Opinion Leaders in Influence Networks and the Integration of Immigrant Communities
Anja Prummer
∗Jan-Peter Siedlarek
†This version: November 22, 2013
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Abstract
We offer a novel explanation for differences in the integration outcomes of immigrant commu- nities through a model which emphasizes the role of group leaders. Group leaders in our models are individuals or organisations that exert social influence in the community, an example being re- ligious organisations. We present a model of integration with distinct channels for social influence and skill acquisition. Skill acquisition leads to higher income, but reduces identification with the immigrant group. A lower group identity in turn makes skill acquisition less costly. Group leaders benefit both from their group maintaining a distinct identity as well as achieving economic success.
In the long run, full integration is achieved only with flexible leaders, which themselves adapt over time. In the presence of rigid leaders that do not adopt, integration can remain incomplete, with long-run integration levels higher for individuals of higher ability. We show that there are incen- tives for leaders to position themselves as rigid and prevent the integration of their community despite this limiting the economic success of their group members. Additionally, we demonstrate how connections among group members can amplify or reduce the influence of group leaders.
∗ap809@cam.ac.uk, Cambridge-INET Institute, University of Cambridge, UK.
†siedlarek@uni-mannheim.de, Department of Economics, University of Mannheim, Germany. We are grateful to Fernando Vega-Redondo and Sanjeev Goyal for helpful discussions and advice. All remaining errors are ours.
1 Introduction
The integration of immigrant communities is a significant challenge in many societies and the issue has received considerable attention by many political institutions including the European Commis- sion which maintains a web site emphasizing the importance of integration.1 Likewise, integration features as a key research question in the work of the OECD, which has published research deal- ing with the integration of immigrants in the labor market.2 Similarly, in the academic literature, the integration experiences of diverse groups in many countries have been studied extensively. A common finding reported is that integration often remains incomplete – even after several decades of residence and across multiple generations – in the sense that immigrant communities remain dis- tinct from their host society across various measures, including economic as well as cultural metrics (Chiswick,1978;Borjas,1994).3
In this paper we offer a novel perspective on the mechanics of economic and cultural integration that provides a new explanation for incomplete integration outcomes: the strong and continuing in- fluence of a group leadership or similar institution, that is an influential part of the community and benefits from it maintaining distinct. We present a model in which a community integrates through two distinct channels: (i) economic integration through investment in host country specific skills such as language, and (ii) cultural integration through adaptation of beliefs and attitudes. Commu- nity members advance economically by investing in host country human capital. Human capital investment leads to better adaptation to the host country labor market and increases labor income to group members in the host society. The returns to such investment activity are affected by the an individual’ identity, that is, the extent of identification with their origin community, reflecting the social costs – familiar e.g. from the “acting white” phenomenon (Austen-Smith and Fryer,2005) – involved in stronger economic integration. A high level of identification with the home community limits economic integration by making it more costly to invest in host country human capital and acquire skills.
Identities of the community members change over time as members assimilate the norms and customs of the host society relative to their own group identity. In our model, these identities evolve on the basis of interaction in the community’s social network, captured in aDeGroot(1974) model of opinion formation, in which each group member updates an opinion by reference to his neigh- bors, the group leader and the host society. Departing from the standard time-invariant updating in DeGroot(1974), our model explicitly allows for interaction of identity formation with the economic integration process in that influence patterns change over time depending.
As another key component of this paper, we account for community leadership. Leaders benefit from both the economic success of their communities as well as their identification with group of origin. In other words, the leader benefits from economic integration and cultural segregation. For an example of such incentive structures consider religious organizations and immigrant churches in European countries, which tend to be based on donations. The amount received will then reflect both the amount of income available to members as well as their propensity to employ that income towards their church, a tendency directly associated with the level of identification with the group the church aims to represent.4
1http://ec.europa.eu/ewsi/en/
2http://www.oecd.org/els/internationalmigrationpoliciesanddata/
3See alsoAlgan et al.(2010) for France, Germany and the United Kingdom,Card and DiNardo(2000) for the US, andHeath and Cheung(2007) for a number of countries as well as the treatment inAlgan et al.(2012).
We show that in such a setting the long-run integration outcome depends on whether leaders adapt themselves or pursue a strategy of not altering their opinion. Whilst in the former setting there is complete integration, in the latter long-run outcomes show incomplete integration with group member identities somewhere between that of the leadership and their host society. The degree to which communities remain distinct depends at the individual level on the ability of group members and at the group level on the cohesiveness of the social network of the community. We also show that leaders have an incentive to use the instruments available to them in a way that prevents integration of their community membership. This mechanic may thus offer an explanation for a particularly conservative and anti-integration stance taken by religious and other leaders.
The following Section2presents related literature and further evidence on the questions we dis- cuss in this paper. Section3 contains the description of the model of the community integration process in the presence of a leadership. Section4presents our main results concerning long-run out- comes and leader incentives. These insights are illustrated further in Section5with a simple worked example. Section6concludes. All proofs are in the appendix.
2 Literature Context
This section summarizes the extant literature on immigration and integration that is relevant to this paper, focussing on the differences in economic outcomes between different ethnic groups, the pro- cess of integration and the role of social networks in immigrant communities. We also briefly sum- marize work on the concepts employed in this paper, in particular concerning models of cultural transmission as well as learning and adaptation in social networks.
The importance of the topic of immigrant integration is reflected in a substantial literature on the issue in particular in sociology but also in economics. We focus here on research most directly relevant to the questions considered in this paper, namely the divergence of integration experiences across groups and the role of social networks.
Differences in Integration between Ethnic Groups
The literature distinguishes between economic and social integration. The former is concerned with integration in the labor market, in education and training in skills which are valued in market inter- actions. Chiswick(1978) focuses on whether immigrants earnings converge to those of natives and he finds this is the case. His findings have been confirmed byBorjas (1994),Algan et al.(2010) for France, Germany and the United Kingdom,Card and DiNardo(2000) for the US, andHeath and Che- ung(2007) for a number of countries. These papers also discuss convergence in terms of education, see additionallyAlgan et al.(2012).
