University managers should be aware of the importance that efficiency has for their own universities, orientating their actions towards research and teaching excellence. This study estimates teaching and research efficiency of the different departments of a university and tests the complementariness versus trade-off between them. The results obtained show high levels of efficiency for the departments analysed, which is important for budget allocation, as one of the main challenges faced by a university is the allocation of budgets to different departments. The results also support the trade-off between teaching efficiency and research activity, the complementariness between teaching efficiency and research efficiency, between research efficiency and time deducted from teaching, as well as the complementariness between research efficiency and administration activity.