In this paper, I analyze the determinants of college enrolment and the changes in these determinants over time. I propose a quantitative life-cycle model with college enrolment. Altruistic parents provide financial support to their children. Using counterfactual experiments, I find that 24 percent of all households are financially constrained in their college decision. Constraints become more severe over time. I show that my model is consistent with a narrow college enrolment gap between students from rich and poor families, as previously reported in the empirical literature. The estimation of enrolment gaps is a popular reduced-form approach for measuring the fraction of constrained households. My results suggest that these reduced-form estimates are misleading, and that a structural model of parental transfers is needed to correctly identify constrained households. Further, I show that parental transfers are an important driver behind the changing role of family income as a determinant of college entry, a fact that is well documented for the US economy.