The design of insurance contracts for home versus nursing home long-term care

We study the design of optimal (private and/or social) insurance schemes for formal home care and institutional care. We consider a three period model. Individuals are either in good health, lightly dependent or heavily dependent. Lightly dependent individuals can buy formal home care which reduces the severity of dependency and reduces the probability to become severely dependent in the next period. Severely dependent individuals pay for nursing home care. In both states of dependency individuals can receive a (private or public) insurance benefit (transfers). These benefits can be flat or depend on the formal care consumed (or a combination of the two). These benefits are financed by a premium (or a tax). Individuals may be alive until the end of period 2 or die at the beginning of periods 1 or 2 with a certain probability which may depend on their state of health. The laissez faire is inefficient because individuals consume a too low level of formal home care and are not insured. The first-best insurances scheme requires a transfer to lightly dependent individuals that, (under some conditions) increases with the amount of formal home care consumed. Severely dependent individuals, on the other hand, must receive a flat transfer (from private or social insurance). The theoretical analysis is illustrated by a calibrated numerical example which show that the expressions have the expected sings under plausible conditions.