Cultural Distance 2007

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Culture Formation and

Endogenous Cultural Distance

Michael M. Pichler

Institute of Mathematical Economics, Bielefeld University

Postfach 100131, D-33501 Bielefeld, Germany

Email: [email protected]

Phone: +49 521 106 2573, Fax: +49 521 106 2997

Second DraftJuly 2009

Abstract

Based on the economics of cultural transmission literature, thispaper introduces a generalized representation of the formation of con-tinuous cultural values (traits) in the socialization process. We definethe culture of a person as a set of cultural values and introduce a cul-

tural formation of preferences framework that models the culture ofa child as the collective outcome of all socialization influences that itexperiences. Thereby, a socialization influence is constituted by thechosen cultural behavior (demonstrated culture) of a person and the(relative) success that it has in the socialization process of the child.By restricting the set of agents from which a child socially learns inthe socialization period to the adult generation only, we can endoge-nize the cultural formation of preferences process as resulting out ofthe optimal parental choices of their demonstrated culture (which con-stitutes a strategic interaction choice) and their (relative) success inthe socialization process of their child. We apply this framework toan OLG environment, where the society is populated by two distinctcultural groups and analyze the endogenous evolution of the distances

between the cultures of the groups.

Keywords: Cultural transmission; Socialization; Preference EvolutionJELClassification numbers: C72, D10, J13, Z13

This research project has been generously supported by the Deutsche Forschungsge-sellschaft (DFG). The author thanks Herbert Dawid and Walter Trockel for invaluableadvice. I am further grateful for useful comments from Fabrizio Panebianco and StefanZeugner. Special thanks are due to the members of the International Research TrainingGroup Economic Behavior and Interaction Models (EBIM).


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1 Introduction

1.1 Motivation

In recent years, the question of assimilation and integration of immigrantswith different cultural backgrounds into hosting societies has attained in-creasing attention, both in media and on the political agenda. In Europe,the debate has centered around an alleged lack of assimilation and inte-gration of Muslim immigrants, compared to other immigrant groups (seee.g. Bisin et al., 2008, who also confirm this allegation empirically, using aU.K. dataset and finding significantly less and slower integration of Mus-lims compared to nonMuslims). Compared to the European nation states,the debate in the U.S.A. has been less pronounced given its historical pre

disposition as an immigrant country. Still influential is the melting pottheory, which assumes that new immigrants will integrate into the society,thereby contributing to the creation of a new cultural identity (Han, 2006,p. 32). While this theory is based on, and successful to describe, the experi-ence of old immigration from western and northern European immigrants(which are culturally close to anglo-saxian Americans), it fails to explain thetendency towards cultural pluralism that the U.S. society has experiencedthereafter (Gordon, 1964, pp. 115119, 132136).

The aim of the present paper is to present a theoretical frameworkthat contributes to the understanding of the process of cultural assimila-tion and behavioral assimilation. We define cultural assimilation as theintertemporal decrease in the distance between the cultures of two groups,underlying the concept of culture as a set of cultural values. Similarly, weinterpret behavioral assimilation as the intertemporal decrease in the dis-tance of the (homogeneous) choices ofdemonstrated cultures of the membersof two cultural groups1. Thereby, a demonstrated culture is defined as theset of cultural values that a person demonstrates through its socioeconomicactions.

The analysis is embedded in a cultural formation of preferences frame-work, in which the culture of a child is formed as the collective outcomeof all socialization influences that it experiences. Thereby, a socializationinfluence is constituted by the demonstrated culture of a person and the(relative) success that it has in the socialization process of the child. Byrestricting the set of agents, from which a child socially learns in the social-ization period to the adult generation only, we can endogenize the culturalformation of preferences process as resulting out of the optimal parentalchoices of their demonstrated culture (which can also be observed by un-related children) and their (relative) success in the socialization process of

1Cultural assimilation and behavioral assimilation in this context are hence mutualconcepts in that they can originate from just one, or both of the cultural groups underscrutiny.

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their child.

1.2 Existing Literature

The present analysis stands in a natural context to the existing literature onthe economics of cultural transmission. This literature is based on the workof Cavalli-Sforza and Feldman (1973, 1981) and Boyd and Richerson (1985)in evolutionary anthropology. Bisin and Verdier (2000, 2001) presented ageneral framework to study the population dynamics of the distribution ofcultural traits under an endogenous intergenerational cultural transmissionmechanism. In this framework, which is now standard in the literature, theendogeneity stems from the purposeful parental choice of socialization inten-

sity, which effectively determines the probability that the child will directlyadopt the culture of the parents. Parents engage into the cost of purpose-ful socialization in order to avoid (decrease the probability) that their childwill not adopt their culture which causes subjective utility losses for theparents. The properties of the model framework have been used in theanalysis of several different issues, such as e.g. preferences for social status(Bisin and Verdier, 1998), voting and political ideology (Bisin and Verdier,2000), corruption (Hauk and Saez-Mart, 2002), hold up problems (Olcinaand Penarrubia, 2004).2 For a more complete overview on the foundationsand theoretical and empirical contributions of the literature see Bisin andVerdier (2008).

Related to this strand of literature is the earlier contribution of Stark(1995) who argues that parents might choose altruistic behavior in front oftheir children even though they are themselves not altruistic. This comes ina (purely egoistic) attempt to instrument the demonstration (or preferenceshaping) effect of altruistic behavior in order to increase the probability ofthe childs adoption of this value and to profit from the childs eventualfuture care for the parents.

While as the above representations of the cultural transmission or pref-erence adoption process as a probabilistic adoption rule have their merits inthe context of discrete traits or preferences, it is less suitable to model theformation of continuous traits or preferences. Bisin and Topa (2003) high-

lighted this issue by reinterpreting the socialization influence of the family asdetermining its weight in the process of the formation of the traits relativeto the weight of the general social environment3. The parental motivation

2The models mentioned restrict the socialization environment of a child to the adultgeneration only. See Saez-Mart and Sjogren (2007) for an analysis on the role that peersplay in the socialization process of children.

3For an earlier treatment on the issue of inheritance and formation of continuous traitsin an empirical context, see Otto et al. (1994). For a treatment of the dynamic evolutionof continuous p opulation attributes in a pure social interactions framework see Bruecknerand Smirnov (2007, 2008).

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to invest in the socialization weight then stems from the aim to avoid sub-

jective utility losses that are increasing in the distance of the childs traitto the parents own trait. Panebianco (2009) uses this framework to studythe dynamic evolution of (continuous) interethnic attitudes and the rise ofethnicbased social hierarchies. The focus is on the study of the effect ofdifferent oblique socialization schemes on the integration process of ethnicgroups. He considers two ethnic groups as fully integrated if they share thesame attitudes toward any other ethnic group.

Also recently, Bisin et al. (2006) introduced parental identitychoice4

into the economics of cultural transmission framework to study the (behav-ioral) assimilation process of cultural minorities. Parents of the minoritygroup can choose among a continuum of lifestyles, all of which represent

different degrees of assimilation to the social norm of the majority. A lackof assimilation causes social costs that positively depend on the populationshare of the majority group and negatively on the degree of ethnic identity(which is a decision variable for the parents5). The parental problem ofchoosing the optimal level of assimilation and ethnic identity is neverthelessseparated from the socialization problem (i.e. both choices have no sociallearning impact on the child), which is organized along the standard linesof the economics of cultural transmission literature. The dynamic evolutionof the degree of assimilation of the cultural minority then derives from thedynamics of the population distribution of cultures and the evolution of thedegree of ethnic identity, which determines the parentally perceived utility

cost of the anticipated lifestyle of the child.

1.3 Paper Contributions

The core of the present paper is the introduction of an (endogenous) culturalformation of preferences framework that provides for a general treatment ofthe formation of continuous cultural values (or preferences or traits) in thesocialization process of a child.

Socialization is the process by which individuals learn their culture. Thisincludes the values, norms and attitudes, as well as the beliefs, religious faith,knowledge, customs, language, skills and practices of the society that the

individual is embedded in. In the present paper, we focus on the socializa-tion of children with respect to the first three categories mentioned. Thereason for this is that they have an inherent continuous structure and canbe represented numerically (the importance of a value, the strength of anorm, how positive/negative a certain attitude is) and can furthermore

4The concept of identity formation is based on the work of Akerlof and Kranton (2000).5Bisin et al. (2006) motivate the construction of ethnic identity as a psychological

mechanism that is rationally employed to reduce the psychological costs associated withbehaving in a nonassimilated way relative to the social norm of the majority culturalgroup.

