Long-term care social insurance: How to avoid big losses?

Abstract

Long-term care (LTC) needs are expected to rapidly increase in the next decades, and at the same time, the main provider of LTC, namely the family, is stalling. This calls for more involvement of the state that today covers <20% of these needs and most often in an inconsistent way. Besides the need to help the dependent poor, there is a mounting concern in the middle class that a number of dependent people are incurring costs that could force them to sell all their assets. In this paper, we study the design of a social insurance program that meets this concern. Following Arrow (Am Econ Rev 53:941–973, 1963), we suggest a policy that is characterized by complete insurance above a deductible amount.

Appendices

Appendix 1: Comparative statics in the laissez-faire

Fully differentiating Eqs. (8–10) with respect to $w_{i},$ we get, respectively,

$$\begin{aligned}&\frac{\partial z_{i}^{1}}{\partial w_{i}}\left[ (q^{0})^{2}\pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) (q^{1})^{2}\right] \nonumber \\&\qquad -\frac{\partial y_{i}}{\partial w_{i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{1}+\frac{\partial z_{i}^{2}}{\partial w_{i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}q^{2}=0, \nonumber \\\end{aligned}$$

(37)

$$\begin{aligned}&\frac{\partial z_{i}^{2}}{\partial w_{i}}\left[ (q^{0})^{2}\pi _{2}u^{\prime \prime }\left( c_{i}^{D_{2}}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) (q^{2})^{2}\right] \nonumber \\&\qquad -\frac{\partial y_{i}}{\partial w_{i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{2}+\frac{\partial z_{i}^{1}}{\partial w_{i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{1}q^{2}=0\nonumber \\ \end{aligned}$$

(38)

and

$$\begin{aligned}&\frac{\partial y_{i}}{\partial w_{i}}\left[ \left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) -(q^{0})^{2}\frac{v^{\prime \prime }\left( \frac{y_{i}}{w_{i}}\right) }{w_{i}^{2}}\right] \nonumber \\&\qquad -\frac{\partial z_{i}^{1}}{\partial w_{i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{1}-\frac{\partial z_{i}^{2}}{\partial w_{i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{2}\nonumber \\&\qquad +(q^{0})^{2}\frac{v^{\prime \prime }\left( \frac{y_{i}}{w_{i}}\right) y_{i}}{w_{i}^{3}}+(q^{0})^{2}\frac{v^{\prime }\left( \frac{y_{i}}{w_{i}}\right) }{w_{i}^{2}}=0 \end{aligned}$$

(39)

Solving the system of Eqs. (37–39) for $\frac{\partial z_{i}^{1}}{\partial w_{i}}$, $\frac{\partial z_{i}^{2}}{\partial w_{i}}$ and $\frac{\partial y_{i}}{\partial w_{i}}$, we obtain

$$\begin{aligned}&\frac{\partial y_{i}}{\partial w_{i}}=\frac{\left[ 1\right] \cdot \left[ 2\right] }{\left[ 3\right] }>0,\\&\frac{\partial z_{i}^{1}}{\partial w_{i}}=\frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{1}\pi _{2}u^{\prime \prime }\left( c_{i}^{D_{2}}\right) \frac{\partial y_{i}}{\partial w_{i}}}{\left[ 1\right] }>0,\\&\frac{\partial z_{i}^{2}}{\partial w_{i}}=\frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{2}\pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) \frac{\partial y_{i}}{\partial w_{i}}}{\left[ 1\right] }>0 \end{aligned}$$

where

$$\begin{aligned}&\left[ 1\right] \equiv \pi _{2}u^{\prime \prime }\left( c_{i}^{D_{2}}\right) (q^{0})^{2}\pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) +\pi _{2}u^{\prime \prime }\left( c_{i}^{D_{2}}\right) \left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) (q^{1})^{2}\\&\qquad \quad \quad +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) (q^{2})^{2}\pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) >0,\\&\left[ 2\right] \equiv -\left( q^{0}\right) ^{2}\frac{v^{\prime \prime }\left( \frac{y_{i}}{w_{i}}\right) y_{i}}{w_{i}^{3}}-\left( q^{0}\right) ^{2}\frac{v^{\prime }\left( \frac{y_{i}}{w_{i}}\right) }{w_{i}^{2}}<0,\\&\left[ 3\right] \equiv \left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) \pi _{2}u^{\prime \prime }\left( c_{i}^{D_{2}}\right) (q^{0})^{2}\pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) -(q^{0})^{2}\frac{v^{\prime \prime }\left( \frac{y_{i}}{w_{i}}\right) }{w_{i}^{2}}\left[ 1\right] <0. \end{aligned}$$

Using Eq. (7), we have

$$\begin{aligned} \frac{\partial z_{i}^{0}}{\partial w_{i}}=\frac{1}{q^{0}}\left[ \frac{\partial y_{i}}{\partial w_{i}}-q^{1}\frac{\partial z_{i}^{1}}{\partial w_{i}}-q^{2}\frac{\partial z_{i}^{2}}{\partial w_{i}}\right] =\frac{q^{0}\pi _{2}u^{\prime \prime }\left( c_{i}^{D_{2}}\right) \pi _{1}u^{\prime \prime } \left( c_{i}^{D_{1}}\right) \frac{\partial y_{i}}{\partial w_{i}}}{\left[ 1\right] }>0. \end{aligned}$$

Since from Eqs. (8) and (9) we have $u'(c_{i}^{D_{1}})=u'(c_{i}^{D_{2}})\,\Leftrightarrow c_{i}^{D_{1}}=c_{i}^{D_{2}}$, it is also true that $u^{\prime \prime }(c_{i}^{D_{1}})=u^{\prime \prime }(c_{i}^{D_{2}})$. Using this and the definitions of $q^{1}$ and $q^{2}$, it is straightforward to see that $\frac{\partial z_{i}^{1}}{\partial w_{i}}=\frac{\partial z_{i}^{2}}{\partial w_{i}}$. Moreover, note that $\frac{\partial z_{i}^{0}}{\partial w_{i}}=\frac{\partial c_{i}^{I}}{\partial w_{i}}$ and, since $L_{1i}$ and $L_{2i}$ remain unchanged, we also have that $\frac{\partial z_{i}^{1}}{\partial w_{i}}=\frac{\partial c_{i}^{D_{1}}}{\partial w_{i}}$ and $\frac{\partial z_{i}^{2}}{\partial w_{i}}=\frac{\partial c_{i}^{D_{2}}}{\partial w_{i}}$.

