Cultural Dynamics, Social Mobility and Urban Segregation∗ Emeline BEZIN†and Fabien MOIZEAU‡ January 2017 Revised version

Abstract We consider the relationship between cultural dynamics, urban segregation and inequality. To this end, we develop a model of neighbourhood formation and cultural transmission. The tension between culture preservation and socioeconomic integration drives the pattern of segregation in the city. We study the dynamics of culture and urban configurations. In the long run, the city may end-up segregated or integrated depending on cultural distance and the initial cultural composition of the population. We also show that segregation fosters the influence of family background on economic fate. Finally, segregation has ambiguous effects for long-run efficiency. Keywords: cultural transmission, peer effects, residential segregation, human capital inequality. JEL Classification: D31, I24, R23.

∗ We thank the Editor, Gilles Duranton, and two anonymous referees for very thoughtful and detailed comments. We are very grateful to David de la Croix, Luisa Gagliardi, Victoire Girard, Fran¸cois Salani´e and Thierry Verdier for their helpful comments. We thank participants at the summer school “Social Interactions and Urban Segregation” (Rennes, 2014), the 9th Meeting of the Urban Economics Association (Washington, 2014), the 64th AFSE congress (Rennes, 2015), the 30th Annual congress of the EEA (Mannheim, 2015), the 12th annual conference of TEPP (Paris), and seminar participants at ENS Cachan, BETA (Strasbourg), EconomiX (Nanterre), Universit´e Saint-Louis (Bruxelles), the CREM-SMART workshop (Rennes), PUCA (Urban Development Construction and Architecture, 2015) and IDEJETRO (Chiba). Financial support from the Agence Nationale de la Recherche (ANR-12-INEG-0002) is gratefully acknowledged. † Paris School of Economics (PSE), E-mail: [email protected] ‡ CREM (Condorcet Center), Universit´e de Rennes 1 and Institut Universitaire de France, E-mail: [email protected].

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1

Introduction

During recent decades, most Western democracies have become more ethnically and culturally diverse. The average proportion of foreign-born individuals in OECD countries rose from 9.5% in 2000 to 13% in 2014 (OECD International Migration Outlook, 2016). This movement is likely to continue, given demographic and migration trends. Increasing diversity challenges social cohesion and puts issues of social integration and national identity at the forefront of the political debate. In the host country, ethnic minorities often live in the less affluent neighborhoods of metropolitan areas. Living in ethnic enclaves produces both benefits and costs for inhabitants suggesting that the choice of place of residence results from a variety of incentives. It responds to the desire to live close to the native population in order to acquire the mainstream culture and become socially-integrated, but also both the wish to cluster with peers and retain the cultural attitudes of the country of origin, sometimes at the expense of social integration. Understanding how residential segregation affects the incentives to socially integrate and preserve the home-country culture is essential for the understanding of potential policies to reduce the ethnic gap. This paper analyzes the interdependency between cultural transmission, urban segregation and economic inequality. Our framework allows us to consider, on the one hand, how segregation influences the way in which cultural traits are passed on from one generation to the next and, on the other hand, how cultural transmission drives the incentives to segregate. We are thus able to answer the following questions: How does segregation contribute to cultural diversity within the society? How does the existence of diverse cultures regarding personal achievement affect segregation and urban inequality? How can we design public policies to affect both segregation and cultural transmission in order to improve societal economic performance? It is well-documented that urban segregation interacts with culture (regarded as preferences, beliefs and social norms). Urban segregation influences ethnic identity, although there is no consensus on the sign of this relationship (see Bisin, Patacchini, Verdier and Zenou, 2011a, and Constant, Sch¨ uller and Zimmerman, 2013). Segregation of ethnic minorities in poor neighborhoods creates a ‘culture of poverty’ by socially isolating individuals from mainstream norms of behavior (see Wilson, 1987, Anderson, 1999, Lamont and Small, 2008). The choice of the social arenas in which children interact such as schools or neighborhoods is also a concern for parents who care about the transmission of desired cultural traits (see, for the particular case of school choice, Ioannides and Zanella, 2008,

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or Tinker and Smart, 2012). Following on from this empirical evidence, we develop a theoretical model based on the following three blocks. First, the population consists of two different cultural types: the culture of the majority (say that of natives) and the minority culture (say that of the foreign-born). We assume that agents who adopt different cultures do not have the same prospects of economic success (i.e. being educated), with the majority culture performing better as it produces better knowledge of the codes of behavior and the functionings of the schooling system. Second, cultural traits are transmitted intergenerationally following a process `a la Bisin and Verdier (2001). Interactions within the family and within society are involved here, and parents have an incentive to socialize children into their own culture. Third, parents choose the place where they wish to live. This choice is not only motivated by the desire to transmit one own’s culture but also by the existence of local peer effects in children’s education (see, for instance, B´enabou, 1993, 1996a, b). Local spillovers matter as, whatever their cultural trait, all parents value having educated children. To capture the influence of culture on socioeconomic outcomes, one crucial feature of our model is that the (subjective) benefits of education and the gain associated with the transmission of cultural traits are linked. More precisely, for mainstream parents, we consider that the benefit of having an educated child rises when the child has acquired the parents’ own culture (i.e. the mainstream culture). The mainstream cultural trait and education are thus complements. For minority families, we consider two cases: complementarity or substitutability. Under substitutability, having the minority cultural trait reduces the benefits of education. These two cases capture the cultural distance between mainstream and minority groups. Complementarity for both groups reflects cultural proximity. While substitutability for the minority group corresponds to cultural polarization. There is empirical support that cultural distance matters for differences in socioeconomic outcomes (for the impact of religion on economic decisions, see Weber, 1958, or Botticini and Eckstein, 2005, 2007, for the influence of cultural origin on social integration of immigrants, see Domingues Dos Santos and Wolff, 2011, for french evidence, or Gang and Zimmerman, 2000, for german evidence, and Borjas, 1995, for US evidence, for the influence of oppositional identities see Akerlof and Kranton, 2002, Fryer and Torelli, 2010, Battu and Zenou, 2010, and Battu et al. 2007). The non-separability between the benefit of education and the gain associated with culturaltrait transmission means that the incentives parents face to transmit cultural trait and to make 3

their offspring educated are intertwined and influence the integration and segregation forces. The main insight of our theory is then that the urban equilibrium and the cultural composition of the population are co-determined. We show that cultural distance has crucial implications for the nature of the long-run equilibrium. When there is cultural polarization, the desires to preserve the minority culture and socially integrate are contradictory, making minority parents less willing to pay to live in better-quality neighborhoods. The segregation force is then strong enough so that the city ends-up segregated. When there is cultural proximity, there are multiple types of long-run urban configurations. We show that the long run urban equilibrium depends on society’s initial cultural composition. We show that the spatial separation of cultural groups adds further glue to the intergenerational transmission of cultural traits. Consistent with the findings in Borjas (1995) and Chetty et al. (2014), segregation thus strengthens the influence of family background on economic fate. We show that segregation has ambiguous effects on the long-run level of education. The initial population cultural composition is key to assess the efficiency of the urban equilibrium.

Related literature. Our paper is related to the literature on cultural transmission launched by Bisin and Verdier (2001). The transmission of the traits such as identities, time preferences and beliefs, which impact educational outcomes, has been analyzed theoretically (see Bisin, Patacchini, Verdier, and Zenou, 2011b, for oppositional identities, Doepke and Zilliboti, 2008, for time preferences and the spirit of capitalism, Guiso, Sapienza and Zingales, 2008, for beliefs and trust in other people, and Lindbeck and Nyberg, 2006, for the transmission of working norms). Our paper is relatively close to some theoretical and empirical studies suggesting that assimilation policies can lead to a cultural backlash from the minority (Bisin, Patacchini, Verdier, and Zenou, 2011b, Carvalho, 2013, and Fouka, 2016). In the same vein, Verdier and Zenou (2017) shows how cultural distance (defined as the degree of centrality in a network) affects choices of assimilation. None of the previous studies consider location choices and is able to show how cultural choices interact with the degree of segregation. Our paper also contributes to the literature on neighborhood effects and endogenous socioeconomic segregation explaining how local interactions drive spatial segregation and persistent income inequality (see for instance, Loury, 1977, B´enabou, 1993, 1996a,b, Borjas, 1998, and Durlauf, 1996). In these analyses, the dynamics of income inequality rely on human-capital accumulation, and individual human capital is determined by both that of their parents and local spillovers. In particular,

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B´enabou’s works emphasize that incentives to segregate into distinct communities are driven by the desire to enhance human-capital accumulation. Departing from B´enabou’s works, Borjas (1998) introduces ethnic spillovers in the human-capital accumulation process that lead ethnic groups to sort across neighborhoods. In the same vein, we consider that cultural aspects are crucial for the emergence of the urban configuration. Moizeau (2015) also studies the influence of culture on residential choices. His analysis considers how in a city either opposing social norms persist or a particular code of behavior spreads and ultimately prevails. The dynamics of cultural traits follow a particular diffusion process proposed by Akerlof (1980). We differ from these previous works as the cultural composition of the population evolves over time as a result of individual decisions. Our approach allows us to take into account the tension between the desire to preserve one’s own culture and the need to integrate in order to improve one’s prospect of economic success. To the best of our knowledge, our paper is the first to emphasize how this tension between culture and economic integration impacts cultural diversity and residential segregation in the long run. Our paper is also related to Card, Mas and Rothstein (2007, 2008), who build a model ` a la Schelling where individuals have preferences over the social environment. Unlike most theoretical models of neighborhood composition, they find that tipping dynamics may lead to multiple long-run equilibria, with integration being a stable outcome. Our cultural explanation of multiple long-run urban configurations here relates the degree of segregation to cultural distance, as well as the cultural composition of the population. It is thus consistent with the empirical findings in Cutler, Glaeser and Vigdor (2008) that (i) the cultural distance between an immigrant group and the native population significantly affects the degree of segregation, and (ii) the group share in the population also matters for the urban configuration. The remainder of our paper is organized as follows. The following section sets out the model. Section 3 then provides a characterization of the segregation that emerges at each date t, and looks at the dynamics of urban segregation and cultural traits. In Section 4, we present the result that both integration and segregation can be long-run outcomes, and Section 5 addresses the issue of efficiency in the urban equilibrium. Last, Section 6 concludes.