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be instrumented to reinterpret inherently discrete cultural elements in a

continuous way6. We will in the present paper collectively refer to the threeconcepts values, norms and attitudes as cultural values 7 and define the fullset of cultural values that are subject to the socialization process as theculture (that an individual adopts).

Sociological theory emphasizes the fact that the major force of the social-ization process is the social learning (of children) from rolemodels that aredemonstrated in the social environment (see e.g. the social learning theoriesof Bandura and Walters (1963), Bandura (1977), Rotter (1954), Festinger(1957), Skinner (1938, 1953) and others). Within the context of the presentpaper, the rolemodel that a person in the social environment of a childrepresents corresponds to the demonstration of one specific set of cultural

values, i.e. one specific culture8. We assume that this is realized throughthe choice of a set of socioeconomic actions out of a feasible set of socioeconomic actions9. For obvious reasons, we will from now on refer to thisconcept of a rolemodel as demonstrated culture, i.e. the set ofdemonstratedcultural values.

As far as the process of culture formation of a child is concerned, weassume that the cultural values that it will adopt through the socializa-tion process realize as the collective outcome of all different demonstratedcultures it socially learns from (as in Bisin and Topa, 2003). Thereby, we re-strict the socialization environment of the children to the adult society only.The weight that any of these adults has in the culture formation process

depends on the relative success that it had in the socialization of the child6More precisely, values define the valuations that a person assigns to different aspects

of life, like the importance of family, friendship, career, or social status. Notably, thisconcept also includes the valuations of (respectively the importance of) discrete culturalelements like a specific religious faith, the acquisition of certain skills, the conduct ofcustoms and certain practices or the own language. Norms reflect the values of a societyor certain groups and define the rules of behavior therein. These rules are transmittedthrough a system of social rewards (for conformity) and punishment (for nonconformity).Attitudes include the views on socioeconomic or political issues, like how much a personis in favor of equal gender treatment, integration of minority groups or an egalitariansociety, as well as direct attitudes like the prejudices against members of other culturalgroups.

7The concept of cultural values is thus closely related to what Sen (1977) terms com-

mitments and Harsanyi (1982) moral preferences.8Stark (1995) discusses a model where parents instrument the demonstration effect of

altruistic b ehavior to increase the probability that children will adopt altruism as a value.9Typical elements of such a set that would correspond to the demonstration of val-

ues would include the extensity (time and frequency) and intensity (effort and devotion)of the pursuit of different aspects of life, different ways of selfexpression or to achieveand demonstrate status. Elements that would correspond to the demonstration of (thestrength) of norms would primarily concern the intensity of the social rewards and pun-ishments for conformity and nonconformity of behavior, while elements that translateinto the demonstration of attitudes would be the possible behaviors in different socialsituations and interactions. Notably, the verbal expression of opinion and thought is ameans to demonstrate all categories of cultural values.

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compared to the aggregated socialization successes of all other socializing

persons.Given that the culture that a person adopts after the socialization process

has been formed by the social learning from rolemodels, viz. demonstratedcultures, we assume that it induces preferences over choices of demonstratedcultures in the adult lifeperiod of that person. These preferences have thebasic property that parents prefer demonstrated cultures that feature in-dividual demonstrated cultural values that are as close as possible to theadopted cultural values (this assumption is in line with the cognitive dis-sonance theory of Festinger, 1957). The interdependence between demon-strated cultures, cultures and preferences induced by cultures constitutesthe motivation to label the representation of the socialization process that

this paper proposes as cultural formation of preferences.

As part of their culture, adults do also adopt socialization values. Theseestablish relevance of the future demonstrated culture of their children fortheir own lives, and can both be of altruistic or egoistic nature. Given thatthe choice of demonstrated culture crucially depends on the adopted cul-ture (since it induces preferences over demonstrated cultures), the parentshave an incentive to actively engage in the culture formation process of theirchildren.

However, the active engagement in the culture formation process of thechild causes socialization costs. Besides the utility loss that parents expe-

rience when choosing a demonstrated culture that does not coincide withtheir adopted culture, they encounter cost of investments (especially timeand effort) in their relative socialization success. The parental optimizationproblem in a strategic interactions setting is then the following. They choosea demonstrated culture and socialization success rates such as to maximizetheir lifetime utility (i.e. the utility that they obtain out of their demon-strated culture and the utility related to the future demonstrated culture oftheir children), subject to the demonstrated cultures of the other adults andtheir socialization cost. Since the set of agents, from which a child sociallylearns in the socialization period is restricted to the adult generation only10,we obtain through the parental optimization choices an endogenous cultural

formation of preferences process.

10In this respect, we apply here an adoption of the direct vertical and oblique trans-mission framework that has been introduced into the economics literature by Bisin andVerdier (2000, 2001) and is a standard application in the literature on the economicsof cultural transmission. The terminology originates from Cavalli-Sforza and Feldman(1981). Vertical transmission denotes the parental socialization contribution of a childand oblique transmission the socialization of the general adult society. These concepts aredistinguished from horizontal transmission, which corresponds to socialization influencesfrom members of the same generation.

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Finally, we embed this framework in a society that is populated by dis-

tinct cultural groups such as to analyze the endogenous process of behavioraland cultural assimilation between the groups (endogenous cultural distance).The most important results of this analysis are summarized in the conclusionof this paper (section 5).

1.4 Outline

The remainder of this paper is organized as follows. Section 2 introducesin a general way the concept of cultural formation of preferences, while assection 3 endogenizes this process in a vertical and oblique transmissionframework. Section 4 and 4.1 present the analysis on the endogenous behav-

ioral and cultural assimilation of cultural groups, and section 5 concludes.All proofs of the propositions in this paper can be found in the appendix(which also contains an extension that did not find place in the main bodyof the text).

2 Cultural Formation of Preferences

Consider a society that is populated by a set of children, c C, and adults,a A (both sets either finite or infinite). As far as the socialization process isconcerned, we assume in the present paper that the childrens social learningis only from the rolemodels of the adult members of the society, hence A

does also constitute the (potential) socialization environment of the child.Let us assume for simplicity, that all adults have available the same set offeasible socioeconomic choices, X RM, and let us denote with n N :={1, . . . , N } the cultural values that are demonstrated through any choicetherein11.

Subject to its available options, the specific socioeconomic choice ofany adult a A, xa X, constitutes a demonstrated culture (the set ofdemonstrated cultural values), which will be represented by the mappingd : X d RN, where d(xa) :=

1d, . . . , Nd

(xa) is the set of

demonstrated cultural values, and where d = d(X) denotes the demon-strated culture set. The latter does hence constitute the frame for the pos-

sible cultural behaviors of the adults12. For ease of notation, we will for11In general, the two dimensions M and N do not coincide. There can be interde-

pendencies between different socioeconomic actions both in terms of common (physical)constraints, and in terms of their joint meaning for one or more cultural values and inthe latter case, the set of cultural values for which two or more socioeconomic actionshave a joint meaning might not coincide with the sets of cultural values for which eachof the socioeconomic actions individually contributes meaning (the union of these setsmight be larger than their intersection).

12One important issue to emphasize at this point is the fact that a A, the position ofany d(xa)

d is pinned down by the position of xa X. Under this property, the scal-ing of the culture set is arbitrary (unless zero or infinite) and the mappings bd : X bd,

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subsequent use define, a A, da := d(xa).

As in the literature on the economics of cultural transmission, we assumethat children are born without defined culture. Following and extending thebasic idea of Bisin and Topa (2003) for the treatment of continuous traits13,we will introduce a formal approach that represents the final culture of thechildren as the collective outcome of all socialization influences that the childhas been exposed to during the socialization process. Thereby, we define asocialization influence as being constituted by a) the demonstrated culture,and b) the relative socialization success that this socialization agent couldhave achieved14.

Let the demonstrated cultures of any of the adult persons a A, i.e. thesocialization environment of any child c C, be da := (

1da , . . . ,

Nda )

d,

and for any cultural value n N, let the socialization successes of thesepersons with the children be nac R+. Then, the most general representa-tion (at least up to the authors assessment) of the formation of continuouscultural values is the following15.

