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“Honor thy father and thy mother” or not: uncertain family aid and the design of social long term care insurance

Abstract

We study the role and the design of long-term care insurance programs when informal care is uncertain; with and without active actuarially-fair private insurance markets against dependency. Three types of public insurance policies are considered: (1) a topping-up scheme, (2) an opting-out scheme, and (3) an opting-out-cum-transfer scheme which combines elements of the first two. A topping-up scheme can never do better than private insurance; opting out and opting-out-cum-transfer schemes can because they provide some insurance against the default of informal care. Long-term care policies have different implications for crowding out. A topping-up policy entails crowding out at both intensive and extensive margins and an opting-out policy leads to crowding out solely at the extensive margin. The opting-out feature of an opting-out-cum-transfer policy too leads to crowding out at the extensive margin, but its transfer element leads to crowding out at the intensive margin and crowding in at the extensive margin.

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Notes

  1. On the LTC programs’ crowding out of family provision or the purchase of private insurance, see, e.g., Cremer et al. (2012) and Grabowski et al. (2012).

  2. In the US, Medicare does not cover LTC; Medicaid, which is offered to families with minimal private sources, does.

  3. One can also think of this parameter as indicating the inverse of a child’s cost of providing care.

  4. This latter paper also allows for the parents to affect the children’s caregiving decisions thus making the probability that children provide care is endogenous.

  5. Our paper also contributes to the general literature on in-kind versus cash transfers which has extensively studied the properties of TU and OO schemes both from a positive and a normative perspective. On the normative side, for instance, Blomquist and Christiansen (1998) show that both regimes can be optimal (to supplement an optimal income tax) depending on whether the demand for the publicly provided good increases or decreases with labor. From a positive perspective, TU regimes may emerge from majority voting rules, as shown by Epple and Romano (1996). For a review of the literature, see Currie and Gahvari (2008).

  6. For an overview of different policies and financing models in the EU, see Lipszyc et al. (2012) and European Commission (2013).

  7. For a survey of these policies in OECD countries, see Gori et al. (2016).

  8. Transfers under this scheme differ from cash transfers under TU in that the latter gives them to everyone.

  9. In our setup, parents always find it optimal to accept the informal care that their children are willing to provide regardless of their participation in any public scheme.

  10. We rule out \(\beta <0\). A negative \(\beta \) implies that children will become happier if their parents are worse off.

  11. This condition is sufficient (but not always necessary) for most of the second-order condition of the paper to be satisfied and for some comparative statics results. We shall point out explicitly where and how it is used.

  12. There is no first stage in the laissez faire. However, to be consistent with the sections that follow, we refer to the last and the next-to-last stages as 3 and 2.

  13. A corner solution at \(a=y\) cannot be ruled out. To avoid a tedious and not very insightful multiplication of cases we assume throughout the paper that the constraint \(a\le y\) is not binding in equilibrium.

  14. The “marginal child” will be a different child depending on the considered economic setting.

  15. The function m is not differentiable at \(\beta =\beta _{0}\). To avoid cumbersome notation we use \(m^{\prime }(\beta )\) for the right derivative at this point.

  16. A corner solution at \(s=0\) can be excluded by the assumption that \(U^{\prime }(0)=\infty \). However, a corner solution at \(s=w{\overline{T}}\), yielding \(c=0\), cannot be ruled out. To avoid a tedious and not very insightful multiplication of cases we assume throughout the paper that the constraint \(c\ge 0\) is not binding in equilibrium (even when first period income is taxed to finance social LTC).

  17. The second-order condition is given by

    $$\begin{aligned} \left( 1-\pi \right) U^{\prime \prime }\left( s\right) +\pi F(\beta _{0})H^{\prime \prime }\left( s\right) +\pi f(\beta _{0})H^{\prime }\left( s\right) \frac{\partial \beta _{0}}{\partial s}<0. \end{aligned}$$

    Or, substituting for \(d\beta _{0}/ds\),

    $$\begin{aligned} \left( 1-\pi \right) U^{\prime \prime }\left( s\right) +\pi H^{\prime \prime }\left( s\right) \left[ F(\beta _{0})-\beta _{0}f(\beta _{0})\right] <0, \end{aligned}$$

    for which the concavity of \(F(\beta )\) represents a sufficient condition.

  18. See Ameriks et al. (2019) and Lillard and Weiss (1997) who find that “a fall into poor health raises the marginal utility of consumption”.

  19. Assume the contrary so that \(H^{\prime }(s^{LF})\le 1\). This implies \(F(\beta _{0}(s^{LF}))H^{\prime }\left( s^{LF}\right) <1\) resulting in \(U^{\prime }\left( s^{LF}\right)<H^{\prime }\left( s^{LF}\right) <1\). Hence the left-hand side of (8), a weighted average of \(U^{\prime }\left( s^{LF}\right) \) and \(H^{\prime }\left( s^{LF}\right) \), must also be less than one. And we have a contradiction.

  20. Full insurance is achieved when \(H^{\prime }\left( e\right) =1\); i.e. when the benefit of one extra dollar of consumption when dependent is equal to its cost.

  21. We continue to assume that the constraint \(c\ge 0\) is not binding in equilibrium.

  22. The second-order conditions are

    $$\begin{aligned} \pi H^{\prime \prime }\left( s+\delta \right) \left[ F(\beta _{0})-\beta _{0}f(\beta _{0})\right]&<0,\\ \pi (1-\pi )U^{\prime \prime }\left( s\right) H^{\prime \prime }\left( s+\delta \right) \left[ F(\beta _{0})-\beta _{0}f(\beta _{0})\right]&>0, \end{aligned}$$

    which are satisfied due to the concavity of \(H(\cdot ),U(\cdot )\) and \(F(\beta )\).