We can thus write

$$\begin{aligned} \frac{\partial c_{i}^{D_{1}}}{\partial w_{i}}-\frac{\partial c_{i}^{I}}{\partial w_{i}}=\frac{\partial c_{i}^{D_{2}}}{\partial w_{i}}-\frac{\partial c_{i}^{I}}{\partial w_{i}}=\frac{\frac{\partial y_{i}}{\partial w_{i}}\pi _{2}u^{\prime \prime } \left( c_{i}^{D_{2}}\right) \left[ 4\right] }{\left[ 1\right] } \end{aligned}$$

(40)

where

$$\begin{aligned} \left[ 4\right] \equiv \left( 1-\pi _{1}-\pi _{2}\right) (1+\lambda )\pi _{1}u^{\prime \prime }\left( c_{i}^{I}\right) -\left[ 1-(1+\lambda )(\pi _{1}+\pi _{2})\right] \pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) . \end{aligned}$$

The sign of $\left[ 4\right]$ is ambiguous in the general case and differs depending on the absolute risk aversion (ARA) exhibited by the utility function. In particular, we are now going to show that $\left[ 4\right] >0$ under decreasing absolute risk aversion (DARA), $\left[ 4\right] <0$ under increasing absolute risk aversion (IARA) and $\left[ 4\right] =0$ under constant absolute risk aversion (CARA).

To see this, let us first note that DARA (resp. IARA and CARA) means that

$$\begin{aligned} ARA(c)=\frac{-u^{\prime \prime }(c)}{u^{\prime }(c)}<(\text {resp.}>\text { and }=)\text { }ARA(d)=\frac{-u^{\prime \prime }(d)}{u^{\prime }(d)}\text { for }c>d, \end{aligned}$$

where $\frac{-u^{\prime \prime }(x)}{u^{\prime }(x)}$ is the Arrow-Pratt measure of absolute risk aversion at wealth x.

Thus, noting that from (8) we have $c_{i}^{I}>c_{i}^{D_{1}},$ under DARA (resp. IARA and CARA) preferences we can write

$$\begin{aligned}&\frac{-u^{\prime \prime }\left( c_{i}^{I}\right) }{u^{\prime }\left( c_{i}^{I}\right) }<(\text {resp.}>\text { and }=)\text { }\frac{-u^{\prime \prime }\left( c_{i}^{D_{1}}\right) }{u^{\prime }\left( c_{i}^{D_{1}}\right) }\\&\Longleftrightarrow \\&u^{\prime \prime }\left( c_{i}^{I}\right) >(\text {resp.}<\text { and }=)\text { } \frac{u^{\prime \prime }\left( c_{i}^{D_{1}}\right) }{u^{\prime }\left( c_{i}^{D_{1}}\right) } \text { }u^{\prime }\left( c_{i}^{I}\right) \end{aligned}$$

We can then multiply both sides by $\left( 1-\pi _{1}-\pi _{2}\right) (1+\lambda )\pi _{1}$ and subtract $\left[ 1-(1+\lambda )(\pi _{1}+\pi _{2})\right] \pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})$ from both sides, which gives

$$\begin{aligned}&\left( 1-\pi _{1}-\pi _{2}\right) (1+\lambda )\pi _{1}u^{\prime \prime } \left( c_{i}^{I}\right) -\left[ 1-(1+\lambda )(\pi _{1}+\pi _{2})\right] \pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) \nonumber \\&\qquad >(\text {resp.}<\text { and }=)\text { }\frac{u^{\prime \prime } \left( c_{i}^{D_{1}}\right) }{u^{\prime }\left( c_{i}^{D_{1}}\right) } \left[ \text { }u^{\prime }\left( c_{i}^{I}\right) \left( 1-\pi _{1}-\pi _{2}\right) (1+\lambda )\pi _{1}\right. \nonumber \\&\qquad \quad \left. -\left[ 1-(1+\lambda )(\pi _{1}+\pi _{2})\right] \pi _{1}u^{\prime }\left( c_{i}^{D_{1}}\right) \right] =0 \end{aligned}$$

(41)

noting that the expression in the last big bracket is zero from Eq. (8).

The left-hand side of inequality (41) is exactly the definition of $\left[ 4\right] ;$ we therefore indeed have that under DARA (resp. IARA and CARA), $\left[ 4\right] >($resp. < and $=)$ 0. We can now use this in (40), which gives that

$$\begin{aligned} \frac{\partial c_{i}^{D_{1}}}{\partial w_{i}}-\frac{\partial c_{i}^{I}}{\partial w_{i}}=\frac{\partial c_{i}^{D_{2}}}{\partial w_{i}}-\frac{\partial c_{i}^{I}}{\partial w_{i}}<(\text {resp.}>\text { and }=)\text { }0\text { } \end{aligned}$$

under DARA (resp. IARA and CARA) preferences.

