Optimal taxation in non-life insurance markets

Abstract

Insurance markets all over the world are subject to taxes, which can be collected through various tax schemes. The most commonly used tax schemes in non-life insurance include taxes on the premium income as well as taxes on corporate profits. The mix of those taxes differs substantially across regions. In Europe, for instance, premium taxes are at much higher levels than those in the United States. This paper shows that the particular mix of taxes matters for insurance companies’ solvency levels and insurance premiums, even when the expected tax revenue is fixed. The analysis provides comparative statics that reveal which combination of premium taxation and corporate taxation maximizes welfare. Numerical examples suggest that relatively high premium tax rates are welfare optimal. In general, a low volatility of insurance claims, a strong reaction of insurance demand to insurer default risk, as well as substantial agency issues between shareholders and management make corporate taxation more favorable in comparison to premium taxation.

Appendices

Appendix 1: Derivation of Eq. (6)

The equity capital endowment that corresponds to the asset–liability ratio s needs to fulfill

$$\begin{aligned}&s\overset{\text{Eq. 1}}{=}\frac{(1-\tau _p) \cdot y \cdot p + (1- \delta ) \cdot K}{y \cdot \mu } \nonumber \\\Rightarrow & {} \, K = y \cdot \frac{ s \cdot \mu - (1-\tau _p) \cdot p }{1-\delta } \end{aligned}$$

(34)

Due to $\max \{A_1 - L_1;0\} = A_1-L_1 + \max \{L_1 - A_1;0\}$ we have

$$\begin{aligned} \text{SHV} = \, & {} e^{-r} \cdot {\mathbb{E}} [\max \{A_1 - L_1;0\} - \tau _c \cdot \bigl (\max \{A_1 - L_1;0\}-K\bigr )] -K\\ = \, & {} A_0-y \cdot \mu \cdot (1-dr) - \tau _c \cdot \bigl ( A_0-y \cdot \mu \cdot (1-dr) - e^{-r} \cdot K \bigr ) -K \end{aligned}$$

The rest follows by inserting Eqs. 1 and 34 and simplifying.

Appendix 2: Derivation of Eq. (9)

The derivation is in line with Kay and Keen (1991, p. 240). Let $\tilde{p}(y,dr)$ denote the inverse of the demand function y(dr, p) with respect to p. Then the first condition is formalized as

$$\begin{aligned} \frac{\partial \text{SHV}}{\partial dr}\overset{\text{Eq. 6}}{=}y \cdot \biggl ( \tilde{p}_{dr} + \mu - \frac{\partial t_0}{\partial dr} - \frac{\partial t_0}{\partial p} \cdot \tilde{p}_{dr} \biggr ) = 0, \end{aligned}$$

(35)

where $-\tilde{p}_{dr}$ can be interpreted as consumers’ willingness-to-pay for a marginal decrease in dr, such that demand stays constant. According to the implicit function theorem, $-\tilde{p}_{dr}=y_{dr}/y_p=:S$. Equation (9) follows from solving Eq. (35) for $-\tilde{p}_{dr}$.

Appendix 3: Derivation of Ineq. (16)

To collect a constant tax revenue, a marginal adjustment of $\tau _i$ must be compensated by an adjustment of $\tau _j$ such that

$$\begin{aligned}&\frac{\text{d} }{\text{d} \tau _i} \bigl ( y \cdot t_0^{tax} \bigr ) + \frac{\text{d} }{\text{d} \tau _j} \bigl ( y \cdot t_0^{tax} \bigr ) \cdot \frac{\text{d}\tau _j}{\text{d}\tau _i}=0,\nonumber \\ \Leftrightarrow \frac{\text{d}\tau _j}{\text{d}\tau _i}= & {} - \frac{\frac{\text{d} }{\text{d}\tau _i} \bigl ( y \cdot t_0^{tax} \bigr ) }{ \frac{\text{d} }{\text{d} \tau _j} \bigl ( y \cdot t_0^{tax} \bigr )}= - \frac{ \bigl ( y_p \frac{\text{d} p}{\text{d}\tau _i} + y_{dr} \frac{\text{d} dr}{\text{d}\tau _i} \bigr ) \cdot t_0^{tax} + y \cdot \frac{\text{d} t_0^{tax}}{\text{d}\tau _i}}{ \bigl ( y_p \frac{\text{d} p}{\text{d}\tau _j} + y_{dr} \frac{\text{d} dr}{\text{d}\tau _j} \bigr ) \cdot t_0^{tax}+ y \cdot \frac{\text{d} t_0^{tax}}{\text{d}\tau _j}} \end{aligned}$$

