Equilibrium and Welfare in Insurance Markets with Time-Inconsistent Consumers

This paper studies the implications of unobservable consumer naivete on selection and welfare in a competitive insurance market with moral hazard. We identify two distinct forces of sophistication driving the formation of market equilibriums. One is that sophistication helps individuals internalize the incentive to take high effort by adopting some commitment devices, and hence sophisticates are willing to pay a lower premium to increase insurance coverage than naifs. The other is that sophistication helps prevent the optimistic underestimation of loss probability, and thus sophisticates are willing to pay a higher premium to increase insurance coverage than naifs. We show that the above forces of sophistication can produce adverse selection, advantageous selection and the absence of risk-coverage correlation observed in various insurance markets. We also show that sophisticates can adopt some commitment devices to alleviate the externality caused by hidden naivete, and certain incentive insurance clauses such as rebate check are welfare-improving when advantageous selection appears.