We study the dynamics of continuous cultural traits (as a specific type of continuous opinions) in an OLG (overlapping generation) structure and in an endogenous social network, where the network changes are inherited. Children learn their cultural trait from their parents and their social environment modelled by network. Parents want their children to adopt a cultural trait that is similar to their own and engage in the socialization process of their children by forming new links or deleting connections. Changing links from the inherited network is costly, but having many links is beneficial. We propose three ways to endogenize the process of network formation. In the first one, the network is supposed to be directed and each dynasty can either form or delete a directed link unilaterally with another dynasty. Therefore, at each period, each family faces a utility opitimization problem where a trade-off between own utility losses and the improvements of child's cultural trait. We have shown that if the cost of network changes is greater than the cost of child care, extremists will never add links, and in the case of sufficiently low cost, extremists may cut all ties with the society. In the second and third models, the network is supposed to be undirected. In the second model, we assume that after each period, a pairwise stable network (PS network for short) is reached. In this case, there always exist sufficiently small cost parameters such that the empty network is the unique PS network. In the third model, we assume that after each period, a pairwise stable network with transfers (PST network for short) is reached. We have shown the existence of the PST network for each period, however, it is not necessary to be unique. Moreover, a necessary and sufficient condition is given such that a network is PST for given V(t) and G(t). The convergence of cultural traits in this case is guaranteed. Regarding the efficiency of the network, we show that there always exist sufficiently small cost parameters such that the empty network is the unique efficient network, and sufficiently large costs of child care such that the complete network is the unique efficient network. Moreover, more detailed dynamics of cultural traits are studied when the costs of network changes and benefits from integration are low, intermediate, and large, respectively
