A feature present in countries with a National Health Service is the coexistence of a public and a private sector. Often, the public payer contracts with private providers while holding idle capacity. This is often seen as inefficiency from the management of public facilities. We present here a different rationale for the existence of such idle capacity: the public sector may opt to have idle capacity as a way to gain bargaining power vis-à-vis the private provider, under the assumption of a more efficient private than the public sector.