+ϵvk>(X,t)

Given the outgoing rate α (the fraction of agents leaving the inside per unit of time), and vk(X,t) being the density of newly housed k-agents, the

Thresholds and segregation

Let us now determine the conditions for having the WTP thresholds within the range [P0,PK1], i.e., determine where people can be housed depending on their WTP category. In other words, let us characterize the sets Ωk.

The absence of segregation in the stationary regime would mean that, at any location, there are transactions involving any WTP category. If, on the contrary, at some locations the market dynamics make the distribution of offer prices higher than the distribution of demand prices,

Numerical simulations

We now simulate the agent-based model as presented in Section 2, illustrating the theoretical results in the case of a pure monocentric city.

Application to the Paris housing market

This section proposes a first comparison between the results of the model – analytically demonstrated and validated by simulations – and empirical observations. The aim here is to see how far the price distribution in Paris can be explained by the phenomena described above.

Conclusion

Going beyond the simple Schelling segregation model, this paper studies social segregation through the spatial distribution of income resulting from dynamic price formation and agents' localization in a particular housing market model. In doing so, it actually introduces a new general framework for studying such markets and the resulting socio-spatial segregation. A specific feature of the model is the specification of the attractiveness of each location, composed of both an intrinsic part and

Acknowledgments

We are grateful to Marc Barthélémy and Jean Vannimenus for fruitful discussions at an early stage of the present work—and to Jean Vannimenus for the Russian proverb. We thank the participants of the Cambridge seminar on networks organized by Sanjeev Goyal for helpful remarks. Errors remain ours. We thank two anonymous referees for their helpful comments and advice, and Nicolas Bernigaud for the map shown in Fig. 8. This work is part of the project “DyXi” (in which all authors are involved)

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