The paper describes the MilaNLP team's submission (Bocconi University, Milan) in the WASSA 2021 Shared Task on Empathy Detection and Emotion Classification. We focus on Track 2 - Emotion Classification - which consists of predicting the emotion of reactions to English news stories at the essay-level. We test different models based on multi-task and multi-input frameworks. The goal was to better exploit all the correlated information given in the data set. We find, though, that empathy as an auxiliary task in multi-task learning and demographic attributes as additional input provide worse performance with respect to single-task learning. While the result is competitive in terms of the competition, our results suggest that emotion and empathy are not related tasks - at least for the purpose of prediction.