Fully differentiating Eqs. (8–10) with respect to $L_{1i},$ we get, respectively,

$$\begin{aligned}&\frac{\partial z_{i}^{1}}{\partial L_{1i}}\left[ \left( q^{0}\right) ^{2}\pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) \left( q^{1}\right) ^{2}\right] \nonumber \\&\qquad -\frac{\partial y_{i}}{\partial L_{1i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{1}\nonumber \\&\qquad +\frac{\partial z_{i}^{2}}{\partial L_{1i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{1}q^{2}-\left( q^{0}\right) ^{2}\pi _{1}u^{\prime \prime }\left( c_{i}^{D_{1}}\right) =0, \end{aligned}$$

(42)

$$\begin{aligned}&\frac{\partial z_{i}^{2}}{\partial L_{1i}}\left[ \left( q^{0}\right) ^{2}\pi _{2}u^{\prime \prime }\left( c_{i}^{D_{2}}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) \left( q^{2}\right) ^{2}\right] \nonumber \\&\qquad -\frac{\partial y_{i}}{\partial L_{1i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{2}+\frac{\partial z_{i}^{1}}{\partial L_{1i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }\left( c_{i}^{I}\right) q^{1}q^{2}=0\nonumber \\ \end{aligned}$$

(43)

and

$$\begin{aligned}&\frac{\partial y_{i}}{\partial L_{1i}}\left[ \left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})-(q^{0})^{2}\frac{v^{\prime \prime }\left( \frac{y_{i}}{w_{i}}\right) }{w_{i}^{2}}\right] \nonumber \\&\qquad -\frac{\partial z_{i}^{1}}{\partial L_{1i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}-\frac{\partial z_{i}^{2}}{\partial L_{1i}}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{2}=0\nonumber \\ \end{aligned}$$

(44)

Solving the system of Eqs. (42–44) for $\frac{\partial z_{i}^{1}}{\partial L_{1i}}$, $\frac{\partial z_{i}^{2}}{\partial L_{1i}}$ and $\frac{\partial y_{i}}{\partial L_{1i}}$, we obtain

$$\begin{aligned} \frac{\partial y_{i}}{\partial L_{1i}}= & {} \frac{q^{1}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})(q^{0})^{2}\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})}{\left[ 3\right] }>0,\nonumber \\ \frac{\partial z_{i}^{1}}{\partial L_{1i}}= & {} \frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\frac{\partial y_{i}}{\partial L_{1i}}}{\left[ 1\right] }\nonumber \\&+\frac{\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\left[ (q^{0})^{2}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})+(q^{2})^{2}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})\right] }{\left[ 1\right] }>0,\nonumber \\ \frac{\partial z_{i}^{2}}{\partial L_{1i}}= & {} \frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{2}\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\left[ \frac{\partial y_{i}}{\partial L_{1i}}-q^{1}\right] }{\left[ 1\right] }<0. \end{aligned}$$

(45)

The last sign follows from the fact that $\frac{\partial y_{i}}{\partial L_{1i}}<q^{1}$. This can be easily seen from (45) using the above definition of $\left[ 3\right]$ and noting that

$$\begin{aligned} \frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})(q^{0})^{2}\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})}{\left[ 3\right] }<1. \end{aligned}$$

Using Eq. (7), we have

$$\begin{aligned} \frac{\partial z_{i}^{0}}{\partial L_{1i}}=\frac{1}{q^{0}}\left[ \frac{\partial y_{i}}{\partial L_{1i}}-q^{1}\frac{\partial z_{i}^{1}}{\partial L_{1i}}-q^{2}\frac{\partial z_{i}^{2}}{\partial L_{1i}}\right] =\frac{q^{0}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\left[ \frac{\partial y_{i}}{\partial L_{1i}}-q^{1}\right] }{\left[ 1\right] }<0. \end{aligned}$$

Note that $\frac{\partial z_{i}^{0}}{\partial L_{1i}}=\frac{\partial c_{i}^{I}}{\partial L_{1i}}$ and, since $L_{2i}$ remains unchanged, $\frac{\partial z_{i}^{2}}{\partial L_{1i}}=\frac{\partial c_{i}^{D_{2}}}{\partial L_{1i}}$. On the other hand, we have

$$\begin{aligned} \frac{\partial c_{i}^{D_{1}}}{\partial L_{1i}}=\frac{\partial z_{i}^{1}}{\partial L_{1i}}-1=\frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\left[ \frac{\partial y_{i}}{\partial L_{1i}}-q^{1}\right] }{\left[ 1\right] }<0. \end{aligned}$$

Using the fact that $u^{\prime \prime }\left( c_{i}^{D_{1}}\right) =u^{\prime \prime }\left( c_{i}^{D_{2}}\right)$ and the definitions of $q^{1}$ and $q^{2}$, it is straightforward to see that $\frac{\partial c_{i}^{D_{1}}}{\partial L_{1i}}=\frac{\partial c_{i}^{D_{2}}}{\partial L_{1i}}$.

We can thus write

$$\begin{aligned} \frac{\partial c_{i}^{D_{1}}}{\partial L_{1i}}-\frac{\partial c_{i}^{I}}{\partial L_{1i}}=\frac{\partial c_{i}^{D_{2}}}{\partial L_{1i}}-\frac{\partial c_{i}^{I}}{\partial L_{1i}}=\frac{\left[ \frac{\partial y_{i}}{\partial L_{1i}}-q^{1}\right] \pi _{2}u^{\prime \prime }\left( c_{i}^{D_{2}}\right) \left[ 4\right] }{\left[ 1\right] } \end{aligned}$$

(46)

Recalling from above that $\left[ 4\right] >($resp. < and $=)$ 0 under DARA (resp. IARA and CARA) preferences, we have that

$$\begin{aligned} \frac{\partial c_{i}^{D_{1}}}{\partial L_{1i}}-\frac{\partial c_{i}^{I}}{\partial L_{1i}}=\frac{\partial c_{i}^{D_{2}}}{\partial L_{1i}}-\frac{\partial c_{i}^{I}}{\partial L_{1i}}>(\text {resp.}<\text { and }=)\text { }0\text { } \end{aligned}$$

under DARA (resp. IARA and CARA).

