“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.

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.

1.1 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.

1.2 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 have⁴²

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

(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 A, B,  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.

1.3 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 G, s, 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 A, B,  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.