While college enrollment has more-than doubled since 1970, elite colleges have barely increased supply, instead reducing admit rates. We show that straightforward reasons cannot explain this behavior. We propose a model where colleges compete on prestige, measured using relative selectivity or relative admit rates. A key comparative static of the model is that higher demand decreases [increases] the admit rate when the weight on prestige is above [below] a critical value, consistent with experience in elite [non-elite] colleges. A calibrated version of the model closely replicates the pattern in the data of declining admit rates at elite colleges while counter-factual simulations without prestige fail. Prestige competition is inefficient. Allowing elite colleges to collude on admissions strategy internalizes the non-pecuniary prestige externality and is Pareto improving.