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2 2.1

The set-up The city

The city is comprised of two residential areas indexed by j = 1, 2. We consider that the land rent is paid to absentee landlords, and without loss of generality normalize the opportunity cost of building a house to 0. Houses are identical across the city. The inelastic supply of houses within a residential area is of mass 1. This land-market is a closed-city model where the population of the city is a continuum of families of mass 2. Each family, comprised of a parent and a child, lives in one and only one house. The city can accommodate the entire population. Agents live two periods. As a child , the individual is subject to socialization and attends school. As an adult, the individual has to decide in which neighborhood her family will live, and the effort to exert to transmit her cultural trait.

2.2

Cultural transmission, peer effects and preferences

Parents differ with respect to their cultural trait. They have either trait a which refers to the mainstream culture group or trait b which refers to the minority culture group. At any date t, the city population is comprised of Qt type a parents and 2 − Qt type b parents. We denote by qtj , the number, resp. fraction, of agents with trait a in area j.

Cultural transmission. The transmission of preferences follows the lines of the model introduced by Bisin and Verdier (2001). The intergenerational transmission of trait i ∈ {a, b} is the result of social interactions which arise at two levels. The child born at date t is first exposed to vertical socialization by her/his parents. The probability that the latter directly transmits her/his trait is τ i . If not socialized within the family (with probability 1 − τ i ), the child adopts the trait of some role model met in neighborhood j. This second socialization process is called oblique transmission. The probability of being obliquely socialized into trait a (resp. b) in neighborhood j is f (qj ) (resp. 1 − f (qj )). We assume that oblique transmission of trait a increases with the fraction of role models with trait a, i.e. f 0 (qj ) > 0, and that f (0) = 0 and f (1) = 1. As stated by Saez-Marti and Sj¨ogren (2008) and Saez-Marti and Zenou (2012), the shape of f (.) captures the degree of conformism, that is how much does the child finds attractive the trait a acquired by role models. When f 0 (0) < 1 and f 0 (1) < 1 (resp. f 0 (0) > 1 and f 0 (1) > 1), the child is more inclined to imitate (resp. distinguish from) role models who are more frequent. 6

The transition probabilities are given by

ab aa = (1 − τ a )(1 − f (qj )) = τ a + (1 − τ a )f (qj ) and Pj,t Pj,t

(1)

bb ba Pj,t = τ b + (1 − τ b )(1 − f (qj )) and Pj,t = (1 − τ b )f (qj ).

(2)

aa denotes the probability that a child from a type a family be socialized into type a In particular, Pj,t

at time t.

Peer effects. The young generation attends school. We do not consider any educational effort but we allow the probability to get education to differ across cultural type i. Further, the probability e¯|i

e|i

e¯|i

of educational success depends on neighborhood j. Denoting by Pj,t (resp. Pj,t = 1 − Pj,t ) the probability of getting education (resp. no education), we assume: e¯|i

Assumption 1 Suppose that Pj,t is a function of qj,t and for any qj,t and j = 1, 2 e¯|a

e¯|b

(i) Pj,t > Pj,t , e¯|a

e¯|b

dPj,t dPj,t (ii) ≥ > 0. dqj,t dqj,t Children do not face the same educational opportunities whatever their type due to cognitive and non-cognitives abilities. Further, trait a is more favorable to education than trait b so that the fraction of mainstream agents who get education is higher than the fraction of minority agents who do so. Several reasons can explain this educational gap. The minority population lacks the knowledge of the social codes and the characteristics of the school system such as the quality of teaching program, the contents and type of educational curricula. The fact that the minority population may have to cope with a foreign language also contributes to this gap (see Dustmann, Frattini and Lanzara, 2012). Our assumption is also consistent with Borjas (1995) result that ethnicity continues to be correlated with children’s human capital once parental skills and average human capital in the neighborhood have been controlled for. Also we assume peer effects in education: the higher the fraction of type a individuals, the higher the fraction of people who educate in the population (due to (i)) and the higher the incentives to educate whatever one’s cultural trait1 . 1 There is an extensive literature on the impact of neighborhood effects on individual socioeconomic outcomes (see, for instance, surveys of Durlauf, 2004, and Topa and Zenou, 2014). We should mention here that there is not yet a consensus about the size of neighborhood effects on educational outcomes. Some experimental or quasi-experimental

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Preferences. Parental preferences depend on private consumption and an altruistic component which depends on both offspring’s educational attainment and cultural type. Specifically, V e¯|ii

0

denotes the gain from the type i parent’s point of view that a child gets educational level e and 0

acquires trait i0 . V e|ii is this gain when the child gets educational level e which is lower than its 0

counterpart e. For the sake of simplicity, V e|ii is exogenous. W We denote by Uti (ρj,t , τ i ), the utility at date t of a parent with trait i and income w, who lives in neighborhood j and exerts socialization effort τ i . We have for a trait-a parent e¯|a

e|a

e¯|b

e|b

aa ab Uta (ρj,t , τ a ) = w − ρj,t + Pj,t (Pj,t V e¯|aa + Pj,t V e|aa ) + Pj,t (Pj,t V e¯|ab + Pj,t V e|ab ) − Θ(τ a ),

(3)

where Θ(.) is the socialization cost which is increasing and convex with respect to τ .2 A parent with trait b and income w who lives in neighborhood j ∈ {1, 2} has utility e¯|b

e|b

e¯|a

e|a

ba bb Utb (ρj,t , τ b ) = w − ρj,t + Pj,t (Pj,t V e¯|ba + Pj,t V e|ba ) − Θ(τ b ) (Pj,t V e¯|bb + Pj,t V e|bb ) + Pj,t

(4)

We make the following assumption. Assumption 2 r (i) Cultural intolerance: V e|ii > V e|ii 0

0

0

∀e ∈ {e, e}, ∀i, i0 ∈ {a, b} with i0 6= i, 0

(ii) Educational gain: ∆V ii ≡ V e¯|ii − V e|ii > 0

∀i, i0 ∈ {a, b},

(iii) Complementarity between education and mainstream cultural trait: ∆V aa > ∆V ab . Following the literature on cultural transmission (Bisin and Verdier, 2001), item (i) assumes that preferences embody cultural intolerance: for a given educational level, a parent prefers a child with her own cultural trait. Item (ii) amounts to say that whatever her trait, the parents prefers that her child gets education. This assumption captures the idea that, eventhough educational gain measured 0

by the magnitude of ∆V ii differs across cultural traits, there is a widespread view that education pays. Item (iii) considers complementarity between education and cultural trait a meaning that, for trait-a parents, educational gain is magnified when the child acquires trait a. We will say that the work finds little evidence of neighbourhood effects on educational outcomes (see Kling et al. 2007, Oreopoulos, 2003). Topa and Zenou (2014) provide an interesting discussion of why these experimental analyses may lead to insignificant treatment effects on economic outcomes. A growing literature in sociology emphasises the duration of exposure to neighbourhoods, which helps explain why experimental work finds little evidence of neighbourhood effects (see Sharkey and Elwert, 2011, Wodtke et al. 2011). Chetty, Hendren and Katz (2015) consider the Moving To Opportunity Experiment and show that treatment effects are substantial when considering the duration of exposure to a better neighbourhood. 2 In order to focus on the impact of cultural traits and rule out any income heterogeneity effect, we assume a linear utility function of private consumption w − ρj,t .

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mainstream culture values education. We make no assumption regarding trait b and we will examine alternatively two cases, that is complementarity (∆V bb > ∆V ba ) and substitutability between trait and education (∆V bb < ∆V ba ). Both cases capture cultural distance, we will say that: Definition 1 (i) Cultural proximity refers to complementarity between culture and education for both mainstream and minority, (ii) cultural polarization refers to complementarity for the mainstream and substitutability for the minority. This non-separability assumption is a key feature of our framework and implies that incentives to exert socialization effort and incentives to make the offspring get education are intertwined. The crucial feature is that parents have two concerns regarding the welfare of their child. First, given cultural intolerance, parents value to have a child with the same cultural trait. Second, they also prefer to have an educated child. Both concerns may be compatible when cultural trait and education are complements while coming into contradiction under substitutability. Hence, when choosing socialization effort and location, parents face a trade-off between cultural transmission and education that hinges on the degree of substitutability between culture and education and also on the endogenous composition of the population.

2.3

Parental choices: socialization and location

Parents make two decisions. They choose both the location j where they pay the land rent ρj,t and their socialization effort. Socialization choice. Direct transmission is the result of a choice. Parents exert the effort τ i ∈ [0, 1] in order to transmit their trait. Let us first consider trait a parents. At date t, given her place of residence j, a type a parent chooses her optimal effort τ ∗a that solves

max Uta (ρj,t , τ a ) subject to (1). a τ

For the sake of presentation, we omit the time index t and the neighborhood index j when not 0

necessary. Given that P e¯|i = 1 − P e|i , and P ii = 1 − P ii for any i, i0 ∈ {a, b}, i 6= i0 , we get the following first-order condition:


(1 − f (q)) P e¯|a ∆V aa − P e¯|b ∆V ab + V e|aa − V e|ab = Θ0 (τ ∗a ).

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(5)

We assume that for any q

P e¯|a ∆V aa − P e¯|b ∆V ab + V e|aa − V e|ab ≥ 0

(6)

implying that there is always an incentive to exert a socialization effort, i.e. τ ∗a (q) ≥ 0.3 From (5), applying the implicit function theorem leads to: e ¯|a

−f 0 (q)(P e¯|a ∆V aa − P e¯|b ∆V ab + V e|aa − V e|ab ) + (1 − f (q)) dPdq dτ ∗a = dq Θ00 (τ ∗a )

∆V aa − ∆V ab


.