Definition 1 (Formation of Cultural Values). The culture of a child, c :=(1c , . . . ,

Nc )

, is formed according to the successrateweighted demon-strated cultural values, i.e. n N

nc =

aA

nacnda da

wherenac :=

nacnac +

aA\{a}

nacda

denote the socialization success rates of the respective persons.

By Definition 1, we assume that the relative impact that any of thedemonstrated cultural values of an adult has in the socialization process of

b R \{, 0, } would be equally suitable to represent the same demonstrated cultures.Notably, the properties of the mapping d are in general only sustained under linear scal-ing, but not under general affine transformations, since the demonstrated cultures couldalso contain entries that represent relative cultural values (especially in the case where

two or more socioeconomic actions that contribute meaning to different cultural valuesdraw on the same resource, e.g. the time spent on different aspects of life).Further note that if b R, then this corresponds to a negative formulation of the

original (formulation of the) cultural values.13For an investigation of the issue of inheritance and formation of continuous traits in

an empirical context see Otto et al. (1994).14The logic of the socialization process that we present here could straightforwardly be

extended to the socialization influence of peers as well as the societys institutions, likethe legal and educational system or the media and marketing sector.

15We do deliberately not specify at this point whether the set of adult agents is finite orinfinite, and use an integral representation since it is more general and encompasses alsoan eventual finite sum representation.

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a child depends on the socialization success that the person can achieve for

this demonstrated cultural value relative to the aggregated socialization suc-cesses that the socialization environment exerts on the child. Given the basicassumption that (a, c) A C, nac R+ we have that

nac [0, 1] (they

are really socialization success rates)16 and thenaA acda = 1. Thus, the

final cultural values of a child are convex combinations of all demonstratedcultural values it is exposed to. Intuitively, this means that the differentdemonstrated cultural values are mutually mediating in the culture forma-tion process of the children. Also, the space where the final culture of anychild can (in general) be located in, which we call the culture set, , coincideswith the convex hull of the demonstrated culture set, = con

d

.In the main body of this paper, we will for ease of the exposition refrain

from a formal specification of potential inputs that lead to the achievementof socialization success. This stems from the fact that in general, thereis a multitude of factors that determine the cognitive impact on the child(among which are notably the socialization interaction times as well as theeffort and devotion a person invests into socializing a child) and notably,these factors might differ for different cultural values17. Thus, one can notaim at including all potential factors into a single formal representation.The interested reader can, however, find in Appendix A.1.1 a proposal fora specification of the socialization success functions that rests on gener-alizable determining factors, as well as a discussion of crucial properties ofthese functions.

To close the analysis in this subsection, let us consider the implications of thechilds adoption, respectively internalization, of a specific culture after thecompletion of the socialization process. We just introduced this culture asbeing formed by the childs social learning from rolemodels (demonstratedcultures), realized through socioeconomic choices out of the adults feasiblesocioeconomic choice sets. It seems then sensible to assume that, in theadult lifetime of a child, its culture defines preferences over choices of socioeconomic actions in the (same) feasible choice set and, thus, preferences overchoices of demonstrated cultures in the (same) demonstrated culture set.

Assumption 1 (Preferences induced by Cultures). Let the adopted culture

of an adult a A be

1a, . . . , Na

:= a . Then, a A,

16Given this property, the socialization success rates could also be interpreted as theprobabilities that a child adopts precisely the demonstrated culture of one specific adultas its own culture, i.e. (a, c) A C, P

c =

da

= ac. This constitutes the linking

bridge between the treatment of discrete cultural traits or preferences and continuouscultural traits or preferences since the way we constructed cultures makes both theprobabilistic and convex mixing approach meaningful.

17In this respect, Panebianco (2009) shows the crucial dependence of the process offormation of interethnic attitudes on whether the children use reciprocity and/or eth-nocentrism as a filter for evaluating the other groups attitudes toward the own ethnicgroup.

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(a) cultures a induce complete and transitive preference relations a

over demonstrated cultures da d, and

(b) the preferences a satisfy the following condition. Let da, d

a d

such that sign

na nda

0, sign

na nd

a

, n N. Then

da a d

a na nda na nda n N, and na nda


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propose a framework to endogenize this process as resulting out of optimal

socialization decisions of parents. These decisions will be embedded in aframework of direct vertical transmission (socialization inside the family)and oblique transmission (socialization by the general social environment).This terminology stems from Cavalli-Sforza and Feldman (1981), which, inthe context of the present paper, could straightforwardly be adopted todirect vertical formation and oblique formation (of preferences).

For ease of the exposition, we will assume here and throughout the rest ofthe paper that reproduction is asexual and that every adult has exactly oneoffspring. Under this assumption, we can denote with a A the children ofthe parents a A and the respective set of children, and with Aa := A\{a}the unrelated adult socialization environment of a child. Further, let us

denote with na := naa, n N, the success rates that the parents have inthe socialization of their child and with

ndAa :=

aAa

nacnac +

aAa\{a}

nacda

da da (1)

the relativesuccessweighted demonstrated culture of the unrelated social-ization environment. Then, we obtain the formation of cultural values (Def-inition 1) under direct vertical and oblique formation as

na = nda + (1

na )(

ndAa

nda ) (2)

which we will also refer to as the parental socialization technique19. It em-bodies the view that parents set a cultural benchmark and can invest intotheir socialization success rates to countervail deviating cultural influencesthat the child is exposed to in the general (adult) social environment.

The motivation for parents to actively employ their socialization tech-nique stems from the fact that the future demonstrated culture (which par-ents expect to depend on the future culture of the children; see below) hasrelevance for their own life. This relevance can stem both from altruis-tic and egoistic reasons and is established by socialization values (whichcould also be termed intergenerational cultural values).

Assumption 2 (Socialization Values). {1s, . . . , S s} N, such that a A the socialization values, sa :=

1sa , . . . , Ssa

, induce complete and tran-

sitive preferences sa over the future expected demonstrated culture of the

own child, Ea

da

d.

19Equation (2) is a generalization of the representation of the formation of continu-ous traits under vertical and oblique transmission in Bisin and Topa (2003). The latterwas termed by Panebianco (2009) as CavalliSforza Socialization Dynamics (referring tothe pathbreaking work of Cavalli-Sforza and Feldman (1973, 1981) on issues related todifferent forms of cultural transmission and highlighting the intergenerational culturaldynamics that this kind of representation embodies).

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Corollary 1 (Utility). Let Assumptions 1 and 2 hold. Then, a A,

(a) the preferences a can be represented by a utility function Ua : d R, where Ua (a) > U

a

da

, da d\{a}, and

(b) the preferences sa can be represented by an (expected) socialization

utility function Ssa : d R.

Assuming that the own utility function and the (expected) socializationutility function are additive separable, we may express the (expected) lifetime utility of any adult a A as

Ua da + S

sa Ea da

As far as the parental expectations of the future demonstrated culture oftheir children are concerned, we make here the standard assumption of theeconomics of cultural transmission literature on a form of parental myopia:Although parents have socialization values and thus take into account theimpact of their chosen demonstrated culture on the future culture, and henceon the expected future demonstrated culture of their children, we assumethat they do not realize this form of behavior changing impact that the so-cialization values impose on their children. Thus, parents expect their chil-dren to unrestrictedly maximize through their chosen demonstrated culturetheir own utility, and hence Ea

da arg maxdad U

a

da. Under the

following convexity assumption (the compactness assumption will be neededlater in this section), the culture set and demonstrated culture set coincide, = d, which then guarantees by Corollary 1 (a) that Ea

da

= a,a A.

Assumption 3 (Convexity and Compactness). The demonstrated cultureset d is convex and compact.

