Cultural Dynamics, Social Mobility and Urban Segregation Emeline BEZINyand Fabien MOIZEAUz 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 con…gurations. 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 in‡uence of family background on economic fate. Finally, segregation has ambiguous e¤ects for long-run e¢ ciency. Keywords: cultural transmission, peer e¤ects, residential segregation, human capital inequality. JEL Classi…cation: 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, François Salanié 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), Université 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. y Paris School of Economics (PSE), E-mail: [email protected] z CREM (Condorcet Center), Université de Rennes 1 and Institut Universitaire de France, E-mail: [email protected].

1

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. According to US Census Bureau population projections, the share of the foreign-born population in the US will rise from 13.4% in 2015 up to 18.8% in 2060. 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 a- uent neighborhoods of metropolitan areas. Living in ethnic enclaves produces both bene…ts and costs for inhabitants. On the one hand, spatial separation from good schools, better-quality local public goods and job opportunities produces the ethnic di¤erences in socioeconomic outcomes. On the other hand, the spatial concentration of minorities provides a sheltered environment and helps the production of cultural amenities, the preservation of the culture of origin, and the sharing of information on co-ethnic business and job opportunities. The costs and bene…ts of segregation suggest 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 a¤ects 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. We develop a theoretical framework where (i) culture results from interactions within the family and the neighborhood, and (ii) individuals choose their place of residence, so that the cultural composition of the neighborhood is endogenous. Our framework allows us to consider, on the one hand, how segregation in‡uences 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 a¤ect segregation and urban inequality? How can we design public policies to a¤ect

2

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). A number of empirical contributions have shown that urban segregation plays a role in attitudes and values such as ethnic identity, although there is no consensus on the sign of this relationship. Using data from the European Social Survey, Bisin, Patacchini, Verdier and Zenou (2011a) show that strong ethnic identities are not fostered in ethnically-segregated neighborhoods, while Constant, Schüller and Zimmerman (2013), using German data, …nd that immigrants living in counties with a strong concentration of their own ethnic group feel greater identi…cation with their country of origin. Traditional work in sociology on the ‘culture of poverty’has pointed out that the segregation of ethnic minorities in poor neighborhoods create an underclass culture by socially isolating individuals from mainstream norms of behavior (see Wilson, 1987, Anderson, 1999, Lamont and Small, 2008). Empirical work by Borjas (1992, 1995, 1998) has underlined that ethnic capital, de…ned as the whole set of ethnic characteristics including culture, attitudes and economic opportunities, may account for half of the persistence in ethnic skill di¤erentials across generations, and is more in‡uential in disadvantaged neighborhoods. The choice of the social arenas in which children interact (schools, neighborhoods, friendship networks, and places of worship) is also a concern for parents who care about the transmission of desired cultural traits. In particular, Ioannides and Zanella (2008) analyze the determinants of location choices using PSID data. Their …ndings suggest that parents with children search for neighborhoods with attributes that are favorable to human-capital production and the transmission of parents’cultural traits. Ethnic studies also show that parental school choice is motivated by their desire to choose peers who will best transmit their preferred cultural traits (see Tinker and Smart, 2012, for Muslim schools in Britain, and Sikkink and Emerson, 2008, for the e¤ect of school choice on racial segregation in the US). Following on from this empirical evidence, we develop a theoretical model based on the following three blocks. First, the population consists of two di¤erent 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 di¤erent 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 à la Bisin and 3

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 e¤ects in children’s education (see, for instance, Bénabou, 1993, 1996a, b). Local spillovers matter as, whatever their cultural trait, all parents value having educated children. To capture the in‡uence of culture on socioeconomic outcomes, one crucial feature of our model is that the (subjective) bene…ts of education and the gain associated with the transmission of cultural traits are linked. More precisely, for mainstream parents, we consider that the bene…t 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 bene…ts of education. These two cases capture the cultural distance between mainstream and minority groups. Complementarity for both groups re‡ects cultural proximity. While substitutability for the minority group corresponds to cultural polarization. Some work, for example, has emphasized that certain religions attach a particular importance to education, so that there is complementarity between education and culture. Max Weber’s classic (1958) sociological work stressed that the Protestant ethic attaches great importance to the value of hard work. More recently, Botticini and Eckstein (2005, 2007) have provided historical evidence that Judaïsm favors the transmission of education. The immigration literature also documents substantial di¤erences in socioeconomic outcomes between ethnic immigrant groups, suggesting that cultural origin matters for social integration (see for instance, Domingues Dos Santos and Wol¤, 2011, for France, or Gang and Zimmerman, 2000, for Germany, and Card et al., 2000, for the US). Noticeably, Borjas (1995), shows 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. This cultural in‡uence can also be interpreted as an identity phenomenon, which prescribes a particular educational behavior (see Akerlof and Kranton, 2002). The ‘Acting White’ phenomenon, in which ethnic minorities oppose majority values of studiousness and hard work, is one example of cultural polarization (see Fryer and Torelli, 2010, Battu and Zenou, 2010, and Battu et al. 2007 for empirical evidence on oppositional identities). The non-separability between the bene…t of education and the gain associated with culturaltrait transmission means that the incentives parents face to transmit cultural trait and to make 4

their o¤spring educated are intertwined and in‡uence 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, whatever the cultural composition of the society, the city ends-up segregated. Speci…cally, some neighborhoods remain culturally homogenous. When there is cultural proximity, there are multiple types of long-run urban con…gurations, depending on society’s initial cultural composition. When the desires for cultural transmission and education are compatible, minority parents are willing to live with mainstream individuals so that their child bene…ts from larger peer e¤ects in education, provided that their child has a high probability of adopting their culture. When this probability is higher for minority parents, which is true when they form a larger share of the population, minority families bid more for land than do mainstream families in the best-quality neighborhoods. Hence, depending on the initial composition of the population, the city will end-up either segregated or integrated. We show that the spatial separation of cultural groups adds further glue to the intergenerational transmission of cultural traits. In a segregated city, a given child is more likely to be surrounded by role models from the parental cultural group. Segregation makes transmission by role models an additional means of obtaining the family trait, reducing cultural change across generations. Consistent with the …ndings in Borjas (1995) and Chetty et al. (2014), segregation thus strengthens the in‡uence of family background on economic fate. We address the fundamental issue of e¢ ciency, and ask whether segregation is desirable for the long-run level of education in a city. We show that segregation has ambiguous e¤ects on the long-run level of education. On the one hand, segregation may generate strong peer e¤ects by concentrating one population in a certain neighborhood, favoring the transmission of the mainstream cultural trait that values economic success. On the other hand, the spatial concentration of individuals with the same trait lowers incentives for parents to socialize their children as they expect the local environment to transmit the desired trait. This negatively a¤ects the spread of mainstream culture and so the long-run level of education.