Moreover, there is convergence regarding cultural habits, values and beliefs, and language. This is termed as social or cultural integration. Although there seems to be convergence in general, there are some immigrant groups whose education, wages as well as cultural habits and values remain distinct even for second-generation immigrants.
Chiswick(1988) describes how early Japanese and Chinese immigrant groups started out similar to Mexican-Americans and blacks but managed to change into a high-skilled population within a relatively short time-frame of one or two generations whilst those other groups did not progress as quickly.Bisin et al.(2008) shows that there seem to be differences between muslim and non-muslim immigrants in the United Kingdom. In particular, muslim immigrants tend to identify strongly with
the values of their immigrant group into the second generation.5 At the same time, there is still a substantial wage difference between Pakistani and Bangladeshi, both groups that are mainly Muslim, and native whites, seeAlgan et al.(2010,2012).6 This difference between muslim and non-muslim immigrants has also been documented for Germany (Constant et al.,2006). They are additionally able to distinguish between different ethnic groups that are muslim, in particular, immigrants from Ex-Yugoslavia and Turks, finding that Turkish muslims identify less with German culture than Ex- Yugoslav muslims.
Networks and Integration
So far we have documented that although many second-generation immigrants are fully integrated, there exist exceptions. The question is in what way the groups, who do not integrate well or only slowly, differ from the groups that do. In order to understand what differences between the groups matter, it is instructive to understand how the integration process works.
A standard explanation of this process is built upon the notion of immigrants building capital that is specific to the country they arrive in and which they do not bring on arrival, a good example being language skills. Once an investment in these skills has been made, i.e. immigrants are proficient in the host country’s language, they will earn higher wages. In particular, their wages will converge to that of natives, meaning economic integration will be attained. That there is indeed a wage differen- tial for immigrants proficient in the host country’s language has been shown byGrenier(1984) for Hispanics in the US, for a summary seeBorjas(1994).
Not being proficient in English leads to a 17% wage penalty, after adjusting for differences in education and other socioeconomic characteristics. This wage differential implies a $ 96,600 (in 1993 dollars) increase in lifetime earnings for a Hispanic immigrant to the US who becomes proficient in English. Despite this, in 1990, 47% of immigrants in the US did not speak English very well, which raises the question why given these high returns to English proficiency immigrants do not invest more in this skill. One possible explanation is that the wage differential is due to selection with more able workers learning English and earning more. However, investment decisions are also connected to the existence of enclaves, such as Cubans in Miami (seePortes(1987)) or Mexicans in Los Angeles.
Immigrants living in such enclaves face lower returns to learning English as most economic exchange takes place within a community of the same ethnic and linguistic background. This is supported by evidence that returns for Hispanics are much lower when living in enclaves (McManus(1990)). But immigrants do not only live in enclaves upon first entering the country, but they also continue to do so i.e. immigrants prefer to reside in areas where there are other immigrants. Moreover, their internal migration decision are much less sensitive to regional wage differentials than those of natives, see Bartel(1989).
These enclaves or immigrant communities therefore seem to make those living in them integrate less. But immigrant communities differ in many respects. First of all there are larger and smaller groups which has an impact on the returns to language. For example, Hispanic immigrants living in small enclaves have higher returns to language than those in bigger enclaves (McManus(1990)).
Another feature of the immigrant community is, according toBreton(1964) the "institutional com- pleteness" of a community.
5This is in contradiction toAlgan et al.(2012) where it is shown that in particular Pakistani and Bangladeshi immigrants tend to identify with the British national identity, see p. 278 Table 8.13.Bisin et al.(2008) have a different measure of integra- tion, namely they focus on differences in terms of attitude towards religion, marriage and schooling.
This is the extent to which an immigrant community provides its members with the social insti- tutions they require, without having to resort to the host society. The institutional completeness of groups varies widely, though: from lose collections of social ties based on acquaintances and friend- ships to the fully fledged social organizations that provide welfare, schools and religious institutions, which perform virtually all services required by its members.
The impact of social structure on economic action is explored further inPortes and Sensenbrenner (1993). They discuss several examples of how networks among immigrant groups emerge due to the adversity faced in the new host country and how these networks help find jobs or lead to access to credit. Interestingly, the authors also show how networks may get in the way of economic progress of immigrant communities and their individual members, which is documented in more detail in the next subsection. Additional papers that find a negative impact of networks includeMunshi(2003), which reports a reduction in investment in education and consequently wages with stronger net- works, andHoff and Sen(2005), which focusses in a developing country context on the way kinship networks may slow the integration of group members into the market economy. Finally,Sanders (2002) offers further insights on the interaction of networks and assimilation, based on research in sociology.
The Role of Group Leaders
Breton(1964) highlights the role of leaders and institutions in keeping membership and maintain- ing boundaries around the group. He finds that religious institutions have the greatest effect out of the sample considered, followed by group specific publications, such as newspapers or periodicals.
The existence of welfare institutions has the least effect on group identity. Portes and Sensenbren- ner(1993) document the following striking examples that show how certain group members have a special interest in keeping the group together:
“The Spanish-language media, so instrumental in maintaining community controls among Latins in South Florida (Forment 1989) also imposes, in the opinion of many ob- servers, a virtual censorship. Joan Didion reports the views on the matter of a dissident exile banker: “This is Miami. . . . A million Cubans are blackmailed, totally controlled by three radio stations. I feel sorry for the Cuban community in Miami. Because they have imposed on themselves, by way of the right, the same condition that Castro has imposed in Cuba. Total intolerance” (Didion 1987, p. 113). Until a few years ago, San Francisco’s Chinatown was a tightly knit community where the family clans and the Chi- nese Six Companies ruled supreme. These powerful associations regulated the business and social life of the community, guaranteeing its normative order and privileged access to resources for its entrepreneurs. Such assets came, however, at the cost of restrictions on most members’ scope of action and access to the outside world. In their study of Chi- natown, Nee and Nee (1973) report on the continuing power of the clans and the Chinese companies and their strong conservative bent. What put teeth in the clans’ demands was their control of land and business opportunities in the Chinese enclave and their will- ingness to exclude those who violated normative consensus by adopting a “progressive”
stance. One of the Nees’ informants complained about this conservative stronghold in terms similar to those of the Miami banker above:
And not only the Moon Family Association, all the family associations, the Six Companies, any young person who wants to make some changes, they call
him a communist right away. He’s redcapped right away. They use all kinds of tricks to run him out. You see, in old Chinatown, they didn’t respect a scholarly person or an intelligent person . . . They hold on to everything the way it was in China, in Kwangtung. Even though we’re in a different society, a different era.