(7)

The above expression shows that q has two effects on the socialization effort. First, when a parent is surrounded by more people with the same cultural trait, she benefits from a more effective oblique transmission and is incited to reduce her socialization effort. This negative effect, captured by the first parenthesis in the numerator of (7), is called the cultural substitution property (see Bisin and Verdier, 2001). Our model involves a second effect: when the fraction of type a agents rises, peer effects in education are stronger and the child is more likely to educate. As cultural trait a and education are complements, this positively affects the incentives to socialize children. This second effect may counteract the cultural substitution effect. Hence, the impact of q on τ ∗a is ambiguous. Further, when the neighborhood population is only comprised of trait a individuals, the cultural substitution effect prevails as τ ∗a (1) = 0. Following the same reasoning, we obtain the socialization effort of trait b parents (see Appendix 7.1). The socialization choice of minority parents depends on a cultural substitution effect. Parents with trait b have less incentive to socialize their child when there are more type b individuals in the neighborhood, i.e. when q is lower. The socialization choice also depends on educational gains weighted by peer effects. An increase in peer effects (captured by a rise of q) has a positive (resp. negative), impact on socialization effort if cultural trait b and education are complements (resp. substitutes). Hence, the overall impact of q on τ ∗b has an ambiguous sign under substitutability while it is positive under complementarity. Further, when the neighborhood is inhabited by only type b individuals, only cultural substitution matters yielding τ ∗b (0) = 0. Vertical and oblique transmissions drive the dynamics of the mainstream culture in neighborhood We could relax this assumption and have values of q such τ ∗a (q) = 0. This would not add any interesting insights to our discussion. 3

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j between t and t + 1

qt+1 − qt = (1 − τ ∗b (qt ))f (qt ) − qt (1 − τ ∗a (qt )) + qt f (qt )(τ ∗b (qt ) − τ ∗a (qt )).

Parents’ Location Choice.

(8)

In this framework, the location choice by determining the socioeco-

nomic composition of the neighborhood is another mean used by parents to influence both cultural transmission and educational outcome of their child. At any date t, the location choice of parents with trait i solves the following program

max Uti (ρj,t , τ ∗i (qj,t )). j

Without loss of generality, we impose that q1,t ≥ q2,t = Qt − q1,t and ρ2,t = 0. Following the literature, the urban equilibrium is defined as follows: ∗ ∗ ), , τ ∗a (q1,t Definition 2 At any date t, given Qt , the urban configuration characterized by ρ∗t , q1,t ∗ ∗ ∗ τ ∗a (Qt − q1,t ), τ ∗b (q1,t ), τ ∗b (Qt − q1,t ) is an equilibrium if no one wants to move and change their

socialization choice. The urban equilibrium is spatially stable if, after a move of a small number of trait-a individuals from neighborhood 1 to neighborhood 2 and a migration of the same number of trait-b individuals in the reverse direction, the highest bidders for neighborhood 1 are trait-a individuals. To obtain the urban equilibria, we need to know who is eager to bid more for land in a particular neighborhood. Differing from the monocentric city benchmark (Duranton and Puga, 2015), the bid curve refers to the willingness to live in urban area 1 as a function of the share of the mainstream group. Specifically, the willingness to pay to live in urban area 1, denoted by ρi for i ∈ {a, b}, is such that a trait-i parent is indifferent between both neighborhoods. For a mainstream individual, given e¯|a

that Pjab = 1 − Pjaa and Pj e¯|a

e|a

= 1 − Pj , we have e¯|b

e¯|a

e¯|b

ρa = P1aa (P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab ) − P2aa (P2 ∆V aa − P2 ∆V ab + V e|aa − V e|ab ) e¯|b

e¯|b

+(P1 − P2 )∆V ab − (Θ(τ ∗a (q1 )) − Θ(τ ∗a (q2 ))).

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(9)

e¯|b

e|b

For a minority individual, using Pjba = 1 − Pjbb and Pj = 1 − Pj , we obtain e¯|b

e¯|a

e¯|b

e¯|a

ρb = P1bb (P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba ) − P2bb (P2 ∆V bb − P2 ∆V ba + V e|bb − V e|ba ) e¯|a

e¯|a

+(P1 − P2 )∆V ba − (Θ(τ ∗b (q1 )) − Θ(τ ∗b (q2 ))).

(10)

The equilibrium in the land market is segregated (resp. integrated) when trait-a parents are more (resp. less) willing to pay to live in urban area 1 than trait-b parents. Formally, the urban equilibrium can be derived from the ranking of the slopes of the bid-rent curves (available in Appendix 7.2). We thus highlight the forces that lead to either segregation or integration. They are triggered by the desire to transmit the cultural trait and the concern to have an educated child. First, cultural intolerance is a force leading to segregation. Because of transmission by role models, the probability to have a child with the same trait is higher when many people share this cultural trait in the neighborhood. Cultural intolerance creates an incentive for each type of agents to cluster together in part. Second, given the concern to have an educated child, parents search for peer effects in education, thus generating an incentive to live in the neighborhood where the mainstream cultural trait prevails. How this motive affects the incentives to segregate depends on cultural distance.

3

Segregated urban equilibria

We first study the conditions that allow segregation to arise at each date t. We make a symmetry assumption ∆V aa = ∆V ba and ∆V ab = ∆V bb in order to focus on the consequences of cultural distance in terms of segregation (for an asymmetric case see Proof of Proposition 1 in the Appendix). Proposition 1 If there is cultural polarization then the unique stable urban equilibrium is always segregated: if Q < 1 then q1∗ = Q and q2∗ = 0, if Q ≥ 1, q1∗ = 1 and q2∗ = Q − 1. Cultural polarization leads to segregation. The reason is that cultural transmission of the minority trait and education are conflicting objectives implying that when they live in neighborhood 1 (searching for peer effects), parents with the minority trait gain more from having a child with the mainstream cultural trait. However, the probability to have a child with the mainstream cultural trait is always lower for minority parents than for mainstream ones (i.e. P1aa > P1ba ) as the latter positively impact this probability by exerting a direct socialization effort. This reduces the gain to live in neighborhood 1 for parents b compared to parents a. Hence, when there is cultural polarization,

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mainstream parents are willing to bid more than minority ones and the city is segregated4 . e¯|a

e¯|b

Remark that in the absence of peer effects, i.e. dPj /dqj =dPj /dqj = 0, only cultural intolerance matters leading the urban equilibrium to be segregated. This highlights a new segregation force which results from the desire to preserve one’s own culture differing from the ones studied in the literature on neighborhood effects (see B´enabou, 1993, 1996a,b or de Bartolome, 1990). The segregated equilibrium exhibits spatial disparities in terms of cultural transmission and educational rate. This comes from the fact that chances to acquire the mainstream cultural trait for children living in urban area 1 are better than for those living in urban area 2. Hence, it can be checked that the probability that a child of trait-a parents be socialized into this trait is always higher in urban area 1. Furthermore, chances to acquire trait a and have greater probability to educate for children of trait-b parents are better in urban area 1. This difference translates into higher education in urban area 1. When Q < 1, urban area 1 is characterized by a positive rate of education while urban area 2 is populated only by minority group reaching a minimal rate of education. When Q > 1, urban area 1 is populated by the mainstream group reaching a maximal rate of education while the mixed urban area 2 is characterized by a lower fraction of youth population who gets education. Further, our framework allows to get insights on the influence of segregation on persistence of the cultural trait across generations. We measure intergenerational cultural mobility by the gap between probabilities to get trait a conditional on the family trait:

P aa − P ba =

 1  1 [q1 P1aa + q2 P2aa ] − (1 − q1 )P1ba + (1 − q2 )P2ba . Q 2−Q

(11)

with transition probabilities depending on equilibrium socialization efforts. In order to have a tractable result of the impact of segregation on cultural mobility, we work with the following specific forms P e¯|i = αi q, with αa > αb , f (q) = q and Θ(τ ) = (1/2θ)τ 2 and show that Corollary 1 When the cost of socialization is high, cultural family background is more persistent across generations under segregation than under integration. As in standard models of cultural transmission, our framework gives rise to persistence of cultural traits across generations, i.e. P aa − P ba > 0. This is due to the existence of a bias in the transmission process induced by the socialization effort of parents. What is more is that segregation adds further 4

When a segregated equilibrium exists, it is always spatially stable. Indeed, there is always a trait i population which strictly prefers her place of residence, i.e. trait a population when Q < 1 and trait b population otherwise. A small perturbation of the segregated equilibrium does not change the identity of the highest bidder for land in neighborhood 1.

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glue in the cultural transmission process. The intuition goes as follows. Segregation (as compared to integration) has two distinct effects on the probability differential, P aa − P ba . First, segregation increases oblique transmission of trait a for both types of child. However, this effect is stronger for children from mainstream family background. Hence, the oblique transmission effect favours the rise of the probability differential P aa −P ba . Second, a rise in q affects socialization efforts. More precisely, due to cultural substitution, an increase in q decreases the effort of parents a which negatively affects the probability differential. When the cost of socialization is high, the cultural substitution effect is low. Hence, the oblique transmission effect prevails and the rise in segregation positively impacts the probability differential. Given Assumption 1, cultural mobility is positively related to social mobility. Corollary 1 is supported by empirical evidence. Borjas (1995) finds that the rate of mean convergence of skills between ethnic groups is reduced when people live in segregated neighborhoods and are more exposed to ethnicity influence. Recent work on intergenerational mobility in the US has emphasized the role of neighborhoods for individual prospects of social mobility (see Chetty, Hendren, Kline and Saez, 2014, Chetty and Hendren, 2015). In particular, Chetty, et al. (2014) use data from Federal income tax records over the 1996-2012 period to show the striking spatial variation in social mobility. They also show that high-mobility areas have less residential segregation and income inequality, better primary schools, and greater social capital and family stability. We now turn to the cultural dynamics that emerge when the city is segregated. Assumption 3 Suppose that

(i) τ a (0) < 1 − f 0 (0), (ii) τ b (1) > 1 − f 0 (1), (iii) ∃ q ∈]0, 1[ such that f (q) ≥ q and τ a (q) ≥ τ b (q).

