We model competitive insurance markets with continuous cost-types. A regula tor sets minimum and maximum coverage levels and a fee for non-buyers. Equilib rium is unique if the type distribution is log-concave. Increasing the non-purchase
fee increases welfare if the density of types is decreasing. The optimal level of the
minimum coverage is positive, below full insurance and induces some pooling at
the minimum coverage contract. The optimal level of the maximum coverage is full
insurance, even in an extension that allows for ex-post moral hazard.