  23. The second-order condition, upon substitution for \(\partial {\widetilde{\beta }}/\partial (s+g)\) from (17), is given by

    $$\begin{aligned} \left( 1-\pi \right) U^{\prime \prime }\left( s\right) +\pi H^{\prime \prime }\left( s+g\right) \left[ F({\widetilde{\beta }})-{\widetilde{\beta }}f\left( {\widetilde{\beta }}\right) \right] <0, \end{aligned}$$

    which is satisfied due to the concavity of \(F\left( \cdot \right) \).

  24. In the US, eligibility for Medicaid, whose services include LTC, is based on having minimal private resources. This creates a perverse incentive for “not-quite-rich” people to transfer their savings to their relatives to become eligible. The OO policy we are considering allows all parents to participate as long as they are prepared to “transfer” their savings to the government.

  25. The derivative of the left-hand side with respect to \(\beta \) is

    $$\begin{aligned} \left[ H(m(\beta ))-H\left( G\right) \right] +\left[ \beta H^{\prime }\left( m(\beta )\right) -1\right] \frac{\partial m}{\partial \beta }. \end{aligned}$$

    If children do not provide care, \(\partial m/\partial \beta =0\). If they do, \(\beta H^{\prime }\left( m(\beta )\right) -1=0\). The above expression thus reduces to

    $$\begin{aligned} \left[ H(m(\beta ))-H\left( G\right) \right] , \end{aligned}$$

    which is positive for all \(m(\beta )>G\).

  26. This also explains why, whenever children are willing to provide care, their parents accept it and forego G. Intuitively, while the children are altruistic they have to pay for the cost of care that comes free to the parents.

  27. We assume that the second-order condition for maximizing \(EU^{OO}\) with respect to s, found from differentiating the left-hand side of (29),

    $$\begin{aligned} \left( 1-\pi \right) U^{\prime \prime }\left( s\right) +\pi f^{\prime }({\widehat{\beta }})\frac{\partial {\widehat{\beta }}}{\partial s}<0, \end{aligned}$$

    is satisfied. Unlike the second-order conditions previously encountered, this is not guaranteed with a concave \(F\left( \cdot \right) \) which implies \(f^{\prime }=F^{\prime \prime }<0\).

  28. The derivative of the parents’ objective function with respect to s is zero. Consequently, the terms pertaining to the induced variation of s, including \(\partial {\widehat{\beta }}/\partial G\), vanish for the parents’ objective function but not for the budget constraint. This explains why we have \(\partial {\widehat{\beta }}/\partial G\) in term B but \(d\widehat{\varvec{\beta }}/dG\) in term C.

  29. Generally speaking, not targeting is more wasteful when the targeted group is small because it entails unnecessary transfers to a larger group of people. In our model, \(1-F({\widetilde{\beta }})\) is the proportion of the not-targeted group in the population.

  30. The tax rate is \(\tau =\pi g^{TU}/w{\overline{T}}\) under TU and \(\tau =\pi F\left( \beta ^{A}\right) g^{TU}/w{\overline{T}}\) under OO.

  31. This argument is purely illustrative of the tradeoff as the areas cannot directly be compared. First, area B does not account for the distribution of \(\beta \). To obtain the effective cost savings one has to multiply area B by \([1-F(\beta ^{A})]\). Second, area C represents the loss in consumption and not in utility. Furthermore, the sum is not weighted by the density.

  32. The wage tax is not an independent instrument; the magnitude of LTC provision determines it through the government’s budget constraint.

  33. Gahvari and Mattos (2007) use a similar argument to rationalize conditional cash transfer programs in developing countries (such as Bolsa-Escola in Brazil and PROGRESA in Mexico). There the transfers are given to those who participate in a “free” publicly-provided program in order to encourage opting in; here it is given to those who opt out in order to encourage opting out.

  34. This second effect is absent in the existing conditional cash transfer programs referred to above.

  35. Since these parents keep their savings, if the policy consists of a tax, they will be paying it from their savings.

  36. We assume that the second-order condition is satisfied:

    $$\begin{aligned} (1-\pi )U^{\prime \prime }(s)-\pi \frac{f^{\prime }({\overline{\beta }})}{H(m({\overline{\beta }}))-H\left( G\right) }<0. \end{aligned}$$
  37. Differentiating (39) with respect to \({\overline{\beta }}\) yields

    $$\begin{aligned} \frac{ds}{d{\overline{\beta }}}=\frac{-\pi f^{\prime }\left( \overline{\beta }\right) }{\left( 1-\pi \right) U^{\prime \prime }\left( s\right) }<0, \end{aligned}$$

    where \(f^{\prime }\left( {\overline{\beta }}\right) =F^{\prime \prime }\left( {\overline{\beta }}\right) <0\) due to the concavity of \(F\left( \cdot \right) \).

  38. The equality is due to the fact that neither G nor g appear directly in Eq. (35).

  39. Another way of looking at the gain is that an increase in g reduces \({\overline{\beta }}\) resulting in a higher number of children assisting their parents. Consequently, with a smaller number of parents on public LTC, one can help the remaining ones more.

  40. When \(\gamma =1\) this term vanishes because of the envelope theorem.

  41. By writing children’s utility as \(U_{c}(c-a)+\beta H(s+g+a)\), \(y-\phi (a)+\beta H(s+g+a)\) or \(y-a+\beta H(s+g+\varphi (a)).\)

  42. There can only exist a unique \(\beta \) that satisfies condition (B1). This follows from the concavity of \(F(\cdot )\). Obviously, if there does not exist any \(\beta \) that satisfies (B1), only Case (i) can arise.