5

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 (de…ned as the degree of centrality in a network) a¤ects 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 e¤ects and endogenous socioeconomic segregation explaining how local interactions drive spatial segregation and persistent income inequality (see for instance, Loury, 1977, Bénabou, 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, Bénabou’s works emphasize that incentives to segregate into distinct communities are driven by the desire to enhance human-capital accumulation. Departing from Bénabou’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 con…guration. Moizeau (2015) also studies the in‡uence 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 di¤usion process proposed by Akerlof (1980). We di¤er 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 …rst 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 à la Schelling where individuals have preferences over the social environment. Unlike most theoretical 6

models of neighborhood composition, they …nd that tipping dynamics may lead to multiple long-run equilibria, with integration being a stable outcome. Our cultural explanation of multiple long-run urban con…gurations 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 …ndings in Cutler, Glaeser and Vigdor (2008) that (i) the cultural distance between an immigrant group and the native population signi…cantly a¤ects the degree of segregation, and (ii) the group share in the population also matters for the urban con…guration. 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 e¢ ciency in the urban equilibrium. Last, Section 6 concludes.

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 e¤ort to exert to transmit her cultural trait.

2.2

The parents’transmission decision and location choice

Parents di¤er 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. Parents make two decisions. They choose

7

both the location j where they pay the land rent

j;t

and their socialization e¤ort.

Cultural Transmission. The transmission of preferences follows the lines of the model introduced by Bisin and Verdier (2001). The intergenerational transmission of trait i 2 fa; bg is the result of social interactions which arise at two levels. The child born at date t is …rst exposed to vertical socialization by her/his parents. The probability that the latter directly transmits her/his trait is i

i

. If not socialized within the family (with probability 1

), 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 Sjögren (2008) and Saez-Marti and Zenou (2012), the shape of f (:) captures the degree of conformism, that is how much does the child …nds 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. The transition probabilities are given by

aa Pj;t =

a

+ (1

a

bb Pj;t =

b

+ (1

b

ab )f (qj ) and Pj;t = (1

)(1

a

)(1

ba f (qj )) and Pj;t = (1

b

f (qj ))

(1)

)f (qj ):

(2)

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

at time t. Preferences. Direct transmission is the result of a choice. Parents exert the e¤ort

i

2 [0; 1] in

0

order to transmit their trait. Let us denote by V ejii , the gain from the type i parent’s point of view 0

that a child gets educational level e and acquires trait i0 . V ejii denotes this gain when the child gets educational level e which is lower than its counterpart e. We will thus say that a child who gets e is 0

a child who gets no education. For the sake of simplicity, V ejii is exogenous. We make the following assumption.

8

Assumption 1 r (i) Cultural intolerance: V ejii > V ejii (ii) Educational gain:

V ii

0

V ejii

0

0

8e 2 fe; eg; 8i; i0 2 fa; bg with i0 6= i; 0

V ejii > 0 8i; i0 2 fa; bg;

(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

V ii di¤ers across cultural traits, there is a widespread view that education

by the magnitude of

pays. Item (iii) considers complementarity between education and cultural trait a meaning that, for trait-a parents, educational gain is magni…ed when the child acquires trait a. We will say that the mainstream culture values education. We make no assumption regarding trait b and we will examine alternatively two cases, that is complementarity ( V bb > and education ( V bb <

V ba ) and substitutability between trait

V ba ). Both cases capture cultural distance: cultural proximity under

complementarity and cultural polarization under substitutability. This non-separability assumption is a key feature of our framework and implies that incentives to exert socialization e¤ort and incentives to make the o¤spring get education are intertwined. We do not formalize the choice of education made by the young generation. Whatever the cultural trait of the child, we consider that, due to some di¤ering cognitive and non-cognitive abilities, a eji

eji

fraction Pj;t (resp. Pj;t = 1

eji

Pj;t ) of type-i children living in neighborhood j gets education (resp.

no education). eji

Assumption 2 Suppose that Pj;t is a function of qj;t and for any qj;t and j = 1; 2 eja

ejb

(i) Pj;t > Pj;t ; eja

ejb

dPj;t (ii) dqj;t

dPj;t > 0: dqj;t

Children do not face the same educational opportunities. 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 9

the minority population may have to cope with a foreign language also contributes to this gap (see Dustmann, Frattini and Lanzara, 2012). Also we assume peer e¤ects 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 . We denote by Uti (

j;t ;

i

); the utility at date t of a parent with trait i and income w, who lives in

neighborhood j and exerts socialization e¤ort

Uta (

where

j;t ;

a

)=w

eja

j;t

i

: We have for a trait-a parent ejb

eja

ejb

aa ab (Pj;t V ejaa + Pj;t V ejaa ) + Pj;t (Pj;t V ejab + Pj;t V ejab ) + Pj;t

( a );

(3)

( b)

(4)

(:) is the socialization cost which is increasing and convex with respect to .2

A parent with trait b and income w who lives in neighborhood j 2 f1; 2g has utility Utb (

j;t ;

b

)=w

ejb

j;t

ejb

eja

eja

bb ba + Pj;t (Pj;t V ejbb + Pj;t V ejbb ) + Pj;t (Pj;t V ejba + Pj;t V ejba )

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 e¤ort and location, parents face a trade-o¤ 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. The socialization choice of parents. Let us …rst consider trait a parents. At date t, given her place of residence j; a type a parent chooses her optimal e¤ort

max Uta ( a

j;t ;

a

a

that solves

) subject to (1).

1 There is an extensive literature on the impact of neighborhood e¤ects 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 e¤ects on educational outcomes. Some experimental or quasi-experimental work …nds little evidence of neighbourhood e¤ects 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 insigni…cant treatment e¤ects on economic outcomes. A growing literature in sociology emphasises the duration of exposure to neighbourhoods, which helps explain why experimental work …nds little evidence of neighbourhood e¤ects (see Sharkey and Elwert, 2011, Wodtke et al. 2011). Chetty, Hendren and Katz (2015) consider the Moving To Opportunity Experiment and show that treatment e¤ects 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 e¤ect, we assume a linear utility function of private consumption w j;t .

10

For the sake of presentation, we omit the time index t and the neighborhood index j when not necessary. Given that P eji = 1

P eji ; and P ii = 1

0

P ii for any i; i0 2 fa; bg, i 6= i0 ; we get the

following …rst-order condition:

(1

f (q)) P eja V aa

P ejb V ab + V ejaa

V ejab =

0

(

a

(5)

):

We assume that for any q

P eja V aa

P ejb V ab + V ejaa

V ejab

(6)

0

implying that there is always an incentive to exert a socialization e¤ort, i.e.

a

0.3

(q)

From (5), applying the implicit function theorem leads to: d a = dq

f 0 (q)(P eja V aa

P ejb V ab + V ejaa

V ejab ) + (1 00 (

eja

f (q)) dPdq

V aa

V ab

a)

:

(7)

The above expression shows that q has two e¤ects on the socialization e¤ort. First, when a parent is surrounded by more people with the same cultural trait, she bene…ts from a more e¤ective oblique transmission and is incited to reduce her socialization e¤ort. This negative e¤ect, captured by the …rst parenthesis in the numerator of (7), is called the cultural substitution property (see Bisin and Verdier, 2001). Our model involves a second e¤ect: when the fraction of type a agents rises, peer e¤ects in education are stronger and the child is more likely to educate. As cultural trait a and education are complements, this positively a¤ects the incentives to socialize children. This second e¤ect may counteract the cultural substitution e¤ect. Hence, the impact of q on

a

is ambiguous.

