Should health care provision be public, private, or both? We look at this question in a setting where people differ in their earnings capacity and express an inelastic demand for health care. We assume that illness reduces a person’s health status when not receiving immediate treatment. Treatment can be obtained in a competitive private sector or in the National Health Service (NHS) where it is provided free of charge but after some (endogenous) waiting time. The equilibrium in the health care sector consists of a fee for private consultations and a contract offered to NHS physicians such that no physician has an incentive to switch place of work, and in addition a waiting time in the NHS such that no patient wants to switch health care provider. This equilibrium is governed by three public policies: the income tax system, the subsidy of private health care, and the terms of the contract offered to NHS physicians. Our findings are threefold. First, a mixed system with a small public health care sector gives a lower social welfare level than a pure private system. Second, a mixed system with a sufficiently large NHS may improve upon a pure private system if the distribution of earnings capacities in society is sufficiently wide. And finally, whatever the size of the NHS, the doctors working there should be given a contract that specifies a heavier work load than the one their private colleagues choose. This is the way the government can exert its monopsony power on the market for physicians to improve social welfare.