Persistent Private Information Revisited

This paper revisits Williams’ (2011) continuous-time model of optimal dynamic insurance with persistent private information and corrects several errors in that paper’s analysis. We introduce and study the class of self-insurance contracts that are implementable as consumption-saving problems for the agent with constant taxes on savings chosen by the principal. We show that the contract asserted to be optimal in Williams (2011) is the special self-insurance contract with zero taxes. When the agent’s private endowment is mean-reverting, that contract is strictly dominated by the optimal self-insurance contract, which imposes a strictly positive tax, induces immiseration when the rate of mean-reversion is high, and sends the agent to bliss when the rate of mean-reversion is low. When the agent’s endowment is not mean-reverting, the contract derived in that paper is, in fact, optimal among all incentive compatible contracts; we provide a new explanation for its properties in terms of the agent’s indifference among all reporting strategies. These results extend to the natural discrete-time analogue of the model. Separately, Williams’ (2011) first-order approach to incentive compatibility relies on an erroneous and unjustified assumption on the space of feasible reporting strategies; our analysis does not. (∗) Subsections 3.3 and 4.2 of this paper build on Strulovici (2011) (the working paper version of Strulovici 2020) and subsume its analysis of the long-run properties of state-consistent contracts. Subsection 5.2 of this paper expands on and subsumes Section 6.2 of Bloedel, Krishna, and Leukhina (2018) (the working paper version of Bloedel, Krishna, and Leukhina 2020). We thank Arash Fahim and Ilya Segal for several helpful conversations. (†) Stanford University. Email: abloedel@stanford.edu (‡) Florida State University. Email: rvk3570@gmail.com (§) Northwestern University. Email: b-strulovici@northwestern.edu