Assumption 3 clarifies the strength of the different forces at stake in the dynamics of cultural traits. According to (8), the dynamics of cultural traits is driven by three distinct forces: (i) the degree of conformism captured by the shape of f , (ii) peer effects in education, and (iii) cultural substitution where the two last forces are embodied in the socialization efforts τ a and τ b . High conformism and peer effects favor homogeneity within the population. Reversely, cultural 14

substitution is a force leading to heterogeneity. Item (i) makes statement about these forces when the mainstream population is low (i.e. q = 0). Both peer effects and conformism are then sufficiently high to overcome the cultural substitution effect (meaning that the socialization effort of mainstream agents is below some threshold). Item (ii) focuses on large fractions of mainstream agents (i.e. q = 1). Conformism and peer effects are then not sufficiently high to overcome cultural substitution (implying that the socialization effort of minority agents is above some threshold). Item (iii) imposes that for some intermediate value of the population composition, conformism and peer effects are high. Although empirical evidence on socialization rates and cultural transmission is still sparse, we can find some support for Assumption 3 in Bisin, Topa and Verdier (2004). From a structural estimation of a cultural transmission model, they find that socialization efforts of some religious group is positively related to the share of this group (what they call cultural complementarity). In other words, socialization rates are lower the smaller are religious shares, thus justifying item (i). Nonetheless, they find that socialization choices of minorities (Jews) is much higher than dominant groups (Protestants and Catholics) when the minority share is close to 0 giving support for item (ii). By Assumption 3, we consider a case where the different forces at stake matter for the dynamics of culture. Since this framework generates homogeneity forces as well as heterogeneity forces, this allows for a general situation where, in the long run, a culturally diverse as well as a culturally homogenous society can emerge. Proposition 2 Under Assumption 3, the population dynamics in the segregated city exhibits at least two stable stationary equilibria: Q = 0 and Q = q ∗ < 1. In the long run, population dynamics converge to either cultural homogeneity or cultural diversity. This is due to the interplay between opposite forces that drive cultural transmission. With Assumption 3, we consider a general case where both types of forces can prevail depending on the composition of the population. Remarkably, because we assume that the cultural substitution effect is high, there exists a longrun segregated equilibrium (Q = q ∗ ) such that neighborhood 1 is culturally diverse. By contrast, in neighborhood 2 with only the minority culture, given item (i) of Assumption 3, the dynamics of cultural trait push toward homogeneity provided. We can show that the persistence of cultural diversity in one neighborhood has important implications for long-run economic inequalities. Let us define inequalities by the educational gap between

15

a mainstream individual and its counterpart, i.e. P e|a − P e|b . It is expressed as follows e|a

P

e|a

−P

e|b

e|a

q 1 P 1 + q 2 P1 = Q

e|b

e|b

(1 − q1 )P1 + (1 − q2 )P2 − . (2 − Q)

(12)

At the long-run equilibrium Q = q ∗ < 1, the educational gap equals e|b

e|a P1 (q ∗ )

1 − q ∗ e|b ∗ P (0) − P1 (q ) − 2 ∗ . ∗ 2−q 2−q

(13)

Let us compare this gap with the one which would be achieved at a long-run equilibrium where both neighborhoods are culturally homogeneous, i.e. q ∗ = 1 which would arise if we would assume that cultural substitution is low5 . In the long run, the educational gap is e|a

e|b

P1 (1) − P1 (0). e|b

(14)

e|b

From (13) and (14) and as P1 (0) = P2 (0) , the long-run equilibrium with Q = q ∗ < 1 leads to lower inequality than the one with q ∗ = 1 if and only if 1 − q ∗ e|b e|b e|a e|a (P1 (0) − P1 (q ∗ )) < P1 (1) − P1 (q ∗ ), ∗ 2−q which is true given the existence of peer effects. From the inequality perspective, the long-run stationary equilibrium where one neighborhood is culturally diverse is thus more desirable than the long-run equilibrium with homogeneous neighborhoods. Hence, our framework emphasizes that the cultural substitution property which leads to cultural diversity is crucial for the long-run level of inequalities. However, it is also likely that segregation pushes to cultural homogeneity in the city. This case occurs when in neighborhood 1 where both cultural groups live cultural substitution is not strong enough. Let us remark that Proposition 2 does not precisely characterize the basin of attraction of these two equilibria because it requires specification of the oblique transmission function f . We could set conditions on f , allowing to avoid cycles and then say that for low (resp. high) values of q0 the city converges to Q = 0 (resp. Q = q ∗ ). 5

More formally, this amounts to assume that item (ii) in Assumption 3 does not hold anymore, i.e. for high fractions of type a individuals, conformism outweighs the effect of cultural substitution i.e. τ b (1) < 1 − f 0 (1).

16

4

Integration and multiple long run cities

Our framework also allows integration to arise. As in Section 3, to make clearer where the main results come from, we impose symmetry, that is ∆V aa = ∆V bb and ∆V ab = ∆V ba . Proposition 3 At each date t, the integrated equilibrium q1∗ = q2∗ = Q/2 exists and is spatially stable only if: (i) there is cultural proximity, (ii) transition probabilities are such that P1aa < P1bb . This Proposition highlights that an integrated city can emerge, at each date t, provided that the minority culture values education. A necessary condition for the integrated equilibrium to be stable is that cultural trait and education are complements and that the probability to transmit one’s cultural trait is relatively high for minority individuals (as compared to mainstream). Compared to the case of cultural polarization, when there is cultural proximity, benefits from education are high for minority parents as having an educated child does not necessarily implies a cultural loss. Since the benefit of education increases when the child acquires the parental trait, parents who benefit more from higher peer effects are those who are more likely to transmit their own trait in neighborhood 1. This is the case for minority parents when P1aa < P1bb (note that this generally arises when parents of type b form a large share of the population).6 Both Propositions 1 and 3 deliver the message that cultural distance matters for the type of urban equilibrium. This echoes empirical findings of Cutler, Glaeser and Vigdor (2008) that cultural distance (as measured either by linguistic differences, regional development gap or geographical proximity) accounts for a substantial part of the degree of segregation (for instance the changing country-of-origin composition of the immigrant population explains virtually all of the post-1970 increases in segregation). As at the integrated equilibrium, both neighborhoods have the same cultural composition, the education rate as well the cultural transmission process do not vary spatially. Still, there is persistence of the cultural trait within dynasties due to the fact that a cultural group has the possibility As it can be easily checked from bid rents (9) and (10), an integrated equilibrium with q1∗ = q2∗ always exists. However, it is not always spatially stable. Under complementarity, parents with trait b prefer to have educated children with their own cultural trait. After a perturbation of the integrated equilibrium that marginally increases q1∗ , neighborhood 1 provides a better educational environment with higher peer effects. Trait-b parents are able to bid for land higher than trait-a parents only if they value more than their counterparts educational gains obtained in neighborhood 1. This possibility arises when the probability to have a child of their own type is higher for parents b given that both cultural traits benefit from similar peer effects and that the surplus of educational gains obtained when the child acquires the parental cultural trait is the same among cultural traits (as we assume symmetry). 6

17

to transmit her own trait by vertical as well as horizontal transmission. Contrary to segregation, integration does not generate any spatial bias in the oblique transmission process. According to both items (i) and (ii) of Proposition 3, segregation as well as integration can emerge in the short run. The following proposition provides a characterization of the urban configurations arising in the long run. Proposition 4 Suppose that there is cultural proximity, the integrated equilibrium with (q1∗ , q2∗ ) = (q ∗ , q ∗ ) and the segregated equilibrium (q1∗ , q2∗ ) = (q ∗ , 0) with q ∗ < 1/2 may be both stable steady states. The result stems from the fact that urban equilibrium and the cultural composition of the population are co-determined. For some parameters’ values, a given population composition gives rise to some urban equilibrium which can be either integrated or segregated (see Proposition 3), and in turn the urban equilibrium provides the cultural environment that enables to preserve this population composition.7 For instance, if Q0 is low, an integrated equilibrium can emerge (cf. item (ii) of Proposition 3). This equilibrium arises due to the high probability with which type b agents transmit their trait in both neighborhoods. Therefore the integrated equilibrium impedes the collapse of population b. Furthermore, due to cultural substitution, parents of type a exert a socialization effort which counters the high transmission of trait b and avoids the extinguishment of population a. When these two forces counterbalance each other, the integrated equilibrium and the associated distribution of cultural traits in the population can both be sustained over time. For higher values of Q0 , a segregated equilibrium emerges (again, Proposition 3 states that under complementarity the urban equilibrium depends on the value of Q). When homogeneity forces, i.e. conformism and peer effects, are high enough, urban segregation maintains this distribution of traits over time. In the homogeneous neighborhood, the cultural uniformity remains due to strong homogeneity forces. In the culturally mixed neighborhood, the incentives of both types parents to preserve their own trait are such that both cultures persist. In such a case, the segregated equilibrium and the associated distribution of cultural traits may be sustained in the long run. Proposition 4 is illustrated in Figure 1. When the intial fraction of majority agents, Q0 , is low ¯ the urban equilbrium is integrated. This equilibrium gives rise to a particular dynamics (i.e., Q0 < Q) of cultural traits which is represented by the bold line. When the dynamics of Qt follows the bold 7

We proceed in the proof by showing that, for some functional forms of transition probabilities, oblique transmission technology, socialization cost and peer effects, there exists a set of parameters such that both the integrated equilibrium and segregated equilibrium exist in the long run and are spatially and dynamically stable. Let us mention that the condition q ∗ < 1/2 is equivalent to item (ii) of Proposition 3 given the particular case we consider in the proof.

18

Qt+1 − Qt

Integrated equilibrium

Segregated equilibrium

Dynamics under integration Dynamics under segregation Stable steady state Qt

Q

Figure 1: Example of the dynamics of cultural traits under cultural proximity ¯ (due to the intuition that given above). line, there exists a stable stationary equilibirum Q < Q Therefore, for low Q0 , the city can converge to a long run equilibirum which is integrated. ¯ the urban equilbrium is When the initial fraction of majority agent, Q0 , is high (i.e., Q0 > Q) segregated. The dynamics of cultural traits is then represented by the dashed line. In this case, ¯ Hence, for higher values of Q0 , the city can converge to a there exists a stable steady state Q > Q. segregated equilibirum. Our result is consistent with empirical findings of Card, Mas and Rothstein (2007, 2008) who show, using Census tract data for the 1970-2000 Censuses, that segregation is not the end of the city history and integrated neighborhoods are sustainable in the long run.8 However, by contrast to the approach of Card, Mas and Rothstein (2007, 2008), our model of cultural transmission allows to shed light on the relationship between the long-run degree of segregation and group share. Interestingly, Cutler, Glaeser and Vigdor (2008) provide evidence for a substantial variation of 8

In their study of american segregation history, Cutler, Glaeser and Vigdor (1999) document that segregation fell from 1970 to 1990 throughout the country, providing evidence that there are some forces pushing away from extreme segregation. From a case studies of segregation in Atlanta, Sacramento and Cleveland, they also stress the diversity of segregation patterns, with some cities (Atlanta and Cleveland) remaining highly segregated and others (Sacramento) with sustainable patterns of integration.