[Nee and Nee 1973, p. 190]
Like Chinatown in San Francisco, the Korean community of New York is undergirded by a number of associations-from traditional extended family groups and various types of gye (rotating credit associations) to modern businesses and professional organizations.
The role of this associational structure in generating social capital for collective advance- ment follows closely the pattern of enforceable trust already described. The flip side of this structure takes, however, a peculiar form among Koreans. As described by Illsoo Kim (1981), the South Korean government, represented by its consulate general, has played a very prominent role in the development of the ethnic community. "Partly because Korean immigrants have a strong sense of nationalism and therefore identify with the home gov- ernment, the Korean Consulate General in New York City . . . has determined the basic tone of community-wide politics" (Kim 1981, p. 227).”
The Spanish-language media, the family clans of Chinatown as well as the Korean government profit in different ways from keeping their community together. Kuran and Sandholm(2008) men- tion the role of leaders who encourage their followers to differentiate themselves from the majority group. The leaders are particularly successful with those, who lack the resources to succeed when integrating. These individuals create separate cultures in which their skills are more valued and their behavior is more accepted preventing integration.
Cutting Group Ties
We are also interested which group members are more likely to cut ties to their group. It seems to be the case that these are specifically the high ability individuals, who do not want to face the obligation to contribute to their group. This is documented inPortes and Sensenbrenner(1993):
"In the indigenous villages surrounding the town of Otavalo in the Ecuadorian Andes, male owners of garment and leather artisan shops are often Protestant (or “Evangelicals”
as they are known locally) rather than Catholic. The reason is not that the Protestant ethic spurred them to greater entrepreneurial achievement nor that they found Evangelical doctrine to be more compatible with their own beliefs, but a rather more instrumental one. By shifting religious allegiance, these entrepreneurs remove themselves from the host of social obligations for male family heads associated with the Catholic church and its local organizations. The Evangelical convert becomes, in a sense, a stranger in his own community, which insulates him from free riding by others who follow Catholic-inspired norms.”
The constraints appear to apply in particular to high ability members who could strive for success outside the group. Furthermore, there is some evidence that there are beneficiaries of the constraints that have an interest in maintaining them.7
7Note the parallels to the literature on “extractive institutions” and their impact on the growth of countries (Acemoglu and Robinson(2012). We focus our attention to similar mechanisms in immigrant groups.
Cultural Transmission and Integration
Our paper also relates to the literature on transmission processes and in particular the transmission of cultural traits and influence in social networks. In economics there is now a significant literature on the transmission of cultural traits, summarized inBisin and Verdier(2010). Relevant recent work in sociology includesFriedkin(2004) which deals with the creation and maintenance of group cohesion focussing on endogenous mechanisms of interpersonal influence. Such influence is found to depend inter alia on the network structure within the group.
In the context of immigration,Borjas(1995) asks how ethnic capital is forwarded through gen- erations. He documents that ethnicity is a good proxy for the socioeconomic background of the neighborhood in which immigrants grow up and neighbourhood effects account for a significant share of the “ethnic externality” observed inBorjas (1992). However, the paper also documents a residual role for ethnicity in intergenerational mobility showing that “ethnic capital” matters and affects children exposed to it.
Moreover, there are several papers that look at cultural integration and assimilation. InKuran and Sandholm(2008) a community’s culture is defined by the preferences and equilibrium behaviors of its group members. Culture undergoes a change through contacts among communities. When communities do not grow because of the arrival of new immigrants, multiculturalism and social integration are opposing goals, i.e. multiculturalism prevents social integration. On the other hand, when new immigrants join a community there will be multiculturalism even in the long run.
Opinion Formation and Learning in Networks
The paper also relates to the literature on opinion formation and learning in networks. Specifically, our model is in the spirit ofDeGroot(1974) and subsequent papers based on learning by averaging over others’ beliefs.8 More recently, the model has been extensively studied byGolub and Jackson (2010). They focus in their own results on wisdom – the extent to which beliefs are correct given the underlying information – and convergence speeds for “large societies”, i.e. as the number of agents involved grows large. Our paper is distinct from these references in the two specific ways. First, in our paper agents assign weights to different elements of their neighbourhood in a way that evolves over time. Both the original work byDeGroot(1974) and the more recentGolub and Jackson(2010) study a setting with time-homogenous influence matrices. Second, the focus on these papers is on a community achieving consensus, that is equal opinions, and wisdom, that is beliefs which are correct relative to underlying information. Whilst we discuss the question of consensus, which corresponds to complete integration in our setting, our model additionally analyses the case where integration is incomplete and varies across agents, that is, where there is an absence of consensus.
In economics, a further key reference using the DeGroot approach isDeMarzo et al.(2003). De- Marzo et al.(2003) cast the DeGroot model as one of learning with “persuasion bias”, i.e. they give economic meaning to the averaging process. Within this setup, they derive results which show how network structure translates into social influence as well as how differences in multi-dimensional beliefs converge towards differences that are uni-dimensional – that is, belief differences can be char- acterised by a position on a scalar in the long run. The model is applied to various applications in economics, political science and business. Two elements of their paper are related to our model: first,
8Note that there are important other papers that investigate learning in networks which use different approaches. For example,Bala and Goyal(1998) andGale and Kariv(2003) adopt a Bayesian perspective on inference by agents. This approach is quite distinct from the setting we adopt here and we do not explore this area further.
