## 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.

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## Notes

See the discussion below, where the recent results in the field are discussed and related to our finding.

This is not equivalent to exclude

*life expectancy*to have a causal effect on education; the impact is just not working through an increase in the expected lifetime working hours. We thank Moshe Hazan for pointing this out.By assuming that individuals cannot borrow during youth does not exclude the possibility of positive savings to smooth consumption across periods. However, in the schooling period, we regard this as a theoretical curiosity, since higher earnings later in life and a desire to smooth consumption will pull in the direction of borrowing rather than saving.

For evidence of credit constraints hampering education, see Flug et al. (1998). Furthermore, the assumption of no annuity markets implies that individuals cannot die in debt. This is true since a lender will always prefer a safe return in the capital market instead of lending money to a mortal individual unless he is compensated for the mortality risk, i.e., if annuity markets exist.

We make these assumptions to focus on the effect on the retirement choice. Considering uncertain survival to the second period would not change our result. Introducing a choice between labor and leisure in the second period would only blur our main result. In fact, a constant labor at supply at the intensive margin is consistent with the empirical finding in Hazan (2009). As he notes, expected lifetime labor supply mainly declined from later entry and earlier exit of the labor market, whereas the intensive margin remained relatively constant.

Equation 4 shows that we assume no depreciation of human capital from the second to the third period of life. Introducing depreciation into the model does not change the results.

Because the following relation would apply: \(u^{\prime }\left( c_{1}\right) =\beta Ru^{\prime }\left( c_{2}\right) .\)

We thank an anonymous referee for pointing this out.

Including accidental bequests would make the effect on the retirement age from an increase in the survival rate ambiguous (see, e.g., Hansen and Lønstrup 2010).

For example, the increasing demand for educated labor caused by technological progress (Galor and Weil 2000)

Actually, in our model, lifetime labor supply shrinks both because of earlier retirement

*and*later entry into the labor market. We focus here how changed mortality rates affect the retirement decision whereas the effect on the entry decision is analyzed in more detail in Sheshinski (2009) and Cervellati and Sunde (2010).

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## Acknowledgements

We thank Oded Galor, Per Svejstrup Hansen, Moshe Hazan, Jens Iversen, Peter Sandholt Jensen, and seminar participants at Brown University, University of Southern Denmark and 2nd LEPAS Workshop on the Economics of Ageing for useful comments and suggestions. We are also grateful to two anonymous referees whose comments greatly improved the paper.

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## Appendix

### Appendix

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

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

### 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:

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:

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

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:

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:

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:

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:

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:

Then determinant of the Hessian matrix is given by:

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

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:

which completes the proof.□

### Proof of Proposition 4

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Hansen, C.W., Lønstrup, L. Can higher life expectancy induce more schooling and earlier retirement?.
*J Popul Econ* **25**, 1249–1264 (2012). https://doi.org/10.1007/s00148-011-0397-1

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DOI: https://doi.org/10.1007/s00148-011-0397-1

### Keywords

- Life expectancy
- Schooling
- Retirement

### JEL CLassification

- J24
- J26
- O11