Opportunities such as higher education can promote intergenerational mobility, leading individuals to achieve levels of socioeconomic status above that of their parents. In this work, which is an extended abstract of a longer paper in the proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, 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.