Family altruism and long-term care insurance

Abstract

The aim of this paper is to analyse long-term care (LTC) insurance purchase decisions by parents who expect to receive assistance from altruistic children. We first propose a simple theoretical model in which we show that the effect of children’s altruism on parents’ insurance decisions is ambiguous and depends on a number of factors: the degree of substitutability between informal and formal care, the degree of parental altruism, and the curvature of the utility functions. We then run an empirical test using data from the U.S., France, Spain, Germany and Israel, and proxy altruism with assistance provided to healthy parents. We find that the effect of children’s altruism is negative in Germany and Israel but not significant in the United States, France and Spain, which possibly suggests that the different forces identified in the theoretical model offset each other.

Appendices

Appendix 1

In this appendix we derive the comparative statics result $\frac{\partial P}{\partial \gamma }$. We have

$$\frac{\partial P}{\partial \gamma } = \frac{{\frac{{\partial EU_{P} }}{\partial P\partial \gamma }}}{{ - SOC_{P} }},$$

(4)

where SOC_P < 0 is the second-order condition for P and $\frac{{\partial EU_{P} }}{\partial P\partial \gamma }$ is obtained by deriving the FOC (2) with respect to γ. In particular, we have

$$\begin{aligned} \frac{{\partial EU_{P} }}{\partial P\partial \gamma } = & - \pi H_{ma} \frac{\partial a}{\partial \gamma } + \frac{{H_{ma} \frac{\partial a}{\partial \gamma }}}{1 + \lambda } + \pi H_{aa} \frac{\partial a}{\partial \gamma }\frac{\partial a}{\partial m}\left( {\frac{1}{{\left( {1 + \lambda } \right)\pi }} - 1} \right) + \pi H_{a} \frac{{\partial \left( {\frac{\partial a}{\partial m}} \right)}}{\partial \gamma }\left( {\frac{1}{{\left( {1 + \lambda } \right)\pi }} - 1} \right) \\ & \; + \pi \beta u^{\prime\prime}\left( c \right)w^{2} \frac{\partial a}{\partial \gamma }\frac{\partial a}{\partial m}\left( {\frac{1}{{\left( {1 + \lambda } \right)\pi }} - 1} \right) - \pi \beta u^{\prime}\left( c \right)w\frac{{\partial \left( {\frac{\partial a}{\partial m}} \right)}}{\partial \gamma }\left( {\frac{1}{{\left( {1 + \lambda } \right)\pi }} - 1} \right). \\ \end{aligned}$$

Using the fact that $\frac{\partial a}{\partial m} = \frac{{\gamma H_{am} }}{{ - \Delta_{a} }}$ with $\Delta_{a} = \gamma H_{aa} + w^{2} u^{\prime\prime}\left( c \right)$ and that $H_{am} = H_{ma}$ and rearranging the first three terms, together with (4), we get the result in Eq. (3).

The expression for $\frac{\partial a}{\partial \gamma }$ is derived using (1), while $\frac{{\partial \left( {\frac{\partial a}{\partial m}} \right)}}{\partial \gamma }$ is obtained by deriving $\frac{\partial a}{\partial m} = \frac{{ - \gamma H_{am} }}{{\gamma H_{aa} + w^{2} u^{\prime\prime}\left( c \right)}}$ with respect to γ.

Appendix 2

Probit models of LTC insurance with dependence (quid pro quo motive)

	U.S.	France	Germany	Spain	Israel
Children’s help	− 0.015 (0.019)	− 0.021 (0.031)	− 0.033** (0.013)	0.002 (0.017)	− 0.132*** (0.047)
In couple	0.029*** (0.006)	0.000 (0.020)	0.001 (0.011)	0.003 (0.010)	− 0.069** (0.034)
Woman	0.034*** (0.006)	0.027* (0.016)	0.002 (0.007)	0.009 (0.007)	− 0.013 (0.025)
Age
50/64	− 0.096*** (0.009)	− 0.058*** (0.026)	0.020 (0.020)	0.023* (0.013)	0.064 (0.043)
65/79	− 0.009 (0.008)	0.024 (0.025)	0.037* (0.020)	0.021* (0.013)	0.066 (0.041)
80+	Ref.	Ref.	Ref.	Ref.	Ref.
Years of education	0.014*** (0.001)	− 0.002 (0.002)	0.002 (0.001)	0.005*** (0.001)	0.015*** (0.003)
Income
1st quartile	− 0.065*** (0.008)	− 0.005 (0.027)	− 0.034*** (0.012)	− 0.004 (0.003)	− 0.169*** (0.037)
2nd quartile	− 0.060*** (0.008)	0.012 (0.023)	− 0.051*** (0.011)	− 0.033*** (0.010)	− 0.176*** (0.033)
3rd quartile	− 0.046*** (0.007)	0.002 (0.021)	− 0.014 (0.008)	− 0.027*** (0.009)	− 0.098*** (0.032)
4th quartile	Ref.	Ref.	Ref.	Ref.	Ref.
Financial transfer	− 0.024* (0.014)	0.003 (0.026)	0.013 (0.009)	0.049** (0.021)	− 0.021 (0.076)
N	14,946	2792	3357	3927	1439
Pseudo-R2	0.055	0.012	0.048	0.083	0.052

1. Heteroscedasticity-consistent standard errors in parentheses
2. *p < 0.1, **p < 0.05, ***p < 0.01