(36)

Equation (36) is well-defined if the denominator of the right-hand side is positive. This is ensured by assuming that $\tau _i$ and $\tau _j$ are in the increasing part of the Laffer curve, i.e., $\frac{\text{d} T_0^{tax}}{\text{d} \tau }=\bigl ( y_p \frac{\text{d} p}{\text{d}\tau } + y_{dr} \frac{\text{d} dr}{\text{d}\tau } \bigr ) \cdot t_0^{tax} + y \cdot \frac{\text{d} t_0^{tax}}{\text{d}\tau }>0$, for $\tau =\tau _i, \tau _j$. A compensated increase in $\tau _i$ enhances consumer surplus if

$$\begin{aligned}&\frac{\text{d}CS}{\text{d}\tau _i} + \frac{\text{d}CS}{\text{d}\tau _j}\cdot \frac{\text{d}\tau _j}{\text{d}\tau _i}=CS_p \cdot \frac{\text{d}p}{\text{d}\tau _i} + CS_{dr} \cdot \frac{\text{d} dr}{\text{d}\tau _i} + \bigl ( CS_p \cdot \frac{\text{d}p}{\text{d}\tau _j} + CS_{dr} \cdot \frac{\text{d} dr}{\text{d}\tau _j} \bigr ) \cdot \frac{\text{d}\tau _j}{\text{d}\tau _i} >0 \end{aligned}$$

The rest follows from Eq. (36) together with $CS_p=-y$ (cf. Eq. 14).

Appendix 4: Calculation of term 2 in line 20

Suppose that $y(dr,p)=a(p+ b(dr))$. For this demand function, we have $y_p = a'(p+ b(dr))$ and $y_{dr} = b'(dr)\cdot a'(p+ b(dr))$. Therefore, we have

$$\begin{aligned} CS_{dr}+\frac{y_{dr}}{y_p}\cdot y(dr,p) =\, & {} \frac{\partial }{\partial dr} \int _p^\infty y(dr,\tilde{p})\text{d} \tilde{p} + b'(dr) \cdot y(dr,p)\\ = \, & {} b'(dr)\cdot \underbrace{\int _p^\infty a'(\tilde{p} + \, b(dr)) \text{d}\tilde{p}}_{=y(dr,p)} + \, b'(dr) \cdot y(dr,p)= 0 \end{aligned}$$

Appendix 5: Optimality condition for consumer surplus analysis with endogenous default ratio

Inserting Term 1 from Eq. (20) into Eq. (16) implies

$$\begin{aligned}&-y \frac{ \frac{\text{d}p}{\text{d}\tau _i} + \frac{y_{dr}}{y_p} \cdot \frac{\text{d} dr}{\text{d}\tau _i}}{\bigl ( y_p \frac{\text{d} p}{\text{d}\tau _i} + y_{dr} \frac{\text{d} dr}{\text{d}\tau _i} \bigr ) \cdot t_0^{tax} + y \cdot \frac{\text{d} t_0^{tax}}{\text{d}\tau _i} }> -y \frac{ \frac{\text{d}p}{\text{d}\tau _j} + \frac{y_{dr}}{y_p} \cdot \frac{\text{d} dr}{\text{d}\tau _j}}{\bigl ( y_p \frac{\text{d} p}{\text{d}\tau _j} + y_{dr} \frac{\text{d} dr}{\text{d}\tau _j} \bigr ) \cdot t_0^{tax} + y \cdot \frac{\text{d} t_0^{tax}}{\text{d}\tau _j} } \nonumber \\\Leftrightarrow & {} \frac{\bigl ( y_p \frac{\text{d} p}{\text{d}\tau _i} + y_{dr} \frac{\text{d} dr}{\text{d}\tau _i} \bigr ) \cdot t_0^{tax} + y \cdot \frac{\text{d} t_0^{tax}}{\text{d}\tau _i} }{ \frac{\text{d}p}{\text{d}\tau _i} + \frac{y_{dr}}{y_p} \cdot \frac{\text{d} dr}{\text{d}\tau _i}} > \frac{\bigl ( y_p \frac{\text{d} p}{\text{d}\tau _j} + y_{dr} \frac{\text{d} dr}{\text{d}\tau _j} \bigr ) \cdot t_0^{tax} + y \cdot \frac{\text{d} t_0^{tax}}{\text{d}\tau _j} }{ \frac{\text{d}p}{\text{d}\tau _j} + \frac{y_{dr}}{y_p} \cdot \frac{\text{d} dr}{\text{d}\tau _j}} \end{aligned}$$