Further, when the neighborhood population is only comprised of trait a individuals, the cultural substitution e¤ect prevails as

a

(1) = 0.

Following the same reasoning, we obtain the socialization e¤ort of trait b parents (see Appendix 7.1). The socialization choice of minority parents depends on a cultural substitution e¤ect. 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 e¤ects. An increase in peer e¤ects (captured by a rise of q) has a positive (resp. negative), impact on socialization e¤ort if cultural trait b and education are complements (resp. 3

We could relax this assumption and have values of q such to our discussion.

11

a

(q) = 0: This would not add any interesting insights

b

substitutes). Hence, the overall impact of q on

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

b

(0) = 0.

Vertical and oblique transmissions drive the dynamics of the mainstream culture in neighborhood j between t and t + 1

qt+1

b

qt = (1

(qt ))f (qt )

a

qt (1

(qt )) + qt f (qt )(

b

a

(qt )

(8)

(qt )):

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

Parents’Location Choice.

nomic composition of the neighborhood is another mean used by parents to in‡uence 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 ( j

i

j;t ;

Without loss of generality, we impose that q1;t

(qj;t )):

q2;t = Qt

q1;t and

2;t

= 0. Following the

literature, the urban equilibrium is de…ned as follows: De…nition 1 At any date t, given Qt ; the urban con…guration characterized by a

(Qt

b

q1;t );

(q1;t );

b

t;

q1;t ;

a

(q1;t );

q1;t ) is an equilibrium if no one wants to move and change their

(Qt

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. The willingness to pay to live in urban area 1, denoted by

i

for i 2 fa; bg, is such

that a trait-i parent is indi¤erent between both neighborhoods. For a mainstream individual, given eja

that Pjab = 1 a

Pjaa and Pj eja

= P1aa (P1 ejb

+(P1

V aa ejb

=1 ejb

P1

P2 ) V ab

eja

Pj , we have

V ab + V ejaa ( (

a

(q1 ))

V ejab ) (

12

a

eja

P2aa (P2

(q2 ))):

V aa

ejb

P2

V ab + V ejaa

V ejab ) (9)

For a minority individual, using Pjba = 1 b

ejb

= P1bb (P1

eja

+(P1

V bb

eja

P1

eja

P2 ) V ba

V ba + V ejbb ( (

b

(q1 ))

ejb

Pjbb and Pj = 1 ejb

V ejba ) (

P2bb (P2

b

ejb

Pj , we obtain V bb

eja

P2

V ba + V ejbb

V ejba ) (10)

(q2 ))):

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 e¤ects in education, thus generating an incentive to live in the neighborhood where the mainstream cultural trait prevails. How this motive a¤ects the incentives to segregate depends on cultural distance.

3

Segregated urban equilibria

We …rst 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 . It prevents from playing with the magnitude of

educational gains to obtain segregation in a trivial way and enables to focus on the consequences of cultural transmission and peer e¤ects in terms of segregation. 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 con‡icting objectives implying that when they live in neighborhood 1 (searching for peer e¤ects), 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 e¤ort. This reduces the gain to live

13

in neighborhood 1 for parents b compared to parents a. Hence, when there is cultural polarization, mainstream parents are willing to bid more than minority ones and the city is segregated4 . eja

ejb

Remark that in the absence of peer e¤ects, 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 di¤ering from the ones studied in the literature on neighborhood e¤ects (see Bénabou, 1993, 1996a,b or de Bartolome, 1991). 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 di¤erence 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 in‡uence 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 ] Q

2

Q

(1

q1 )P1ba + (1

q2 )P2ba :

(11)

with transition probabilities depending on equilibrium socialization e¤orts. In order to have a tractable result of the impact of segregation on cultural mobility, we work with the following speci…c forms P eji =

i

q, with

a

>

b

, f (q) = q and

( ) = (1=2 )

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

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.

14

process induced by the socialization e¤ort of parents. What is more is that segregation adds further glue in the cultural transmission process. The intuition goes as follows. Segregation (as compared to integration) has two distinct e¤ects on the probability di¤erential, P aa

P ba . First, segregation

increases oblique transmission of trait a for both types of child. However, this e¤ect is stronger for children from mainstream family background. Hence, the oblique transmission e¤ect favours the rise of the probability di¤erential P aa P ba . Second, a rise in q a¤ects socialization e¤orts. More precisely, due to cultural substitution, an increase in q decreases the e¤ort of parents a which negatively a¤ects the probability di¤erential. When the cost of socialization is high, the cultural substitution e¤ect is low. Hence, the oblique transmission e¤ect prevails and the rise in segregation positively impacts the probability di¤erential. Given Assumption 2, cultural mobility is positively related to social mobility. Corollary 1 is supported by empirical evidence. Borjas (1995) …nds 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 in‡uence. 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) (ii)

a

(0) < 1

b

(1) > 1

f 0 (0); f 0 (1);

(iii) 9 q 2]0; 1[ such that f (q)

q and

a

(q)

b

(q):

Assumption 3 clari…es the strength of the di¤erent 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 e¤ects in education, and (iii) cultural substitution where the two last forces are embodied in the socialization e¤orts 15

a

and

b

.

High conformism and peer e¤ects favor homogeneity within the population. Reversely, cultural 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 e¤ects and conformism are then su¢ ciently high to overcome the cultural substitution e¤ect (meaning that the socialization e¤ort of mainstream agents is below some threshold). Item (ii) focuses on large fractions of mainstream agents (i.e. q = 1). Conformism and peer e¤ects are then not su¢ ciently high to overcome cultural substitution (implying that the socialization e¤ort of minority agents is above some threshold). Item (iii) imposes that for some intermediate value of the population composition, conformism and peer e¤ects are high. Although empirical evidence on socialization rates and cultural transmission is still sparse, we can …nd some support for Assumption 3 in Bisin, Topa and Verdier (2004). From a structural estimation of a cultural transmission model, they …nd that socialization e¤orts 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 …nd 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 di¤erent 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 e¤ect 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 de…ne inequalities by the educational gap between 16

a mainstream individual and its counterpart, i.e. P eja eja

P

eja

P

ejb

eja

q 1 P 1 + q 2 P1 = Q

P ejb . It is expressed as follows ejb

(1

q1 )P1 + (1 (2 Q)

ejb

q2 )P2

:

(12)

At the long-run equilibrium Q = q < 1, the educational gap equals eja P1 (q

1 2

)

q ejb P (q ) q 1

ejb

P2 (0) : 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 eja

ejb

P1 (1) ejb

(14)

P1 (0):

ejb

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 2

q ejb (P1 (0) q

ejb

eja

P1 (q )) < P1 (1)

eja

P1 (q );

which is true given the existence of peer e¤ects. 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 speci…cation 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 e¤ect of cultural substitution i.e. b (1) < 1 f 0 (1).

