This paper introduces a new approach to "cultural transmission of preferences" (see Bisin and Verdier, 2000, 2001). It is based on the conceptualization of the culture of a person as a set of cultural values and attitudes, represented as an n-dimensional tuple in Euclidean space. The culture of a person is formed through social learning and imitation from role-models, which correspond to the chosen "displayed cultures" of parents ("vertical transmission") and the society at large ("oblique transmission"). Parents might choose a "displayed culture" that does not coincide with their true culture, since they aim at countervailing negative cultural influences that their children are exposed to in the society at large. Additionally, they can invest into the success that their displayed culture has in the socialization process of their children. We will consider in the present paper an OLG model with two cultural groups, and where in any period, the members of each of the cultural groups have identical culture. We show that if parents have a desire for cultural closeness to their children (e.g. "imperfect empathy"), then they will always behave culturally more "radical" relative to the culture of the other cultural group. Furthermore, they will always invest into their socialization success. Nevertheless, these investments are never sufficient to let the distance between the future cultures of the children of both cultural groups be larger or equal than the cultural distance of the parental generation. As a consequence, the cultures of both groups converge to a homogeneous steady state culture, which can be interpreted as a mixture of the two initial cultures. This result corresponds to the "melting pot" theory of integration of cultural groups.