References

  • Ameriks J, Briggs J, Caplin A, Shapiro MD (2019) Long-term-care utility and late-in-life saving. J Polit Econ (forthcoming)

  • Blomquist S, Christiansen V (1995) Public provision of private goods as a redistributive device in an optimum income tax model. Scand J Econ 97:547–567

    Article  Google Scholar 

  • Blomquist S, Christiansen V (1998) Topping up or opting out? The optimal design of public provision schemes. Int Econ Rev 39:399–411

    Article  Google Scholar 

  • Brown J, Finkelstein A (2009) The private market for long term care in the U.S. A review of the evidence. J Risk Insur 76(1):5–29

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Cremer H, Gahvari F, Pestieau P (2017) Uncertain altruism and the provision of long term care. J Public Econ 151:12–24

    Article  Google Scholar 

  • Cremer H, Gahvari F, Pestieau P (2014) Endogenous altruism, redistribution, and long term care. B.E J Econ Anal Policy Adv 14:499–525

    Article  Google Scholar 

  • Cremer H, Pestieau P, Ponthière G (2012) The economics of long-term care: a survey. Nordic Econ Policy Rev 2:107–148

    Google Scholar 

  • Currie J, Gahvari F (2008) Transfers in cash and in-kind: theory meets the data. J Econ Liter 46:333–83

    Article  Google Scholar 

  • Diamond P (2006) Optimal tax treatment of private contributions for public goods with and without warm glow preferences. J Public Econ 90:897–919

    Article  Google Scholar 

  • Epple D, Romano RE (1996) Public provision of public goods. J Polit Econ 104:57–84

    Article  Google Scholar 

  • European Commission (2013) Long term care in ageing societies: challenges and policy options, commission staff working document

  • Gahvari F, Mattos E (2007) Conditional cash transfers, public provision of private goods and income redistribution. Am Econ Rev 97:491–502

    Article  Google Scholar 

  • Gori C, Barbarella F, Campbell J, Ikegami N, d’Amico F, Holder H, Ishibashi T, Johansson L, Komisar H, Theobald H (2016) How different countries allocate long-term care resources to older users: changes over time. In: Gori C, Fernandez J-L, Wittenberg R (eds) Long-term care reforms in OECD Countries, Ch 4. Policy Press, Bristol, UK

    Google Scholar 

  • Grabowski DC, Norton EC, Van Houtven CH (2012) Informal care. In: Jones A (ed) The elgar companion to health economics, 2nd edn, ch. 30. Edward Elgar Publishing, Cheltenham, UK

    Google Scholar 

  • Hammond P (1987) Altruism. In: Eaton J, Milgate M, Newman P (eds) The New Palgrave: a dictionary of economics. Macmillan Press, London

    Google Scholar 

  • Karlsson M, Iversen T, Øyen H (2012) Scandinavian long-term care financing. In: Costa-Font J, Courbage C (eds) Financing long-term care in Europe. Springer, Berlin, pp 254–278

    Chapter  Google Scholar 

  • Lillard L, Weiss Y (1997) Uncertain health and survival: effects on end-of-life consumption. J Bus Econ Stat 15:254–268

    Google Scholar 

  • Lipszyc B, Sail E, Xavier A (2012) Long term care: need, use and expenditures in the EU-27. European Commission: European Economy, Economic Papers 469

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

  • Norton EC (2016) Health and long-term care. In: Piggott J, Woodland A (eds) Handbook of the economics of population aging, vol 1B. North Holland.

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Correspondence to Helmuth Cremer.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Chiara Canta and Helmuth Cremer gratefully acknowledge the financial support from Chaire “Marché des risques et creation de valeur” of the FdR/SCOR. Helmuth Cremer acknowledges funding received by TSE from ANR under Grant ANR-17-EURE-0010 (Investissements d’Avenir program). We wish to thank the Associate Editor and the reviewer for their detailed, thoughtful and constructive comments and suggestions. We thank Motohiro Sato whose suggestions inspired the representation of uncertain altruism we use.

Appendices

Appendix A

Proof of \(s^{FI}<s^{LF}<s^{FI}+\delta \): Substitute \(s^{FI}\left( \delta \right) \) for \(s^{FI}\) in Eq. (10), differentiate it totally with respect to \(\delta \), and simplify to get

$$\begin{aligned}&\left( 1-\pi \right) U^{\prime \prime }\left( s^{FI}\right) \frac{ds^{FI} }{d\delta }+\pi \left( \frac{ds^{FI}}{d\delta }+1\right) \\&\quad \times \left[ -f\left( \beta _{0}\right) \frac{H^{\prime \prime }\left( s^{FI}+\delta \right) }{H^{\prime }\left( s^{FI}+\delta \right) }+F\left( \beta _{0}\left( s^{FI}+\delta \right) \right) H^{\prime \prime }\left( s^{FI}+\delta \right) \right] =0. \end{aligned}$$

Then collect the terms and “solve” for \(ds^{FI}/d\delta \). This results in

$$\begin{aligned} \frac{ds^{FI}}{d\delta }=-\frac{\pi H^{\prime \prime }\left( s^{FI} +\delta \right) \left[ F\left( \beta _{0}\right) -f\left( \beta _{0}\right) \beta _{0}\right] }{\left( 1-\pi \right) U^{\prime \prime }\left( s^{FI}\right) +\pi H^{\prime \prime }\left( s^{FI}+\delta \right) \left[ F\left( \beta _{0}\right) -f\left( \beta _{0}\right) \beta _{0}\right] }<0, \end{aligned}$$
(A1)

where the sign of (A1) follows from the concavity of \(H(\cdot ),U(\cdot )\) and \(F(\beta )\). Now observe that setting \(\delta =0\) in (10) simplifies it to Eq. (8) in the laissez faire so that \(s^{FI}\left( 0\right) =s^{LF}\). This allows us to deduce, for \(\delta >0,\)

$$\begin{aligned} s^{FI}<s^{LF}. \end{aligned}$$

Next, \(s^{FI}<s^{LF}\) implies that \(U^{\prime }\left( s^{FI}\right) >U^{\prime }\left( s^{LF}\right) \). Comparing (10) with (8) then tells us that

$$\begin{aligned} F\left( \beta _{0}\left( s^{FI}+\delta \right) \right) H^{\prime }\left( s^{FI}+\delta \right) <F(\beta _{0}(s^{LF}))H^{\prime }\left( s^{LF}\right) . \end{aligned}$$
(A2)