Notably, any of the properties of d actually stems from the properties ofthe socioeconomic choice set X under the mapping d. For guaranteeingcompactness, it suffices that X is compact and the mapping d continuous.However, convexity of X together with continuity of d is not in general

sufficient to assure convexity of d (but e.g. linearity of d would be).Let us now discuss the (effort and devotion) costs associated with in-

vestments into the socialization successes. First, we will represent thesecosts by an indirect cost function of a choice of the parental socializationsuccess rates. Second, we assume that the costs of a choice of some fixedna depends also on how credible the respective chosen demonstrated cul-tural value of the parents is. This is motivated from the fact that it is wellknown that persons that seem to be more satisfied with their chosen wayof life have a more pronounced social learning impact (on children). Sincethe level of satisfaction (or utility) out of choices of demonstrated cultural

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values is determined by their position relative to the adopted cultural values

(Assumption 1 (b)), we propose a cost function C : [0, 1]n d R+,C

a, da, a

, where a :=

1a, . . . ,

Na

.20

The parental optimization problem is it then to choose a demonstratedculture and the socialization success rates such as to maximize the expectedlifetime utility subject to socialization cost. We obtain, a A

max(da,a)d[0,1]n

Ua

da

+ S

sa (a) C

a,

da, a

(3)

subject to the parental socialization technique (equations (2)). The opti-mization problems induce a game where parents have to decide about both

their strategic interaction variables,

d

a, and their nonstrategic interactionvariables, a. As far as the latter are concerned, we assume that throughtheir choice of the socialization success rates, the parents scale the abso-lute socialization successes that all unrelated adults have with their childrensymmetrically, i.e. the relative socialization successes of equation (1) stayfixed.

We will introduce below two suitable equilibrium concepts, where thefollowing notation will be useful. Denote dA :=

daaA

and A :=a

aA

; further let dA := aAd and [0, 1]nA := aA[0, 1]

n, as well

as da :=

da

aAa

. Finally, denote the net (expected) lifetime utility(own and socialization utility net of socialization cost) of any adult a A

as M

da, a, da

, and with (, , ) a game where the first argument de-

notes the set of players, the second the utility of the individual players andthe third the strategy space of the individual players.

Definition 2 (Equilibrium of a Period).

(a) Call a tuple

d

A , A

dA [0, 1]

nA a weak equilibrium of a period

(WEP) if d

A is a pure strategy Nash Equilibrium of the game

A, M

da,

a,

da

, d

and a A, a arg maxa[0,1]n M

d

a , a, da

.

(b) Call a tuple

d

A , A

dA [0, 1]

nA an equilibrium of a period (EP)

if it is a pure strategy Nash Equilibrium of the game

A, M

da, a, da

, d [0, 1]n

.

20We silently made the assumption here that parents can choose all individual socializa-tion success rates within the full unit interval. The deeper implications of this assumptionare again discussed in Appendix A.1.1.

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The WEP concept effectively decomposes the simultaneous parental op-

timization problems (3) into a sequential problem. Parents choose plannedsocialization success rates and then they choose their optimal demonstratedculture subject to this plan and the demonstrated cultures of the otheradults. If the planned socialization success rates are then also optimal inthe context of the demonstrated culture profile, a WEP is reached. Forparents to play such strategies (that are not also EP strategies), we re-quire a certain form of myopia, since they do not necessarily constitute localmaximizers, let alone global maximizers, of the simultaneous optimizationproblems. But since the set of weak equilibria of a period obviously containsthe set of equilibria of a period, any characterization of the chosen strategiesof the first concept (which we will do in section 4) must also hold for the

latter concept. Since the conditions for the existence of a WEP are weakerthan that for the existence of an EP, the distinction of the two concepts issensible.

Proposition 1 (Equilibrium Existence). Let Assumption 3 hold. Then,

a) if the (hypothetical) optimization problems maxdad M

da, a, da

and maxa[0,1]n M

da, a,

da

are continuous and strictly concave

a A, then a WEP exists.

b) if the target functions of the optimization problems (3), max(da,a)d[0,1]n

M

da, a,

da

, are continuous and quasiconcave a A, then an EPexists.

Proof. In Appendix A.2.1.

At a first glance, the condition for the existence of a WEP does not lookweaker than that for the existence of a EP though it is on a closer look.The condition for existence of a WEP requires that Ua , S

sa and C are

jointly strictly concave in both a and da.

21 To discuss the conditions on theindividual maps for quasiconcavity of problem (3) to hold, first note thatthe Hessian matrices of the parental socialization technique (2) are indefi-nite22. Thus, to guarantee at least quasiconcavity of the full optimization

problem, the maps Ua and C must jointly be concave enough (thus,effectively also strictly concave) to compensate for the eventual resulting

nonconcavities of Ssa

a

da, a

.23

21If Ssa is (strictly) concave, then it is also (strictly) concave in both a and

da. This

holds since when fixing the entries of one vector, the parental socialization technique (2)is linear in the entries of the other vector.

22a A, n N,Hn

a (nda ,na )

= 1.23In the following section, we will introduce a corresponding assumption for twice con-

tinuously differentiable functions.

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The aim of the present section was to introduce a general framework to

endogenize the cultural formation of preferences process and to provide forconditions for a (weak) equilibrium of a period to exist. For a characteriza-tion of the choices in a (weak) equilibrium of a period, as well as an analysisof the intergenerational evolution of the cultures of the agents, we will haveto further specify the model. This is the subject of the following section.

4 Endogenous Cultural Distance

In this section, we will embed the endogenous cultural formation of prefer-ences framework in an environment where the society is populated by twodistinct cultural groups. We define a cultural group as the collection of fam-

ilies for which it holds that the members of the parental generation haveequal culture. We will provide for a characterization of weak equilibria ofa period (hence also for equilibria of a period), a discussion of comparativestatics effects of the model framework (both in the present section 4), aswell as we will represent the dynamics of the cultural distance between thetwo cultural groups in section 4.1 (in general, we define the cultural dis-tance as the Euclidean distance between the cultures of the groups). Toease the analytical tractability, the exposition will be based on a number ofsimplifications compared to the general framework of section 3.

Consider an overlapping generations society that is (initially) populatedby two distinct cultural groups, indexed G {L, H}, with the populationshares denoted qG. In the present setting, it will be convenient to indexthe (sets of) adults and children as g G and g G, and to denote G :={L, H}\{G}. In any given period, all adult members of a cultural group haveequal culture (we will introduce below conditions that assure this property),which we reduce for the present analysis to consist of only one own culturalvalue and one socialization value, i.e. G {L, H}, g G, g = G :=(G,

sG).

Since parents are myopic and do not consider the behavior changingimpact of their childrens adopted socialization value(s), it is immediatethat their childrens future demonstrated socialization value(s) do also notenter into their socialization utility function (since if they would enter, wecould also not sensibly maintain the parental myopia assumption), so thatS

sG (g) = s

sG (g). Since parents do effectively not care for the formationof their childrens socialization values, we assume for simplicity that theyare intertemporarily fixed.

Continuing this line of assumptions, we do in the present analysis alsonot consider the own utility contribution of the parents chosen displayedsocialization values, which we denote sdg . In this context, we assume forconvenience that total own utility is additive separable in the utility outof the endogenous demonstrated cultural value and the utility out of the

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demonstrated socialization value, UG dg = uG dg + usG sdg , and

consider subsequently only the first component24.Finally, two additional specifications shall be introduced. The first con-

cerns the socialization values. Within the present analysis, it is interestingto consider different classes of socialization values, which we specify by the(usual) strength of the socialization value, sG, and the classspecific targetfor the future demonstrated own cultural value of the children (which ac-cords to the future culture of the children under the parental myopia), G.

25

The according complete notation for the (myopic) socialization utility is then

ssG

g, G

.

The second specification concerns the role that the position of the cho-sen demonstrated own cultural value relative to the adopted one plays in

the indirect socialization cost functions. Remember from section 3 thatwe motivated their relevance as stemming from their determination of thelevel of satisfaction that an adult can achieve through its choice of demon-strated culture. It is then immediate to represent the indirect cost (of theonedimensional case) as a function c : [0, 1] R R+, c

g, u

G

dg

.

Denoting with d the choice set of demonstrated own cultural values,we finally obtain the (reduced) parental optimization problems (3) as

max(dg,g) d[0,1]

uG

dg

+ s

sG

g, G

c

g, u

G

dg

(4)

subject to the parental socialization techniques (2).Given the fixed socialization value (and the disregarding of the demon-

strated socialization values of the adults), we will concentrate the analysisbelow on the formation and evolution of the endogenous cultural value. Inthis light, we will (in an inconsistency with the previous representation of thecultures of the adults) subsequently refer to the single endogenous culturalvalue as the culture of a group (unless otherwise noted). Accordingly, we willdenote as cultural distance the Euclidean distance between the endogenouscultural values of the two groups26, := |L H|.

24Since parents do not consider the impact of their demonstrated socialization value onthe future demonstrated socialization value of their children, they can unrestrictedly max-

imize the second utility component, i.e. choose sdg = sG we leave however undiscussedat this point the foundations and implications of such a choice.

