Standard insurance models predict that people with high 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 in an insurance model where people choose treatment intensity. We analyse different market structures and show the following: if insurers have market power, the violation of single crossing caused by income differences and endogenous treatment choice can explain the empirically observed outcome. Our results do not rely on differences in risk aversion between types.