Recurrence relationships are used to relate breeding values of age-sex classes from different time periods. Their application to single-stage (progeny) and multi-stage (parent) selection is demonstrated. These relationships enable definition of the effect of age structure, initial genetic differences between age groups, and the extent to which allowances are made for these or later genetic differences between age groups. The expressions derived show that, given initial genetic differences between age groups, subsequent progeny means will fluctuate even under completely random selection. Using these means as a basis for measuring responses to selection, it is shown that there can be selective effects where selection is at random within parental age classes. A careful definition of the alternative programme is therefore important in interpreting results of selection experiments and in investment appraisal of selection programmes.

These models were then used to describe economic returns from parent and progeny selection programmes and from programmes in which returns are realized in more than one age group. A further extension of the model accommodates the effects of finite population size on returns through its effects on genetic variance.

By separating the (constant) within- and (fluctuating) between-group components of the selection differential within the recurrence relationships a number of computational problems are overcome.