17

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, bene…ts from education are high for minority parents as having an educated child does not necessarily implies a cultural loss. Since the bene…t of education increases when the child acquires the parental trait, parents who bene…t more from higher peer e¤ects 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 …ndings of Cutler, Glaeser and Vigdor (2008) that cultural distance (as measured either by linguistic di¤erences, 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 6

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 e¤ects. 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 bene…t from similar peer e¤ects 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).

18

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 con…gurations 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 e¤ort 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 e¤ects, 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. Our result is consistent with empirical …ndings 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 7

We proceed in the proof by showing that, for some functional forms of transition probabilities, oblique transmission technology, socialization cost and peer e¤ects, 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.

19

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 segregation throughout ethnic groups, space and time. They identify group share as a signi…cant determinant of the degree of segregation and …nd 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 bene…t 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 bene…t 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 bene…t 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 e¢ cacy 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. 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. 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 de…nition and speci…cation).

20

5

Is social segregation optimal?

We now turn to the issue whether it is e¢ cient to let people sort themselves into urban areas, or should we implement particular urban policies ? In particular, we consider urban policies which a¤ect 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 e¢ ciency criterion, i.e. the long-run rate of education. Proposition 5 Suppose that Assumption 3 holds and 2^ q < 1. For any Q0 2 [^ q ; 2^ q ], segregation is e¢ cient. For any Q0 2 [2^ q ; 1], integration is e¢ cient. Proposition 5 stresses that the e¢ ciency of the urban policy depends on the cultural composition 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 2 [^ q ; 2^ q ], segregation, which concentrates type a individuals in neighborhood 1, allows for su¢ cient peer e¤ects 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 e¤ects 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 a¤ects 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 e¤ects in this neighborhood. When the city fraction of the mainstream group is su¢ ciently high, i.e. Q0 2 [2^ q ; 1], this provides higher incentives to socialize children to trait a. This bene…ts 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 pro…table 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. 10

One example of a quota policy is the SRU law (loi relative à la Solidarité 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 …gures below this ratio have to pay …nes (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

Whereas Bénabou (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 e¢ ciency of a segregated equilibrium, We emphasize the importance of the population cultural composition. Our result that the e¢ ciency 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 di¢ culty 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 bene…ts 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 bene…ts will be smaller 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 e¤orts should halt, because merely transferring the poor from high- to moderate-poverty neighborhoods is likely to be socially ine¢ cient.”(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 eja

Qt 2

Qt 2

P ejb

:

When the city is segregated, the gap equals

P eja

P ejb =

8 > < > :

eja

P1 (Qt ) eja

P1 (1) Qt

+

Qt 1 eja P2 Qt

ejb

ejb

(1 Qt )P1 (Qt )+P2 (0) (2 Qt )

(Qt

1)

ejb

P2 (Qt

if Qt 1);

1 otherwise:

It is easy to check that under Assumption 2, educational gap is lower under integration than under segregation. Segregation increases the cultural disparity between neighborhoods which magni…es peer e¤ects di¤erences between both traits. It turns out that, for some cultural composition of the 22

population, there is a trade-o¤ between e¢ ciency 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 con…gurations arise depending on initial cultural composition of the society. In 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 in‡uence on the child’s destiny. Finally, we highlight the crucial role of the population composition to assess the e¢ ciency 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 ‡exible enough to introduce some housing market features such as tenure choice, housing depreciation and maintenance, development and redevelopment of housing stock. These features would in‡uence 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 di¤erentiate the social arenas where peer e¤ects and oblique transmission are determined. For instance, considering that peer e¤ects are circumscribed within schools whereas oblique transmission is produced in the urban area would a¤ect the trade-o¤ 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, 23

2015). However, this literature disregards endogenous strati…cation. Fruitful research would be to investigate how culture interacts with both residential segregation and institutions providing local public goods and education services.

References [1] Akerlof, G., 1980, “A Theory of Social Custom of Which Unemployment May Be One Consequence”, Quarterly Journal of Economics, vol. 94, 749-775. [2] Akerlof, G. and R. Kranton, 2002, “Identity and Schooling: Some Lessons for the Economics of Education”, Journal of Economic Literature, vol. 15, 1167-1201. [3] Anderson, E., 1999, The Code of The Street: Decency, Violence, and the Moral Life of the Inner-City, New York: W.W. Norton & Company. [4] Battu, H. and Y. Zenou, 2010, “Oppositional Identities and Employment for Ethnic Minorities. Evidence from England,”Economic Journal, vol.120, 52-71. [5] Battu, H., McDonald, M. and Y. Zenou, 2007, “Oppositional Identities and the Labor Market,” Journal of Population Economics, vol. 20, 643-667. [6] Bénabou, R., 1993, “Workings of a City: Location, Education, and Production”, Quarterly Journal of Economics, vol. 108, 619-652. [7] Bénabou, R., 1996a, “Heterogeneity, Strati…cation and Growth: Macroeconomic Implications of Community Structure and School Finance”, American Economic Review, vol. 86, 584-609. [8] Bénabou, R., 1996b, “Equity and E¢ ciency in Human Capital Investment: the Local Connection”, Review of Economic Studies, vol. 63, 237-264. [9] Bisin, A. and Th. Verdier, 2001, “The Economics of Cultural Transmission and the Evolution of Preferences”, Journal of Economic Theory, vol. 97, 298-319. [10] Bisin, A. and Th. Verdier, 2015, “On the Dynamics of Culture and Institutions”, mimeo. [11] Bisin, A., Topa, G. and Th. Verdier, 2004, “Religious Intermarriage and Socialization in the United States”, Journal of Political Economy, vol. 112, 615-664.