DeMarzo et al.(2003) allow for a limited form of time inhomogeneity in the transition matrix; and second, their analysis includes a perspective on isolated groups, which are not influenced by oth- ers. Regarding time-inhomogeneity,DeMarzo et al.(2003) allow for agents to vary the weight they put on their own beliefs over time. Whilst presenting an interesting deviation from the standard model with constant influence matrix, the variation is limited by the fact that interaction patterns with other agents do not vary over time. In our paper, the time variation allows for more complex dynamics, including variation in the relative weight assigned to different parts of the neighbour- hood (e.g. host society versus group leader). Other papers that study time-inhomogenous consensus processes includeChatterjee and Seneta(1977),Lorenz(2005) andLorenz(2007), which focus on the mathematical relationship between consensus processes and Markov chains.
The second feature of our model which builds onDeMarzo et al.(2003) concerns isolated groups that do not listen to the surroundings. DeMarzo et al.(2003) show how in the presence of isolated groups, the non-isolated entities converge to an opinion which is part of the convex hull of the con- sensus positions of the isolated groups. This feature appears in our analysis of integration with rigid leaders. Another reference which studies isolated groups in a similar way in a time-homogeneous setting isPierre(2012). Neither paper allows for the time variation to arise from economic decisions taken by the agents.
A final strand of related literature considers cultural transmission. Here related papers include Kuran and Sandholm(2008), Büchel et al. (2011) and Büchel et al. (2012). The first, considers a model in which agents trade off their desire to coordinate their actions with others with their own personal preferences. The resulting equilibrium behaviour shows individuals taking actions that reflect an average of their own and others’ preferences.Kuran and Sandholm(2008) use this setting to study the interaction of different groups in society which they introduce by allowing the coordination parameter to be stronger within than across groups. They find strong forces for cultural integration within and across group boundaries, which form a contrast to our results on incomplete integration.
The other two papers on cultural transmission build directly on the DeGroot model by introducing strategic behaviour by agents. Büchel et al.(2011) model sequential generations of agents, where parents (generation t) are interested in transmitting to their children (generation t + 1) their own opinion. They therefore consider the social network which their children are exposed to and make adjustments of their own position in order to compensate for the influence acting on their children from elsewhere. For example, parents that are religious might act more religious than they really are in an environment where their children are in contact with others that are less religious. A second paper, Büchel et al. (2012), analyses the impact of agents having different levels of "conformity”
in the opinion formation model – they distort their position in order not to appear too different from the people around them. Our paper is related in that we allow agents to take decisions which then affect the way they interact with their neighbours. However, instead of intergenerational or social conformity we study the interaction of economic and cultural integration as well as resulting incentives for group leaders.
3 Model
We present a model of the economic and cultural integration process of an community with a leader.
We first outline the interaction of the immigrant community with the host society or environment, before introducing the strategic decision problem of the leader whether or not to support integration.
3.1 Integration Process and Identity Formation of Group Members
There is a set N of agents which represents a distinct community in a host environment. The group is headed by a leader, denoted by L; the residual agents are indexed i = 1, . . . , n and are labeled group members. As the process of integration is an inherently dynamic one, we construct a dynamic model in discrete time. In period t every agent i is described by (i) the level of group identification pi(t) and (ii) a stock of human capital Hi(t).
The level of group identification pi(t) measures how much a group member identifies with his group versus the host society. We let pi(t) ∈ [0, 1]. Full integration or full identification with the norms of the host country is given by pi(t) = 0, the highest possible level of non-integration is pi(t) = 1.
Both p and h change over time as the integration process proceeds. In each period there are two stages: (i) host country specific human capital investment and (ii) belief adaptation.
First, group members decide how much to invest in the acquisition of host country skills. We assume here that group members optimize myopically in every period: group member i chooses in period t an investment level hi(t). He takes his decision with reference to costs and benefits of his decision in the current period. The cost of investment in t not only depends on the magnitude of the human capital investment hi(t) but also on the level of group identification pi(t − 1) and is denoted by c(hi(t); pi(t − 1)). The higher the level of identification with their own group, the more costly it is to invest in the acquisition of host country skills. If group identification is high, individuals find it harder to adopt different norms and social behaviors. Norms that exist within their own group are adhered to to a greater extent and this makes it more difficult, to adapt to the different norms of the host country. And this in turn makes the acquisition of host country specific skills more costly.
We summarise the restrictions on the cost function in Assumption1.
Assumption 1(Cost Function).
(i) c1(hi, pi) > 0, c11(hi, pi) = 0 (ii) c2(hi, pi) > 0
(iii) c12(hi, pi) > 0
For any given level of investment hi, total costs are increasing in group identity pi, as are marginal costs for additional investment. We further assume that costs are linearly increasing in hi. This removes incentives for spreading out investment over time. Agents immediately choose the optimal investment level which reduces complexity of the model without affecting our main results.
Payoffs from investment in human capital depend positively on the accumulated capital stock Hi
in period t through a production function f (Hi). Therefore, the return to labor increases in the level of host country specific skill, f0(Hi) > 0, f00(Hi) < 0.
The human capital stock consists of the depreciated human capital stock from the previous period and current period investment, that is:
Hi(t) = δHi(t − 1) + hi(t)
where δ ∈ (0, 1) denotes the rate of depreciation. The payoff from the investment also depends on the ability, or innate productivity, of a group member, αi > 0 such that the investment in host country specific capital is greater the higher the ability of a group member. Specifically, we assume a multiplicative structure with total payoff given by αif (Hi(t)).