25We will consider in the subsequent analysis both a characterization of a WEP (re-stricted to cases where all parents of a cultural groups make identical decisions; see below)and comparative statics under a general class of socialization values in the present sec-tion, as well as we will consider the dynamics of the cultures for the case where thesocialization target coincides with the cultures of the parents in section 4.1. Within thepresent framework, there arise two other (but unconsidered) natural variants for groupspecific socialization targets, namely an exogenously fixed target, and the representativedisplayed culture of the adult members of a cultural group.

26Obviously, in the present context, the true cultural distance between the two groupswould be d (L, H)E .

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Assumption 4 (Curvature I). G {L, H}, g G,

(a) the maps uG and ssG are continuous27 and concave and c is contin-

uous and concave in both arguments, and

(b) either ssG is strictly concave, and/or uG is strictly concave and c is

strictly concave in g, and/or c is strictly concave in both arguments.

It is immediate that under Assumption 4, the conditions for the existenceof a WEP are satisfied, since then both (hypothetical) optimization problemsin Proposition 1 (a) are strictly concave. At this point, we introduce twofurther simplifications that are crucial for the subsequent analysis: First, weassume that the relative aggregated socialization successes of the unrelated

members of both cultural groups in the socialization process of any childcoincide with the ratio of the population shares of the groups, i.e. G {L, H}, g G

(1 qG)

gGg

gg dg = qG

gG

gg dg (5)

where Gg := G\{g}. Second, we consider only weak equilibria of a period,in which all parents of a cultural group make identical decisions 28. We willcall such a WEP a homogeneous weak equilibrium of a period (HWEP).

Proposition 2 (Existence of HWEP). If Assumption 14 and condition

(5) hold, then a HWEP exists.


Since under the setup of the present section, the optimization problemsof the adult members depend on their own culture, the socialization targetculture and the population shares, we will denote the grouphomogeneousparental decisions in a HWEP as a tuple

d

G

L, H,

sL,

sH, L, H, qH

, G

L, H,

sL,

sH, L, H, qH

G{L,H}

.

Using these in the parental socialization technique (2), we obtain the en-dogenous future homogeneous culture of the children of both groups, G {L, H}

G

(P) := d

G (P)

d

G (P) d

G(P)

(1 G(P))(1 qG)

27Note here that under Assumption 3, the utility functions would also be guaranteed tobe continuous (since then the preferences are defined on a convex set).

28Notably, under the last two simplifying assumptions, it follows that all children of acultural group adopt the same culture, so that the population shares of the two distinctgroups are constant over generations.

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where P := {L, H, sL,

sH, L, H, qH}. Obviously, these HWEP maps do

also endogenously determine the future cultural distance between the twogroups as

(P) : =

L(P)

H(P)

=dL (P) dH(P) (L(P)qH + H(P)(1 qH))

In the rest of this section, we will introduce additional assumptions thatallow for a characterization both of a HWEP and thus also of a homo-geneous equilibrium of a period (HEP) and of the comparative statics ofthe cultural system.

Assumption 5 (Slope). G {L, H}, g G,

(a) the mappings uG and ssG are C1, ands

sG is strictly decreasing on both

sides of G,

(b) the map c is C1, strictly increasing in the first argument g (0, 1],

where c(0,)g

= 0, and decreasing in the second argument.

Notably, under Assumption 1 (b), uG is also strictly decreasing on both

sides ofG. Together with Assumption 5 (a), this implies that uG (G)

dg= 0,

and furthermore

ssG(G,G) g = 0. For the following Proposition, let us

denote with G0 the initial cultures of both groups G {L, H}, and denoted :=

,

.

Proposition 3 (Characterization of a HWEP). Let Assumptions 15 andcondition (5) be satisfied. Then, if H0 > L0 and if L L and H Hholds in any period29, the following properties are satisfied in any evolvingHWEP

1. Case H > L:

(a) G (0, 1], G {L, H}

(b) d

H H > L d

L (with equalities iff H = or L = )

(c) H > H

> L

> L

29Restricting the analysis to cases where the initial cultures of the two groups areunequal, we can without loss of generality assume that H0 > L0 . The condition L Land H H holds per definition in the case of the intergenerational cultural closenesssocialization motivation (section 4.1), since then G G in any period.

Further note for completeness that we consider here only classes of socialization valuesthat are either independent of individual parental decisions of demonstrated cultural valuesby construction, or where parents believe this to hold (i.e. they do not internalize theeventual effect of their decisions on the socialization target).

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2. Case H = L:

(a) G [0, 1], G {L, H}

(b) d

H H = L d

L

(c) H H

H

L


By the results of Proposition 3, 1., it follows that if the cultural systemstarts with initially different cultures, then in any HWEP (as long as the cul-tural system has not converged to a point where the cultures of both groupsare homogeneous), parents of both groups will, if possible, disintegrate be-haviorally (i.e. they choose a more radical demonstrated culture relative

to the demonstrated culture of the members of the other groups, than the

choice of their true culture would be), which implies d

Hd

L =: d

> .This comes in attempt to countervail the negative cultural influence thatthe adult members of the other cultural group exert on the child. Of course,such a deviation from the parents utility maximizing demonstrated culturalvalue (dg = G) must be accompanied with strictly positive investmentsinto the socialization success rate. Since parents do not perceive a socializa-tion utility loss for marginal deviations of the childs culture (respectivelyexpected demonstrated culture) from the socialization target cultures, thefuture cultures of the children of both cultural groups will always lie strictlyin the interior of the cultural space that is formed by the target cultures.

Furthermore, given the strictly positive socialization success rates thatthe parents choose in any HWEP, the final culture of their children is closerto the target culture than the (relativesuccessweighted) representativedemonstrated culture of the unrelated adult environment is (which is equalfor all parents and children under condition (5)). This means that the rela-tive positions of the cultures of the two groups (in terms of a lower or higherendogenous cultural value) are preserved over generations. Thus, althoughthe cultures of both groups are subject to endogenous dynamical change,the labels of the two cultural groups are dynamically persistent. Given theadditional results of Proposition 3, 2., this result holds weakly even when thecultures of both groups should have converged to a homogeneous culture.

To guarantee the existence of a HEP, and for the subsequent comparativestatics results, we have to apply stronger assumptions (in analogy to previousnotation, we denote here with g the tuple of demonstrated cultures of theset of adults g := Gg G).

Assumption 6 (Curvature II). G {L, H},

(a) the maps uG, ssG and c are C2, and

2M(dg ,g ,dg) dg

2

2M(dg ,g ,dg)2g

2M(dg ,g ,dg)

dg g

2, and

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(b) the marginal cost of a deviation of the future (demonstrated) culture

of the children from the target culture is increasing in the socializationvalue, sign(G G)

2 ssG(G,

sG)

G sG

0.

Proposition 4 (Existence of HEP). If Assumptions 14, and 6 (a) as wellas condition (5) hold, then a HEP exists.


In analogy to previous notation, we will denote the grouphomogeneousparental decisions in a HEP as a tuple

d

G

L, H, s

L, s

H, L, H, qH

, G

L, H, s

L, s

H, L, H, qH

G{L,H} .

Proposition 5 (Comparative Statics). Let Assumptions 16 be satisfied,and let us distinguish the cultural groups as G = L/H. Then, in any HEPwhere L = H,

30 (bG1(P), bG2(P)) < / > (0, 0), where bG1(P)bG2(P) 1, such that if G {L, H}

11

2 M

d

G , G ,

d

g

dg g

< / >

bG1(P)

2M

d

G ,

G ,d

g

dg

2

bG2(P)2M

d

G ,

G ,d

g

2g

the following comparative statics results hold31

(a)

sign

d

G

G

d

G

G

d

G

sG

d

G

sG

d

G

G

d

G

G G G

G G

G sG

G s

G

GG

GG

=+/+ / /+ /+ +/+ //+ +/ +/+ +/+ /+ +/

(b)sign

d

L

qH , sign L

qH {(, +), (+, ), (0, 0)}sign

d

H

qH

, sign

H qH

{(+, +), (, ), (0, 0)}


30If L = H is a steady state as in section 4.1, then marginal changes in the states andthe parameters will cause no changes in the optimal choices of the parents.

31Note that in case G G (section 4.1) the comparative statics effects of the firstand third column as well as of the second and fourth column have to be combined, whichleaves the signs unchanged.

