In the case of elderly care services in Sweden, the answer is ‘yes’. In Bergman et al. (2016), the authors…
assemble a large data set on elderly care services in Sweden between 1990 and 2009 and estimate how opening to private provision affected mortality rates – an important and not easily contractible quality dimension – using a difference-in-difference-in-difference approach. The results indicate that privatization and the associated increase in competition significantly improved non-contractible quality as measured by mortality rates.
Clearly, privatization does not guarantee improved quality in all countries and in all care settings, but this study provides some evidence that competition in the health care field results in better care for patients.