Summarising the individual decision problem for each group member i ∈ {1, 2, . . . , n} we thus have:
maxhi(t){αif [Hi(t)] − c [hi(t) , pi(t − 1)]} (1) s.t. Hi(t) = δHi(t − 1) + hi(t)
Agents are myopic regarding their investment decision and do not take into account that the human capital will benefit them in the future. If an immigrant assumes that he will return to his home country in the next period, then this is the correct specification. Otherwise it can be seen as a lower bound on investment in human capital.9 Finally, we assume that the marginal return to investment at Hi= 0 is positive for all agents and levels of group identification.
Assumption 2. Every group member invests in human capital.
αif0(0) − c1(0; 1) > 0 ∀ αi. (2) As we have seen from the maximization problem, the level of group identification influences the investment decision of the group members. However, the level of host country specific human capital acquired also affects the level of group identity.
Group identity changes over time through social interaction between agents in the spirit ofDe- Groot(1974). Each agent starts out with an initial value of group identification pi(0) = 1 reflecting complete identification with the community of origin. In every subsequent period agents update their group identity based on their own group identity and the identity of the other group members.
Additionally, group members are influenced by their group leader as well as the host society. The group leader can influence the level of group identity through his own identity, that is his own values and norms. Formally, the adaptation of group identity is according to the updating matrix T . This is a (n + 2) × (n + 2) row stochastic influence matrix.10 The updating matrix is time varying as it depends on the host country specific capital stock of all workers, i.e. T (H(t)), where H denotes the vector of human capital. Given T (H(t)) and the initial values of the group identity we then obtain the new levels of group identity.
We denote the vector containing all values of group identification by p(t). It contains the val- ues of group identification of the group leader, pL(t), the values of the group members as well as the group identification of the host society, the environment of the group. Therefore, p(t) = (pE(t), pL(t), p1(t), . . . , pn(t)).11
9We intend to show how our results change when the investment decision is forward looking.
10See for exampleDeGroot(1974),DeMarzo et al.(2003) andGolub and Jackson(2010). There are n + 2 elements in the matrix as it includes a row and column for the host society, the group leader as well as all group members.
11Note that the dimensions of p and H differ as the group leader and the environment are taken into account in p, but not in H as the group leader and the environment do not.
Formally, the updating process is given by
p(t) = T (H(t))p(t − 1) (3)
=
1 0 0 . . . 0
0 ωLL ωL1 . . . ωLn
ω1E(t) ω1L(t) . . . ω2n(t) ... ... ... ... ... ωnE(t) ωnL(t) . . . ωnn(t)
p(t − 1)
We assume that T(1,1)= 1, i.e. the environment is not influenced by the immigrant group. Further, we let pE(0) = 0, which then together with T(1,1) = 1 implies that pE(t) = 0 ∀ t. This captures the notion that the immigrant community is exposed to the host society in various forms and to varying degrees, whereas the host society is influenced by the immigrant group only to a very limited degree and remains invariant.12
Likewise, the group leader being a distinct institution does not adapt to the host society itself such that T(2,1)(t) = 0 ∀ t. He may or may not be influenced by the group members around. In any case, T(2,2)(t) ≥ 0 ∀ t, implying the group leader always has strictly positive weight on his own past identity.
For the remaining agents in the network, Ti,1(t) = ωiE(t) = g(Hi(t)) so that the degree of influ- ence of the host society is a function of level of host country skills. The restrictions imposed on the function g are given in Assumption3.
Assumption 3(Weights).
1. g(Hi(t)) ∈ (0, 1) ∀ Hi(t) 2. g0(Hi(t)) > 0, g00(Hi(t)) < 0
The higher the investment in host country skills, the more an individual is exposed to the influ- ence of the host country and the more his group identification will be influenced by its norms, which in turn also implies that there will be less influence exerted by the group. Further, all group members are influenced by their identity last period, ωii(t) > 0, ∀t.
The model thus presents a dynamic system the state of which at each point t is described by a vec- tor pair (Ht, pt). The two vectors present the two interconnected channels of integration: economic integration which takes place through investment by agents and cultural integration by adapting attitudes to the host society.
3.2 Group Leader
Having outlined the dynamics of integration, we next turn to the role of the group leader. The group leader is interested in maximizing his own payoff, which is based on the group identity of the group members as well as their income. The economic situation of religious leaders depend, for example, on the amount of donations they receive from the group. In the empirical literature on charitable contributions it has been shown that contributions are increasing in income. Further, contributions to religious organizations are increasing in church attendance. The more involved individuals are in religious activity, the more they donate to their church (Bekkers and Wiepking,
12This assumption is plausible in a society where the immigrant group is not too large compared to the host society. This appears to be true in many real world immigration settings. One notable exception is Miami, where the number of Cuban immigrants is so large that they gained notable influence on the host society.
2007). However, social norms matter also as shown by evidence that Protestants are more generous than Catholics, which is said to be largely because of stronger social norms and their higher level of church attendance, seeBerger(2006). Similarly, the printers of national language newspapers13 would only be profitable if someone buys their newspaper, which in turn depends on how much they identify as being German as well as whether they can afford the paper. The Korean government has an interest in their members being affluent as this also gives them a higher social standing as well as them identifying with the goals of the government.
We consider two different leadership styles: a rigid (R) and an adaptive (A) group leader. A rigid group leader is not influenced by the identity of the group members, i.e. ωLL(t) = 1, ∀t. An adaptive group leader, on the other hand, assigns strictly positive weight to at least one group member, that is ωLL < 1, ωLi ≥ 0, i ∈ {1, . . . , n} andPn
i=1ωLi = 1 − ωLL > 0. An adaptive group leader tries to accommodate his followers and adjusts to their position. A rigid one, however, does not do so and is not influenced at all by the host society. One can think of this leader as an individual. Another interpretation of the rigid leader is that the group leaders change every few years. Some of the imams, who work in Germany are educated in Turkey and then come to Germany for a couple of years and therefore have poor knowledge of German (Halm et al.(2012),Ceylan(2010)).