24

[12] Bisin, A., Patacchini, E., Verdier, Th. and Y. Zenou, 2011a, “Ethnic Identity and Labor-Market Outcomes of Immigrants in Europe”, Economic Policy, vol. 65, 57-92. [13] Bisin, A., Patacchini, E., Verdier, Th. and Y. Zenou, 2011b, “Formation and Persistence of Oppositional Identities”, European Economic Review, vol. 55, 1046-1071. [14] Borjas, G., 1992, “Ethnic Capital and Intergenerational Mobility”, Quarterly Journal of Economics, vol. 107, 123-150. [15] Borjas, G., 1995, “Ethnicity, Neighborhoods, and Human-Capital Externalities”, American Economic Review, vol. 85, 365-390. [16] Borjas, G., 1998, “To Ghetto or Not to Ghetto: Ethnicity and Residential Segregation”, Journal of Urban Economics, vol. 44, 228-253. [17] Botticini, M. and Z. Eckstein, 2005, “Jewish Occupational Selection: Education, Restrictions, or Minorities?”, Journal of Economic History, vol. 65, 922-948. [18] Botticini, M. and Z. Eckstein, 2007, “From Farmers to Merchants, Conversions and Diaspora: Human Capital and Jewish History”, Journal of European Economic Association, vol. 5, 885926. [19] Brueckner, J., 2011, Lectures on Urban Economics, The MIT Press, Cambridge, Massachusetts. [20] Card, D., DiNardo, J. and E. Estes, ,2000, “The More Things Change. Immigrants and the Children of Immigrants in the 1940s, the 1970s, and the 1990s”, in Borjas G. (Eds), Issues in the Economics of Immigration, 227-270, University of Chicago Press, Chicago. [21] Card, D., Mas, A. and J. Rothstein, 2007, “Tipping and the Dynamics of Segregation”, Quarterly Journal of Economics, vol. 123, 177-218. [22] Card, D., Mas, A. and J. Rothstein, 2008, “Are Mixed Neighborhoods Always Unstable? TwoSided and One-Sided Tipping”, NBER Working Paper 14470. [23] Carvalho, J-P., 2013, “Veiling”, Quarterly Journal of Economics, vol. 128, 337-370. [24] Chetty, R., Hendren, N. and L. Katz, 2015, “The E¤ects of Exposure to Better Neighborhoods on Children: New Evidence from the Moving to Opportunity Experiment”, mimeo.

25

[25] Chetty, R., Hendren, N., Kline, P. and E. Saez, 2014, “Where is the Land of Opportunity? The Geography of Intergenerational Mobility in the United States”, Quarterly Journal of Economics, vol. 129, 1553-1623. [26] Constant, A., S., Schüller, and K. Zimmermann, 2013, “Ethnic Spatial Dispersion and Immigrant Identity”, IZA DP n 7868. [27] Cutler, D., Glaeser, E. and J. Vigdor, 1999, “The Rise and Decline of the American Ghetto”, to. Journal of Political Economy, vol. 107, 455-506. [28] Cutler, D., Glaeser, E. and J. Vigdor, 2008, “Is the Melting Pot Still Hot? Explaining the Resurgence of Immigrant Segregation”, Review of Economics and Statistics, vol. 90, 478-497. [29] de Bartolome, Ch., 1990, “Equilibrium and Ine¢ ciency in a Community Model with Peer Effects”, Journal of Political Economy, vol. 98, 110-133. [30] Doepke, M. and F. Zilibotti, 2008, “Occupational Choice and the Spirit of Capitalism”, Quarterly Journal of Economics, vol. 123, 747-793. [31] Domingues Dos Santos, M. and F.C., Wol¤, 2011, “Human Capital Background and the Educational Attainment of Second-Generation Immigrants in France”, Economics of Education Review, vol. 30, 1085-1096. [32] Durlauf, S., 1996, “A Theory of Persistent Income Inequality”, Journal of Economic Growth, vol. 1, 75-93. [33] Durlauf, S., 2004, “Neighborhood E¤ects”, in J. V. Henderson and J. F. Thisse (eds), Handbook of Regional and Urban Economics, Amsterdam: Elsevier Publisher. [34] Dustmann, Ch., Frattini, T. and G. Lanzara, 2012, “Educational Attainment of SecondGeneration Immigrants: An International Comparison”, Economic Policy, vol. 27, 143-185. [35] Fouka, V., 2016, “Backlash: The Unintended E¤ects of Language Prohibition in US Schools after World War I”, mimeo. [36] Fryer, R. and P. Torelli, 2010, “An Empirical Analysis of Acting White”, Journal of Public Economics, vol. 94, 380-396.

26

[37] Galster, G., 2002, “An Economic E¢ ciency Analysis of Deconcentrating Poverty Populations”, Journal of Housing Economics, vol. 11, 303-329. [38] Gang, I. N. and K. F. Zimmermann, 2000, “Is Child Like Parent? Educational Attainment and Ethnic Origin”, Journal of Human Resources, vol. 35, 550-569. [39] Gobillon, L. and B. Vignolles, 2016, “Evaluation de l’e¤et d’une politique spatialisée d’accès au logement : la loi SRU”, La Revue Economique, vol. 67, 615-638. [40] Guiso, L., Sapienza, P. and L. Zingales, 2008, “Social Capital as Good Culture”, Journal of the European Economic Association, vol. 6, 295-320. [41] Ioannides, Y. and G. Zanella, 2008, “Searching for the Best Neighborhood: Mobility and Social Interactions”, mimeo. [42] Kling, J. R., Liebman, J. B. and L. F. Katz, 2007, “Experimental Analysis of Neighborhood E¤ets”, Econometrica, vol. 75, 83-119. [43] Lamont, M. and M. L. Small, 2008, “How Culture Matters for the Understanding of Poverty: Enriching our Understanding”, in L. A. Chih and D. R. Harris (eds), The Colours of Poverty: Why Racial and Ethnic Disparities Exist, New York: Russell Sage Foundation. [44] 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. [45] Loury, G., 1977, “A Dynamic Theory of Racial Income Di¤erences”, in A. Le Mond (eds), Women, Minorities and Employment Discrimination, Lexington, MA: Lexington Books. [46] Moizeau, F., 2015, “Dynamics of Social Norms in the City”, Regional Science and Urban Economics, vol. 51, 70-87. [47] Oreopoulos, P., 2003, “The Long-Run Consequences of Living in a Poor Neighborhood”, Quarterly Journal of Economics, vol. 118, 1533-1575. [48] 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.

27

[49] Saez-Marti, M., and A. Sjögren, 2008, “Peers and Culture”, Journal of Economic Theory, vol. 143, 153-176. [50] Saez-Marti, M. and Y. Zenou, 2012, “Cultural Transmission and Discrimination ”, Journal of Urban Economics, vol. 72, 137-146. [51] Sharkey, P., 2008, “The Intergeneration Transmission of Context”, American Journal of Sociology, vol. 113, 931-969. [52] Sharkey, P. and F. Elwert, 2011, “The Legacy of Disadvantage: Multigenerational Neighborhood E¤ects on Cognitive Ability”, American Journal of Sociology, vol. 116, 1934-1981. [53] 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. [54] 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. [55] Topa, G. and Y. Zenou, 2014, “Neighborhood versus Network E¤ects”, in G. Duranton, V. Henderson and W. Strange (eds), Handbook of Regional and Urban Economics, Amsterdam: Elsevier Publisher. [56] Verdier, Th. and Y. Zenou, 2017, “The Role of Social Networks in Cultural Assimilation”, Journal of Urban Economics, vol. 97, 15-39. [57] Weber, Max, 1958, The Protestant Ethic and the Spirit of Capitalism, New York: Charles Scribner’s Sons. [58] Wilson, W.J., 1987, The Truly Disadvantaged: The Inner City, the Urban Underclass and Public Policy, Chicago: University of Chicago Press. [59] Wodtke, G.T., Harding, D.J. and F. Elwert, 2011, “Neighborhood E¤ects in Temporal Perspective: The Impact of Long-Term Exposure to Concentrated Disadvantage on High School Graduation”, American Sociological Review, vol. 76, 713-736.