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Proposition 5 requires that for both cultural groups, in any HEP, the two

socialization instruments (decision variables) must not be too strong substi-tutes in the parental socialization problem (4) the strength of which isdetermined by an expression that weights the concavities of problem (4) inthe two socialization instruments. If this condition together with Assump-tions 16 are fulfilled, then the parents will always alter the use of the twoinstruments in a symmetric way. This means that an increase of behavioraldisintegration will be adjoined by an increase of the socialization successrate. To understand this behavior, one has to be aware of the combined im-pact of two different effects. The first and direct effect is caused by a changein the own culture or own parameters of a cultural group. The second,strategic interaction, effect is caused by the mutual reactions of parents of

both cultural groups to changes in the demonstrated culture of the othergroup.

If the condition in Proposition 5 is fulfilled for one cultural group G {L, H}, then the parents of that group would react to the following changesby increasing both socialization instruments (the direct effects): An increaseof the distance of the own culture to the (demonstrated) culture of the othergroup (since the own utility loss for more radical cultural behavior is de-creased); an increase of the socialization target culture to the (demonstrated)culture of the other group (since the increased distance of the future cul-ture of the children to the socialization target culture decreases expectedutility)32; an increase of the socialization motivation intensity (since equal

distances of the culture of the children to the target culture cause lower ex-pected utility); and an increase of the distance of the demonstrated cultureof the other group to the own socialization target culture (which increasesthe distance of the future culture of the children to the socialization targetculture).

The central question now is whether Proposition 5 is also fulfilled for theother cultural group G. This then would imply that the parents of thatgroup would react to an increase of the distance of the demonstrated cultureof group G to their socialization target culture by increasing the distanceof their demonstrated culture to the (demonstrated) culture of group G. Inthis case, both the direct effect of a change in the culture or parameters of

a group and the strategic interaction effect work in the same direction andthe overall effect of changes in the cultures or parameters is unambiguous.Since the direct effects as described above for cultural group G are also validfor group G, the remaining comparative statics effects of Proposition 5 (a)for changes in the culture or parameters of the other group (G) are due tothe analogous reasoning.

The only parameterchange that has a direct impact on the optimal deci-sions of both cultural groups concerns the population share qH = 1 qL. An

32In the case G G the first two effects are combined.

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increase (decrease) of the own population share improves (disimproves) the

cultural composition of the public social space for the parents of a culturalgroup. This means that it becomes more (less) attractive as a substitutefor own socialization investments and parents would, under the conditionof Proposition 5, react by decreasing (increasing) them. Nevertheless, thestrategic interaction effect works in exactly the other direction since par-ents would react to less (more) distant demonstrated cultures of the othercultural group by decreasing (increasing) the distance of their own demon-strated culture to the demonstrated culture of the other group. The relativestrength of the direct and strategic interaction effects will then determinethe direction of the overall changes in the use of the socialization instru-ments (which could also be zero), which illustrates Proposition 5 (b).

The following section specifies the properties of the model presented so farunder a class of socialization values that is related to the parental desirefor intergenerational cultural closeness. The analysis is extended to deriveproperties of the dynamics of the cultural system.

4.1 Intergenerational Cultural Closeness

It is an undisputed fact that family is one of the most central aspects inhuman life. This section introduces a class of socialization values that isbased on intrafamily considerations of the parents. Specifically, this class

represents a desire of the parents for the future culture of their children tobe close to their own culture which hence serves as the target culturein the socialization process. Subsequently, we will shortly introduce threejustifications for this assumption.

The first justification is based on a special form of parental altruismcalled imperfect empathy. This concept has been introduced into the eco-nomics literature by Bisin and Verdier (2000, 2001). Parents are altruisticto their children and want to maximize the utility out of the childrensfuture socioeconomic choices, based on the expectations of their culture.Nevertheless, parents can assess this utility only through the filter of theirown preferences (culture). In the present framework, this introduces a de-

sire for cultural closeness, since the parents are myopic and expect theirchildren to choose a lifestyle that accords to their culture. A second in-terpretation of family related values is what we call family cohesion, andis based on social interactions within the family, thus on the demonstratedcultures of the generations. If the demonstrated cultures of parents (in theirthird lifeperiod) and children (in their adultperiod) are more distant, thenthis causes more frictions, misunderstandings and (mutual) disappointment.Again given the parental myopia, and under the assumption that the par-ents expect to set, in their third lifeperiod, a demonstrated culture that

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equals their true culture33, this then justifies the assumption of the parental

desire for intergenerational cultural closeness. The third, and purely philo-sophical, interpretation is based on the assumption that parents simply havean intrinsic desire for their children to develop a personality, hence culture,that is similar to their own one (e.g. if they view the life of their childrenas a continuation of their own life)34.

Given these motivations for a desire for intergenerational cultural close-ness, we have that G G, and the expected socialization utility functionis s

sG (g, G).

35 Using this to specify optimization problem (4), we havethe following characterization of a HWEP under the parental desire forintergenerational cultural closeness.

Proposition 6 (Characterization of a HWEP). Let Assumptions 15 andcondition 5 be satisfied. Then if H0 > L0, the following properties aresatisfied in any evolving HWEP

1. Case H > L

(a) G (0, 1), G {L, H}

(b) d

H H > H

> L

> L d

L

(with equalities iff H = or L = )

2. Case L = H

(a) G = 0, G {L, H}

(b) d

H = H = H

= L

= L = d

L


33In the third lifeperiod, adults have no more need to engage in socialization, and thuscan unrestrictedly maximize their utility out of socioeconomic choices.

34Thus, the latter justification for intergenerational cultural closeness class is directlyrelated to the future culture of the children (while as the other justifications were onlyindirectly related given the parental myopia). We nevertheless refrained from introducinga separate notation for such cases in section 3, given their exceptionality.

35In the present case, sG can be considered as a composed value out of the usual timediscount rate (since the relevant socioeconomic actions of the children take place one

period after the parents take the socialization actions), and a value that represents thestrength of the parental desire for intergenerational cultural closeness.

In the case of imperfect empathy, it is immediate to specify ssG (g, G) :=

sGu

G (g),where sG then corresponds to the (discounted) degree of parental altruism (i.e. thevaluation that parents put on the future utility of their children relative to their ownutility). The formulations in the literature on the economics of cultural transmission,notably Bisin and Verdier (2000, 2001), Bisin and Topa (2003) and Panebianco (2009),correspond to the special case sG = 1. Given imperfect empathy, the utility loss thatparents perceive if their children do not exactly adopt the parental culture is thenV

sG (g, G) :=

sG

uG (G) u

G (g)

.

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Compared to the general results of Proposition 3, in case of the class of

intergenerational cultural closeness socialization values, where the social-ization target coincides with the culture of the parents, they would neverchoose a combination of socioeconomic actions and socialization successrate such that their children adopt exactly the parental culture. This comesfrom the fact that if the future culture of the children only marginally de-viates from the parents culture, this is not perceived as costly. As a result,the final cultures of the children of both cultural groups lie in the interior ofthe interval that is constituted by the cultures of the parental generation,G {L, H},

G (L, H). This then implies that over generations, the

cultures of the groups assimilate (but not completely)

< . Asymp-totically, though, the cultures converge to a homogeneous culture, where

H = L, and parents have no more need to engage into active socializa-tion, so that this homogeneous culture is also a rest point. While as theexact location of the homogeneous steady state depends on the initial lo-cation of the cultures (and the socialization values and population shares),the corresponding cultural distance of zero in a homogeneous steady stateis globally asymptotically stable.

This result will be stated more formally in a Corollary. First, considerthe present overlapping generations model in the continuous time limit36.Let the paths of cultures that evolve out of the parental problems (4) underthe socialization utility function s

sG (g, G) and under the conditions of

a HWEP be denoted H(t, H0) and L(t, L0), where t is the timeindex.

The corresponding path for the endogenous cultural distance is denoted(t, 0 ) := H(t, H0) L(t, L0), where

0 := H0 L0 .

Corollary 2 (Dynamics of Cultural Distance). If Assumptions 15 are

fulfilled, then limt (t, 0 ) = 0,

0

0,

. Moreover, G

{L, H}, limt G(t, G0) = (L0 , H0).

Proof. By Proposition 6, 1.(b), the dynamics of the cultural distance is acontraction mapping and the system converges to a homogeneous culture forany initial cultural distance. By Proposition 6, 2., any homogeneous cultureis a rest point. The last result follows directly from Proposition 6.