So, although a single group leader might be willing to adapt, the change that occurs during the four years this imam is in Germany is not so dramatic as to approximate it by an adaptive group leader. Instead, the succession of slightly adaptive group leaders should be modeled as a rigid leader.
Formally, to simplify notation, we denote the income of individuals by πi(αi, Hi(t)) = αif (Hi(t)).
We use here the gross income without investment costs as these costs should be thought of as psycho- logical costs. It is not expensive to learn a language, or to learn about the culture in the country one lives in, but rather a matter of time and willingness to be exposed to something new and different.
We then denote the vector that only contains the level of group identification of the group members as ˜p(t) and the vector of incomes as π. Both vectors are of dimension n × 1. We denote the economic payoff of the group leader as Π(˜p(t), π) in period t and for our purposes let
Π(˜p(t), π) =
n
X
i=2
ζ (pi(t), πi(αi, Hi(t))) , (4)
where ζ is a function that positively depends on both group identity and income. We now turn to the analysis of our model and its implications.
4 Analysis
We are interested both in the integration process and its properties as well as in whether there is long-run integration or not. We start out discussing whether there will be long-run integration or not and if not, what the long-run group identity of immigrants is. In particular, we are interested in the differences in group identity if the leader is rigid or adaptive. To address this question we look at the steady state of the group identity levels as well as of the human capital stock.
13The situation of German newspapers in the United States is described in detail inBreton(1964)
4.1 Steady State
The steady state h, p is characterized by constant levels of human capital H and group affiliation p.
It thus describes the point in the system at which there are no changes in subsequent periods. Once a steady state is reached, investment and identity updating decisions do not lead to further changes.
Therefore, if a steady state exists, then the steady state levels of human capital and beliefs satisfy
H = δH + h (5)
p = T H p (6)
where h represents the vector of optimal investment levels at the steady state as given by
c1(hi(t), pi(t − 1)) = αif1(Hi(t)). (7) Equation7is also sufficient given our assumptions on c(.) and f (.). Replacing the h term in that equation with (1 − δ)H reduces the system to one expressed in terms of the vector pair H, p char- acterized by:
c1 (1 − δ)Hi, pi = αif0(Hi) ∀ i ∈ {1, 2, . . . , n} (8)
p = T H p (9)
Note that given assumptions on the cost and benefit functions Hi > 0 ∀ i implies that in the steady state workers put a strictly positive amount of weight on the host society. But so far we still have not established that a steady state exists, we have only shown what properties it has given it exists. We now show existence both for the case with an adaptive and a rigid leader. We can establish convergence in the adaptive model in which the leader institution is influenced to some extent by the community. We can also characterize the steady state integration vector as one of full integration as established in the following proposition. This proof, as well as all others, can be found in the appendix.
Proposition 1(Steady State A). In the model with an adaptive leader, the system converges to a unique steady state in which each group member’s identity reaches full integration. That is
p = limt→∞p(t) = 0.
The steady state belief vector is thus one of consensus and complete integration. Human capital levels are given by the first order condition in equation (8), solved for each i at pi= 0, ∀i.
This is in contrast to what happens if the group leader is rigid. In the case of a rigid leader, or a fixed institution not updating its position, there are now two poles of fixed opinion which exert a constant influence on group members. We can show that in the steady state, these two components balance and group members adopt levels of integration that are a convex combination of the two.
In order to show our result, we define g(p) ≡ g(H∗(p)), where H∗(p) is the optimal level of human capital given a group identity of p. To ensure a steady state we also impose additional assumptions on g(p).
Assumption 4(Properties of g(p)).
|g0(p)| < 1, (10)
g0(p) p g(p)
< 1, (11)
∂ p
1−g(p)g0(p)
∂p < 0. (12)
This implies that an increase in group identity leads to a decrease of the optimal level of human capital, which in turn affects the weight on the environment. But this weight decreases less than group identity increases (equation (10)). Further, the elasticity of the weight on the environment is below one (equation (11) ).14 Last, the derivative of1−g(p)p g0(p) with respect to p is negative (equation 12).
Proposition 2(Steady State R).
Suppose that Assumption4holds. Then, in the model with an autonomous leader, the system converges to a unique steady state in which each group member’s identity is a convex combination of the position of the host society and that of the autonomous leader.
We show that Assumption4provides conditions under which the one period updating process is a contraction. Given it is a contraction, it then follows that there exists a unique steady state. The fact that each group member’s position is a convex combination follows from the fact that in the limit, as t → ∞, the weight that each group member assigns to himself and other group members converges to zero. But the network structure still influences the steady state weights. In order to characterize the steady state group identity in more detail, we consider the two extreme cases, namely the case where the group members only put positive weights on the environment, the group leaders and themselves and the case in which group members listen to all other members within the group.
To clarify, consider the following two updating matrices, where TCr denotes the updating matrix when the group members are connected and TN Cr the matrix if the group members are not connected:
TN Cr (H(t)) =
1 0 0 . . . 0
0 1 0 . . . 0
g(H1(t)) (1 − g(H1(t))) γ (1 − g(H1(t))) (1 − γ) . . . 0
... ... ... ... ...
g(Hn(t)) (1 − g(Hn(t))) γ 0 . . . (1 − g(Hn(t))) (1 − γ)
TCr(H(t)) =
1 0 0 . . . 0
0 1 0 . . . 0
g(H1(t)) (1 − g(H1(t))) γ (1 − g(H1(t)))1−γn . . . (1 − g(H1(t)))1−γn
... ... ... ... ...
g(Hn(t)) (1 − g(Hn(t))) γ (1 − g(Hn(t)))1−γn . . . (1 − g(Hn(t)))1−γn
,
where γ ∈ (0, 1) gives the relative influence of the leader compared to the group members.
Proposition 3(Steady State Weights on Environment and Leader).