But,

$$\begin{aligned} \frac{d}{ds}F\left( \beta _{0}(s)\right) H^{\prime }\left( s\right)&=f\left( \beta _{0}\right) \frac{d\beta _{0}(s)}{ds}H^{\prime }\left( s\right) +F\left( \beta _{0}(s)\right) H^{\prime \prime }\left( s\right) \\&=\left[ F\left( \beta _{0}(s)\right) -\beta _{0}(s)f\left( \beta _{0}\right) \right] H^{\prime \prime }\left( s\right) <0. \end{aligned}$$

so that \(F\left( \beta _{0}(s)\right) H^{\prime }\left( s\right) \) is a decreasing function of s. It then follows from (A2) that

$$\begin{aligned} s^{FI}+\delta >s^{LF}. \end{aligned}$$

\(\square \)

Proof of Proposition 4:

We first prove that \({\widetilde{\beta }}={\widetilde{\beta }}\left( s^{TU} +g^{TU}\right) <{\widehat{\beta }}\left( g^{TU}+s^{TU},s^{TU}\right) \equiv \beta ^{A}\). Start from the optimal policy under TU and examine under what conditions it can be replicated under OO. Consider the optimal policy under TU, \(g^{TU}\), which yields \(s^{TU}\) and an expected utility for the parent given by

$$\begin{aligned} EU^{TU}\equiv w{\overline{T}}-\pi g^{TU}-s^{TU}+\left( 1-\pi \right) U\left( s^{TU}\right) +\pi \left[ \int _{{\widetilde{\beta }}}^{\infty }H\left( m\left( \beta \right) \right) dF(\beta )+F({\widetilde{\beta }})H\left( s^{TU} +g^{TU}\right) \right] , \end{aligned}$$
(A3)

Replace this policy by an OO policy in which G is set equal to \(g^{TU}+s^{TU}\) and s is set equal to \(s^{TU}\). The expected utility of parents under this alternative policy is, from (32),

$$\begin{aligned}&EU^{OO}\left( g^{TU}+s^{TU},s^{TU}\right) =w{\overline{T}}-s^{TU}+\left( 1-\pi \right) U\left( s^{TU}\right) +\nonumber \\&\qquad \pi F(\beta ^{A})\left[ H\left( g^{TU}+s^{TU}\right) -g^{TU}\right] +\pi \int _{\beta ^{A}}^{\infty }H\left( m\left( \beta \right) \right) dF(\beta ). \end{aligned}$$
(A4)

Subtracting (A3) from (A4) and simplifying

$$\begin{aligned}&EU^{OO}\left( g^{TU}+s^{TU},s^{TU}\right) -EU^{TU}=\pi \left[ \int _{\beta ^{A}}^{\infty }H\left( m\left( \beta \right) \right) dF(\beta )-\int _{\beta ^{A}}^{\infty }H\left( m\left( \beta \right) \right) dF(\beta )\right] \nonumber \\&\qquad +\pi g^{TU}\left[ 1-F(\beta ^{A})\right] +\pi H\left( g^{TU}+s^{TU}\right) \left[ F(\beta ^{A})-F(\widetilde{\beta })\right] . \end{aligned}$$
(A5)

Next compare \({\widetilde{\beta }}\) with \(\beta ^{A}\) to determine if a child with \(\beta ={\widetilde{\beta }}\) provides assistance under this alternative OO policy. Recall from (25) that, under OO and for a given G and s, the threshold level of \(\beta \) below which no assistance is provided is implicitly defined by \(\widehat{\beta }\left[ H(m({\widehat{\beta }}))-H\left( G\right) \right] -\left( m({\widehat{\beta }})-s\right) =0\). Hence at \(G=g^{TU}+s^{TU}\) and \(s=s^{TU}\), this threshold level, \(\beta ^{A}\equiv {\widehat{\beta }}\left( g^{TU} +s^{TU},s^{TU}\right) \), is given by

$$\begin{aligned} {\widehat{\beta }}\left[ H(m(\beta ^{A}))-H\left( g^{TU}+s^{TU}\right) \right] -\left( m(\beta ^{A})-s^{TU}\right) =0. \end{aligned}$$

But, from the definition of \({\widetilde{\beta }}\), we have that \(m({\widetilde{\beta }})=g^{TU}+s^{TU}\). Hence at \(\beta ={\widetilde{\beta }}\), the left-hand side of the above expression is

$$\begin{aligned} {\widetilde{\beta }}\left[ H(m({\widetilde{\beta }}))-H\left( g^{TU} +s^{TU}\right) \right] -\left( m({\widetilde{\beta }})-s^{TU}\right) =-g^{TU}<0, \end{aligned}$$

implying that

$$\begin{aligned} {\widetilde{\beta }}<\beta ^{A}\text {.} \end{aligned}$$

Hence a child with \(\beta ={\widetilde{\beta }}\) will not provide aid under this alternative OO policy.

With \({\widetilde{\beta }}<\beta ^{A}\), we rewrite Eq. (A5) as

$$\begin{aligned}&EU^{OO}\left( g^{TU}+s^{TU},s^{TU}\right) -EU^{TU}=\nonumber \\&\pi \left\{ \left[ 1-F(\beta ^{A})\right] g^{TU}-\int _{{\widetilde{\beta }} }^{\beta ^{A}}\left[ H\left( m\left( \beta \right) \right) -H\left( g^{TU}+s^{TU}\right) \right] dF(\beta )\right\} , \end{aligned}$$
(A6)

which is non-negative if the right-hand side of (A6) is non-negative. Now since the optimal OO values of G and s are generally different from \(g^{TU}+s^{TU}\) and \(s^{TU}\), it must be the case that

$$\begin{aligned} EU^{OO}\ge EU^{OO}\left( g^{TU}+s^{TU},s^{TU}\right) \ge EU^{TU}, \end{aligned}$$

if the right-hand side of (A6) is non-negative. \(\square \)