28

7

Appendix

7.1

Socialization Choice of Trait b Parents

At date t, a trait-b parent chooses her optimal e¤ort

max U b ( ;

b

b

b

that solves

) subject to (2).

This leads to the following …rst-order condition:

f (q)(P ejb V bb

P eja V ba + V ejbb

V ejba ) =

P eja V ba + V ejbb

V ejba

0

(

b

(15)

)

We assume that for any q

P ejb V bb

(16)

0:

b

which amounts to say that trait-b parents have an incentive to transmit their own trait, i.e.

(q)

0.

By the implicit function theorem, we get f 0 (q) P ejb V bb d b = dq

7.2

P eja V ba + V ejbb 00 (

eja

V ejba + f (q) dPdq

V bb

V ba

b)

:

(17)

Bid-Rent Slopes

Using the envelope theorem, the slope of the trait-a bid curve is given as d a = (1 dq1

a

(1

eja

(q1 ))f 0 (q1 )(P1 a

(q2 ))f 0 (q2 )

V aa

ejb

P1

dq2 eja P2 V aa dq1

V ab + V ejaa ejb

P2

V ejab )

V ab + V ejaa

V ejab

eja

+

P1aa

V

aa

+ (1

P1aa )

V

ab

dP1 e dq1 eja

P2aa V aa + (1

P2aa ) V ab

29

dP2 dq2 : dq2 dq1

(18)

For type b individuals, we have d b = dq1

ejb

(1

b

(q1 ))f 0 (q1 )(P1

+(1

b

(q2 ))f 0 (q2 )

eja

V bb

V ba + V ejbb

P1

dq2 ejb (P V bb dq1 2

eja

V ejba )

V ba + V ejbb

P2

V ejba )

ejb

+ P1bb V bb + 1

P1bb

dP1 dq1

V ba

ejb

P2bb V bb + (1

7.3

dP2 dq2 : dq2 dq1

P2bb ) V ba

(19)

Proof of Proposition 1

Given (18) and (19), let us write the bid-rent slope di¤erential for any given q1 d a dq1

d b = (1 dq1 +(1 (1 (1

a

eja

b

ejb

V aa

(q1 ))f 0 (q1 )(P1

ejb

P1

V ejab )

V ba + V ejbb

V ejba )

dq2 eja ejb P2 V aa P2 V ab + V ejaa V ejab dq1 dq 2 ejb eja b (q2 ))f 0 (q2 ) (P V bb P2 V ba + V ejbb V ejba ) dq1 2 a

(q2 ))f 0 (q2 )

eja

+P1aa

eja

V bb

(q1 ))f 0 (q1 )(P1

V ab + V ejaa

P1

dP1 dq1

ejb

V aa + (1

ejb bb dP1 V bb 1 P1 dq1 eja dP dq2 P2aa 2 V aa dq2 dq1 ejb dP dq2 V bb + +P2bb 2 dq2 dq1

P1aa )

dP1 dq1

V ab

eja

P1bb

dP1 dq1

(1

P2aa )

V ba ejb

dP2 dq2 V ab dq2 dq1 eja

(1

P2bb )

dP2 dq2 V ba : dq2 dq1

Due to the oblique transmission e¤ect, the …rst four terms of the above expression are positive. Hence, a su¢ cient condition for the bid-rent slope di¤erential to be positive is that the following inequality is satis…ed eja aa dP1 P1 V aa + (1 dq1 eja aa dP2 dq2 P2 V aa dq2 dq1 ejb dP P1bb 1 V bb 1 dq1 ejb bb dP2 dq2 +P2 V bb + dq2 dq1

ejb

P1aa )

dP1 dq1

V ab

ejb aa dP2 dq2 (1 P2 ) dq2 dq1 eja dP1 P1bb V ba dq1 eja bb dP2 dq2 (1 P2 ) dq1 dq1

30

V ab

V ba > 0:

If

V aa >

V ab ;

V ba >

V bb ;

V aa =

V ba and

V ab =

V bb ; the LHS of the above

inequality is equivalent to ejb

eja

P1aa + P1bb

dP1 dq1

1

V ba

dP1 dq1

V bb

eja

P2aa + P2bb

ejb

dP2 dq2 V ba dq2 dq1

1

!

dP2 dq2 V bb dq2 dq1

!

> 0:

Given (1) and (2) and dq2 =dq1 =

1, we have ejb

eja

a 1 (1

q 1 ) + q1

dP1 dq1

b 1

V ba

dP1 dq1

eja

+

a 2 (1

q 2 ) + q2

dP2 dq2

b 1

V bb

!

ejb

V ba

dP2 dq2

V bb

!

> 0:

Given Assumption 2 and if

V ba >

V bb , we have any j = 1; 2 eja

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 q1 = 1 and q2 = Q

1;

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 con…gurations, trait-a individuals remain the highest bidders for neighborhood 1 after the small perturbation of the equilibrium.

31

7.4

Proof of Corollary 1

We consider the following functional forms: P eji =

i

a

q, with

b

>

, f (q) = q and

( ) = (1=2 )

2

.

From (5) and (15), we get

a

=

q2(

a

V aa

b

V ab ) + q((

a

V aa

b

V ab )

(V ejaa

V ejab )) + (V ejaa

V ejab );

and b

b

= q2(

V bb

a

V ba ) + q(V ejbb

V ejba ):

Let us consider the probability di¤erential

P aa

P ba =

q1 Q q1 a ( a (q1 ) + (1 (q1 ))f (q1 )) + ( Q Q 1 + q1 Q 1 q1 b (1 (q1 ))f (q1 ) (1 2 Q 2 Q

a

(Q b

a

q1 ) + (1

(Q

(Q

q1 ))f (Q

q1 ))

q1 ) :

q1 ))f (Q

Let us compare this di¤erential 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 di¤erential of probabilities at the segregated equilibrium writes as

P aa

P ba = (

a

a

(Q) + (1

1 2

(Q))Q)

Q (1 Q

b

(20)

(Q))Q:

Under integration, we have

P aa

P ba =

a

Q 2

a

+ 1

Q 2

Q 2

Q 2

b

1

Q : 2

(21)

We have a higher di¤erential under segregation if (20) is greater than (21) that is

( >

Since the function

1 2 b

a

a

(Q) + (1 Q (1 Q

b

a

(Q))Q)

(Q))Q

b

1

Q 2 Q 2 b

is increasing in q; we have (1 Q Q Q

32

Q 2

0;

a

Q 2

Q 2

Q ; 2

(Q)) < 1

if 1 2

+ 1

b

Q 2

. The RHS is negative

which is equivalent to 2Q

Q;

which is true. Therefore, if the LHS is positive then (20) is greater than (21). The LHS is equal to zero at Q = 0. If P aa =

a

a

(Q) + (1

(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 = dq

a

2 (

+ q2(

V aa

a

b

V aa

b

V ab )(1

a

q) + (

V ab )

(V ejaa

V aa

b

V ab

(V ejaa

V ejab ))(1

2q)

V ejab ) + 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

a

(

V aa

b

V ab )

2 (V ejaa

V ejab ) + 1

a

b

0

leading to 2(V

ejaa

V

ejab )

1 +(

V aa

e1 :

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

(qt ) = (f (qt )

b

qt )(1

Steady states are such that

(qt )) + ( a (qt )

q = 0 and q = 1 are steady states of the map

0

0

(qt ) =(f (qt )

b

f (qt )) + qt :

(0)

0 = 0,

(1)

1 = 0 so that

.

