University access has significantly expanded in OECD countries, and further growth figures prominently in political agendas. We study possible consequences of historical and future expansions in a stochastic, general equilibrium Roy model where tertiary educational attainment is determined by cognitive ability and socioeconomic disadvantage. In our analysis, individual productivity depends not only on education but also directly on cognitive ability. The expansion of university access in the UK that started in the 1960s provides an ideal case study to draw lessons for the future. We find that this expansion led to the selection into college of progressively less talented students from advantaged backgrounds. Appropriate counterfactual policies existed that would have achieved the dual goal of increasing college graduates’ cognitive ability while improving tertiary education opportunities for the disadvantaged.