By Corollary 2, the social(ization) interactions of the two cultural groupsinduce a dynamic cultural assimilation process, which results in the creationof a new homogeneous equilibrium culture that can be interpreted as amixture of the two initial cultures. This theoretical result corresponds tothe melting pot theory of cultural assimilation (see Han, 2006, p. 32).

36We take the derivation of the continuous time limit from Bisin and Verdier (2001): Inan OLG economy where the agents live units of time and have children 1 h units oftime after birth, one obtains the continuous time limit by taking the limit for , h 0with h

0.

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Figure 1: Dynamics of the Cultural System

To close the analysis of the cultural system under the intergenerational cul-tural closeness socialization motivation, we illustrate the analytical resultsby means of a numerical simulation37. Figure 1 shows the dynamics of thecultural system for two majority/minority cases. In the upper graph of eachcase, the solid lines represent the demonstrated culture of the groups and thedotted lines represent the path of the cultures of the groups (and the dashdotted line locates the populationshare weighted convex combination of theinitial cultures; this would equate to the cultural steady state if parents ofboth cultural groups would not invest into the socialization instruments).In both graphs, the low value cultural group is the minority, whose path ofthe socialization success rate is represented by the dotted lines in the lowergraph of each case.

In both parts of Figure 1, one can see that parents choose a behavioraldisintegration as well as strictly positive socialization success rates as longas the cultural distance between the groups is strictly positive. Nevertheless,parents never choose high enough investments into the socialization instru-ments to assure that the cultural distance between the two groups wouldbe eventually nondecreasing, which leads to an intertemporal assimila-tion process toward a homogeneous steady state culture (Proposition 6, andCorollary 2). Since (if the conditions of Proposition 5 hold) the demon-strated cultural distance as well as the socialization success rates are strictlydeclining for decreases in the cultural distance (this follows directly fromProposition 5 when G G), they are also strictly declining throughout

37For the numerical simulation, we used the following specifications: uG

dg

=

G dG2, ssG (g, G) = sG |G g|2, c g, uG dg = 2g 1 uG dg /20.

It is immediate that these functions satisfy all relevant Assumptions of this section. Fur-thermore, we used an initial cultural distance of 28 units, and the total length of thetimeaxis in all graphs corresponds to 100 periods. Nevertheless, the choice of the initialcultural distance and the resulting length of the convergence path could be arbitrarilychanged, so that they are not indicated in the graphs.

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the transitory path, i.e. the cultural groups do also assimilate behaviorally.

What can also be seen in the present application, although the share ofthe minority group is only one fifth of the total population, the socializationinvestments of its parents succeed those of the majoritys parents to a highenough extent for the cultural system to converge to a steady state cul-ture that is located in the interior of the populationshare weighted convexcombination of the two initial cultures and the initial culture of the minor-ity. The higher incentives of the parents of the minority group to investinto their socialization instruments stem from the more unfavorable culturalcomposition of the general socialization environment of their children (theunrelated adults), which makes it less suitable to use for substitution of ownsocialization investments. Finally, compared to the left case of Figure 1,

the right case features a higher socialization value for both cultural groups.This leads into strictly larger socialization investments of the parents of bothcultural groups for any given cultural distance (Proposition 5), so that theprocess of cultural and behavioral assimilation is accordingly slower.

5 Conclusions

This paper extended and generalized existing approaches in the culturaltransmission literature to the formation of (continuous) cultural values (cul-tural preferences or traits) that we interpret as collectively defining the cul-ture of a person. Under this property, we can define a collection of (adult)

individuals that have a homogeneous culture as a cultural group. Within theendogenous cultural formation of preferences framework, that this paper hasintroduced as its first main part (which is introduced and summarized insection 1.3), there then arise the natural questions of qualitative propertiesof the behavioral and cultural assimilation processes between different cul-tural groups. To analyze these issues has been the task of the second mainpart of the present paper.

Thereby, the analysis is restricted to the case of two different culturalgroups, where the homogeneous cultures of the adults of a group are spec-ified to one collective endogenous cultural value and one exogenously fixedsocialization value (i.e. the collective value captures all cultural informationother than the socialization value). As far as the latter is concerned, weconsider different classes of socialization values that determine the targetfor the future (demonstrated) culture of the children.

Since one of the main objects of the analysis is the endogenous evolutionof the cultural distance between the endogenous cultural values of the twogroups (which we define as the respective Euclidean distance), we introducesuitable conditions that guarantee that in any equilibrium of a given period,all parents of a cultural group take identical decisions with respect to thedemonstrated culture and the socialization success rate. The conditions also

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ensure that all children of a cultural group do adopt the same culture and,

thus, we obtain a continuous intergenerational representation of the twocultural groups.

The most important results that hold in a (weak) equilibrium of a periodunder a general class of socialization values are the following. In case of astrictly positive cultural distance between the two groups, parents alwayschoose a behavioral disintegration, complemented with a strictly positivesocialization success rate. This comes in an attempt to countervail thenegative cultural influence that the social learning from the other groupsadults has on the own child. Further, the relative positions of the twocultural groups in terms of a lower or higher endogenous cultural value aresustained over generations, i.e. the labels of the two cultural groups are

dynamically persistent.As far as the comparative statics are concerned, we can show that if the

two socialization instruments are not too strong substitutes in the parentaloptimization problem, then both the behavioral disintegration, as well asthe investments in the socialization success rate, are increasing in the so-cialization value. The same effect holds for (symmetric) increases in thecultural distance between the two groups. In both cases then, the culturalassimilation process between the two groups is slower (or an eventual cul-tural dissimilation process is faster). This theoretical result might deliver anexplanation for the empirical observation of different paths of the integra-tion and assimilation processes of cultural groups, since it both depends on

the (perceived) cultural distance between the groups, as well as the strengthof intergenerational considerations (the socialization value).

Notably, these theoretical results constitute empirically testable hypothe-ses. In this respect, the main explanatory variables for a cultural and behav-ioral assimilation process are the type and strength of the socialization value,as well as the (initial) cultural distance. As far as the first is concerned, goodproxies to represent a general class of socialization values would e.g. be theimportance of family, combined with the strength of behavioral normsin a cultural group38. Since we constructed demonstrated cultures as theimage of socioeconomic choices, an eventual behavioral assimilation pro-cess between cultural groups is directly empirically observable. Given the

theoretical results of the present paper (notably the characterizations of anequilibrium of a period and the comparative statics results), the respectivedata can further serve as a good proxy for the underlying cultural distancesand, thus, for the cultural assimilation or dissimilation process in casethat explicit data on the cultures of groups are not or not to a satisfactory

38Arnett (1995) labels the two polar cases of the strength of the behavioral norms inthe socialization process as broad and narrow socialization. Cultures characterized bybroad socialization encourage individualism, independence, and selfexpression [. . . ] . Incontrast, cultures characterized by narrow so cialization hold obedience and conformity asthe highest values and discourage deviation from cultural expectations [. . . ] . (p. 617)

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extent available.

To analyze the endogenous evolution of the cultural distance in a dy-namic setting, we applied a special class of socialization values, which isbased on a parental desire for intergenerational cultural closeness. Thecharacteristic of this class is that, under a certain parental myopia, thesocialization target for the childrens future culture coincides with the cul-ture of the parents. Notably, this class also covers the concept of imperfectempathy, as used by Bisin and Verdier (2000, 2001) and throughout theliterature on the economics of cultural transmission. We can show that un-der the intergenerational cultural closeness socialization values, in case ofa strictly positive cultural distance, the cultures that the children of bothcultural groups adopt lie in the interior of the interval that is constituted by

the cultures of both groups. Further, in case of equal cultures, the parentsdo have no need to invest into their socialization instruments, so that sucha homogeneous culture is a steady state. These results imply that if the cul-tural system starts out with initially different cultures, the cultural distancestrictly declines over generations (cultural assimilation). Even, the two cul-tures converge to a homogeneous steady state, and the resulting steady stateculture can be interpreted as a mixture of the two original cultures. Thiscorresponds to the melting pot theory of integration of cultural groups.

