Will all people become alike if they are alike?

We have examined the proposition that initial differences in per capita wealth are eliminated across generations if people face the same opportunities and behave alike. Although we may not rule out that this intrinsic identity of individuals is, indeed, an equalizing force under certain circumstances we have analysed a model in which this is not necessarily true. Apart from differences in initial endowments all agents in this model receive the same wage and the same rental, they have the same number of children, and they possess the same utility function. This function reflects the satisfaction agents derive from both life time consumption and the legacy they leave to their children. Even if this structure remains unchanged for a long sequence of generations the offsprings of the poor may stay poor relative to the offsprings of the rich. The society may end up with as many classes as it started with. We have identified a family of multi class societies which exhibit long run stability. This family is characterized by a large average per capita wealth and relatively small differences between rich and poor. On the other hand poor societies with large differences in individual wealth are dynamically unstable. They have a chance of eventually belonging to the former family of economies.