19

segregation throughout ethnic groups, space and time. They identify group share as a significant determinant of the degree of segregation and find that the impact of group share may be positive or negative depending on the segregation index considered. Groups forming higher shares of the metropolitan population tend to be both more isolated and less dissimilar than other groups9 . As pointed out by the authors, this result provides support for larger group spreading out in more neighborhoods while maintaining a high concentration in other neighborhoods. By contrast, smaller groups may need to cluster in order to benefit from shared cultural amenities. Our theoretical framework provides a rationale for this empirical result. Here, individuals from a larger group have a high chance to transmit their trait whatever the neighborhood they live in. Since they benefit from living next to mainstream individuals (which is the case because there exist some local spillovers increasing the prospects of economic success), they have an incentive to spread out in more neighborhoods, so as to benefit from the local externalities, while still maintaining a high transmission of their trait. By contrast, individuals from smaller groups do not have this opportunity otherwise they would reduce drastically the probability to transmit their trait. As a consequence, individuals from smaller groups prefer to cluster in few areas in order to ensure the persistence of their culture at the expense of better economic prospects. An interesting implication of Proposition 4 is that it provides conditions for the efficacy of policies that would aim to favor integration. Integration policies would more easily help the cultural mixing to maintain in neighborhoods provided the cultural distance is low and the size of the minority group is relatively large in the city population. Finally, results of Section 3 and 4 which both echoe empirical findings by Cutler, Glaeser and Vigdor (2008) can be summarized in the following graph. Urban equilibrium in the long run, and then the degree of segregation, depends on both cultural distance and the population composition. The x-axis captures cultural distance as given by ∆V bb − ∆V ba . When ∆V bb − ∆V ba is negative, there is cultural polarization, while if positive there is cultural proximity. The y-axis captures the differential of probabilities P bb − P ba which is determined by the population composition (it is decreasing in the fraction of type-a agents). When this differential is negative (resp. positive), the distribution of cultural traits is biased toward the majority (resp. minority) trait. As illustrated in Figure 2, when there is cultural polarization and/or the population composition is biased toward the majority cultural trait, the city ends-up segregated. However, when there is 9

Dissimilarity is high when some ethnic group disproportionately resides in some area of a city relative to mainstream group while isolation measures the degree of exposure that individuals of some ethnic group have to other members of their group (see Cutler, Glaeser, Vigdor, 1999, for more precise definition and specification).

20

Integrated equilibrium

Segregated equilibrium Pbb − Pba

Segregated equilibrium

Segregated equilibrium

Cultural polarization

Cultural proximity

Cultural distance: ∆Vbb − ∆Vba

Figure 2: Urban equlibria as a function of cultural distance and the population composition both cultural proximity and the distribution of cultural traits is biased toward the minority, the long-run city is integrated.

5

Is social segregation optimal?

We now turn to the issue whether it is efficient to let people sort themselves into urban areas, or should we implement particular urban policies ? In particular, we consider urban policies which affect the location of agents within the city. Enforcing quotas of inhabitants from a given social category is one way of promoting social mixing in a given urban area10 . We discuss whether integration or segregation is more desirable in view of our efficiency criterion, i.e. the long-run rate of education. Proposition 5 Suppose that Assumption 3 holds and 2ˆ q < 1. For any Q0 ∈ [ˆ q , 2ˆ q ], segregation is efficient. For any Q0 ∈ [2ˆ q , 1], integration is efficient. Proposition 5 stresses that the efficiency of the urban policy depends on the cultural composition 10

One example of a quota policy is the SRU law (loi relative `a la Solidarit´e et au Renouvellement Urbains) in force in France since 2000. In French municipalities with at least 3,500 inhabitants (1,500 inhabitants in the Paris administrative region, Ile-de-France), 20% of the available housing stock must be public housing. Municipalities with figures below this ratio have to pay fines (see Gobillon and Vignolles, 2016, for an evaluation of this policy, and also Brueckner, 2011, for a presentation and an analysis of various housing policies promoting integration).

21

of the population. The intuition is that, depending on the composition of the city population, either segregation or integration provides higher incentives to socialize children to cultural trait a which favors education. For some relatively low initial fractions of the mainstream group, i.e. Q0 ∈ [ˆ q , 2ˆ q ], segregation, which concentrates type a individuals in neighborhood 1, allows for sufficient peer effects in this neighborhood to provide high incentives to transmit cultural trait a compared to trait b. By way of contrast, were the city to be perfectly integrated, it would reduce the fraction of mainstream agents, and thus peer effects in neighborhood 1, in such a way that the rate of transmission of trait b would be higher in both urban areas. This would negatively affects education in the long run. However, for higher initial fractions of the mainstream group, integration, which increases the fraction of agents a in neighborhood 2 (compared to segregation), rises the intensity of peer effects in this neighborhood. When the city fraction of the mainstream group is sufficiently high, i.e. Q0 ∈ [2ˆ q , 1], this provides higher incentives to socialize children to trait a. This benefits to education in the long run. On the contrary, segregation, by reducing the fraction the mainstream inhabitants in neighborhood 2 would decrease the incentives to transmit trait a in this neighborhood favoring the spreading of trait b. Note that due to cultural substitution (cf. Assumption 3), it is never profitable for the long-run rate of education to have a too large fraction of type a agents, whatever the urban equilibrium. Incentives to transmit trait a would become low as mainstream parents would rely more intensively on oblique transmission to transmit their trait. Whereas B´enabou (1996a) stresses that the degree of complementarity between individuals’ levels of human capital at the community and the society levels is key to assess the efficiency of a segregated equilibrium, We emphasize the importance of the population cultural composition. Our result that the efficiency of segregation depends on the distribution of culture in the whole population has important implications. It suggests that poverty deconcentration and integration policies must circumvent the difficulty to identify the degree of neighborhood social mix most favorable to education. According to Galster (2002)’s meta-analysis of the empirical evidence on the impact of poverty concentration on socioeconomic success, if behavioral problems are related to neighborhood poverty rates within a range of approximately 15-40% of poverty rate, “This implies that net social benefits will be larger if neighborhoods with greater than roughly 15% poverty rates are replaced with (an appropriately larger number) of neighborhoods having less than 15% poverty rates. However, net social benefits will be smaller

22

if neighborhoods with greater than about 40% poverty rates are replaced with (an appropriately larger number) of neighborhoods having between about 15-40% poverty rates. Put more bluntly in policy terms, unless very low-poverty neighborhoods can be opened up for occupation by the poor, deconcentration efforts should halt, because merely transferring the poor from high- to moderate-poverty neighborhoods is likely to be socially inefficient.” (p. 322, Galster, 2002) What are the consequences of urban policies on economic inequalities? To answer that question, let us compare the educational gaps under integration and segregation. Suppose that the city is integrated at date t, i.e. qt1 = qt2 = Qt /2, given (12) the educational gap, is given by

P

e¯|a



Qt 2

 −P

e¯|b



Qt 2

 .

When the city is segregated, the gap equals

P e|a − P e|b =

    

e¯|a

P1 (Qt ) − e ¯|a P1 (1) Qt

+

Qt −1 e¯|a P2 Qt

e ¯|b

e ¯|b

(1−Qt )P1 (Qt )+P2 (0) (2−Qt )

if Qt ≤ 1

e¯|b

(Qt − 1) − P2 (Qt − 1),

otherwise.

It is easy to check that under Assumption 1, educational gap is lower under integration than under segregation. Segregation increases the cultural disparity between neighborhoods which magnifies peer effects differences between both traits. It turns out that, for some cultural composition of the population, there is a trade-off between efficiency and equity: while segregation promotes long-run education, it widens the educational gap between the two cultural groups.

6

Conclusion

How does segregation impact the transmission of traits which are critical for economic success? When does cultural heterogeneity leads to residential segregation? This paper provides some answer by developing a model where neighborhood formation and cultural transmission interact. A key feature of our framework is that the parental choice of the place of residence relies on the need to socially integrate and the concern to preserve own culture. This tension between culture and economic integration impacts cultural diversity and residential segregation in the long run. When the minority culture shares the mainstream view that education is valuable, multiple types of long-run urban configurations arise depending on initial cultural composition of the society. In 23

particular, an integrated city can emerge in the long run provided that cultural mixing in neighborhoods allows for the preservation of the minority culture. We also show that segregation strengthens the parental influence on the child’s destiny. Finally, we highlight the crucial role of the population composition to assess the efficiency of urban policies. The model could be extended along several lines. First, there is a consensus that housing market dynamics impact segregation (see the review of Rosenthal and Ross, 2014). The model is flexible enough to introduce some housing market features such as tenure choice, housing depreciation and maintenance, development and redevelopment of housing stock. These features would influence locations choices and allow us to explore their implications on the pattern of segregation and cultural dynamics. Second, our model considers that cultural distance is binary: there is either cultural proximity or cultural polarization. Integrating identity behaviors that can lead to cultural clash or cultural assimilation would allow us to endogenize cultural distance and study how it responds to assimilation policies. Third, relaxing the assumption that children attend the school of their urban area would allow to differentiate the social arenas where peer effects and oblique transmission are determined. For instance, considering that peer effects are circumscribed within schools whereas oblique transmission is produced in the urban area would affect the trade-off faced by parents when deciding the place of residence. This extension could shed new light on the consequences of school choice systems on segregation and inequality dynamics. Finally, the literature on cultural transmission has addressed the issue of the design and dynamics of institutions (see Bisin and Verdier, 2015). However, this literature disregards endogenous stratification. Fruitful research would be to investigate how culture interacts with both residential segregation and institutions providing local public goods and education services.