(37)

Using $S=\frac{y_{dr}}{y_p}$ allows for rewriting the LHS (and the RHS accordingly) of the last line as

$$\begin{aligned}&\frac{\bigl ( y_p \frac{\text{d} p}{\text{d}\tau _i} + y_{dr} \frac{\text{d} dr}{\text{d}\tau _i} \bigr ) \cdot t_0^{tax} + y \cdot \frac{\text{d} t_0^{tax}}{\text{d}\tau _i} }{ \frac{\text{d}p}{\text{d}\tau _i} + S \cdot \frac{\text{d} dr}{\text{d}\tau _i}} \nonumber \\= \, & {} \frac{ y_p \cdot \bigl ( \frac{\text{d} p}{\text{d}\tau _i} + S \cdot \frac{\text{d} dr}{\text{d}\tau _i} \bigr ) \cdot t_0^{tax} + y \cdot \bigl ( \frac{\partial t_0^{tax}}{\partial p} \cdot \frac{\text{d} p}{\text{d}\tau _i} + \frac{\partial t_0^{tax}}{\partial dr} \cdot \frac{\text{d} dr}{\text{d}\tau _i} +\frac{\partial t_0^{tax}}{\partial \tau _i}\bigr ) }{\frac{\text{d}p}{\text{d}\tau _i} + S \cdot \frac{\text{d} dr}{\text{d}\tau _i}} \nonumber \\= \, & {} \frac{ \bigl ( y_p \cdot t_0^{tax} + y \cdot \frac{\partial t_0^{tax}}{\partial p} \bigr ) \cdot \bigl ( \frac{\text{d} p}{\text{d}\tau _i} + S\cdot \frac{\text{d} dr}{\text{d}\tau _i} \bigr ) + y \cdot \bigl ( \frac{\partial t_0^{tax}}{\partial dr} - S \cdot \frac{\partial t_0^{tax}}{\partial p} \bigr ) \cdot \frac{\text{d} dr}{\text{d}\tau _i} +y \cdot \frac{\partial t_0^{tax}}{\partial \tau _i}}{\frac{\text{d}p}{\text{d}\tau _i} + S \cdot \frac{\text{d} dr}{\text{d}\tau _i}} \nonumber \\= \, & {} y_p \cdot t_0^{tax} + y \cdot \frac{\partial t_0^{tax}}{\partial p} + y \cdot \frac{ \frac{\partial t_0^{tax}}{\partial \tau _i} + \bigl ( \frac{\partial t_0^{tax}}{\partial dr} - S \cdot \frac{\partial t_0^{tax}}{\partial p} \bigr ) \cdot \frac{\text{d} dr}{\text{d}\tau _i}}{\frac{\text{d}p}{\text{d}\tau _i} + S \cdot \frac{\text{d} dr}{\text{d}\tau _i} } \end{aligned}$$

(38)

Using Eqs. (9) and (12) the denominator of the last ratio of line 38 can be rewritten as

$$\begin{aligned}&\frac{\text{d}p}{\text{d}\tau _i} + S \cdot \frac{\text{d} dr}{\text{d}\tau _i}\\= \, & {} \frac{ \frac{\partial t_0}{\partial \tau _i} - \bigl (\mu - \frac{\partial t_0}{\partial dr} \bigr ) \cdot \frac{\text{d} dr}{\text{d} \tau _i} }{ 1-\frac{\partial t_0}{\partial p} } + \frac{\mu -\frac{\partial t_0}{\partial dr}}{1-\frac{\partial t_0}{\partial p}} \cdot \frac{\text{d} dr}{\text{d} \tau _i} \\= \, & {} \frac{ \frac{\partial t_0}{\partial \tau _i} }{ 1-\frac{\partial t_0}{\partial p} } \end{aligned}$$

Inserting this term for the denominator of the last ratio of Eq. (38), removing all items that do not depend on $\tau _i$, and taking the reciprocal gives the result.

