This paper analyzes the impact of two competing strategies (Cournot vs Bertrand competition) in a context where universities compete for students and can enroll beyond capacity at strictly convex costs. Universities interact in a two-stage game to decide on capacity and tuition fees/number of students enrolled. In Bertrand competition, when costs are sufficiently convex, universities adopt low capacities in the first stage in order to sustain high fees in the second stage. Conversely, Cournot competition leads to a higher capacity for each university and to a larger number of students enrolled. Under certain conditions, the equilibrium adopted in Bertrand competition may be more efficient in terms of cost minimization but it leads to a lower social welfare level.