Fully differentiating Eqs. (8–10) with respect to $\lambda ,$ we get, respectively,

$$\begin{aligned}&\frac{\partial z_{i}^{1}}{\partial \lambda }\left[ (q^{0})^{2}\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})+\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})(q^{1})^{2}\right] -\frac{\partial y_{i}}{\partial \lambda }\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}\nonumber \\&\qquad +\frac{\partial z_{i}^{2}}{\partial \lambda }\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}q^{2}-q^{0}\pi _{1}u^{\prime }(c_{i}^{D_{1}})\left( \pi _{1}+\pi _{2}\right) -q^{0}\pi _{1}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})\nonumber \\&\qquad +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}\left[ \pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) +\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right) \right] =0, \end{aligned}$$

(47)

$$\begin{aligned}&\frac{\partial z_{i}^{2}}{\partial \lambda }\left[ (q^{0})^{2}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})+\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})(q^{2})^{2}\right] -\frac{\partial y_{i}}{\partial \lambda }\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{2}\nonumber \\&\qquad +\frac{\partial z_{i}^{1}}{\partial \lambda }\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}q^{2}-q^{0}\pi _{2}u^{\prime }(c_{i}^{D_{2}})\left( \pi _{1}+\pi _{2}\right) -q^{0}\pi _{2}\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})\nonumber \\&\qquad +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{2}\left[ \pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) +\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right) \right] =0 \end{aligned}$$

(48)

and

$$\begin{aligned}&\frac{\partial y_{i}}{\partial \lambda }\left[ \left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})-(q^{0})^{2}\frac{v^{\prime \prime }\left( \frac{y_{i}}{w_{i}}\right) }{w_{i}^{2}}\right] -\frac{\partial z_{i}^{1}}{\partial \lambda }\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}\nonumber \\&\qquad -\frac{\partial z_{i}^{2}}{\partial \lambda }\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{2}+q^{0}\left( \pi _{1}+\pi _{2}\right) \frac{v^{\prime }\left( \frac{y_{i}}{w_{i}}\right) }{w_{i}}\nonumber \\&\qquad -\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})\left[ \pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) +\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right) \right] =0 \end{aligned}$$

(49)

Solving the system of Eqs. (47–49) for $\frac{\partial z_{i}^{1}}{\partial \lambda }$, $\frac{\partial z_{i}^{2}}{\partial \lambda }$ and $\frac{\partial y_{i}}{\partial \lambda }$, we obtain

$$\begin{aligned} \frac{\partial y_{i}}{\partial \lambda }= & {} \frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})(q^{0})^{2}\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\left[ \pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) +\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right) \right] }{\left[ 3\right] }\nonumber \\&\qquad +\frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})q^{0}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\left( \pi _{1}+\pi _{2}\right) \left[ 4\right] }{\left[ 3\right] }, \end{aligned}$$

(50)

$$\begin{aligned}&\frac{\partial z_{i}^{1}}{\partial \lambda }=\frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{1}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\left[ \frac{\partial y_{i}}{\partial \lambda }-\pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) -\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right) \right] }{\left[ 1\right] }\nonumber \\&\qquad +\frac{q^{0}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\pi _{1}\left[ u^{\prime }(c_{i}^{D_{1}})\left( \pi _{1}+\pi _{2}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})\right] }{\left[ 1\right] }, \end{aligned}$$

(51)

$$\begin{aligned}&\frac{\partial z_{i}^{2}}{\partial \lambda }=\frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})q^{2}\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\left[ \frac{\partial y_{i}}{\partial \lambda }-\pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) -\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right) \right] }{\left[ 1\right] }\nonumber \\&\qquad +\frac{q^{0}\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\pi _{2}\left[ u^{\prime }(c_{i}^{D_{2}})\left( \pi _{1}+\pi _{2}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})\right] }{\left[ 1\right] } \end{aligned}$$

(52)

Let us first discuss $\frac{\partial y_{i}}{\partial \lambda }$. It is easy to see that its first term is always positive, while the sign of the second term depends on the sign of $\left[ 4\right]$. Therefore, the second term is positive (resp. negative and equal to zero) under DARA (resp. IARA and CARA) preferences. This implies that $\frac{\partial y_{i}}{\partial \lambda }$ is clearly positive under DARA and CARA, but its sign is ambiguous under IARA. Moreover, it can be easily seen that $\frac{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime \prime }(c_{i}^{I})(q^{0})^{2}\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})}{\left[ 3\right] }<1$. This means that the first term of $\frac{\partial y_{i}}{\partial \lambda }$ is smaller than $\pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) +\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right)$. We therefore have $\frac{\partial y_{i}}{\partial \lambda }<\pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) +\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right)$ under CARA and IARA preferences, while this comparison is ambiguous under DARA.

Let us now turn to $\frac{\partial z_{i}^{1}}{\partial \lambda }$ and $\frac{\partial z_{i}^{2}}{\partial \lambda }$. First, using the fact that $u^{\prime \prime }(c_{i}^{D_{1}})=u^{\prime \prime }(c_{i}^{D_{2}})$ and the definitions of $q^{1}$ and $q^{2}$, it is straightforward to see that $\frac{\partial z_{i}^{1}}{\partial \lambda }=\frac{\partial z_{i}^{2}}{\partial \lambda }$. It can then be noted that the second terms of $\frac{\partial z_{i}^{1}}{\partial \lambda }$ and $\frac{\partial z_{i}^{2}}{\partial \lambda }$ are always negative (substitution effects as explained in the main text). As for the first terms (income effects), it follows from the above discussion of $\frac{\partial y_{i}}{\partial \lambda }$ that they are also negative under CARA and IARA, but their sign is ambiguous under DARA. Thus, under CARA and IARA we clearly have $\frac{\partial z_{i}^{1}}{\partial \lambda }=\frac{\partial z_{i}^{2}}{\partial \lambda }<0$, while under DARA the sign is undetermined.