, we obtain

b

1)(1

+ ( a (qt )

0

(qt ))qt (1

qt = 0. First, we have

(qt )

Let us perform the derivative

b

: [0; 1] ! [0; 1] de…ned such that

(qt ))

(qt ))(1

(f (qt )

qt )

( a (qt )

f (qt ))

33

d a dqt

d b + dqt b

d b dqt 0

qt (1

(qt ))qt f (qt ) + 1:

f (qt ))

We deduce, 0

0

(0) =

a

(0) + f (0) > 0;

(0) =

b

(1) + f (1) > 0:

and 0

With item (i) and (ii) of Assumption 3 we have that there exists some q 2]0; 1[ such that q^, q , q^ < q , such that (i)

(^ q ) = q^,

0

0

0

(0) < 1,

(1) > 1. Also, with item (iii), we known

(q) > q. As we have 0

(q ) = q ,

0

(^ q ) > 1,

2 C 2 , we deduce that there exist

(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 )=

(qt1 ) + (qt2 ):

Fixed points of the map Q are such that

(qt1 ) + (qt2 ) = qt1 + qt2 :

When the city is segregated, steady states are Q = 0, Q = q^, Q = q , Q = 1, Q = 1 + q^, Q = 1 + q , Q = 2 with urban urban equilibria respectively given by (q 1 ; q 2 ) = (0; 0), (q 1 ; q 2 ) = (^ q ; 0), (q 1 ; q 2 ) = (q ; 0), (q 1 ; q 2 ) = (1; 0), (q 1 ; q 2 ) = (1; q^), (q 1 ; q 2 ) = (1; q ), (q 1 ; q 2 ) = (1; 1).

A necessary and su¢ cient condition for stability of any steady state Q when the segregated urban equilibrium is (q 1 ; q 2 ) is11 . d(Qt+1 Qt ) jQ < 0; dQt , Since item (i) of Assumption 3 implies

0

0

(q 1 ) +

0

(q 2 ) < 2:

(0) < 1 we deduce that the equilibrium (0; 0) is stable.

Furthermore, from the previous part of this proof, we have

0

(q ) < 1 so that

0

(0) +

0

(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 su¢ cient condition for stability because one can easily check that

34

dQt+1 dQt

> 0.

0

(1) > 1, we deduce that

0

0

(1) +

0

(1) > 2 and

0

(1) +

(^ q ) > 2 (since

0

(^ q ) > 1). Equilibria

(1; 1) and (1; q^) are unstable. For equilibria (^ 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 d a dq1

d b dq1

q1 =Q=2

< 0: q1 =Q=2 ejj

At the integrated equilibrium, as q1 = q2 = Q=2, we have P1aa = P2aa ; P1bb = P2bb ; dP1 =dq1 = ejj

dP2 =dq1

f 0 (q1 ) = f 0 (q2 )(dq2 =dq1 ). Hence, the bid-slope rent

(dq1 =dq2 ) for j = a; b and

di¤erential becomes d a dq1

q1 =Q=2

d b dq1

Q 2

a

= 2 1 q1 =Q=2

+2 1

b

eja aa dP1 +2P1 dq1 ejb dP 2P1bb 1 dq1

Given the symmetry assumption,

V aa =

f0

Q 2

Q 2 Q 2

f0

eja

V aa

(P1

ejb

(P1

V bb

ejb

P1

eja

P1

V ab + V ejaa V ba + V ejbb

V ejab ) V ejba )

ejb

V

aa

+ 2 (1

P1aa )

dP1 dq1

V ab

eja

V bb

2 1

V bb and

35

P1bb V ab =

dP1 dq1

V ba :

V ba and Assumption 2 the bid-rent

slope di¤erential can be written as follows: d a dq1

d b dq1

q1 =Q=2

Q 2

a

= 2 1 q1 =Q=2

+2 1

Q 2

b

Q ; r; 2

f0 f0

eja

(P1

V aa

Q ; r; 2

(P1

V bb

V ba :

ejb

V bb

ejb

P1

eja

P1

V ab + V ejaa V ba + V ejbb

V ejab ) V ejba )

eja

+2

P1aa

P1bb

dP1 dq1

Considering that the …rst two terms being positive due to cultural transmission motives, a necessary condition for the bid-rent slope di¤erential to be negative is

P1aa If P1aa

P1bb

V bb

P1bb > 0; it is equivalent to V bb

which is impossible as must have P1aa

7.7

V ba < 0:

V bb

V ba < 0

V ba < 0 is a su¢ cient condition for segregation to emerge. We then

P1bb < 0 and

V bb

V ba > 0.

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 speci…cations P eja = q + a, P ejb = q + ba with a > b, ; a; b > 0 and ( )=

2

=2, and f (qt ) = rqt =(rqt + (1

qt ) ) with ; r;

+ a < 1. Let

> 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

is then necessary to keep some degree of liberty. It is not neces-

sary, 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

a

b

1 (qt ) = (1

1 (qt ) = f (qt )

f (qt ))

qt ( V aa

qt ( V aa

V ejab ) + a V aa

V ab ) + b V aa

36

b V ab + V ejaa

a V ab + V ejbb

V ejba :

V ejab ;

Steady states are such that Qt+1 = Qt which is true if q1;t+1 (8) and the above functional forms, for j 2 f1; 2g, qj;t+1

q1;t = 0 and q2;t+1

q2;t = 0. Given

qj;t = 0 admits at least three solutions: 0,

1 and q 2]0; 1[ such that h(qt ) = 0; where h(qt ) = f (qt )

q t + qt

a

(qt )

b

f (qt )

b

(qt ) + qt f (qt )(

a

(qt )

(qt )):

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

a

b

(q )

f (q )

(q ) = 0;

and

q = 0;

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

C1: (1

q ( V aa

q C2:

q ( V aa

q )

=q +

V ab ) + a V aa

V ab ) + b V aa

a V ab + V ejbb

V ejab

V ejba = 0;

q (1 q ) : rq

Without loss of generality, let us normalize V ejaa

q =

b V ab + V ejaa

( V aa

V ejab )

with D = ( V aa

( V aa V ab )(a + b) 4 ( V aa V ab )

V ab )(b + a

+ 8 ( V aa

V ejab = 0. The expression for q is then

V ab )(a V aa

) + (V ejbb

V ejba )

(V ejbb

V ejba ) +

p D

;

2

b V ab ):

Hence, both urban con…gurations (q1 ; q2 ) = (q ; q ) and (q1 ; q2 ) = (0; q ) are steady states. Let us now check dynamic stability. Let us …rst consider the integrated equilibrium (q1 ; q2 ) = (q ; q ): It is a stable steady state if and only if dh(q) dq

<0 q=q

which is equivalent to

(1

a

0

(q ))(f (q )

1) + q (1

q )

37

d a dq

q=q

d b dq

q=q

!

