  Opportunities such as higher education can promote intergenerational
mobility, leading individuals to achieve levels of socioeconomic status above
that of their parents. We develop a dynamic model for allocating such
opportunities in a society that exhibits bottlenecks in mobility; the problem
of optimal allocation reflects a trade-off between the benefits conferred by
the opportunities in the current generation and the potential to elevate the
socioeconomic status of recipients, shaping the composition of future
generations in ways that can benefit further from the opportunities. We show
how optimal allocations in our model arise as solutions to continuous
optimization problems over multiple generations, and we find in general that
these optimal solutions can favor recipients of low socioeconomic status over
slightly higher-performing individuals of high socioeconomic status -- a form
of socioeconomic affirmative action that the society in our model discovers in
the pursuit of purely payoff-maximizing goals. We characterize how the
structure of the model can lead to either temporary or persistent affirmative
action, and we consider extensions of the model with more complex processes
modulating the movement between different levels of socioeconomic status.
