We focus on analysing cost minimising production behaviour in multi-output settings. We distinguish between two approaches for modelling the use of joint inputs. The cooperative approach assumes cost minimisation at the aggregate firm level, while the non-cooperative approach assumes cost minimisation at the level of the individual output division. Our framework extends the existing non-parametric framework for analysing single output production. An empirical application to the English and Welsh drinking water and sewerage sector shows the practical usefulness of our framework. Specifically, we compare the empirical validity of the cooperative and non-cooperative models for describing the observed production behaviour.