To answer this question, a recent NBER working paper by Frankes, Gruber and Justicz (2020) examine the U.S. military health system. What is the Military Health Service (MHS)?
The MHS is a $50.6 billion/year program that provides care to active-duty military, their dependents, and military retirees, covering over 9 million eligible individuals. Crucially, MHS beneficiaries have access to government-owned and run facilities on military bases, as well as private providers that are contracted to the military through an insurance company. Care is split roughly equally between the two sources; 49 percent of outpatient encounters and more than 67 percent of hospitalizations for MHS beneficiaries take place with private providers
An appropriate identification of the impact of public vs. private provision of health service would require both variation in the receipt of services across public and private providers and a mechanism to randomize people across these two provider types. The authors use exogenous moves (i.e., base transfers) as the source of this variation as captured in the Military Health System Data Repository. Specifically, they use this source of variation to compare quality of inpatient care for childbirth:
We consider every mother in our sample who gives birth at least twice. We then restrict our sample to all mothers who move to a new base between births, and we compare those mothers who see a change in the availability of military hospitals. That is, we compare mothers who move but both before and after the move do (or do not) have a military hospital nearby, to mothers who move and who see a change in their nearby access to military hospitals.
The authors then find the private care–while slightly more expensive–is higher quality and cost-effective.
We find that mothers delivering off-base use more resources than those delivering on-base; total resource utilization appears to be about 1% higher for those using the private- rather than public-care system, primarily driven by higher Cesarean section rates off-base. At the same time, we find that the quality of care appears to be significantly higher for mothers delivering and receiving prenatal care off-base. We find that mothers and babies receiving off-base care have fewer complications and incidence of maternal or neonatal trauma. Our results suggest that, at current levels, shifting childbirth from on- to off-base is likely to be cost effective.
Although more research is needed beyond just the study of childbirth, this study represents and important finding regarding the debate over whether public vs. private provision of health care services is preferred.