Proof of (40) and (41):

To prove (40), differentiate (38 )–(39) partially with respect to G and g, and “solve” using Cramer’s rule:

$$\begin{aligned} \frac{\partial s^{OC}}{\partial G}|_{g}&=\frac{-\pi f^{\prime } ({\overline{\beta }}){\overline{\beta }}H^{\prime }(G)}{(1-\pi )U^{\prime \prime }(s^{OC})\Delta H-\pi f^{\prime }({\overline{\beta }})}<0, \end{aligned}$$
(A7)
$$\begin{aligned} \frac{\partial s^{OC}}{\partial g}|_{G}&=\frac{\pi f^{\prime } ({\overline{\beta }})}{(1-\pi )U^{\prime \prime }(s^{OC})\Delta H-\pi f^{\prime }({\overline{\beta }})}>0, \end{aligned}$$
(A8)
$$\begin{aligned} \frac{\partial \varvec{{\overline{\beta }}}}{\partial G}|_{g}&=\frac{(1-\pi )U^{\prime \prime }(s^{OC}){\overline{\beta }}H^{\prime }(G)}{(1-\pi )U^{\prime \prime }(s^{OC})\Delta H-\pi f^{\prime }({\overline{\beta }} )}>0, \end{aligned}$$
(A9)
$$\begin{aligned} \frac{\partial \varvec{{\overline{\beta }}}}{\partial g}|_{G}&=\frac{-(1-\pi )U^{\prime \prime }(s^{OC})}{(1-\pi )U^{\prime \prime }(s^{OC})\Delta H-\pi f^{\prime }({\overline{\beta }})}<0, \end{aligned}$$
(A10)

where \(\Delta H=H(m({\overline{\beta }}))-H\left( G\right) \). The signs follow from the negativity of the denominator in each of the equations (second-order condition of the parents’ optimization problem with respect to s), and the concavity of \(F\left( \cdot \right) \) which implies \(f^{\prime }\left( \cdot \right) =F^{\prime \prime }\left( \cdot \right) <0\). To prove (41), divide (A7) by (A8 ) and (A9) by (A10). \(\square \)

Proof of (44):

Rearrange Eqs. (42)–(43) and divide the former by the latter to get

$$\begin{aligned} \frac{\frac{1}{\pi }\frac{\partial \pounds ^{OC}}{\partial G}-F(\overline{\beta })H^{\prime }\left( G\right) +f({\overline{\beta }})\frac{\partial {\overline{\beta }}}{\partial G}\Delta H+\left[ F({\overline{\beta }} )-F({\overline{\beta }})\frac{\partial s^{OC}}{\partial G}\right] }{\frac{1}{\pi }\frac{\partial \pounds ^{OC}}{\partial g}+f({\overline{\beta }} )\frac{\partial {\overline{\beta }}}{\partial g}\Delta H+\left[ 1-F(\overline{\beta })-F({\overline{\beta }})\frac{\partial s^{OC}}{\partial g}\right] } =\frac{\frac{\partial \varvec{{\overline{\beta }}}}{\partial G}}{\frac{\partial \varvec{{\overline{\beta }}}}{\partial g}}=-\overline{\beta }H^{\prime }\left( G\right) , \end{aligned}$$

where we have used (41). ex Multiplying through

$$\begin{aligned}&\frac{1}{\pi }\frac{\partial \pounds ^{OC}}{\partial G}-F(\overline{\beta })H^{\prime }\left( G\right) +f({\overline{\beta }})\frac{\partial {\overline{\beta }}}{\partial G}\Delta H+\left[ F({\overline{\beta }} )-F({\overline{\beta }})\frac{\partial s^{OC}}{\partial G}\right] =\\&\quad -{\overline{\beta }}H^{\prime }\left( G\right) \left\{ \frac{1}{\pi } \frac{\partial \pounds ^{OC}}{\partial g}+f({\overline{\beta }})\frac{\partial {\overline{\beta }}}{\partial g}\Delta H+\left[ 1-F(\overline{\beta })-F({\overline{\beta }})\frac{\partial s^{OC}}{\partial g}\right] \right\} . \end{aligned}$$

Or

$$\begin{aligned}&\frac{1}{\pi }\frac{\partial \pounds ^{OC}}{\partial G}-F(\overline{\beta })H^{\prime }\left( G\right) +f({\overline{\beta }})\frac{\partial {\overline{\beta }}}{\partial G}\Delta H+\left[ F({\overline{\beta }} )-F({\overline{\beta }})\frac{\partial s^{OC}}{\partial G}\right] \\&\quad + \left\{ \frac{1}{\pi }\frac{\partial \pounds ^{OC}}{\partial g} +f({\overline{\beta }})\frac{\partial {\overline{\beta }}}{\partial g}\Delta H+\left[ 1-F({\overline{\beta }})-F({\overline{\beta }})\frac{\partial s^{OC} }{\partial g}\right] \right\} {\overline{\beta }}H^{\prime }\left( G\right) =0. \end{aligned}$$

or,

$$\begin{aligned}&\frac{1}{\pi }\frac{\partial \pounds ^{OC}}{\partial G}+\frac{1}{\pi }{\overline{\beta }}H^{\prime }\left( G\right) \frac{\partial \pounds ^{OC} }{\partial g}=-f({\overline{\beta }})\Delta H\left[ \frac{\partial {\overline{\beta }}}{\partial G}+\frac{\partial {\overline{\beta }}}{\partial g}{\overline{\beta }}H^{\prime }\left( G\right) \right] \\&\quad +F({\overline{\beta }})\left[ \frac{\partial s^{OC}}{\partial G} +{\overline{\beta }}H^{\prime }\left( G\right) \frac{\partial s^{OC}}{\partial g}\right] +F({\overline{\beta }})\left[ H^{\prime }\left( G\right) -1\right] -\left[ 1-F({\overline{\beta }})\right] {\overline{\beta }}H^{\prime }\left( G\right) . \end{aligned}$$