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[43] Lindbeck, A. and S. Nyberg, 2006, “Raising Children to Work Hard: Altruism, Work Norms, and Social Insurance”, Quarterly Journal of Economics, vol. 121, 1473-1503. [44] Loury, G., 1977, “A Dynamic Theory of Racial Income Differences”, in A. Le Mond (eds), Women, Minorities and Employment Discrimination, Lexington, MA: Lexington Books. [45] Moizeau, F., 2015, “Dynamics of Social Norms in the City”, Regional Science and Urban Economics, vol. 51, 70-87. [46] Oreopoulos, P., 2003, “The Long-Run Consequences of Living in a Poor Neighborhood”, Quarterly Journal of Economics, vol. 118, 1533-1575. [47] Rosenthal, S. and and S. Ross, 2014, “Change and Persistence in the Economic Status of Neighborhoods and Cities”, in G. Duranton, V. Henderson and W. Strange (eds), Handbook of Regional and Urban Economics, Amsterdam: Elsevier Publisher. [48] Saez-Marti, M., and A. Sj¨ogren, 2008, “Peers and Culture”, Journal of Economic Theory, vol. 143, 153-176. [49] Saez-Marti, M. and Y. Zenou, 2012, “Cultural Transmission and Discrimination ”, Journal of Urban Economics, vol. 72, 137-146. [50] Sharkey, P., 2008, “The Intergeneration Transmission of Context”, American Journal of Sociology, vol. 113, 931-969. [51] Sharkey, P. and F. Elwert, 2011, “The Legacy of Disadvantage: Multigenerational Neighborhood Effects on Cognitive Ability”, American Journal of Sociology, vol. 116, 1934-1981. [52] Sikkink, D. and M.O. Emerson, 2008, “School Choice and Racial Segregation in US Schools: The Role of Parents’ Education”, Ethnic and Racial Studies, vol. 31, 267-293. [53] Tinker, C. and A. Smart, 2012, “Constructions of Collective Muslim Identity by Advocates of Muslim Schools in Britain”, Ethnic and Racial Studies, vol. 35, 643-663. [54] Topa, G. and Y. Zenou, 2014, “Neighborhood versus Network Effects”, in G. Duranton, V. Henderson and W. Strange (eds), Handbook of Regional and Urban Economics, Amsterdam: Elsevier Publisher.

28

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29

7

Appendix

7.1

Socialization Choice of Trait b Parents

At date t, a trait-b parent chooses her optimal effort τ ∗b that solves

max U b (ρ, τ b ) subject to (2). τb

This leads to the following first-order condition:

f (q)(P e¯|b ∆V bb − P e¯|a ∆V ba + V e|bb − V e|ba ) = Θ0 (τ ∗b )

(15)

We assume that for any q

P e¯|b ∆V bb − P e¯|a ∆V ba + V e|bb − V e|ba ≥ 0.

(16)

which amounts to say that trait-b parents have an incentive to transmit their own trait, i.e. τ ∗b (q) ≥ 0. By the implicit function theorem, we get

e ¯|a f 0 (q) P e¯|b ∆V bb − P e¯|a ∆V ba + V e|bb − V e|ba + f (q) dPdq ∆V bb − ∆V ba dτ ∗b = . dq Θ00 (τ ∗b )

7.2

(17)

Bid-Rent Slopes

Using the envelope theorem, the slope of the trait-a bid curve is given as dρa e¯|a e¯|b = (1 − τ ∗a (q1 ))f 0 (q1 )(P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab ) dq1  dq2  e¯|a e¯|b −(1 − τ ∗a (q2 ))f 0 (q2 ) P2 ∆V aa − P2 ∆V ab + V e|aa − V e|ab dq1 e¯|a
aa aa aa ab dP1 + P1 ∆V + (1 − P1 )∆V e dq1
dP2e¯|a dq2 − P2aa ∆V aa + (1 − P2aa )∆V ab . dq2 dq1

30

(18)

For type b individuals, we have dρb e¯|b e¯|a = −(1 − τ ∗b (q1 ))f 0 (q1 )(P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba ) dq1 dq2 e¯|b e¯|a (P2 ∆V bb − P2 ∆V ba + V e|bb − V e|ba ) +(1 − τ ∗b (q2 ))f 0 (q2 ) dq1
dP1e¯|b
+ P1bb ∆V bb + 1 − P1bb ∆V ba dq1
dP2e¯|b dq2 . − P2bb ∆V bb + (1 − P2bb )∆V ba dq2 dq1

7.3

(19)

Proof of Proposition 1

Given (18) and (19), and using dq2 /dq1 = −1, the bid-rent slope differential for any given q1 is dρa dρb e¯|a e¯|b − = (1 − τ ∗a (q1 ))f 0 (q1 )(P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab ) dq1 dq1 e¯|b

e¯|a

e¯|b

e¯|a

+(1 − τ ∗b (q1 ))f 0 (q1 )(P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba )   e¯|a e¯|b e|ab ∗a 0 aa ab e|aa −V +(1 − τ (q2 ))f (q2 ) P2 ∆V − P2 ∆V + V +(1 − τ ∗b (q2 ))f 0 (q2 )(P2 ∆V bb − P2 ∆V ba + V e|bb − V e|ba ) e¯|a

e¯|b

dP1 dP ∆V aa + (1 − P1aa ) 1 ∆V ab dq1 dq1 e¯|b e¯|a
bb dP1 bb bb dP1 −P1 ∆V − 1 − P1 ∆V ba dq1 dq1 +P1aa

e¯|a e¯|b aa dP2 aa aa dP2 +P2 ∆V + (1 − P2 ) ∆V ab dq2 dq2 e¯|b e¯|a dP dP −P2bb 2 ∆V bb − (1 − P2bb ) 2 ∆V ba . dq2 dq2

(20)

First, let us consider that ∆V ab = α∆V aa with α ∈ [0, 1] . One can see that the bid-rent slope differential is, other things being equal, a linear function of ∆V aa with slope     e¯|a e¯|b e¯|a e¯|b (1 − τ ∗a (q1 ))f 0 (q1 ) P1 − αP1 + (1 − τ ∗a (q2 ))f 0 (q2 ) P2 − αP2 ! ! e¯|a e¯|a e¯|b e¯|b dP dP dP dP + P1aa 1 + (1 − P1aa ) 1 α + P2aa 2 + (1 − P2aa ) 2 α dq1 dq1 dq2 dq2 which is positive. Hence, one could choose ∆V aa high enough so that dρa /dq1 >dρb /dq1 . This amounts to say that if mainstream parents attach much more importance to education than the minority ones whatever the type of the child (∆V aa > ∆V ab > ∆V ba > ∆V bb ), then segregation arises. As we aim to capture the key role of cultural distance, we do not put emphasis on this case 31

in the main text eventhough it could arise in practice. Let us consider now that ∆V aa > ∆V ab , ∆V ba > ∆V bb , ∆V aa = ∆V ba and ∆V ab = ∆V bb . Due to the oblique transmission effect, the first four terms of (20) are positive. Hence, a sufficient condition for the bid-rent slope differential to be positive is that the following inequality is satisfied e¯|b

e¯|a


P1aa + P1bb − 1

dP dP1 ∆V ba − 1 ∆V bb dq1 dq1 e¯|a

!

e¯|b

dP2 dP ∆V ba − 2 ∆V bb dq2 dq2


+ P2aa + P2bb − 1

!

> 0.

Given (1) and (2), we have e¯|a

e¯|b

dP1 dP ∆V ba − 1 ∆V bb dq1 dq1


b

τ1a (1 − q1 ) + q1 τ1

e¯|a

+

τ2a (1

− q2 ) +

q2 τ1b

e¯|b

!

dP dP2 ∆V ba − 2 ∆V bb dq2 dq2




!

> 0.

Given Assumption 1 and if ∆V ba > ∆V bb , we have any j = 1, 2 e¯|a

dPj dqj


∆V ba − ∆V bb > 0 for j.

Hence, the segregated equilibrium exists. Spatial stability requires that, after a move of a small number of trait-a individuals from neighborhood 1 to neighborhood 2 and a migration of the same number of trait-b individuals in the reverse direction, the highest bidders for neighborhood 1 are trait-a individuals. At the segregated equilibrium when Q < 1, q1∗ = Q and q2∗ = 0, the bid-rent equilibrium is the willingness of trait-b individuals. At this equilibrium price level, trait-a individuals strictly prefer to live in urban area 1. When Q ≥ 1, q1∗ = 1 and q2∗ = Q − 1, the bid-rent equilibrium is the willingness of trait-a individuals. At this equilibrium price level, trait-b individuals strictly prefer to live in urban area 2. Whatever one of these equilibrium configurations, trait-a individuals remain the highest bidders for neighborhood 1 after the small perturbation of the equilibrium.

32

7.4

Proof of Corollary 1

We consider the following functional forms: P e¯|i = αi q, with αa > αb , f (q) = q and Θ(τ ) = (1/2θ) τ 2 . From (5) and (15), we get

τ ∗a = −θq 2 (αa ∆V aa − αb ∆V ab ) + θq((αa ∆V aa − αb ∆V ab ) − (V e|aa − V e|ab )) + θ(V e|aa − V e|ab ),

and τ ∗b = θq 2 (αb ∆V bb − αa ∆V ba ) + θq(V e|bb − V e|ba ). Let us consider the probability differential q1 ∗a Q − q1 ∗a (τ (q1 ) + (1 − τ ∗a (q1 ))f (q1 )) + (τ (Q − q1 ) + (1 − τ ∗a (Q − q1 ))f (Q − q1 )) Q Q
1 + q1 − Q 1 − q1 − (1 − τ ∗b (q1 ))f (q1 ) − (1 − τ ∗b (Q − q1 ))f (Q − q1 ) . 2−Q 2−Q

P aa − P ba =

Let us compare this differential under segregation and integration. Suppose that Q < 1 (similar arguments hold for the case Q > 1 so that we skip the proof in order to lighten the exposition). The differential of probabilities at the segregated equilibrium writes as

P aa − P ba = (τ ∗a (Q) + (1 − τ ∗a (Q))Q) −

1−Q (1 − τ ∗b (Q))Q. 2−Q

(21)

Under integration, we have

P

aa

−P

ba

          Q Q Q Q Q ∗a ∗a ∗b = τ + 1−τ − 1−τ . 2 2 2 2 2

(22)

We have a higher differential under segregation if (21) is greater than (22) that is        Q Q Q ∗a ∗a + 1−τ (τ (Q) + (1 − τ (Q))Q) − τ 2 2 2    1−Q Q Q > (1 − τ ∗b (Q))Q − 1 − τ ∗b , 2−Q 2 2 ∗a

∗a

Since the function τ ∗b is increasing in q, we have (1 − τ ∗b (Q)) < 1 − τ ∗b if 1−Q Q Q − ≤ 0, 2−Q 2

33

Q 2





. The RHS is negative

which is equivalent to 2Q ≥ Q, which is true. Therefore, if the LHS is positive then (21) is greater than (22). The LHS is equal to zero at Q = 0. If P aa = τ ∗a (Q) + (1 − τ ∗a (Q))Q is an increasing function of Q, then the LHS is strictly positive for any Q > 0. Let us perform the derivative, one gets dP aa = −2θ(αa ∆V aa − αb ∆V ab )(1 − q) + θ(αa ∆V aa − αb ∆V ab − (V e|aa − V e|ab ))(1 − 2q) dq + θq 2 (αa ∆V aa − αb ∆V ab ) − θ(V e|aa − V e|ab ) + 1.