Using Eq. (7), we have

$$\begin{aligned} \frac{\partial z_{i}^{0}}{\partial \lambda }= & {} \frac{1}{q^{0}}\left[ \frac{\partial y_{i}}{\partial \lambda }-q^{1}\frac{\partial z_{i}^{1}}{\partial \lambda }-q^{2}\frac{\partial z_{i}^{2}}{\partial \lambda }\right] \\= & {} \frac{q^{0}\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\pi _{1}u^{\prime \prime }(c_{i}^{D_{1}})\left[ \frac{\partial y_{i}}{\partial \lambda }-\pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) -\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right) \right] }{\left[ 1\right] }\\&-\frac{\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\pi _{1}\left( q^{1}+q^{2}\right) \left[ u^{\prime }(c_{i}^{D_{1}})\left( \pi _{1}+\pi _{2}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})\right] }{\left[ 1\right] } \end{aligned}$$

The sign of $\frac{\partial z_{i}^{0}}{\partial \lambda }$ is ambiguous. While its second term is always positive (substitution effect), the ambiguity comes from the first term (income effect). Under CARA and IARA preferences the first term is negative, which makes the sign of the whole expression undetermined. Under DARA preferences the sign of the first term is itself undetermined.

Note that $\frac{\partial z_{i}^{0}}{\partial \lambda }=\frac{\partial c_{i}^{I}}{\partial \lambda }$ and, since $L_{1i}$ and $L_{2i}$ remain unchanged, we have that $\frac{\partial z_{i}^{1}}{\partial \lambda }=\frac{\partial c_{i}^{D_{1}}}{\partial \lambda }$ and $\frac{\partial z_{i}^{2}}{\partial \lambda }=\frac{\partial c_{i}^{D_{2}}}{\partial \lambda }$.

We can thus write

$$\begin{aligned}&\frac{\partial c_{i}^{D_{1}}}{\partial \lambda }-\frac{\partial c_{i}^{I}}{\partial \lambda }=\frac{\partial c_{i}^{D_{2}}}{\partial \lambda }-\frac{\partial c_{i}^{I}}{\partial \lambda }=\frac{\left[ \frac{\partial y_{i}}{\partial \lambda }-\pi _{1}\left( z_{i}^{1}-z_{i}^{0}\right) -\pi _{2}\left( z_{i}^{2}-z_{i}^{0}\right) \right] \pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})[4]}{\left[ 1\right] }\nonumber \\&\qquad +\frac{\pi _{2}u^{\prime \prime }(c_{i}^{D_{2}})\pi _{1}\left[ u^{\prime }(c_{i}^{D_{1}})\left( \pi _{1}+\pi _{2}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})\right] }{\left[ 1\right] } \end{aligned}$$

(53)

The second term of the RHS of (53) (substitution effect) is always negative. The first term (income effect) is also negative under IARA preferences but is equal to zero under CARA and undetermined under DARA. We thus have that

$$\begin{aligned} \frac{\partial c_{i}^{D_{1}}}{\partial \lambda }-\frac{\partial c_{i}^{I}}{\partial \lambda }=\frac{\partial c_{i}^{D_{2}}}{\partial \lambda }-\frac{\partial c_{i}^{I}}{\partial \lambda }<0\text { } \end{aligned}$$

under IARA and CARA, but the sign is undetermined under DARA.

Appendix 2: FOCs of the government’s problem (general case)

The FOCs of the government’s problem in the general case write as follows:

$$\begin{aligned}&\frac{\partial \mathcal {L}}{\partial z_{h}^{1}}=\pi _{1}u^{\prime }(c_{h}^{D_{1}})\left[ n_{h}+\gamma \right] -\mu n_{h}p^{1}=0 \end{aligned}$$

(54)

$$\begin{aligned}&\frac{\partial \mathcal {L}}{\partial z_{h}^{2}}=\pi _{2}u^{\prime }(c_{h}^{D_{2}})\left[ n_{h}+\gamma \right] -\mu n_{h}p^{2}=0 \end{aligned}$$

(55)

$$\begin{aligned}&\frac{\partial \mathcal {L}}{\partial z_{h}^{0}}=(1-\pi _{1}-\pi _{2})u^{\prime }(c_{h}^{I})\left[ n_{h}+\gamma \right] -\mu n_{h}p^{0}=0 \end{aligned}$$

(56)

$$\begin{aligned}&\frac{\partial \mathcal {L}}{\partial y_{h}}=-v'\left( \frac{y_{h}}{w_{h}}\right) \frac{1}{w_{h}}\left[ n_{h}+\gamma \right] +\mu n_{h}=0 \end{aligned}$$

(57)

$$\begin{aligned}&\frac{\partial \mathcal {L}}{\partial z_{l}^{1}}=n_{l}\pi _{1}u^{\prime }(c_{l}^{D_{1}})-\mu n_{l}p^{1}-\gamma \pi _{1}u^{\prime }(\widetilde{c}_{l}^{D_{1}})=0 \end{aligned}$$

(58)

$$\begin{aligned}&\frac{\partial \mathcal {L}}{\partial z_{l}^{2}}=n_{l}\pi _{2}u^{\prime }(c_{l}^{D_{2}})-\mu n_{l}p^{2}-\gamma \pi _{2}u^{\prime }(\widetilde{c}_{l}^{D_{2}})=0 \end{aligned}$$

(59)

$$\begin{aligned}&\frac{\partial \mathcal {L}}{\partial z_{l}^{0}}=(1-\pi _{1}-\pi _{2})u^{\prime }(c_{l}^{I})\left[ n_{l}-\gamma \right] -\mu n_{l}p^{0}=0 \end{aligned}$$

(60)

$$\begin{aligned}&\frac{\partial \mathcal {L}}{\partial y_{l}}=-n_{l}v'\left( \frac{y_{l}}{w_{l}}\right) \frac{1}{w_{l}}+\mu n_{l}+\gamma v'\left( \frac{y_{l}}{w_{h}}\right) \frac{1}{w_{h}}=0 \end{aligned}$$

(61)

Appendix 3: Specific examples with identical needs

Recall that $D_{h}=c_{h}^{I}-c_{h}^{D_{1}}=c_{h}^{I}-c_{h}^{D_{2}}$ and $D_{l}=c_{l}^{I}-c_{l}^{D_{1}}=c_{l}^{I}-c_{l}^{D_{2}}$.

