Does increasing diversity in university-intake require sacrificing academic performance, and if so, by how much? We develop an empirical framework to explore this trade-off ex-post, using admissions data matched with post-admission academic outcomes. We propose a simple, theoretical model of admissions for a university that values both future academic performance and diversity, and faces capacity-constraints. We show that the implicit weight on equity vis-a-vis expected future performance in the university's objective-function is captured by the ratio of inter-group difference in the admission-rate and that in the post-entry academic performance of marginal entrants. The problem of identifying marginal entrants can be mitigated using performance data for students admitted from waitlists, leading to bounds for the relative weights. These bounds (a) hold irrespective of whether researchers observe all applicant characteristics known to admission officers and (b) require no information about rejected candidates, who are typically not followed up. We apply this idea to admissions data from Cambridge, using scores on blindly-marked post-admission exams as the performance metric. In mathematical subjects, where female enrolment is relatively low, we and robust evidence that improving gender-balance requires significant performance sacrifice, and conclude an implicit weight of at least 10-20% on gender-equity in the university's objective function. There is no evidence of such trade-off in equally competitive non-mathematical subjects and, contrary to popular perception, for applicants' school-type. Our methods and results illustrate a formal way to quantify ex-post efficiency costs of diversity in a context where societal objective encompasses both equity and efficiency.