This function is a polynomial of order two which is convex. It is equal to 1 at q = 1. It is positive at q = 0 if and only if

−θ(αa ∆V aa − αb ∆V ab ) − 2θ(V e|aa − V e|ab ) + 1 ≥ 0

leading to θ≤

2(V

e|aa

−V

e|ab )

1 ≡ θe1 . + (αa ∆V aa − αb ∆V ab )

The derivative of the polynomial dP aa /dq is positive at q = 0 so that we conclude that when the above inequality holds, dP aa /dq is positive on the whole interval [0, 1]. Hence, the result.

7.5

Proof of Proposition 2

1. From (8), let us study the dynamics described by the map Ψ : [0, 1] → [0, 1] defined such that

Ψ(qt ) = (f (qt ) − qt )(1 − τ b (qt )) + (τ a (qt ) − τ b (qt ))qt (1 − f (qt )) + qt .

Steady states are such that Ψ(qt ) − qt = 0. First, we have Ψ(0) − 0 = 0, Ψ(1) − 1 = 0 so that q = 0 and q = 1 are steady states of the map Ψ. 0

Let us perform the derivative Ψ , we obtain dτ b + Ψ (qt ) =(f (qt ) − 1)(1 − τ (qt )) − (f (qt ) − qt ) dqt 0

0

b



dτ a dτ b − dqt dqt 0

 qt (1 − f (qt ))

+ (τ a (qt ) − τ b (qt ))(1 − f (qt )) − (τ a (qt ) − τ b (qt ))qt f (qt ) + 1.

34

We deduce, 0

0

0

0

Ψ (0) = τ a (0) + f (0) > 0, and Ψ (0) = τ b (1) + f (1) > 0. 0

0

With item (i) and (ii) of Assumption 3 we have Ψ (0) < 1, Ψ (1) > 1. Also, with item (iii), we known that there exists some q ∈]0, 1[ such that Ψ(q) > q. As we have Ψ ∈ C 2 , we deduce that there exist 0

0

qˆ, q ∗ , qˆ < q ∗ , such that (i) Ψ(ˆ q ) = qˆ, Ψ(q ∗ ) = q ∗ , Ψ (ˆ q ) > 1, Ψ (q ∗ ) < 1.

2. Cultural dynamics in the segregated city.

The dynamics of cultural traits in the city is described by the map Q : [0, 2] → [0, 2] which is such that 1 2 Q(Qt+1 ) = Q(qt+1 + qt+1 ) = Ψ(qt1 ) + Ψ(qt2 ).

Fixed points of the map Q are such that

Ψ(qt1 ) + Ψ(qt2 ) = qt1 + qt2 . When the city is segregated, steady states are Q = 0, Q = qˆ, Q = q ∗ , Q = 1, Q = 1 + qˆ, Q = 1 + q ∗ , Q = 2 with urban urban equilibria respectively given by (q 1 , q 2 ) = (0, 0), (q 1 , q 2 ) = (ˆ q , 0), (q 1 , q 2 ) = (q ∗ , 0), (q 1 , q 2 ) = (1, 0), (q 1 , q 2 ) = (1, qˆ), (q 1 , q 2 ) = (1, q ∗ ), (q 1 , q 2 ) = (1, 1).

A necessary and sufficient condition for stability of any steady state Q when the segregated urban equilibrium is (q 1 , q 2 ) is11 . d(Qt+1 − Qt ) |Q < 0, dQt 0

0

⇔Ψ (q 1 ) + Ψ (q 2 ) < 2. 0

Since item (i) of Assumption 3 implies Ψ (0) < 1 we deduce that the equilibrium (0, 0) is stable. 0

0

0

Furthermore, from the previous part of this proof, we have Ψ (q ∗ ) < 1 so that Ψ (0) + Ψ (q ∗ ) < 2 and we deduce that the equilibrium (q ∗ , 0) is stable. Now, from item (iii) of Assumption 3, we also have 11

This is also a sufficient condition for stability because one can easily check that

35

dQt+1 dQt

> 0.

0

0

0

0

0

0

Ψ (1) > 1, we deduce that Ψ (1) + Ψ (1) > 2 and Ψ (1) + Ψ (ˆ q ) > 2 (since Ψ (ˆ q ) > 1). Equilibria (1, 1) and (1, qˆ) are unstable. For equilibria (ˆ q , 0), (1, 0), and (1, q ∗ ), our assumptions do not allow to conclude. We deduce that the population dynamics in the segregated city admits at least to stable long-run equilibria Q = 0 and Q = q ∗ .

7.6

Proof of Proposition 3

Existence. When q1 = q2 = Q/2, both willignesses to pay are equal to 0 implying that nobody has an incentive to move. The integrated city, where both neighborhood cultural compositions are identical, is a urban equilibrium. Spatial stability. The stability condition requires that after a move of a small number of trait-b individuals from neighborhood 1 to neighborhood 2 and a migration of the same number of trait-a individuals in the reverse direction, the highest bidders for neighborhood 1 are trait-b individuals. This amounts to check whether the bid rent is steeper for the trait b-individuals. Formally, stability requires that dρb dρa − < 0. dq1 q∗ =Q/2 dq1 q∗ =Q/2 1

1

e¯|j

At the integrated equilibrium, as q1∗ = q2∗ = Q/2, we have P1aa = P2aa , P1bb = P2bb , dP1 /dq1 =   e¯|j − dP2 /dq1 · (dq1 /dq2 ) for j = a, b and −f 0 (q1 ) = f 0 (q2 )(dq2 /dq1 ). Hence, the bid-slope rent differential becomes      dρa dρb Q Q e¯|a e¯|b ∗a − = 2 1−τ f0 (P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab ) dq1 q∗ =Q/2 dq1 q1 =Q/2 2 2 1      Q Q e¯|b e¯|a +2 1 − τ ∗b f0 (P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba ) 2 2 e¯|a e¯|b aa dP1 aa aa dP1 +2P1 ∆V + 2 (1 − P1 ) ∆V ab dq1 dq1 e¯|b
dP1e¯|a dP −2P1bb 1 ∆V bb − 2 1 − P1bb ∆V ba . dq1 dq1

Given the symmetry assumption, ∆V aa = ∆V bb and ∆V ab = ∆V ba and Assumption 1 the bid-rent

36

slope differential can be written as follows:      Q dρa Q dρb e¯|a e¯|b 0 ∗a f ; r, γ (P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab ) − = 2 1−τ dq1 q∗ =Q/2 dq1 q1 =Q/2 2 2 1      Q Q e¯|b e¯|a 0 ∗b f ; r, γ (P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba ) +2 1 − τ 2 2 e¯|a

dP1 ∆V bb − ∆V ba . +2 P1aa − P1bb dq1 Considering that the first two terms being positive due to cultural transmission motives, a necessary condition for the bid-rent slope differential to be negative is

P1aa − P1bb





∆V bb − ∆V ba < 0.

If P1aa − P1bb > 0, it is equivalent to ∆V bb − ∆V ba < 0 which is impossible as ∆V bb − ∆V ba < 0 is a sufficient condition for segregation to emerge. We then must have P1aa − P1bb < 0 and ∆V bb − ∆V ba > 0.

7.7

Proof of Proposition 4

We provide an analytical example for which both types of urban equilibria, integrated equilibrium and segregated equilibrium, are spatially and dynamically stable in the long run. Let us consider the following specifications P e¯|a = βq + a, P e¯|b = βq + ba with a > b, β, a, b > 0 and β + a < 1. Let Θ(τ ) = θτ 2 /2, and f (qt ) = γrqtα /(rqtα + (1 − qt )α ) with α, r, γ > 0. Let us stress that, in order to be able to solve the problem analytically, one needs to restrict the model parameters so as to a obtain closed form solution. The parameter γ is then necessary to keep some degree of liberty. It is not necessary, however, for the result exposed in Proposition 4. A numerical example is available upon request.

1. Stationary distribution of cultural traits and dynamic stability. Given the above functional forms, (5) and (15) can be written as follows
1 τ ∗a (qt ) = (1 − f (qt )) βqt (∆V aa − ∆V e|ab ) + a∆V aa − b∆V ab + V e|aa − V e|ab , θ
1 τ ∗b (qt ) = f (qt ) βqt (∆V aa − ∆V ab ) + b∆V aa − a∆V ab + V e|bb − V e|ba . θ

37

Steady states are such that Qt+1 = Qt which is true if q1,t+1 − q1,t = 0 and q2,t+1 − q2,t = 0. Given (8) and the above functional forms, for j ∈ {1, 2}, qj,t+1 − qj,t = 0 admits at least three solutions: 0, 1 and q ∈]0, 1[ such that

h(qt ) = 0, where h(qt ) = f (qt ) − qt + qt τ ∗a (qt ) − f (qt )τ ∗b (qt ) + qt f (qt )(τ ∗b (qt ) − τ ∗a (qt )).

To obtain a closed form solution, let us consider that parameters are such that there exists q ∗ solving

τ ∗a (q ∗ ) − τ ∗b (q ∗ ) = 0,

and

f (q ∗ ) − q ∗ = 0,

which implies h(q ∗ ) = 0. This is equivalent to say that parameters are such that


C1: (1 − q ∗ ) βq ∗ (∆V aa − ∆V ab ) + a∆V aa − b∆V ab + V e|aa − V e|ab −
q ∗ βq ∗ (∆V aa − ∆V ab ) + b∆V aa − a∆V ab + V e|bb − V e|ba = 0, C2: γ = q ∗ +

q ∗ (1 − q ∗ )α . rq ∗ α

Without loss of generality, let us normalize V e|aa − V e|ab = 0. The expression for q ∗ is then β(∆V aa − ∆V e|ab ) − (∆V aa − ∆V ab )(a + b) − (V e|bb − V e|ba ) + q∗ = 4β(∆V aa − ∆V ab )
2 with D = (∆V aa − ∆V ab )(b + a − β) + (V e|bb − V e|ba )

√

D

,

+ 8β(∆V aa − ∆V ab )(a∆V aa − b∆V ab ).