Suppose first, the group members do not put a positive weight on the group identification of any other group
members, i.e. the updating matrix is given by TN Cr . Then,
ωiE = g(Hi)
1 − (1 − g(Hi))(1 − γ) (13)
ωiL= γ(1 − g(Hi))
1 − (1 − g(Hi))(1 − γ) (14)
If the group members assign positive weights on all other group member, i.e. the updating matrix is TCr, then
ωiE= g(Hi) 1 − (n − 1)1−γn +1−γn P
j6=ig(Hi) 1 −Pn
j=1 1−γ
n−1 1 − g(Hj) (15)
ωiL= γ 1 − g(Hi) 1 −Pn
j=1 1−γ
n 1 − g(Hj) (16)
These weights also depend on the level of ability.
Proposition 4(Effect of Ability on Steady State Group Identity).
In both network types, C and N C, higher ability leads to a higher weight on the environment and accordingly to a lower level of group identity in the steady state.
Then, we can also compare under what network structure the weights are higher or lower for an individual with a certain ability.
Proposition 5(Comparison of Group Identity for Different Network Types).
A group member i has a higher group identity in steady state in the complete network than in the incomplete network, whenevern1P
jg(pCj) > g(pN Ci ) and has a lower group identity if n1P
jg(pCj) < g(pN Ci ).
If a group member has a lower weight on the environment (this is a low ability group member) than the average, then going from the incomplete network to the complete one leads him to increase the weight on the environment and to lower his group identity. The opposite holds when the weight on the environment is higher than the average. Then, an group member (this group member will have high ability) will have a higher group identity in the complete network compared to the incom- plete network.
So, what this shows is that the group identities in a complete network are less dispersed, with the group identity of high ability group members being higher compared to the incomplete networks and the the group identity of low ability group members being lower compared to the incomplete network.
Corollary 1(Ability αi= α ∀i).
Whenever all group members have the same level of ability, αi= α ∀i, n1P
jg(pCj ) = g(pN Ci ). Therefore, the network structure does not have an impact on the steady state group identity.
This follows immediately from the proof of Proposition5. The difference in network structure only matters when there are at least two group members who have a different ability.
4.2 Group Identity and Human Capital on the Convergence Path
So far, we only considered what happens in steady state. We are also interested in the levels of group identity and human capital on the convergence path. In particular, we are interested in the difference on the path when the leader is rigid or adaptive. We first show how group identity evolves with a rigid leader compared to an adaptive one.
Proposition 6(Group Identification).
For every i ∈ {1, . . . , n},
pri(1) = pai(1),
pri(t) > pai(t) ∀t > 1
Proposition6shows that in every period, the group identification of a member in a group with an adaptive leader is lower than the identification of a member in a group with a rigid leader. The comparison of human capital stock in case of a rigid leader and an adaptive one is similar.
Proposition 7(Host Specific Human Capital).
For every i ∈ {1, . . . , n},
∀t ≤ 2 Hir(t) = Hia(t),
∀t > 2 Hir(t) < Hia(t)
Proposition7shows that in every period the capital stock of the group member with an adaptive leader is higher than the capital stock of a group member with a rigid leader. This is a consequence of Proposition6.
4.3 Rigid versus Adaptive Leader: Whose Payoff is Greater?
So far, we only considered what happens when a group leader has some given properties. Now, we want to see what happens when the group leader can determine whether he is influenced by the group members, i.e. whether he can choose to be rigid or adaptive. Recall that the group leader has a payoff function that depends both on the income of group members as well as their group identity.
We simplify equation (4) and consider for now
Π(˜p(t), π) =
n
X
i=2
pi(t)πi(αi, Hi(t)). (17)
Then, we can characterize when the group leader prefers to be a rigid leader in every period.
Proposition 8(Rigid vs Adaptive Leader Payoffs).
Whenever
∂p(t + 1)
∂p(t)
p(t)
p(t + 1) > −∂f (p(t))
∂p(t) p(t)
f (p(t)), (18)
the leader prefers to be a rigid leader in each period.
We are further interested in whether the group leader prefers the network to be more or less cohesive. We have not been able to obtain results analytically so far, but can show in a example what the group leader prefers.
5 A Worked Example
In this section, we develop a simple example of a community with a leader and two workers and
state levels of group identity and human capital with given functional forms.
5.1 Description
We consider a very simple example, namely a group with n = 2 and leader L and specify the follow- ing functional forms:
f (Hi) =pHi (19)
c(hi; pi) = 1 + pi
2 hi (20)
g(Hi) =pHi. (21)
Assuming that the two group members are both connected to the leader reduces the set of possible network structures to two: agents are either connected in a ring (which is also the complete network in this setting) or a star. The two possible configurations are illustrate in Figure1.