< 0:

0

Let us perform f (qt ). We obtain (

f (qt ) =

Assume that

1)

rqt (1 qt )( 1) : (rqt + (1 qt ) )2

0

0

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

We have 0

f (q ) = 0

When

= 1, f (q ) =

1

q q

using f (q ) = q : 0

> 1. When

equals q (implying that r is large), f (q ) = 0. We can 0

deduce that there exists ~ such that 8 < ~ , we have f (q ) < 1. Suppose that

0

0

< ~ . Going back to the sign of h (q ), given that we have f (q ) < 1, we deduce

the following. (i) If (ii) If

a

d dq

q=q a

d dq

q=q

1= , then, for

d b dq d b dq

0

< 0, then h (q ) < 0. q=q

> 0, noting that

a

d dq

q=q

d b dq

q=q

and

a

(q ) are proportional to

q=q

su¢ ciently high both terms become su¢ ciently low so that the inequality holds. In

other words, there exists e such that 8 > e, h (q ) < 0. 0

0

Second, let us consider the segregated equilibrium (q1 ; q2 ) = (q ; 0): Given that h (q ) < 0, the urban con…guration (q1 ; q2 ) = (q ; 0) is a stable steady state if and only if 0

h (0) < 0 ,

1+

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 eja

P1aa dP1 =dq1

ejb

P1bb dP1 =dq1 < 0

38

which is equivalent to eja

( eja

ejb

dP )q ) 1 dq1

+ (1

(

+ (1

)(1

dP q )) 1 < 0: dq1

)(1

q )) < 0

ejb

Since dP1 =dq1 >dP1 =dq1 , this inequality holds only if

(

+ (1

)q )

(

+ (1

leading to 1 q < : 2 Given our functional forms, and

a

b

(q ) =

, the condition for the integrated equilibrium

(q ) =

(q1 ; q2 ) = (q ; q ) to be spatially stable, can be written as follows 0

f (q ) 2 ( V aa + ( V aa

V ab ) (2q

0

Since we have f (q ) = 0 when 8 <

0

V ab )q + (a + b)( V aa

V ab ) + V ejbb

V ejba

1) < 0:

= q , as long as q < 12 , we deduce that there exists

0

such that

, 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 Given the above functional forms, we get

(a V aa

b V ab )(

+ (V ejbb

V ejba + b V aa

+(1

) q (2q

+ (1

V aa

1)

39

)q

a

(0))

a V ab )(1 V ab > 0:

)q

a

(q ; 0) >

b

(q ; 0):

Let us de…ne

(a V aa

( ) + (V ejbb + (1

The function

is linear in

b V ab )(

+ (1

V ejba + b V aa ) q (2q

a V ab )(1 V aa

1)

a

)q

(0))

)q

V ab :

. We have

(1) = (a V aa

b V ab )(1

a

(0)) > 0;

therefore, if (0) > 0 then we deduce ( ) > 0 8 . Let us perform (0), (0) = (a V aa

b V ab )(q

+ q (2q

V ejaa

1)

= 2 q 2 ( V aa a

a

V ejba + b V aa

a V ab )q

V ejab :

V ab ) + q

(0)(a V aa

(0)) + (V ejbb

( V aa

V ab )(a + b

) + V ejbb

V ejba

b V ab ):

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

(0)jq

=1

a

= (1 + b V aa

which is positive given that

a

(0))(a V ejaa

b V ejab ) + ( V ejaa

a V ab + V ejbb

b

(0) < 1 and that

V ejab )

V ejba :

(1) > 0. This polynomial is positive if and only

if q > q~, with

q~ =

( V aa

with D0 = ( V aa +8

a

V ab )( (a + b)) 4 ( V ejaa V ab )(a + b

(0)( V ejaa

V ejbb V ejab )

) + (V ejbb

V ejab )(a V ejaa

Then, q > q~

40

V ejba + V ejba )

b V ejab ):

2

p

D0

;

is equivalent to p which is true given that

a

D>

p 0 D

(0) < 1:

4. Conclusion.

Now we conclude that, when parameters are such that C1 and C2 are satis…ed and that 0

(i)

< minf~ ; g;

(ii)

> e;

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 ab = 0:01, V ejbb r = 30,

7.8

V ejba = 0:0005, a = 0:005, b = 0:004,

= 0:008,

V aa = 0:07,

= 0:05,

= 0:37,

= 1:1.

Proof of Proposition 5

1. Suppose that Q0 2]^ q ; 2^ q [. (i) If segregation holds the urban equilibrium is such that q 1 = Q 2]^ q ; 2^ q [ and q 2 = 0. Given the function

(:) in proof of Proposition 2, the dynamics of cultural traits is described by

Qt+1 =

(Qt ) + (0) =

(Qt ):

From proof of Proposition 2, we have that q is a steady state of the map sequence Qt with dynamics captured by the map Qt is decreasing. By the continuity of the map

and for Q0 2]^ q ; q [ the

is increasing, while for Q0 2]q ; 1[, the sequence , we deduce that for any Q0 2]^ q ; 1[, the sequence

Qt converges to Q = q . Hence for any Q0 2]^ q ; 2^ q [, the sequence Qt converges to Q = q . (ii) If integration holds the urban equilibrium is such that q 1 = q 2 = Q=2 < q^. The dynamics

41

of cultural traits is described by Qt+1 = 2

Qt 2

:

From proof of Proposition 2, we know that 0 is a steady state of the map

. Also, we can deduce

that for Q0 < 2^ q or equivalently Q0 =2 < q^, the sequence Qt is decreasing. By the continuity of the map

, we deduce that for any Q0 2]^ q ; 2^ 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 2]2^ q ; 1[. (i) If segregation holds the urban equilibrium is such that q 1 = Q 2]2^ q ; 1[ and q 2 = 0. The dynamics of cultural traits is described by

Qt+1 =

(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 > q^. The dynamics of cultural traits is described by Qt+1 = 2

Qt 2

:

From proof of Proposition 2, we know that q is a steady state of the map

. 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.

42