But we have, from (41),

$$\begin{aligned} \frac{\partial {\overline{\beta }}}{\partial G}+{\overline{\beta }}H^{\prime }\left( G\right) \frac{\partial {\overline{\beta }}}{\partial g}=\frac{\partial s^{OC} }{\partial G}+{\overline{\beta }}H^{\prime }\left( G\right) \frac{\partial s^{OC}}{\partial g}=0. \end{aligned}$$

Substituting in the expressions above results in

$$\begin{aligned} \frac{1}{\pi }\frac{\partial \pounds ^{OC}}{\partial G}+\frac{1}{\pi } {\overline{\beta }}H^{\prime }\left( G\right) \frac{\partial \pounds ^{OC} }{\partial g}=F({\overline{\beta }})\left[ H^{\prime }\left( G\right) -1\right] +\left[ 1-F({\overline{\beta }})\right] \left[ -\overline{\beta }H^{\prime }\left( G\right) \right] . \end{aligned}$$

Evaluating this expression at the optimal solution to the OO scheme (assuming it has an interior solution) and \(g=0\), we arrive at Eq. (44). \(\square \)

Appendix B

Let \(\delta \) denote the amount of private insurance against dependency purchased and \(\pi \delta \) its actuarially fair premium.

Topping up

We have previously examined the implications of actuarially fair insurance markets for the laissez faire solution in Sect. 2.2. Comparing the market outcome there with the TU solution engineered by the government in Sect. 3, one immediately observes that the two solutions are identical. The implication of this result is that a TU policy offers only full insurance against dependency and does nothing by way of providing insurance against the default of altruism.

Opting out

Start with a pure OO policy and examine the changes that private insurance may lead to in each stage of our model. As far as the children are concerned, Eq. (25) changes to

$$\begin{aligned} {\widehat{\beta }}\left[ H(m({\widehat{\beta }}))-H\left( G\right) \right] -\left( m({\widehat{\beta }})-s-\delta \right) =0, \end{aligned}$$

where \({\widehat{\beta }}(G,s+\delta )\) has replaced \({\widehat{\beta }}(G,s)\). Yet partial differentiation of \({\widehat{\beta }}(G,s+\delta )\) with respect to G and s yields equations for \(\partial {\widehat{\beta }}/\partial G\) and \(\partial {\widehat{\beta }}/\partial s\) identical to (26 )–(27). Partial differentiation of \(\widehat{\beta }(G,s+\delta )\) with respect to \(\delta \) results in \(\partial \widehat{\beta }/\partial \delta =\partial {\widehat{\beta }}/\partial s\).

The parents’ expected utility now includes a term for the cost of purchasing insurance:

$$\begin{aligned} EU= & {} w\left( 1-\tau \right) {\overline{T}}-s-\pi \delta +\left( 1-\pi \right) U\left( s\right) \\&+\pi \left[ H\left( G\right) F({\widehat{\beta }} )+\int _{{\widehat{\beta }}}^{\infty }H\left( m\left( \beta \right) \right) dF(\beta )\right] , \end{aligned}$$

which they maximize with respect to s and \(\delta \). The first-order condition with respect to s, assuming an interior solution as previously, yields an equation identical to (28), and then, upon substituting for \(\partial {\widehat{\beta }}/\partial s\), an equation identical to (29), except for \(s+\delta \) replacing s in \({\widehat{\beta }}\). To determine \(\delta \), consider the partial derivative of EU with respect to \(\delta \),

$$\begin{aligned} \frac{\partial EU}{\partial \delta }=-\pi -\pi f({\widehat{\beta }})\left[ H(m({\widehat{\beta }}))-H\left( G\right) \right] \frac{\partial {\widehat{\beta }}}{\partial \delta }. \end{aligned}$$

Substitute for \(\partial {\widehat{\beta }}/\partial \delta \), from the expression for \(\partial {\widehat{\beta }}/\partial s\) in (27), evaluate at \(\delta =0\), and use (29). This yields

$$\begin{aligned} \frac{\partial EU}{\partial \delta }|_{\delta =0}=-\pi +\pi f(\widehat{\beta })=\left( 1-\pi \right) \left[ 1-U^{\prime }\left( s^{OO}\right) \right] . \end{aligned}$$

Two possibilities arise:

Case (i): \(U^{\prime }\left( s^{OO}\right) \ge 1\) so that \(\delta =0\) and nobody purchases any private insurance for dependency even if offered at an actuarially fair premium. This lead us back to the pure OO solution.

Case (ii): \(U^{\prime }\left( s^{OO}\right) <1\). Under this circumstance \(\delta >0\) so that at the optimum \(U^{\prime }\left( s\right) =1\). Consequently, the solution for savings is the same we had under laissez faire with insurance markets: \(s=s^{FI}\). Substituting this value in the first-order condition for s (which continues to be represented by (29)), we haveFootnote 42

$$\begin{aligned} f({\widehat{\beta }}(G,s^{FI}+\delta ))=1. \end{aligned}$$
(B1)

This condition implies that \({\widehat{\beta }}\) only depends on the shape of the distribution function \(F(\beta )\), and not on the public policy. Solving for \(\delta \) then results in \(\delta \left( G\right) \). Substituting \(\delta \left( G\right) \) in (B1), differentiating the resulting identity with respect to G, and simplifying results in

$$\begin{aligned} \frac{d\delta }{dG}=-\frac{\frac{\partial {\widehat{\beta }}}{\partial G}}{\frac{\partial {\widehat{\beta }}}{\partial \delta }}=-\frac{\frac{\widehat{\beta }H^{\prime }\left( G\right) }{H(m({\widehat{\beta }}))-H\left( G\right) } }{-\frac{1}{H(m({\widehat{\beta }}))-H\left( G\right) }}=\widehat{\beta }H^{\prime }\left( G\right) . \end{aligned}$$
(B2)

