Hospital markets are often characterised by price regulation and the existence of different ownership types. Using a Hotelling framework, this paper analyses the effect of heterogeneous objectives of the hospitals on quality differentiation, profits, and overall welfare in a price regulated duopoly with exogenous symmetric locations. In contrast to other studies on mixed duopolies, this paper shows that in this framework privatisation of the public hospital may increase overall welfare. This holds if the public hospital is similar to the private hospital or less efficient and competition is low. The main driving force is the single regulated price which induces under-(over-)provision of quality of the more (less) efficient hospital compared to the first-best. However, if the public hospital is sufficiently more efficient and competition is fierce, a mixed duopoly outperforms both a private and a public duopoly due to an equilibrium price below (above) the price of the private (public) duopoly. This medium price discourages overprovision of quality of the less efficient hospital and - together with the non-profit objective - encourages an increase in quality of the more efficient public hospital.