Let us first assume that $u(x)=\ln x$. Then, from (54) to (55) we have $c_{h}^{D_{1}}=c_{h}^{D_{2}}=\frac{(n_{h}+\gamma )}{\mu (1+\lambda ^\mathrm{g})n_{h}}$ and from (56), $c_{h}^{I}=\frac{\left( 1-\pi _{1}-\pi _{2}\right) (n_{h}+\gamma )}{\mu n_{h}\left[ 1-(1+\lambda ^\mathrm{g})\pi _{1}-(1+\lambda ^\mathrm{g})\pi _{2}\right] }$.

Similarly, from (58) to (59) we have $c_{l}^{D_{1}}=c_{l}^{D_{2}}=\frac{(n_{l}-\gamma )}{\mu (1+\lambda ^\mathrm{g})n_{l}}$ and from (60), $c_{l}^{I}=\frac{\left( 1-\pi _{1}-\pi _{2}\right) (n_{l}-\gamma )}{\mu n_{l}\left[ 1-(1+\lambda ^\mathrm{g})\pi _{1}-(1+\lambda ^\mathrm{g})\pi _{2}\right] }$.

We then get

$$\begin{aligned} D_{h}=\frac{(n_{h}+\gamma )\lambda ^\mathrm{g}}{\mu n_{h}\left[ 1-(1+\lambda ^\mathrm{g})\pi _{1}-(1+\lambda ^\mathrm{g})\pi _{2}\right] (1+\lambda ^\mathrm{g})} \end{aligned}$$

and

$$\begin{aligned} D_{l}=\frac{(n_{l}-\gamma )\lambda ^\mathrm{g}}{\mu n_{l}\left[ 1-(1+\lambda ^\mathrm{g})\pi _{1}-(1+\lambda ^\mathrm{g})\pi _{2}\right] (1+\lambda ^\mathrm{g})}. \end{aligned}$$

Noting that $\frac{(n_{h}+\gamma )}{n_{h}}>1$ and $\frac{(n_{l}-\gamma )}{n_{l}}<1$, we have $D_{h}>D_{l}$.

Let us now consider $u(x)=-e^{-x}$. Now, from (54) to (55) we have $c_{h}^{D_{1}}=c_{h}^{D_{2}}=-\ln \left[ \frac{\mu (1+\lambda ^\mathrm{g})n_{h}}{(n_{h}+\gamma )}\right]$ and from (56), $c_{h}^{I}=-\ln \left[ \frac{\mu n_{h}\left[ 1-(1+\lambda ^\mathrm{g})\pi _{1}-(1+\lambda ^\mathrm{g})\pi _{2}\right] }{\left( 1-\pi _{1}-\pi _{2}\right) (n_{h}+\gamma )}\right]$.

Similarly, from (58) to (59) we have $c_{l}^{D_{1}}=c_{l}^{D_{2}}=-\ln \left[ \frac{\mu (1+\lambda ^\mathrm{g})n_{l}}{(n_{l}-\gamma )}\right]$ and from (60) we have $c_{l}^{I}=-\ln \left[ \frac{\mu n_{l}\left[ 1-(1+\lambda ^\mathrm{g})\pi _{1}-(1+\lambda ^\mathrm{g})\pi _{2}\right] }{\left( 1-\pi _{1}-\pi _{2}\right) (n_{l}-\gamma )}\right]$.

We then obtain

$$\begin{aligned} D_{h}=D_{l}=\ln \left[ \frac{(1+\lambda ^\mathrm{g})\left( 1-\pi _{1}-\pi _{2}\right) }{\left[ 1-\pi _{1}(1+\lambda ^\mathrm{g})-\pi _{2}(1+\lambda ^\mathrm{g})\right] }\right] . \end{aligned}$$

Appendix 4: Implementation of the (non-paternalistic) second-best with different needs

In this appendix, we come back to the initial specification of the individual problem with an explicit modelling of private insurance (as presented in Sect. 2). Individuals thus earn income $y_{i}$, pay private insurance premiums $P_{i}$ and get insurance benefits $\alpha _{1i}L_{1i}$ and $\alpha _{2i}L_{2i}.$ We assume that the government can observe all these variables and consider the following (nonlinear) policy instruments:

• Tax based on income and insurance premiums $T(y_{i},P_{i})$ paid before the realisation of the state of nature;

• Tax based on insurance benefits in the state of low severity dependence $T_{1}(\alpha _{1i}L_{1i});$

• Tax based on insurance benefits in the state of high severity dependence $T_{2}(\alpha _{2i}L_{2i}).$

We also assume here that the government has the same loading costs as private insurers, i.e. $\lambda ^\mathrm{g}=\lambda$.

Given the government’s policy, the Lagrangian of individual i writes as

$$\begin{aligned} \mathcal {L}&=\pi _{1}u\left( c_{i}^{D_{1}}\right) +\pi _{2}u\left( c_{i}^{D_{2}}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u\left( c_{i}^{I}\right) -v\left( \frac{y_{i}}{w_{i}}\right) \\&\quad +\mu _{i}\left[ P_{i}-\pi _{1}(1+\lambda )\alpha _{1i}L_{1i}-\pi _{2}(1+\lambda )\alpha _{2i}L_{2i}\right] \end{aligned}$$

where

$c_{i}^{D_{1}}=y_{i}-P_{i}-T(y_{i},P_{i})-(1-\alpha _{1i})L_{1i}-T_{1}(\alpha _{1i}L_{1i})$

$c_{i}^{D_{2}}=y_{i}-P_{i}-T(y_{i},P_{i})-(1-\alpha _{2i})L_{2i}-T_{2}(\alpha _{2i}L_{2i})$

