This paper studies public provision of long-term care insurance in a world in which family assistance is (i) uncertain and (ii) endogenous, depending on the time parents spend raising their children. Public benefits will be paid in case of disability but cannot be combined with self-insurance or family aid. The benefits are provided equally to all recipients and financed by a proportional payroll tax. The paper shows that tax distortions imply that full insurance is undesirable. It characterizes the optimal tax and identifies the elements that determine its size. Of crucial importance are the extent of under-insurance, the effect of the tax on the probability of altruism, the distortionary effect of the tax, and, with wage heterogeneity, the covariance between the social marginal utility of lifetime income and (i) earnings (positive effect) and (ii) the probability of altruism default (negative effect).
Helmuth Cremer and Pierre Pestieau gratefully acknowledge financial support from the Chaire “Marché des risques et creation de valeur” of the FdR/SCOR. This paper has been presented as the CESifo Summer Institute “The Economics of Long Term Car”. We thank all the participants and, particularly our discussant Gregory Ponthière for their insightful comments and suggestions. We are also grateful to the referees and the editor, Antonio Cabrales, for their remarks.
Derivation of 
First, totally differentiate  with respect to z to get an expression for We have,
Derivation of 
Substitute the expressions for and from  and  in the last bracketed expression on the right-hand side of , while also substituting for from the government’s budget constraint , and for from eq. . We have
The sign of
To obtain , we must have which in turn requires
We also have
which is not necessarily satisfied.
Derivation of expression 
Using the operator E, this yields .
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If s is “too large,” ; this is uninteresting and we will rule it out.
The companion paper Cremer, Gahvari, and Pestieau (2012b) takes a closer look at the case where optimal policy implies a level of , such that even altruistic children decide to provide no aid. This occurs when the parameter of altruism, , is small.
Observe that an interior solution for s is ensured as long as marginal utility of consumption is high at “low” consumption levels. However, to have an interior solution for T, it must be the case that That is,
Means testing otherwise be a way to implement nonlinear policies. The design of nonlinear LTC has been studied in a different setting by Pestieau and Sato (2009).
This term is the same as the expression  in the homogeneous individuals case, only indexed by parent i.
In other words, it is not the second-order derivative of a conventionally defined expenditure function.
Our expressions are valid irrespective of the sign of this term. When it is negative the interpretation changes even though the basic logic remains the same. To avoid repetition and since this case does not appear to be intuitively appealing, we have decided not to address it explicitly.
In the previous section we have shown that is always true with homogenous individuals. With heterogeneous individuals over-insurance () can strictly speaking no longer be ruled out (especially when heterogeneity is very significant so that redistributive benefits of z are very large). However, such overinsurance is in any even hard to reconcile with our assumption that altruistic children help their dependent parents. Which overinsurance in public provision, this would only be possible for a “very large” level of . None of these complications appear to be intuitively or empirically appealing and so we assume .
Recall that we assume .
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