We highlight that convergence in repeated averaging models commonly used to study cultural traits or opinion dynamics is not equivalent to convergence in Markov chain settings if transition matrices are time-varying. We then establish a new proof for the convergence of cultural traits in the model of Panebianco (2014) correcting the existing proof. The new proof provides novel insights on the long-run outcomes for inessential individuals. We close with a discussion of conditions for convergence in repeated averaging models with time-varying transition matrices.
We explain persistent differences in cultural traits of immigrant groups with the presence of community leaders. Leaders influence the cultural traits of their community, which have an impact on the group's earnings. They determine whether a community will be more assimilated and wealthier or less assimilated and poorer. With a leader, cultural integration remains incomplete. The leader chooses more distinctive cultural traits in high-productivity environments and if the community is more connected. Lump-sum transfers to immigrants can hinder cultural integration. These findings are in line with integration patterns of various ethnic and religious groups.