$c_{i}^{I}=y_{i}-P_{i}-T(y_{i},P_{i}).$

Assuming interior solutions, the individual FOCs write as follows:

$$\begin{aligned} \frac{\partial \mathcal {L}}{\partial y_{i}}= & {} \left[ \pi _{1}u^{\prime }\left( c_{i}^{D_{1}}\right) +\pi _{2}u^{\prime }\left( c_{i}^{D_{2}}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }\left( c_{i}^{I}\right) \right] \left( 1-T_{y_{i}}\right) -\frac{v^{\prime }\left( \frac{y_{i}}{w_{i}}\right) }{w_{i}}=0 \nonumber \\\end{aligned}$$

(62)

$$\begin{aligned} \frac{\partial \mathcal {L}}{\partial P_{i}}= & {} \left[ \pi _{1}u^{\prime }\left( c_{i}^{D_{1}}\right) +\pi _{2}u^{\prime }\left( c_{i}^{D_{2}}\right) +\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})\right] \left( -1-T_{p_{i}}\right) +\mu _{i}=0 \nonumber \\\end{aligned}$$

(63)

$$\begin{aligned} \frac{\partial \mathcal {L}}{\partial \alpha _{1i}}= & {} u^{\prime }\left( c_{i}^{D_{1}}\right) \left[ 1-T_{1}^{\prime }(\alpha _{1i}L_{1i})\right] -\mu _{i}(1+\lambda )=0 \end{aligned}$$

(64)

$$\begin{aligned} \frac{\partial \mathcal {L}}{\partial \alpha _{2i}}= & {} u^{\prime }\left( c_{i}^{D_{2}}\right) \left[ 1-T_{2}^{\prime }(\alpha _{2i}L_{2i})\right] -\mu _{i}(1+\lambda )=0 \end{aligned}$$

(65)

where $T_{y_{i}}$ and $T_{p_{i}}$ denote the partial derivatives of T with respect to $y_{i}$ and $P_{i}.$

From (62), we have

$$\begin{aligned} \frac{v'\left( \frac{y_{i}}{w_{i}}\right) }{(1-\pi _{1}-\pi _{2})u'(c_{i}^{I})}=w_{i}\left( 1-T_{y_{i}}\right) \left[ 1+\frac{\pi _{1}u^{\prime }(c_{i}^{D_{1}})}{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})}+\frac{\pi _{2}u^{\prime }(c_{i}^{D_{2}})}{\left( 1-\pi _{1}-\pi _{2}\right) u^{\prime }(c_{i}^{I})}\right] \nonumber \\ \end{aligned}$$

(66)

Combining (64) and (65), we get

$$\begin{aligned} \frac{u^{\prime }(c_{i}^{D_{1}})}{u^{\prime }(c_{i}^{D_{2}})}=\frac{\left[ 1-T_{2}^{\prime }(\alpha _{2i}L_{2i})\right] }{\left[ 1-T_{1}^{\prime }(\alpha _{1i}L_{1i})\right] } \end{aligned}$$

(67)

Combining (64) and (63), we can get

$$\begin{aligned} \frac{u^{\prime }(c_{i}^{I})}{u^{\prime }(c_{i}^{D_{1}})}= & {} \frac{-\pi _{1}}{\left( 1-\pi _{1}-\pi _{2}\right) }+\frac{\left[ 1-T_{1}^{\prime }(\alpha _{1i}L_{1i})\right] }{\left( 1-\pi _{1}-\pi _{2}\right) (1+\lambda )\left( 1+T_{p_{i}}\right) }\nonumber \\&-\frac{\pi _{2}}{\left( 1-\pi _{1}-\pi _{2}\right) }\frac{\left[ 1-T_{1}^{\prime }(\alpha _{1i}L_{1i})\right] }{\left[ 1-T_{2}^{\prime }(\alpha _{2i}L_{2i})\right] } \end{aligned}$$

(68)

Finally, combining (65) and (63), we can get

$$\begin{aligned} \frac{u^{\prime }(c_{i}^{I})}{u^{\prime }(c_{i}^{D_{2}})}= & {} \frac{-\pi _{2}}{\left( 1-\pi _{1}-\pi _{2}\right) }+\frac{\left[ 1-T_{2}^{\prime }(\alpha _{2i}L_{2i})\right] }{\left( 1-\pi _{1}-\pi _{2}\right) (1+\lambda )\left( 1+T_{p_{i}}\right) }\nonumber \\&-\frac{\pi _{1}}{\left( 1-\pi _{1}-\pi _{2}\right) }\frac{\left[ 1-T_{2}^{\prime }(\alpha _{2i}L_{2i})\right] }{\left[ 1-T_{1}^{\prime }(\alpha _{1i}L_{1i})\right] } \end{aligned}$$

(69)

Let us first look at labour supply. For type h, combining (66) with (16), (18) and (19), we have $T_{y_{h}}=0$.

For type l, combining (66) with (17), (20) and (21), we get

$$\begin{aligned} T_{y_{l}}=1-\frac{\left[ n_{l}-\gamma \right] \beta }{\left[ n_{l}-\gamma \frac{w_{l}v'\left( \frac{y_{l}}{w_{h}}\right) }{w_{h}v'\left( \frac{y_{l}}{w_{l}}\right) }\right] }>0 \end{aligned}$$

where

$$\begin{aligned} \beta =\frac{\left[ n_{l}-\gamma \frac{u'(\widetilde{c}_{l}^{D_{1}})}{u'(c_{l}^{D_{1}})}\right] \left[ n_{l}-\gamma \frac{u'(\widetilde{c}_{l}^{D_{2}})}{u'(c_{l}^{D_{2}})}\right] }{\left[ n_{l}-\gamma \frac{u'(\widetilde{c}_{l}^{D_{1}})}{u'(c_{l}^{D_{1}})}\right] \left[ n_{l}-\gamma \frac{u'(\widetilde{c}_{l}^{D_{2}})}{u'(c_{l}^{D_{2}})}\right] +p^{1}\left[ n_{l}-\gamma \frac{u'(\widetilde{c}_{l}^{D_{2}})}{u'(c_{l}^{D_{2}})}\right] \left[ \gamma \frac{u'(\widetilde{c}_{l}^{D_{1}})}{u'(c_{l}^{D_{1}})}-\gamma \right] +p^{2}\left[ n_{l}-\gamma \frac{u'(\widetilde{c}_{l}^{D_{1}})}{u'(c_{l}^{D_{1}})}\right] \left[ \gamma \frac{u'(\widetilde{c}_{l}^{D_{2}})}{u'(c_{l}^{D_{2}})}-\gamma \right] }<1. \end{aligned}$$

