Can higher life expectancy induce more schooling and earlier retirement?

Abstract

In this paper, we show that it may be optimal for individuals to educate more and retire earlier when life expectancy increases. This result reconciles the findings of Hazan (Econometrica 77:1829–1863, 2009) with theory. Further, the paper contributes to a better understanding of the conflicting empirical findings on the causal effect on income per capita from increased life expectancy.

Appendix

The first-order conditions 6–8 are here repeated for convenience:

$$ U_{e}=-\psi u^{\prime }(c_{1})+\beta u^{\prime }(c_{2})h^{\prime }(e)+\beta ^{2}\phi u^{\prime }(c_{3})h^{\prime }(e)\left[ 1-l\right] =0, $$

(19)

$$ U_{s}=-u^{\prime }(c_{2})+\phi \beta Ru^{\prime }(c_{3})=0, $$

(20)

$$ U_{l}=-u^{\prime }(c_{3})wh(e)+\theta u^{\prime }(l)=0. $$

(21)

To prove the propositions, we need the following second-order derivatives:

$$ U_{ss}=u^{\prime \prime }(c_{2})+\phi \beta R^{2}u^{\prime \prime }(c_{3})<0 , $$

(22)

$$ U_{es}=-\beta h^{\prime }(e)u^{\prime \prime }(c_{2})+\beta ^{2}\phi h^{\prime }(e)\left[ 1-l\right] Ru^{\prime \prime }(c_{3})\lessgtr 0, $$

(23)

$$ U_{s\phi }=\beta Ru^{\prime }(c_{3})>0, $$

(24)

$$ U_{e\phi }=\beta ^{2}u^{\prime }(c_{3})h^{\prime }(e)\left[ 1-l\right] >0, $$

(25)

$$ U_{ls}=-wRu^{\prime \prime }(c_{3})h(e)>0, $$

(26)

$$ U_{sl}=-wRu^{\prime \prime }(c_{3})h(e)>0, $$

(27)

$$ U_{el}=-\beta ^{2}\phi u^{\prime }(c_{3})h^{\prime }(e)-\beta ^{2}\phi w \left[ 1-l\right] h^{\prime }(e)h(e)u^{\prime \prime }(c_{3})\gtrless 0, $$

(28)

$$ U_{l\phi }=0, $$

(29)

$$ U_{ee}=\left. \begin{array}{c} w\psi u^{\prime \prime }(c_{1})+\beta h^{\prime \prime }(e)u^{\prime }(c_{2})+\beta wh^{\prime }\left( e\right) h^{\prime }(e)u^{\prime \prime }(c_{2})+ \\ \beta ^{2}\phi \left[ 1-l\right] h^{\prime \prime }(e)u^{\prime }(c_{3})+\beta ^{2}\phi w\left[ 1-l\right] ^{2}h^{\prime }(e)h^{\prime }(e)u^{\prime \prime }(c_{3})<0\text{,} \end{array} \right. $$

(30)

$$ U_{ll}=w^{2}h(e)^{2}u^{\prime \prime }(c_{3})+\theta v^{\prime \prime }(l)<0 . $$

(31)

Proof of Proposition 1

It is to be shown that $\frac{\partial e}{\partial \phi }>0$.

Under the assumption of an exogenous retirement age, the first-order conditions reduces to Eqs. 19 and 20. By taking the total differential of these and solving the subsequent system of equations for $\frac{\partial e}{\partial \phi }$, we obtain:

$$ \frac{\partial e}{\partial \phi }=\frac{\left| \begin{array}{c} U_{ss} -U_{s\phi } \\ U_{es} -U_{e\phi } \end{array} \right|}{\left| H\right| }\text{,} $$

where H is the Hessian matrix. For the problem to have a unique solution, $\left\vert H\right\vert >0$ which is now first proven. The determinant of Hessian matrix is given by:

$$ \left| H\right| =\left| \begin{array}{c} U_{ee} U_{es} \\ U_{se} U_{ss} \end{array} \right| \text{,} $$

Inserting Eqs. 22, 23, 30 and assume, without loss of generality, that w = β = 1 yields:

$$ \begin{array}{rll} \left\vert H\right\vert &=&u^{\prime \prime }(c_{2})\Big[ \psi u^{\prime \prime }(c_{1}) + u^{\prime }(c_{2})h^{\prime \prime }(e)\notag\\&&\left.+\left[ \left[ h^{\prime }(e)\right] ^{2}u^{\prime \prime }(c_{3})+u^{\prime }(c_{3})h^{\prime \prime }(e)\right] \phi \left[ 1-l\right] +R^{2}\left[ h^{\prime }(e)\right] ^{2}\phi u^{\prime \prime }(c_{3})\right] \\ &&+u^{\prime \prime }(c_{3})\phi R\left[ 2\left[ 1-l\right] \left[ h^{\prime }(e)\right] ^{2}u^{\prime \prime }(c_{2})\right.\notag\\&&Ru^{\prime \prime }(c_{1})+R\left[ 1-l\right] u^{\prime }(c_{3})\phi h^{\prime \prime }(e)+Ru^{\prime }(c_{2})h^{\prime \prime }(e)\Big] \\ &>&0\text{,} \end{array} $$

(32)

given the assumption on h(e) and u(c _i ) for i = 1,2,3.

