Let us revisit Exercise 11 and consider a family from the middle class. In which generation on the average will the descendants of this family at the first time belong to the upper class?

Exercise 11

It concerns the transition of individuals between social classes defined on the basis of income or occupation. Sociologists distinguish inter-generational mobility from parents’ class to child’s class and intra-generational mobility over an individuals life course. The particular model below illustrates the former case. For the sake of simplicity, assume that in each generation, the members of any family belong to the same class, so we are talking about family transitions.

Consider three classes: 0 (upper), 1 (middle), and 2 (lower). For a family chosen at random, let X_t be the family’s social class in the t-th generation. Suppose that X_t is Markov chain with the transition matrix

(In the literature, one may find various figures, and the reader should not view the numbers above as real.) Suppose that in a particular year, the percentages of the families belonging to the classes mentioned is given by the vector (20%, 50%, 30%). Estimate the same vector for the great-grandchildren of the children born in this year.