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28.03.2021 | Original Paper

# Fair long-term care insurance

verfasst von: Marie-Louise Leroux, Pierre Pestieau, Gregory Ponthiere

Erschienen in: Social Choice and Welfare | Ausgabe 3/2021

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## Abstract

The study of optimal long-term care (LTC) social insurance is generally carried out under the utilitarian social criterion, which penalizes individuals who have a lower capacity to convert resources into well-being, such as dependent elderly individuals or prematurely dead individuals. This paper revisits the design of optimal LTC insurance while adopting two egalitarian social criteria—ex ante and ex post egalitarianism—which give priority to the worst-off either in expected or in realized terms. Using a lifecycle model with risk about the duration of life and risk about old-age dependence, it is shown that the optimal second-best insurance under the ex ante egalitarian criterion involves, in comparison to utilitarianism, higher LTC and pension benefits and higher tax rates on savings and on labor earnings. In comparison to ex ante egalitarianism, ex post egalitarianism involves lower LTC and pension benefits, a higher tax on savings and a lower tax rate on labor earnings, in order to increase consumption of the young, who include persons who will turn out to be short-lived.

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Fußnoten
1
According to Norton (2000), about 2/3 of LTC is provided informally. However, the share of the informal care may decrease over time, and various scenarios exist regarding the extent of the decrease of informal care with respect to formal care.

2
On the various possible explanations of that puzzle, see Brown and Finkelstein (2011), Pestieau and Ponthiere (2011), Boyer et al. (2020) and Ameriks et al. (2018).

3
Some European countries, like France and Germany, already have implemented a public LTC transfer in case of old-age dependency. However, it remains quite small compared to the amount of LTC spending that agents may have to pay in case of dependency.

4
Note that this limitation of utilitarianism is not specific to an economy with LTC needs. Actually, as stressed in Sen and Williams (1982), utilitarianism also faces problems in its treatment of handicapped persons, who have also a different capacity to convert resources into well-being in comparison with non-handicapped persons.

5
Data are obtained from SHARE (waves 2, 4 and 6). See Lefebvre et al. (2018). Note also that the above inequalities imply that $$p_{H}<p_{L}$$.

6
Assuming time-additive preferences is made for the sake of analytical simplicity. Relying on more general preferences does not prevent the compensation for unequal lifetimes, as long as there exist positive critical consumption levels (i.e. the thresholds $${\bar{c}}$$ and $${\tilde{c}}$$ defined in this section) making individuals indifferent between life and death (see Fleurbaey et al. 2014).

7
For simplicity, we abstract here from time preferences. Note, however, that the survival probability $$\pi _{i}$$ acts as a biological discount factor, so that introducing pure time preferences would be somewhat redundant in this framework.

8
That assumption is made for simplicity. It enables us to eliminate income effects and to have a simple expression for the labor supply as a function of the wage only.

9
This normalisation of the utility of death to zero is standard in the literature. Assuming instead that it is equal to a (finite) constant would not change our results.

10
It should be stressed that this assumption involves a simplification of reality. In real life, there are many consumption goods and services, and one cannot exclude that the marginal utility of consumption of some goods or services may be higher under dependency than under autonomy. However, our model is a one-good economy and, in that simplified context, assuming that dependency reduces the marginal utility of consumption is a plausible proxy.

11
This assumption is standard in models where longevity is risky. Without annuities, there would be accidental bequests, which would have to be dealt with within an intergenerational model. Introducing this would lead us far beyond the scope of this paper.

12
In our framework, a higher hourly wage not only extends consumption possibilities ceteris paribus, but is also associated with a higher survival probability, which decreases the return of savings under perfect annuities, and, hence, consumption possibilities ceteris paribus. A sufficient condition for having the intuitive case where type-H individuals are, ceteris paribus, better off ex ante than type-L individuals, is that the gap in survival probabilities $$\pi _{H}$$ and $$\pi _{L}$$ is relatively small in comparison with the gap in $$w_{H}$$ and $$w_{L}$$.

13
Without that assumption, it could be the case that type-L individuals could turn out to be better off ex post than type-H individuals with the same longevity and old-age health status, a counterintuitive case.

14
Some ethical doctrines assume consequentialism and welfarism, but give up sum-ranking. This is the case for the criterion proposed by Atkinson and Stiglitz (1980), which includes some degree of inequality aversion across individual utilities. One could also sum up utilities renormalized in such a way as to give priority to some (disadvantaged) members of the population. This section focuses on the standard version of utilitarianism, based on postulates (1) to (3), which is the most widespread one.