Appendix 6: Sign of weighting factor in line 24

According to Eq. (9), we have

$$\begin{aligned}&\frac{\partial t_0^{tax}}{\partial dr} - S \cdot \frac{\partial t_0^{tax}}{\partial p} =\frac{\partial t_0^{tax}}{\partial dr} - \frac{\mu - \frac{\partial t_0}{\partial dr}}{1-\frac{\partial t_0}{\partial p}} \cdot \frac{\partial t_0^{tax}}{\partial p}<0 \end{aligned}$$

Based on Eq. (7), these terms are calculated as

$$\begin{aligned}&\tau _c \cdot \bigl [ \mu \cdot (s_{dr}+1) - e^{-r}\cdot s_{dr} \cdot \frac{\mu }{1-\delta } \bigr ]\\&- \frac{\bigl [1-\tau _c - \frac{\tau _c\cdot (1-\delta -e^{-r})+\delta }{1-\delta }\cdot s_{dr}\bigr ] \cdot \mu }{(1-\tau _c \cdot e^{-r})\cdot \frac{1-\tau _p}{1-\delta }} \cdot \left(\tau _p + \tau _c \cdot e^{-r}\cdot \frac{1-\tau _p}{1-\delta }\right)<0\\\Leftrightarrow & {} (\tau _c - \tau _c \cdot e^{-r}-\tau _c \cdot \tau _p \cdot \delta + \tau _p \cdot \delta ) \cdot s_{dr}\\&< \tau _c - \tau _c \cdot e^{-r}-\tau _c \cdot \tau _p \cdot \delta + \tau _p \cdot \delta + \tau _p - e^{-r} \cdot \tau _c \cdot \tau _p\\\Leftrightarrow & {} s_{dr} < -1 + \underbrace{\frac{\tau _p \cdot (1- e^{-r} \cdot \tau _c )}{(1-e^{-r})\cdot \tau _c + (1-\tau _c)\cdot \tau _p \cdot \delta }}_{\ge 0} \end{aligned}$$

which holds since $s_{dr}\le -1$.

Appendix 7: Derivation of the default ratio in Eq. (27)

According to the definition of dr,

$$\begin{aligned} dr= \, & {} \frac{e^{-r}\cdot {\mathbb{E}}[\max \{L_1-A_1;0\}]}{L_0} = \, e^{-r}\cdot {\mathbb{E}}\Bigl [\max \Big \{\frac{L_1}{L_0}-\underbrace{\frac{A_0}{L_0}}_{=s}\cdot e^r;0\Big \}\Bigr ] \nonumber \\= \, & {} {\mathbb{E}}\Bigl [\max \Big \{e^{-r}\cdot \frac{ L_1}{L_0};s\Big \}\Bigr ] - s \nonumber \\= \, & {} {\mathbb{P}}\Bigl ( e^{-r}\cdot \frac{ L_1}{L_0}> s \Bigr )\cdot {\mathbb{E}}\Bigl [e^{-r}\cdot \frac{ L_1}{L_0}\, \Big \vert \, e^{-r}\cdot \frac{ L_1}{L_0}> s\Bigr ] + s \cdot \mathbb{P}\Bigl ( e^{-r}\cdot \frac{ L_1}{L_0} \le s \Bigr )- s \nonumber \\= \, & {} {\mathbb{P}}\Bigl ( e^{-r}\cdot \frac{ L_1}{L_0}> s \Bigr )\cdot {\mathbb{E}}\Bigl [e^{-r}\cdot \frac{ L_1}{L_0}\, \Big \vert \, e^{-r}\cdot \frac{ L_1}{L_0}> s\Bigr ] - s \cdot {\mathbb{P}}\Bigl ( e^{-r}\cdot \frac{ L_1}{L_0} > s \Bigr ) \end{aligned}$$

(39)

