Self-protection and insurance demand with convex premium principles

In economic analysis, rational decision-makers often take actions to reduce their risk exposure. These actions include purchasing market insurance and implementing prevention measures to modify the shape of the loss distribution. Under the assumption that the insureds' actions are fully observed by the insurer, this paper investigates the interaction between self-protection and insurance demand when insurance premiums are determined by convex premium principles within the framework of distortion risk measures. Specifically, the insured selects an optimal proportional insurance share and prevention effort to minimize the risk measure of their end-of-period exposure. We explicitly characterize the optimal combination of prevention effort and insurance demand in a self-protection model when the insured adopts tail value-at-risk or a subclass with strictly concave distortion functions. Additionally, we conduct comparative static analyses to illustrate our main findings under various premium structures, risk aversion levels, and loss distributions. Our results indicate that market insurance and self-protection are complementary, supporting classical insights from the literature regarding corner insurance policies (i.e., null and full insurance) in the absence of ex ante moral hazard. Finally, we consider the effects of moral hazard on the interaction between self-protection and insurance demand. Our findings show that ex ante moral hazard shifts the complementary effect into substitution effect.