Thus, ${\rm sign}\frac{\partial e}{\partial \phi }=\left\vert \begin{array}{cc} U_{ss} -U_{s\phi } \\ U_{es} -U_{e\phi } \end{array} \right\vert$. Inserting the expressions in Eqs. 22–25 yields:

$$ {\rm sign}\frac{\partial e}{\partial \phi }={\rm sign}\text{ }\left[ -u^{\prime \prime }(c_{2})u^{\prime }(c_{3})h^{\prime }(e)\left[ R+\left[ 1-l\right] \right] \right] >0\text{,} $$

which completes the proof.□

Proof of Proposition 2

It is to be shown that $\frac{\partial e}{\partial l}\geq 0$ if Eq. 11 holds.

The proof parallels that of Proposition 1. Thus:

$$ \frac{\partial e}{\partial l}=\frac{\left\vert \begin{array}{cc} U_{ss} -U_{sl} \\ U_{es} -U_{el} \end{array} \right\vert }{\left\vert H\right\vert } $$

In the proof of Proposition 1, it is shown that $\left\vert H\right\vert >0$. Thus, ${\rm sign}\frac{\partial e}{\partial l}=\left\vert \begin{array}{cc} U_{ss} -U_{sl} \\ U_{es} -U_{el} \end{array} \right\vert$. Inserting Eqs. 22, 23, 27, and 28 yields:

$$ \begin{array}{rll} {\rm sign}\frac{\partial e}{\partial l}&=&{\rm sign}\left[ \beta ^{3}\phi wh^{\prime }\left( e\right) u^{\prime \prime }\left( c_{2}\right) \left[ u^{\prime }\left( c_{3}\right) +u^{\prime }\left( c_{3}\right) \frac{u^{\prime \prime }\left( c_{3}\right) }{u^{\prime \prime }\left( c_{2}\right) }\beta \phi R^{2}\right.\right.\notag\\&&\,Rwh\left( e\right) u^{\prime \prime }\left( c_{3}\right) +\left[ 1-l \right] h\left( e\right) wu^{\prime \prime }\left( c_{3}\right) \Bigg] \Bigg] \text{,} \end{array} $$

(33)

because $\beta ^{3}\phi wh^{\prime }(e)u^{\prime \prime }(c_{2})<0\$ we conclude that $\frac{\partial e}{\partial l}>0$ if the following condition holds:

$$ 1+\frac{u^{\prime \prime }\left( c_{3}\right) }{u^{\prime \prime }\left( c_{2}\right) }\beta \phi R^{2}<h\left( e\right) w\frac{\sigma _{3}}{c_{3}} \left[ R+\left[ 1-l\right] \right] \text{,} $$

(34)

which is the condition in Eq. 11 where $\sigma _{3}\equiv -c_{3}\frac{ u^{\prime \prime }(c_{3})}{u^{\prime }(c_{3})}$ is the coefficient of relative risk aversion. This completes the proof.□

Proof of Proposition 3

It is to be shown that $\frac{\partial l}{\partial \phi }>0.$

The proof parallels those of Proposition 1 and 2. Thus:

$$ \frac{\partial l}{\partial \phi }=\frac{\left\vert \begin{array}{cc} U_{ss} -U_{s\phi } \\ U_{ls} -U_{l\phi } \end{array} \right\vert }{\left\vert H\right\vert }\text{.} $$

Then determinant of the Hessian matrix is given by:

$$ \left\vert H\right\vert =\left\vert \begin{array}{cc} U_{ll} U_{ls} \\ U_{sl} U_{ss} \end{array} \right\vert \text{.} $$

By using Eqs. 22, 26, and 31, this yields:

$$ \left\vert H\right\vert =v^{\prime \prime }(l)u^{\prime \prime }(c_{3})\theta R^{2}\phi ^{2}+u^{\prime \prime }(c_{2})u^{\prime \prime }(c_{3})h(e)^{2}\phi +v^{\prime \prime }(l)u^{\prime \prime }(c_{2})\theta \phi >0\text{,} $$

with the assumed increasing and concave functions h(e),u(l), and u(c _i ), i = 1,2,3.

Thus, ${\rm sign}\frac{\partial l}{\partial \phi }=\left\vert \begin{array}{cc} U_{ss} -U_{s\phi } \\ U_{ls} -U_{l\phi } \end{array} \right\vert$. Inserting Eqs. 22, 24, 26, and 29 and obtain:

$$ {\rm sign}\frac{\partial l}{\partial \phi }={\rm sign}\left[ -\phi R^{2}\beta ^{4}wh(e)u(c_{3})u^{\prime \prime }(c_{3})\right] >0\text{,} $$

which completes the proof.□

Proof of Proposition 4

Follows directly from Eqs. 12 and 13.□