Note that $\ln (e^{-r}\cdot L_1/L_0) \sim N(-\sigma _L^2/2,\sigma _L^2)$. Hence,

$$\begin{aligned} \underbrace{{\mathbb{P}}\Bigl ( e^{-r}\cdot \frac{ L_1}{L_0}> s \Bigr )}_{={\mathbb{P}}(L_1>A_1)} = 1-\Phi \Bigl (\frac{\ln (s)+\sigma _L^2/2}{\sigma _L}\Bigr )=\Phi \Bigl (-\frac{\ln (s)+\sigma _L^2/2}{\sigma _L} \Bigr ) \end{aligned}$$

and (cf., e.g., Bebu and Mathew 2009, p. 378)

$$\begin{aligned} {\mathbb{E}}\Bigl [e^{-r}\cdot \frac{ L_1}{L_0}\, \Big \vert \, e^{-r}\cdot \frac{ L_1}{L_0} > s\Bigr ] = \frac{\Phi \Bigl (\sigma _L - \frac{\ln s + \sigma _L^2/2}{\sigma _L}\Bigr )}{\Phi \Bigl (- \frac{\ln s + \sigma _L^2/2}{\sigma _L}\Bigr )} \end{aligned}$$

Therefore, Eq. (39) can be rewritten as

$$\begin{aligned}&{\mathbb{P}}\Bigl ( e^{-r}\cdot \frac{ L_1}{L_0}> s \Bigr )\cdot {\mathbb{E}}\Bigl [e^{-r}\cdot \frac{ L_1}{L_0}\, \Big \vert \, e^{-r}\cdot \frac{ L_1}{L_0}> s\Bigr ] - s \cdot {\mathbb{P}}\Bigl ( e^{-r}\cdot \frac{ L_1}{L_0} > s \Bigr )\\= \, & {} \Phi \Bigl (\sigma _L - \frac{\ln s + \sigma _L^2/2}{\sigma _L} \Bigr ) - s \cdot \Phi \Bigl (-\frac{\ln (s)+\sigma _L^2/2}{\sigma _L} \Bigr ) \end{aligned}$$

Appendix 8: Derivation of the asset–liability ratio in equilibrium (Eqs. 32 and 33)

The derivation is analogous to Proposition 2 in Schlütter (2014). For $t_0$ as defined in Eq. (7), (9) implies

$$\begin{aligned}&S = \frac{1-\tau _c-\frac{\tau _c\cdot (1-\delta -e^{-r})+\delta }{1-\delta }\cdot \frac{\partial s}{\partial dr}}{ \frac{(1-\tau _c \cdot e^{-r}) \cdot (1-\tau _p)}{1-\delta }}\cdot \mu \\\Leftrightarrow & {} S/ \mu \cdot (1-\tau _c \cdot e^{-r}) \cdot (1-\tau _p) = (1-\tau _c)\cdot (1-\delta ) - (\tau _c\cdot (1-\delta -e^{-r})+\delta )\cdot \frac{\partial s}{\partial dr}\\\Leftrightarrow & {} \frac{\partial s}{\partial dr}=- \frac{S/ \mu \cdot (1-\tau _c \cdot e^{-r}) \cdot (1-\tau _p)-(1-\tau _c)\cdot (1-\delta )}{ \tau _c\cdot (1-\delta -e^{-r})+\delta } \end{aligned}$$

According to Eqs. (30) and (28),

$$\begin{aligned}&\Phi \Bigl (-\frac{ln(s)}{\sigma _L}-\frac{\sigma _L}{2} \Bigr )= \frac{ \tau _c\cdot (1-\delta -e^{-r})+\delta }{S/ \mu \cdot (1-\tau _c \cdot e^{-r}) \cdot (1-\tau _p)-(1-\tau _c)\cdot (1-\delta )}\\\Leftrightarrow & {} -\frac{ln(s)}{\sigma _L}-\frac{\sigma _L}{2}=\Phi ^{-1}\Bigl ( \frac{ \tau _c\cdot (1-\delta -e^{-r})+\delta }{S/ \mu \cdot (1-\tau _c \cdot e^{-r}) \cdot (1-\tau _p)-(1-\tau _c)\cdot (1-\delta )}\Bigr )\\\Leftrightarrow & {} s=\exp \Bigl [-\sigma _L \cdot \Phi ^{-1}\Bigl ( \frac{ \tau _c\cdot (1-\delta -e^{-r})+\delta }{S/ \mu \cdot (1-\tau _c \cdot e^{-r}) \cdot (1-\tau _p)-(1-\tau _c)\cdot (1-\delta )}\Bigr ) -\frac{\sigma _L^2}{2} \Bigr ] \end{aligned}$$