L
1
2
Host Society
(a) Star
L
1
2
Host Society
(b) Ring
Figure 1: Network Configurations for Working Example
5.2 Simulations – Convergence to Steady State and Steady State Levels
In this section we present results of a simple simulation exercise which explicitly models the dynam- ics of the model for the example configurations to analyze the convergence process and steady state properties. The parameter values used in the simulations are as follows:
p01 p02 p03 p0ext α1 α2 δ γ
1 1 1 0 .8 .5 .9 .9
Table 1: Parameter Specifications
5.2.1 Rigid Leader
We consider first the case of a rigid group leader who is not influenced by the group members. The properties of the convergence paths can be seen in figure2. The convergence path for the investment
is not monotonically decreasing for the line but spikes at a certain point for both agents. Next, we are
0 5 10 15 20 25 30 35 40
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Time Period
Investment Level
Convergence Path of Investment
Line: Agent 2 Line: Agent 3 Ring: Agent 2 Ring: Agent 3
(a) Convergence in Investment
0 5 10 15 20 25 30 35 40
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time Period
Group Identity
Convergence Path of Group Identity
Line: Agent 2 Line: Agent 3 Ring: Agent 2 Ring: Agent 3
(b) Convergence in Group Identities
Figure 2: Convergence of Investment and Group Identities with Rigid Leader
also interested in the time until convergence and how this differs depending on the influence of the group leader. This can be seen in figure3. We do not interpret the convergence time in absolute terms, as it is unclear what time span a model period corresponds to. But we can consider relative terms, that is we can compare whether convergence is faster in the ring or the star network. Convergence time seems to be higher for intermediate group leader influence although this also differs for the two network structures as well as the ability types. The low ability agent in the star has the highest convergence time when the group leader influence is fairly low but the convergence time decreases as γ increases. Further, the convergence times of the group members in the ring are more balanced than the convergence times in the star that is the group members in the ring have approximately the same convergence time whereas the convergence time in the star differs more. Last, we also
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10 15 20 25 30 35
Influence of Leader on Group
Convergence Time
Convergence of Investment
Star: Agent 2 Star: Agent 3 Circle: Agent 2 Circle: Agent 3
(a) Convergence Time Investment
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10 15 20 25 30 35 40
Influence of Leader on Group
Convergence Time
Convergence of Beliefs
Star: Agent 2 Star: Agent 3 Circle: Agent 2 Circle: Agent 3
(b) Convergence Time Group Identities
Figure 3: Convergence Time of Investment and Group Identities depending on Influence of Rigid Leader
the influence of the group leader, γ. They are given in figure4. It can be seen that the difference between the star nodes in terms of investment and group identity is larger than the difference in the ring. This difference is more pronounced for group identity. In particular, the investment of the high ability individual in the star is higher than the investment of the high ability worker in the ring, the investment of the low ability individual is lower in the star than in the ring. Further, the group identity of the high ability individual in the star is lower than the identity in the ring, and vice versa for the low ability worker. The outcomes for the star structure appear therefore more responsive to changes in ability than the outcomes for the ring structure. The exact values of the steady state
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Influence of Leader on Group
Investment
Steady State Investment
Star: Agent 2 Star: Agent 3 Circle: Agent 2 Circle: Agent 3
(a) Steady State Investment
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Influence of Leader on Group
Investment
Steady State Beliefs
Line: Agent 2 Line: Agent 3 Ring: Agent 2 Ring: Agent 3
(b) Steady State Identities
Figure 4: Steady State with Rigid Leader
values of investment, human capital as well as group identification can be seen in table2. We also show for this case the payoff of the group leader. We find that the leader’s payoff in the ring structure
Network Agent Investment Human Capital Group Identification Payoffs Leader
Star Agent 2 0.0324 .3242 0.4050 .1845
Agent 3 0.0089 .0886 0.6799 .1012
Ring Agent 2 0.0320 .3199 0.4145 .1875
Agent 3 0.0090 .0898 0.6681 .1001
Table 2: Rigid Leader: Steady State for γ = 0.9
is overall higher than the payoff in the star structure. The leader gains less from the low ability type in the ring compared to the star, but this is compensated by the increased gain from the high ability type. Therefore a group leader would prefer a more integrated community where individuals can influence each other instead of a more dispersed one.
5.2.2 Adaptive Leader
Next, we consider the case when the leader is influenced by the group members. In this case beliefs converge to zero independently of the network structure, as can be seen from the convergence paths in figure5. The time until convergence depending on group leader influence is given in figure6.
0 5 10 15 20 25 30 35 40 0.02
0.04 0.06 0.08 0.1 0.12 0.14 0.16
Time Period
Investment Level
Convergence Path of Investment
Line: Agent 2 Line: Agent 3 Ring: Agent 2 Ring: Agent 3
(a) Convergence in Investment
0 5 10 15 20 25 30 35 40
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time Period
Group Identity
Convergence Path of Group Identity
Line: Agent 2 Line: Agent 3 Ring: Agent 2 Ring: Agent 3
(b) Convergence in Group Identities
Figure 5: Convergence of Investment and Group Identities with Adaptive Leader
With an adaptive leader the time of convergence is highest when group leader influence is high and low when group members give little weight to the group leader. As there is full integration in the long run, it follows that a high weight on the environment leads to full integration more quickly. The
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
15 20 25 30 35 40 45 50 55 60
Influence of Leader on Group
Convergence Time
Convergence of Investment
Star: Agent 2 Star: Agent 3 Circle: Agent 2 Circle: Agent 3
(a) Convergence Time Investment
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
15 20 25 30 35 40 45 50
Influence of Leader on Group
Convergence Time
Convergence of Beliefs
Star: Agent 2 Star: Agent 3 Circle: Agent 2 Circle: Agent 3
(b) Convergence Time Group Identities
Figure 6: Convergence Time of Investment and Group Identities depending on Influence of Adaptive Leader
steady state investment level is independent of γ as well as the network configuration, which reflects the fact that all levels of γ result in the same steady state level of integration, determined solely by the influence of the host society. For our specific example we obtain a human capital stock of .64 for the high ability workers and one of .25 for the low ability ones. The investment levels for the high types is .064, for the low types .025.
6 Conclusion
In this paper we develop a model of immigrant integration combining cultural integration through social influence with economic integration through investment in skills acquisition and conduct a study of the properties of the resulting integration process and how it is influenced by the character- istics of individuals as well as the group structure. In the absence of opinion leaders in the immigrant community which we model as autonomous individuals or institutions which do not adapt cultur- ally, the group members eventually integrate fully and adopt a human capital level consistent with that achieved by natives. However, if there are leaders which do not adapt, the integration process remains incomplete as the long run level of group identity is the result of a balance of the influence of both leaders and the wider host society. At this point, economic integration is also incomplete in the sense that human capital levels and consequently wages are lower than they would be for natives.
The analysis indicates that individual characteristics influence steady state outcomes in the expected way: higher ability members have greater incentives to acquire skills and thus end up being more integrated than low ability members. We also consider the impact of social structure on integration by comparing a very centralized shape (star) with a more dispersed configuration (ring). We find that the star structure exaggerates the divergence between the levels of integration of high and low ability agents.
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