There are a number of directions, in which the present theoretical analy-sis could be extended. First, it might be interesting to identify whether

there exist classes of socialization values under which nondecreasing cul-tural distances, and a cultural equilibrium with strictly positive culturaldistance, are possible. Second, the behavior of the cultural system underdifferent classes of socialization values should also be analyzed in a gener-alized culture framework, i.e. one with an ndimensional set of culturalvalues, eventually combined with an endogenous treatment of the socializa-tion value(s).

References

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Quarterly Journal of Economics, 115(3):715753.

Arnett, J. J. (1995). Broad and narrow socialization: The family in thecontext of a cultural theory. Journal of Marriage and the Family, 57:617628.

Bandura, A. (1977). Social Learning Theory. General Learning Press, NewYork, NY.

Bandura, A. and Walters, R. H. (1963). Social Learning and PersonalityDevelopment. Holt, Rinehart & Winston, New York.

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Bisin, A., Patacchini, E., Verdier, T., and Zenou, Y. (2006). bend it like

beckham: Identity, socialization and assimilation. CEPR Discussion Pa-pers 5662, C.E.P.R. Discussion Papers.

Bisin, A., Patacchini, E., Verdier, T., and Zenou, Y. (2008). Are muslimimmigrants different in terms of cultural integration? Journal of theEuropean Economic Association, 6(2-3):445456.

Bisin, A. and Topa, G. (2003). Empirical models of cultural transmission.Journal of the European Economic Association, 1(2-3):363375.

Bisin, A. and Verdier, T. (1998). On the cultural transmission of preferencesfor social status. Journal of Public Economics, 70(1):7597.

Bisin, A. and Verdier, T. (2000). A model of cultural transmission, votingand political ideology. European Journal of Political Economy, 16(1):529.

Bisin, A. and Verdier, T. (2001). The economics of cultural transmission andthe dynamics of preferences. Journal of Economic Theory, 97(2):298319.

Bisin, A. and Verdier, T. (2008). Cultural transmission. In Durlauf, S. N.and Blume, L. E., editors, The New Palgrave Dictionary of Economics.Palgrave Macmillan, Basingstoke.

Boyd, R. and Richerson, P. J. (1985). Culture and the Evolutionary Process.

University of Chicago Press.Brueckner, J. K. and Smirnov, O. (2007). Workings of the melting pot:

Social networks and the evolution of population attributes. Journal ofRegional Science, 47(2):209228.

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Cavalli-Sforza, L. L. and Feldman, M. (1973). Cultural versus biologicalinheritance: phenotypic transmission from parents to children. American

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A Appendix

A.1 Extensions

A.1.1 Properties of the Socialization Success Functions

Let us first discuss the specification of the determinants of the socializa-tion successes in an topdown approach (concerning the level of detail ofthe representation). To the best of the authors assessment, the most gen-eral representation of the latter views the socialization successes as a func-tion of the extensity and intensity with which a person socializes a child,nac := (ac,

nac), (a,c,n) A C N . The extensity of socialization

is represented by ac [0, 1] (without loss of generality, we normalize the

length of the socialization period to one), which coincides with the lengthof the time that the person has spent with the child and is interpreted as aglobal determinant of the socialization successes of all cultural values, whileas the socialization intensities, nac R+, are viewed as cultural value specific(this issue will be discussed in more detail below).

It is immediate to assume that if a child has no contact with a per-son, then it has no socialization influence, (0, ) = 0. In line with thebasic assumption in section 2, we also assume that the socialization suc-cess can not be negative39, (, ) R+. Furthermore, we assume that ifa child merely observes a demonstrated cultural value (which we normal-ize to correspond to a socialization intensity level of zero), this also yields

strictly positive social learning effects, i.e. (, 0) is strictly increasing. To-gether with the assumption that the aggregate exposure of the child tosocialization influences coincides with the length of the socialization period,c C,

aA acda = 1, it follows that the aggregated socialization success

is strictly positive,aA (ac,

nac)da > 0, as well as that the socialization

success rates of all persons lie in the unit interval, nac [0, 1], a A,such that

aA

nacda = 1 (which corresponds to the basic properties of the

socialization successes that we introduced in section 2).Let us now discuss general properties of the cultural value specific in-

tensities of socialization, nac. We assume that they are determined by twobasic factors: The general devotion of a person to a child, ac R+, which

we interpret as a global determinant of all socialization intensities (and italso determines the trust that a child has in a person); and the credibility,nac R+, that a child assigns to the claim that is implied by a chosen

39The assumption of positive socialization successes comes at the cost of a loss of gen-erality. In principle, it is possible that adult agents do cause a socialization unsuccessin the socialization process of a child, e.g. if a demonstrated cultural value is immoral oragainst the human nature (which can thus also be realized by a child with yet undefinedcultural values) and hence their demonstrated cultural value can be interpreted as arolemodel of how not to behave. Nevertheless, we do for simplicity not integrate therelated phenomena into the formal representation of the present paper.

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demonstrated cultural value of a person that this choice is really supe-

rior to all other possible choices of demonstrated cultural values. Thus,nac := (ac,

nac).

As far as the credibility levels are concerned, they themselves dependcrucially on two factors. First, a cultural value specific socialization effort(e.g. the effort to convince a child about the superiority of adopting a certainrealization of a cultural value compared to the other demonstrated culturalvalues that a child is confronted with in the socialization environment),enac R+. And second (as has already been noted in section 3), it is wellknown that individuals that seem to be more satisfied with their chosen lifestyle (demonstrated cultural values) have a more pronounced social learningimpact in the childrens socialization process. In light of the properties of

the adults preferences as introduced in the preceding subsection, the level ofsatisfaction crucially depends on the chosen demonstrated cultural values ofthe adults compared to their adopted cultural values. Thus, we finally obtainthe cultural value specific credibilities as a function nac :=

enac,

na ,

nda

.

Assumption 7 (Boundedness and Continuity). (a, n) A N

(a) the inputs to the socialization success functions, enac andac, are bounded,and

(b) the maps nac and nac are continuous, and (, ) = 0 only at (0, ).

Under the latter assumption, the range of the map is also bounded. Wefinally use the properties of the socialization success functions (and theirinputs) that we introduced in the present section to discuss in more detailthe silent assumption in section 3 that parents can choose all socializationsuccess rates within the full unit interval, i.e. a A, a [0, 1]

n. To dothis, let Assumption 7 hold.

First, a = 0 iff aa =: a = 0 since (, ) = 0 only at (0, ) andaAa

(aa, ) da < ,

aAa

aa = 1 a. This implies that parentscan choose a zero interaction time with their children, i.e. effectively parentsmight give their children away which obviously constitutes a strongassumption.

Second, a = 1 iff a = 1 since then naa = 0, a

Aa, and sincea

Aa (a

a, ) da

> 0, a

Aaa

a = 1 a > 0 and the map is bounded

(so that also the parents can not achieve infinite socialization success). Thisimplies that parents can choose to fully distract their children from the un-related adults which does obviously also constitute a strong assumption.

A.2 Proofs

A.2.1 Proof of Proposition 1

In this proof, the following further notation will be useful for brevity: a :=a

aAa

, da := aAad, [0, 1]na := aAa [0, 1]

n. Further, note that

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under Assumption 3, the set dA [0, 1]nA is compact and convex since it is

the Cartesian product of compact and convex sets (if the set of adults is acontinuum, thus infinite, the product of the choice sets is endowed with theproduct topology).

(a)

Denote maxdad M

da, a, da

as Ma1 and maxa[0,1]n M

da, a,

da

as Ma2. Next, define the bestreply correspondence of problem Ma1 asBa : [0, 1]

n da d and the solution correspondence to problem Ma2

as Sa : dA [0, 1]

n. Denote the corresponding product correspondencesas BA = aABa : [0, 1]

nA

dA

dA, respectively SA = aASa :

dA

[0, 1]nA. Finally, consider the composition of the product correspondences as

SA BA : [0, 1]nA dA dA [0, 1]nA, which can equivalently be writtenas SA BA : [0, 1]nA

dA [0, 1]

nA

dA.

Since Ma1 and Ma2 are continuous and d and [0, 1]n are constant(and therefore continuous), it follows by Berges Theorem of the Maximumthat Ba and Sa are nonempty, compactvalued and upper hemicontinuous.Since both optimization problems are also strictly concave, the solution cor-respondences are even singlevalued, and thus, Ba and Sa are continuousfunctions. These properties are inherited by the product correspondencesBA and SA. Finally, since SA BA is the composition of continuous func-tions, it is also a con

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Cultural Distance 2007