Long term care insurance with state-dependent preferences

We study the demand for Long Term Care (LTC hereafter) insurance in a setting where agents have state-dependent preferences over both a daily life consumption good and LTC expenditures. We assume that dependency creates a demand for LTC expenditures while decreasing the marginal utility of daily life consumption, for any given consumption level. Agents optimize over their consumption of both goods as well as over the amount of LTC insurance (LTCI). We first show that some agents optimally choose not to insure themselves, while no agent wishes to buy complete insurance, in accordance with the so-called LTCI puzzle. At equilibrium, the transfer received from the insurer covers only a fraction of the LTC expenditures. The demand for LTCI need not increase with income when preferences are non state-dependent or insurance is actuarially unfair. Also, preferences have to be state-dependent with no insurance bought to rationalize the empirical observation of a higher marginal utility at equilibrium when autonomous. Finally, focusing on iso-elastic preferences, we recover the empirical observation that health/LTC expenditures are not very sensitive to income, and we show that LTCI as a fraction of income should decrease with income and then become nil above a threshold.