Many optimal contracting papers use quasi-linear preferences. To exclude stochastic mechanisms they impose a (su cient) condition on how the curvature of an agent's objective function varies with type. We show with quasi-linear preferences that an optimal deterministic outcome without bunching...

Standard insurance models predict that people with high (health) risks have high insurance coverage. It is empirically documented that people with high income have lower health risks and are better insured. We show that income differences between risk types lead to a violation of single crossing...