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Fair long-term care insurance

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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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Notes

  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. On this result, see also Fleurbaey et al. (2014).

  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.

References

  • Ameriks J, Briggs J, Caplin A, Shapiro M, Tonetti C (2018) The long-term-care insurance puzzle: modeling and measurement. NBER working paper 22726

  • Atkinson A, Stiglitz J (1980) Lectures on public economics. MacGraw Hill International, New-York

    Google Scholar 

  • Balia S, Jones A (2008) Mortality, lifestyle and socioeconomic status. J Health Econ 27:1–26

    Article  Google Scholar 

  • Beccaria C (1764) Des délits et des peines. Version française (2015). Flammarion, Paris

    Google Scholar 

  • Becker G, Philipson T, Soares R (2005) The quantity and quality of life and the evolution of world inequality. Am Econ Rev 95(1):277–291

    Article  Google Scholar 

  • Bentham J (1789) An introduction into the principles of morals and legislation. In: Mill JS, Warnock M (eds) Utilitarianism. Fontana Press, London

    Google Scholar 

  • Boyer MM, De Donder P, Fluet C, Leroux ML, Michaud PC (2020) Long-term care insurance: information frictions and selection. Am Econ J Econ Policy 12(3):134–169

    Article  Google Scholar 

  • Brown J, Finkelstein A (2007) Why is the market for LTC insurance so small? J Public Econ 91:1967–1991

    Article  Google Scholar 

  • Brown J, Finkelstein A (2011) Insuring long term care in the US. J Econ Perspect 25(4):119–142

    Article  Google Scholar 

  • Cambois E, Laborde C, Romieu I, Robine J-M (2011) Occupational inequalities in health expectancies in France in the early 2000s: unequal chances of reaching and living retirement in good health. Demogr Res 25(12):407–436

    Article  Google Scholar 

  • CNSA—Caisse Nationale de Solidarité pour l’Autonomie (2017) Analyse des tarifs des EPHAD en 2016, numéro 4, Juin 2017

  • Christensen K, Johnson T, Vaupel J (2006) The quest for genetic determinants of human longevity: challenges and insights. Nat Rev Genet 7:436–448

    Article  Google Scholar 

  • Contoyannis P, Jones A (2004) Socio-economic status, health and lifestyle. J Health Econ 23:965–995

    Article  Google Scholar 

  • Cremer H, Pestieau P (2014) Long term care social insurance and redistribution. Int Tax Public Finance 21:955–974

    Article  Google Scholar 

  • Cremer H, Roeder K (2013) Long-term care policy, myopia and redistribution. J Public Econ 108:33–43

    Article  Google Scholar 

  • Farrer L, Cupples L, Haines J, Hyman B, Kukull W, Mayeux R et al (1997) Effects of age, sex, and ethnicity on the association between apolipoprotein e genotype and Alzheimer disease: a meta analysis. JAMA 278(16):1349–1356

    Article  Google Scholar 

  • Ferrer-i-Carbonell A, van Praag B (2002) The subjective costs of health losses due to chronic diseases. An alternative model for monetary appraisal. Health Econ 11(8):709–722

    Article  Google Scholar 

  • Finkelstein A, Luttmer E, Notowidigdo M (2013) What good is wealth without health. The effect of health on the marginal utility of consumption. J Eur Econ Assoc 11:221–258

    Article  Google Scholar 

  • Fleurbaey M (2008) Fairness, Responsibility and Welfare. Oxford University Press, Oxford

    Book  Google Scholar 

  • Fleurbaey M (2010) Assessing risky social situations. J Polit Econ 118:649–680

    Article  Google Scholar 

  • Fleurbaey M, Maniquet F (2004) Compensation and responsibility. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare, 1st edn, vol 2, chapt 22. North-Holland, pp 507–604

  • Fleurbaey M, Ponthiere G (2013) Prevention against equality? J Public Econ 103(1):68–84

    Article  Google Scholar 

  • Fleurbaey M, Leroux ML, Ponthiere G (2014) Compensating the dead. J Math Econ 51(C):28–41

    Article  Google Scholar 

  • Fleurbaey M, Leroux ML, Pestieau P, Ponthiere G (2016) Fair retirement under risky lifetime. Int Econ Rev 57(1):177–210

    Article  Google Scholar 

  • INSEE (2015) Les niveaux de vie en France. INSEE Premiè re 1665

  • Jousten A, Lipszyc B, Marchand M, Pestieau P (2005) Long term care insurance and optimal taxation for altruistic children. Finanz Arch 61:1–18

    Article  Google Scholar 

  • Lefebvre M, Perelman S, Schoenmaeckers J (2018) Iné galités face à la mort et au risque de dépendance. Revue Française d’Economie 33(2):75–112

    Article  Google Scholar 

  • Leroux ML, Ponthiere G (2018) Working time regulation, unequal lifetimes and fairness. Soc Choice Welf 51:437–464

    Article  Google Scholar 

  • Mill JS (1861) Utilitarianism. New version edited by Mary Warnock. Fontana Press, London

    Google Scholar 

  • Norton E (2000) Long term care. In: Cuyler A, Newhouse J (eds) Handbook of health economics, vol 1b, chapt 17. North-Holland

  • OECD (2011) Help wanted? Providing and paying for long-term care. http://www.oecd.org/health/health-systems/help-wanted-9789264097759-en.htm. Accessed Sept 2018

  • Pestieau P, Ponthiere G (2011) The long term care insurance puzzle. In: Costa-Font J, Courbage C (eds) Financing long term care in Europe: institutions, markets and models. Palgrave Macmillan, London, pp 41–52

    Google Scholar 

  • Pestieau P, Sato M (2008) Long term care: the state, the market and the family. Economica 75:435–454

    Article  Google Scholar 

  • Sen AK, Williams B (1982) Utilitarianism and beyond. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Sheshinski E (1972) The optimal linear income-tax. Rev Econ Stud 39(3):297–302

    Article  Google Scholar 

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Correspondence to Gregory Ponthiere.

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We would like to thank C. Goulao, J. Hindricks, Z. Kalamov, M. Kifmann, J-M. Lozachmeur, F. Maniquet, A. Millner, E. Schokkaert, two anonymous reviewers, as well as the participants to the CREPP-CDER workshop in University of Laval, to the CESifo Workshop in Public Sector Economics, to the European Health Economics Workshop (U. of Verona), to the Workshop on Heterogeneity in Life Expectancy and Pensions Design at UCLouvain, and to the CORE seminar for their helpful comments and discussions.

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Leroux, ML., Pestieau, P. & Ponthiere, G. Fair long-term care insurance. Soc Choice Welf 57, 503–533 (2021). https://doi.org/10.1007/s00355-021-01324-z

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