We can now study the government’s optimal choice of G. The government maximizes the parents’ optimized value of EU subject to its budget constraint,

$$\begin{aligned} \tau wT=\pi F({\widehat{\beta }})\left[ G-s^{FI}-\delta \right] . \end{aligned}$$

This leads to the maximization of the following welfare function,

$$\begin{aligned} \pounds =EU\left( G\right) -\pi F(\widehat{\varvec{\beta }})\left[ G-s^{FI}-\delta \left( G\right) \right] , \end{aligned}$$

where \(\widehat{\varvec{\beta }}\equiv {\widehat{\beta }}(G,s^{FI} +\delta \left( G\right) )\). Maximizing \(\pounds \) with respect to G, using the envelope theorem, we have

$$\begin{aligned} \frac{d\pounds }{dG}&=\pi H^{\prime }\left( G\right) F(\widehat{\beta })-\pi f({\widehat{\beta }})\left[ H(m({\widehat{\beta }}))-H\left( G\right) \right] \frac{\partial {\widehat{\beta }}}{\partial G}|_{s,\delta }\\&\quad - \pi \left[ F(\widehat{\varvec{\beta }})\left( 1-\frac{d\delta }{dG}\right) +(G-s^{FI}-\delta \left( G\right) )f({\widehat{\beta }} )\frac{d\widehat{\varvec{\beta }}}{dG}\right] . \end{aligned}$$

Observe that the three terms on the right-hand side correspond to terms AB,  and C in the pure opting out solution (Eq. (33)). The first two terms have identical formulations. In term C, \(d\delta /dG\) has replaced \(ds^{OO}/dG\) and \((G-s^{FI}-\delta \left( G\right) )\) has replaced \((G-s^{OO})\). Moreover, we have \({\widehat{\beta }}={\widehat{\beta }} (G,s^{FI}+\delta \left( G\right) )\) rather than \(\widehat{\beta }={\widehat{\beta }}(G,s^{OO}\left( G\right) )\).

Next substitute for \(\partial {\widehat{\beta }}/\partial G,d\delta /dG\), and \(d\widehat{\varvec{\beta }}/dG\) in the expression for \(d\pounds /dG\) to get

$$\begin{aligned} \frac{d\pounds }{dG}=\pi \left\{ \left[ F({\widehat{\beta }})-f(\widehat{\beta }){\widehat{\beta }}+F({\widehat{\beta }}){\widehat{\beta }}\right] H^{\prime }\left( G\right) -F({\widehat{\beta }})\right\} , \end{aligned}$$

where \(F({\widehat{\beta }})-f({\widehat{\beta }}){\widehat{\beta }}>0\) due to the concavity of \(F(\cdot )\). Two possibilities arise depending on the sign of \(d\pounds /dG\) at \(G=s^{FI}+\delta \left( G\right) \). If \(d\pounds /dG\le 0\), there is no interior solution for G and an OO policy is not helpful. Under this circumstance, we have the laissez faire solution with insurance markets for dependency (as in Sect. 2.2) which is equivalent to the TU solution. Otherwise, if \(d\pounds /dG>0,\) there is an interior solution for G given by

$$\begin{aligned} H^{\prime }\left( G\right) =\frac{F({\widehat{\beta }})}{F(\widehat{\beta })+\left( 1-F({\widehat{\beta }})\right) \left( -{\widehat{\beta }}\right) }>1, \end{aligned}$$
(B3)

where, from (B1), \(f({\widehat{\beta }})\) has been set equal to one. An OO policy is desirable but it still does not offer full insurance.

Opting-out-cum-transfers

The presence of private insurance markets lead to :the following changes. As far as the children are concerned, their threshold level of \(\beta \), \({\overline{\beta }}(G,s+\delta +g)\), changes to

$$\begin{aligned} {\overline{\beta }}\left[ H(m({\overline{\beta }}))-H\left( G\right) \right] -\left( m({\overline{\beta }})-s-\delta -g\right) =0, \end{aligned}$$
(B4)

Partial differentiation of \({\overline{\beta }}(G,s+\delta +g)\) with respect to Gs, and g yields identical equations to (36)–(37 ) for \(\partial {\overline{\beta }}/\partial G\) and \(\partial \overline{\beta }/\partial g=\partial {\overline{\beta }}/\partial s\); partial differentiation of \({\overline{\beta }}(G,s+\delta +g)\) with respect to \(\delta \) results in \(\partial {\overline{\beta }}/\partial \delta =\partial {\overline{\beta }}/\partial g=\partial {\overline{\beta }}/\partial s\).

Turning to the parents’ expected utility, it is now given by

$$\begin{aligned} EU= & {} w\left( 1-\tau \right) {\overline{T}}-s-\pi \delta +\left( 1-\pi \right) U\left( s\right) \\&+\pi \left[ H\left( G\right) F({\overline{\beta }} )+\int _{{\overline{\beta }}}^{\infty }H\left( m\left( \beta \right) \right) dF(\beta )\right] , \end{aligned}$$

which they maximize with respect to s and \(\delta \). The first-order condition with respect to s, assuming an interior solution as previously, yields an equation identical to (38), and then upon substituting for \(\partial {\overline{\beta }}/\partial s\) an equation identical to (39), except for \(s+\delta \) replacing s in \(\overline{\beta }\). To determine \(\delta \), consider the partial derivative of EU with respect to \(\delta \),

$$\begin{aligned} \frac{\partial EU}{\partial \delta }=-\pi -\pi f({\overline{\beta }})\left[ H(m({\overline{\beta }}))-H\left( G\right) \right] \frac{\partial {\overline{\beta }}}{\partial \delta }. \end{aligned}$$

Substitute for \(\partial {\overline{\beta }}/\partial \delta \), from the expression for \(\partial {\overline{\beta }}/\partial s\) in (37), evaluate at \(\delta =0\), and use (39) to get

$$\begin{aligned} \frac{\partial EU}{\partial \delta }|_{\delta =0}=-\pi +\pi f(\overline{\beta })=\left( 1-\pi \right) \left[ 1-U^{\prime }\left( s^{OC}\right) \right] . \end{aligned}$$

Two possibilities arise:

Case (i): \(U^{\prime }\left( s^{OC}\right) \ge 1\) so that \(\delta =0\) and nobody purchases any private insurance even if offered at an actuarially fair premium. This leads us back to the opting-out-cum-transfer solution.