Hence, both urban configurations (q1 , q2 ) = (q ∗ , q ∗ ) and (q1 , q2 ) = (0, q ∗ ) are steady states. Let us now check dynamic stability. Let us first consider the integrated equilibrium (q1 , q2 ) = (q ∗ , q ∗ ). It is a stable steady state if and only if dh(q) <0 dq q=q∗ which is equivalent to 0

(1 − τ ∗a (q ∗ ))(f (q ∗ ) − 1) + q ∗ (1 − q ∗ )

38

! dτ ∗a dτ ∗b − < 0. dq q=q∗ dq q=q∗

0

Let us perform f (qt ). We obtain (α−1)

rq (1 − qt )(α−1) f (qt ) = αγ t α . (rqt + (1 − qt )α )2 0

0

Assume that α > 1 meaning that there is conformism. In such a case, note that we have f (0) = 0.

We have 0

f (q ∗ ) =

α γ − q∗ γ 1 − q∗

using f (q ∗ ) = q ∗ .

0

0

When γ = 1, f (q ∗ ) = α > 1. When γ equals q ∗ (implying that r is large), f (q ∗ ) = 0. We can 0

deduce that there exists γ˜ such that ∀γ < γ˜ , we have f (q ∗ ) < 1.

0

0

Suppose that γ < γ˜ . Going back to the sign of h (q ∗ ), given that we have f (q ∗ ) < 1, we deduce the following. ∗b ∗a 0 − dτdq < 0, then h (q ∗ ) < 0. (i) If dτdq ∗ ∗ q=q q=q  ∗a dτ ∗a dτ ∗b (ii) If dq − dq > 0, noting that dτdq ∗ ∗ q=q

q=q

q=q ∗

−





dτ ∗b dq

q=q ∗

and τ ∗a (q ∗ ) are proportional to

1/θ, then, for θ sufficiently high both terms become sufficiently low so that the inequality holds. In e h0 (q ∗ ) < 0. other words, there exists θe such that ∀θ > θ,

0

Second, let us consider the segregated equilibrium (q1 , q2 ) = (q ∗ , 0). Given that h (q ∗ ) < 0, the urban configuration (q1 , q2 ) = (q ∗ , 0) is a stable steady state if and only if 0

h (0) < 0 ⇔ −1 + τ a (0) < 0.

which is true.

2. Spatial stability of the integrated equilibrium (q1 , q2 ) = (q ∗ , q ∗ ).

Following proof of Proposition 3, a necessary condition for spatial stability of the integrated equilibrium is     e¯|a e¯|b P1aa dP1 /dq1 − P1bb dP1 /dq1 < 0

39

which is equivalent to e¯|a

e¯|b

dP dP (τ + (1 − τ )q ) 1 − (τ ∗ + (1 − τ ∗ )(1 − q ∗ )) 1 < 0. dq1 dq1 ∗

e¯|a

∗

∗

e¯|b

Since dP1 /dq1 >dP1 /dq1 , this inequality holds only if (τ ∗ + (1 − τ ∗ )q ∗ ) − (τ ∗ + (1 − τ ∗ )(1 − q ∗ )) < 0

leading to 1 q∗ < . 2 Given our functional forms, and τ a (q ∗ ) = τ b (q ∗ ) = τ ∗ , the condition for the integrated equilibrium (q1 , q2 ) = (q ∗ , q ∗ ) to be spatially stable, can be written as follows 0

f (q ∗ ) 2β(∆V aa − ∆V ab )q ∗ + (a + b)(∆V aa − ∆V ab ) + V e|bb − V e|ba




+ (∆V aa − ∆V ab )β(2q ∗ − 1) < 0. 0

0

Since we have f (q ∗ ) = 0 when γ = q ∗ , as long as q ∗ < 12 , we deduce that there exists γ such that 0

∀γ < γ , this inequality holds.

3. Existence of the segregated equilibrium (q1 , q2 ) = (q ∗ , 0).

The condition for the segregated equilibrium (q1 , q2 ) = (q ∗ , 0) to exist is ρa (q ∗ , 0) > ρb (q ∗ , 0). Given the above functional forms, we get

(a∆V aa − b∆V ab )(τ ∗ + (1 − τ ∗ )q ∗ − τ a (0)) + (V e|bb − V e|ba + b∆V aa − a∆V ab )(1 − τ ∗ )q ∗
+(1 − τ ∗ )βq ∗ (2q ∗ − 1) ∆V aa − ∆V ab > 0.

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Let us define

Λ(τ ∗ ) ≡ (a∆V aa − b∆V ab )(τ ∗ + (1 − τ ∗ )q ∗ − τ a (0)) + (V e|bb − V e|ba + b∆V aa − a∆V ab )(1 − τ ∗ )q ∗
+ (1 − τ ∗ )βq ∗ (2q ∗ − 1) ∆V aa − ∆V ab .

The function Λ is linear in τ ∗ . We have

Λ(1) = (a∆V aa − b∆V ab )(1 − τ a (0)) > 0,

therefore, if Λ(0) > 0 then we deduce Λ(τ ∗ ) > 0 ∀τ ∗ . Let us perform Λ(0),

Λ(0) = (a∆V aa − b∆V ab )(q ∗ − τ a (0)) + (V e|bb − V e|ba + b∆V aa − a∆V ab )q ∗
+ βq ∗ (2q ∗ − 1) ∆V e|aa − ∆V e|ab . = 2βq ∗ 2 (∆V aa − ∆V ab ) + q ∗ (∆V aa − ∆V ab )(a + b − β) + V e|bb − V e|ba




− τ ∗a (0)(a∆V aa − b∆V ab ).

This is a polynomial function of q ∗ . It is convex, negative at q ∗ = 0 and for q ∗ = 1

Λ(0)|q∗ =1 = (1 − τ ∗a (0))(a∆V e|aa − b∆V e|ab ) + β(∆V e|aa − ∆V e|ab ) + b∆V aa − a∆V ab + V e|bb − V e|ba .

which is positive given that τ ∗a (0) < 1 and that τ ∗b (1) > 0. This polynomial is positive if and only if q ∗ > q˜, with
√ (∆V aa − ∆V ab )(β − (a + b)) − V e|bb − V e|ba + D0 q˜ = , 4β(∆V e|aa − ∆V e|ab )
2 with D0 = (∆V aa − ∆V ab )(a + b − β) + (V e|bb − V e|ba ) + 8βτ ∗a (0)(∆V e|aa − ∆V e|ab )(a∆V e|aa − b∆V e|ab ).

Then, q ∗ > q˜

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is equivalent to √ D>

√ 0 D

which is true given that τ ∗a (0) < 1.

4. Conclusion.

Now we conclude that, when parameters are such that C1 and C2 are satisfied and that 0

(i) γ < min{˜ γ , γ }, e (ii) θ > θ, urban equilibria (q1 , q2 ) = (q ∗ , q ∗ ) and (q1 , q2 ) = (0, q ∗ ) are both stable steady states.

An example of some parameters combination for which these conditions hold is ∆V aa = 0.07, ∆V ab = 0.01, V e|bb − V e|ba = 0.0005, a = 0.005, b = 0.004, β = 0.008, θ = 0.05, γ = 0.37, r = 30, α = 1.1.

7.8

Proof of Proposition 5

1. Suppose that Q0 ∈]ˆ q , 2ˆ q [.

(i) If segregation holds the urban equilibrium is such that q 1 = Q ∈]ˆ q , 2ˆ q [ and q 2 = 0. Given the function Ψ(.) in proof of Proposition 2, the dynamics of cultural traits is described by

Qt+1 = Ψ(Qt ) + Ψ(0) = Ψ(Qt ).

From proof of Proposition 2, we have that q ∗ is a steady state of the map Ψ and for Q0 ∈]ˆ q , q ∗ [ the sequence Qt with dynamics captured by the map Ψ is increasing, while for Q0 ∈]q ∗ , 1[, the sequence Qt is decreasing. By the continuity of the map Ψ, we deduce that for any Q0 ∈]ˆ q , 1[, the sequence Qt converges to Q = q ∗ . Hence for any Q0 ∈]ˆ q , 2ˆ q [, the sequence Qt converges to Q = q ∗ .

(ii) If integration holds the urban equilibrium is such that q 1 = q 2 = Q/2 < qˆ. The dynamics

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of cultural traits is described by  Qt+1 = 2Ψ

Qt 2

 .

From proof of Proposition 2, we know that 0 is a steady state of the map Ψ. Also, we can deduce that for Q0 < 2ˆ q or equivalently Q0 /2 < qˆ, the sequence Qt is decreasing. By the continuity of the map Ψ, we deduce that for any Q0 ∈]ˆ q , 2ˆ q [, the sequence Qt converges to Q = 0.

We conclude that the steady state size of group a under segregation q ∗ is higher than the size reached under integration which is zero. Since the long-run level of education is an increasing function of Q, we deduce that segregation maximizes the long-run level of education.

2. Suppose that Q0 ∈]2ˆ q , 1[.

(i) If segregation holds the urban equilibrium is such that q 1 = Q ∈]2ˆ q , 1[ and q 2 = 0. The dynamics of cultural traits is described by

Qt+1 = Ψ(Qt ).

Following point 1.(i) we deduce that the sequence Qt converges to Q = q ∗ .

(ii) If integration holds the urban equilibrium is such that q 1 = q 2 = Q/2 > qˆ. The dynamics of cultural traits is described by  Qt+1 = 2Ψ

Qt 2

 .

From proof of Proposition 2, we know that q ∗ is a steady state of the map Ψ. Following point 1.(i) and 1.(ii) we deduce that the sequence Qt converges to Q = 2q ∗ .

We conclude that the steady state size of group a under integration 2q ∗ is higher than the size reached under segregation q ∗ . Since the long-run level of education is an increasing function of Q, we deduce that segregation maximizes the long-run level of education.

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