15
This result is related to the use of quasi-linear first-period utility. Other preferences, including a separate term for the disutility from labor, would lead, for a given longevity/health status type, to unequal ex post well-being under utilitarianism.

16
Since pension/LTC benefits are flat, we have a pure Beveridgian insurance system. On the contrary, we could have introduced a link between past contributions and pension/LTC benefits and have a Bismarkian system. This would not change qualitatively our results. Only the size of the redistribution towards type-L agents would be smaller.

17
Since private and public insurance are substitutes, interior solutions for $$s_{i}^{*}$$ and $$a_{i}^{*}$$ will only be observed for low levels of public benefits, $$\psi$$ and g. In the following, we assume that this is the case. This is a strong assumption particularly for low-income individuals, but it allows us to obtain intuitive results. Assuming otherwise would not qualitatively change our results.

18
Note that the objective of equalizing expected lifetime utilities could also be defended without relying on the Principle of Compensation, by defining social justice in terms of equality of well-being prospects (i.e. whatever the nature of the determinants of well-being is).

19
About 1/4 to 1/3 of longevity inequalities within a cohort are due to the genetic background, a pure circumstance (see Christensen et al. 2006). Contoyannis and Jones (2004) and Balia and Jones (2008) showed that about 25% of inequalities in lifetime are due to lifestyles (eating behavior, drinking behavior, etc.), but those lifestyles are acquired through a socialization process within the family, which is, again, a pure circumstance.

20
Regarding the risk of old-age dependence, Farrer et al. (1997) show that there exists a statistical association between apolipoprotein e genotype and the risk of Alzheimer disease.

21
Note that the social objective of equalizing ex post lifetime utilities could also be defended without relying on the Principle of Compensation, for instance by defining social justice in terms of equality of realized well-being among all individuals (independently of the nature of factors—efforts or circumstances—determining realized well-being levels).

22

23
Indeed, the egalitarian constraint is such that $$u\left( d^{e}\right) =v\left( b^{e}\right) =0$$, and, given that $$u\left( d\right) >v\left( d\right) \,\forall d$$, it implies that $$b^{e}>d^{e}$$.

24
It is possible to show that, at the second-best optimum, the egalitarian constraints are binding. We can prove this by contradiction. Assume instead that the egalitarian constraints are not binding. For given levels of $$\psi$$ and g, in order to increase the welfare of the short-lived, it is optimal to increase $$\sigma$$ so as to tax away private savings and LTC insurance, as well as to decrease $$\tau$$. From the resource constraint of the government, the decrease in $$\tau$$ stops when $$\psi$$ and g are fixed at their minimum, that is when the egalitarian constraints are binding. This is true as long as $${\bar{c}}$$ and $${\tilde{c}}$$ are not too large (a weak assumption).

25
Since savings and private LTC insurance are taxed away, individuals choose neither to save not to invest in a private insurance so that taxation of aggregate savings does not appear in the government budget constraint.

26
See Tables 1 and 6, in INSEE (2015). We consider that high-qualification and intermediary occupations as type-H individuals, and clerks and manual workers as type-L individuals.

27
See Table 2 (male column) of Cambois et al. (2011). Using the weights for the different socio-demographic categories used in the previous step, we aggregate life expectancy at 65 and life expectancy at 65 without daily-life restrictions for each type H and L, and obtain probabilities by dividing these by the length of the period (i.e. 40 years).

28
$$\beta$$ solves: $$\beta =\alpha -\delta \log (d)+\log (0.53d)$$ with $$\alpha$$ and $$\delta$$ already obtained in previous steps and where $$d=w_{L}/2$$ (i.e. half of the income of the low wage individual).

29
For simplicity, our calculations assume that individual labor supply is fixed. This implies some minor changes on the government budget constraints and on the first-order condition with respect to $$\tau$$ at the second-best utilitarian optimum. Relaxing that assumption would not modify our results substantially.

30
Note that, at the utilitarian and the ex ante egalitarian second-best optima, we have corner solutions: $$s_{H}^{*}>0$$ while $$s_{L}^{*}=0$$ and $$a_{i}^{*}=0$$ $$\forall i$$.

31
Note also that, in line with our theoretical results, utilitarianism and ex ante egalitarianism give priority to the LTC benefit over the pension benefit.

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Titel
Fair long-term care insurance
verfasst von
Marie-Louise Leroux
Pierre Pestieau
Gregory Ponthiere
Publikationsdatum
28.03.2021
Verlag
Springer Berlin Heidelberg
Erschienen in
Social Choice and Welfare / Ausgabe 3/2021
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-021-01324-z

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