Turning to insurance, from the discussion in Sect. 4.2 it follows immediately that for type h we have $T_{p_{h}}=0,$ $T_{1}^{\prime }(\alpha _{1h}L_{1h})=0$ and $T_{2}^{\prime }(\alpha _{2h}L_{2h})=0.$

For type l, combining the government’s FOCs (58) and (59), we can get

$$\begin{aligned} \frac{u^{\prime }(c_{i}^{D_{1}})}{u^{\prime }(c_{i}^{D_{2}})}=1-\frac{\gamma }{n_{l}}\left[ \frac{u^{\prime }(\widetilde{c}_{l}^{D_{2}})-u^{\prime }(\widetilde{c}_{l}^{D_{1}})}{u^{\prime }(c_{l}^{D_{2}})}\right] . \end{aligned}$$

Combining this with (67), we have

$$\begin{aligned} \frac{\left[ 1-T_{2}^{\prime }(\alpha _{2i}L_{2i})\right] }{\left[ 1-T_{1}^{\prime }(\alpha _{1i}L_{1i})\right] }=1-\frac{\gamma }{n_{l}}\left[ \frac{u^{\prime }(\widetilde{c}_{l}^{D_{2}})-u^{\prime }(\widetilde{c}_{l}^{D_{1}})}{u^{\prime }(c_{l}^{D_{2}})}\right] \end{aligned}$$

(70)

We know from the discussion in the main text that the optimal allocation implies $u^{\prime }(c_{i}^{D_{1}})<u^{\prime }(c_{i}^{D_{2}})$ if $\widehat{L}_{2h}>\widehat{L}_{1h}$, $u^{\prime }(c_{i}^{D_{1}})>u^{\prime }(c_{i}^{D_{2}})$ if $\widehat{L}_{2h}<\widehat{L}_{1h}$ and $u^{\prime }(c_{i}^{D_{1}})=u^{\prime }(c_{i}^{D_{2}})$ if $\widehat{L}_{2h}=\widehat{L}_{1h}$. We thus have that $T_{2}^{\prime }(\alpha _{2l}L_{2l})>T_{1}^{\prime }(\alpha _{1l}L_{1l})$ if $\widehat{L}_{2h}>\widehat{L}_{1h}$, $T_{2}^{\prime }(\alpha _{2l}L_{2l})<T_{1}^{\prime }(\alpha _{1l}L_{1l})$ if $\widehat{L}_{2h}<\widehat{L}_{1h}$ and $T_{2}^{\prime }(\alpha _{2l}L_{2l})=T_{1}^{\prime }(\alpha _{1l}L_{1l})$ if $\widehat{L}_{2h}=\widehat{L}_{1h}$.

Further, combining (68) with (24), using (70) and rearranging, we obtain

$$\begin{aligned}&\frac{\left[ 1-T_{1}^{\prime }(\alpha _{1l}L_{1l})\right] }{\left( 1+T_{p_{l}}\right) }=\frac{\left[ 1-\pi _{1}(1+\lambda )-\pi _{2}(1+\lambda )\right] \left[ n_{l}-\gamma \frac{u'\left( \widetilde{c}_{l}^{D_{1}}\right) }{u'\left( c_{l}^{D_{1}}\right) }\right] }{\left[ n_{l}-\gamma \right] }\nonumber \\&\qquad +\pi _{1}(1+\lambda )+\frac{\pi _{2}(1+\lambda )n_{l}u^{\prime }\left( c_{l}^{D_{2}}\right) }{n_{l}u^{\prime }\left( c_{l}^{D_{2}}\right) -\gamma \left[ u^{\prime }\left( \widetilde{c}_{l}^{D_{2}}\right) -u^{\prime }\left( \widetilde{c}_{l}^{D_{1}}\right) \right] } \end{aligned}$$

(71)

Similarly, combining (69) with (25), we can get

$$\begin{aligned}&\frac{\left[ 1-T_{2}^{\prime }(\alpha _{2l}L_{2l})\right] }{\left( 1+T_{p_{l}}\right) }=\frac{\left[ 1-\pi _{1}(1+\lambda )-\pi _{2}(1+\lambda )\right] \left[ n_{l}-\gamma \frac{u'\left( \widetilde{c}_{l}^{D_{2}}\right) }{u'\left( c_{l}^{D_{2}}\right) }\right] }{\left[ n_{l}-\gamma \right] }\nonumber \\&\qquad +\pi _{2}(1+\lambda )+\frac{\pi _{1}(1+\lambda )\left( n_{l}u^{\prime }\left( c_{l}^{D_{2}}\right) -\gamma \left[ u^{\prime }\left( \widetilde{c}_{l}^{D_{2}}\right) -u^{\prime }\left( \widetilde{c}_{l}^{D_{1}}\right) \right] \right) }{n_{l}u^{\prime }\left( c_{l}^{D_{2}}\right) } \end{aligned}$$

(72)

Equations (70–72) define $T_{p_{l}},$ $T_{1}^{\prime }(\alpha _{1l}L_{1l})$ and $T_{2}^{\prime }(\alpha _{2l}L_{2l}).$ It can be noticed that these marginal taxes can be chosen in several ways as long as the equations are satisfied. For instance, one of the three marginal taxes can be equal to zero as long as the other two are chosen optimally. These equations also show that taxing only the premium is generally not enough: this is only possible if $\widehat{L}_{2h}=\widehat{L}_{1h}$. Otherwise, in addition to the marginal tax on the premium, a marginal tax or subsidy is needed in at least one of the two dependence states.