Case (ii): \(U^{\prime }\left( s^{OC}\right) <1\). Under this circumstance \(\delta >0\) so that at the optimum \(U^{\prime }\left( s\right) =1\). Again, the solution for savings will be the same as we had under laissez faire with insurance markets: \(s=s^{FI}\). Substituting in the first-order condition for s, which continues to be represented by (39), we have

$$\begin{aligned} f({\overline{\beta }}(G,s^{FI}+\delta +g))=1. \end{aligned}$$
(B5)

The system of Eqs. (B4)–(B5) jointly determines the values of \(\delta \) and \({\overline{\beta }}=\overline{\beta }(G,s^{FI}+\delta +g)\) as functions of G and g: \(\delta ^{OC}(G,g)\) and \(\overline{{\varvec{\beta }}}\left( G,g\right) \equiv \overline{\beta }(G,s^{FI}+\delta ^{OC}(G,g)+g)\). Differentiating this system of equations with respect to G and g, we have

$$\begin{aligned} \frac{\partial \overline{\varvec{\beta }}}{\partial G}|_{g}=0,\frac{\partial \delta }{\partial G}|_{g}={\overline{\beta }}H^{\prime }\left( G\right) \text {; and }\frac{\partial \overline{\varvec{\beta }}}{\partial g} |_{G}=0,\frac{\partial \delta }{\partial g}|_{G}=-1. \end{aligned}$$

Next is the determination of the government’s optimal choice of G. The government maximizing the parents’ optimized value of EU subject to its budget constraint

$$\begin{aligned} \tau wT=\pi \left\{ F({\overline{\beta }})\left[ G-s^{FI}-\delta \right] +\left( 1-F({\overline{\beta }})\right) g\right\} . \end{aligned}$$

This leads to the maximization of the following welfare function

$$\begin{aligned} \pounds =EU\left( G,g\right) -\pi \left\{ F(\overline{\varvec{\beta } })\left[ G-s^{FI}-\delta \right] +\left( 1-F(\overline{\varvec{\beta } })\right) g\right\} , \end{aligned}$$

where \(\overline{\varvec{\beta }}=\overline{\varvec{\beta }}\) \(\left( G,g\right) \). Differentiating \(\pounds \) partially with respect to G and g, and using the envelope theorem, one obtains

$$\begin{aligned} \frac{\partial \pounds }{\partial G}&=\pi H^{\prime }\left( G\right) F({\overline{\beta }})-\pi f({\overline{\beta }})\left[ H(m(\overline{\beta }))-H\left( G\right) \right] \frac{\partial {\overline{\beta }}}{\partial G}|_{s,\delta ,g}\\&\quad -\pi \left[ \left( G-s^{FI}-\delta \right) f({\overline{\beta }} )\frac{\partial \overline{\varvec{\beta }}}{\partial G}+F(\overline{\beta })\left( 1-\frac{\partial \delta }{\partial G}\right) -gf(\overline{\beta })\frac{\partial \overline{\varvec{\beta }}}{\partial G}\right] ,\\&=\pi \left\{ H^{\prime }\left( G\right) F({\overline{\beta }})-f(\overline{\beta }){\overline{\beta }}H^{\prime }\left( G\right) -F(\overline{\beta })\left[ 1-{\overline{\beta }}H^{\prime }\left( G\right) \right] \right\} ,\\ \frac{\partial \pounds }{\partial g}&=-\pi f({\overline{\beta }})\left[ H(m({\overline{\beta }}))-H\left( G\right) \right] \frac{\partial {\overline{\beta }}}{\partial g}|_{s,\delta ,G}\\&\quad - \pi \left[ \left( G-s^{FI}-\delta \right) f({\overline{\beta }} )\frac{\partial \overline{\varvec{\beta }}}{\partial g}-F(\overline{\beta })\frac{\partial \delta }{\partial g}-gf({\overline{\beta }})\frac{\partial \overline{\varvec{\beta }}}{\partial g}+\left( 1-F(\overline{\beta })\right) \right] \\&=\pi \left[ f({\overline{\beta }})-1\right] =0. \end{aligned}$$

Observe again that the three terms on the right-hand side of \(\partial \pounds /\partial G\) correspond to terms AB,  and C in the opting-out-cum-transfer solution (Eq. 42), with a slightly different formulation for term C. The terms continue to have the same interpretations. There continue to be two possibilities depending on the sign of \(\partial \pounds /\partial G\) at \(G=s^{FI}+\delta \left( G\right) +g\). If \(\partial \pounds /\partial G\le 0\), there is no interior solution for G and an opting out policy is not desirable (not even a pure one). The solution will then be the same as the laissez faire solution with insurance markets. If \(\partial \pounds /\partial G>0\), there is an interior solution for G characterized by (B3). Either way parents are under-insured.

Similarly, the two terms on the right-hand side \(\partial \pounds /\partial g\) correspond to terms \(B^{\prime }\), and \(C^{\prime }\) in the opting-out-cum-transfer solution (Eq. 43), with a slightly different formulation for \(C^{\prime }\). As before, and for the same reason, there is no term corresponding to A. They continue to have the same interpretations.

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Canta, C., Cremer, H. & Gahvari, F. “Honor thy father and thy mother” or not: uncertain family aid and the design of social long term care insurance. Soc Choice Welf 55, 687–734 (2020). https://doi.org/10.1007/s00355-020-01260-4

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