## Abstract

This paper analyzes the effects of population aging on the preferred legal retirement age. What is revealed is the crucial role that the indirect ‘macro’ effects resulting from a change in the legal retirement age play in the optimal decision. Two social security systems are studied. Under a defined contribution scheme, aging lowers the preferred legal retirement age. However, under a defined pension scheme, the retirement age is delayed. This result shows the relevance of correctly choosing the parameter affected by the dependency ratio in the design of the social security programme.

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## Notes

In a further analysis, we shall extend the model to the more general case where the utility of leisure may not be independent of age. That is, the utility of leisure while retired will be

*v*(*l*^{t})=*v*(*t*), with*v*′(*t*)>0.Our analysis should be interpreted from a long-run perspective.

Since 0<

*τ*(*n*)<1, then \(0 < p < \frac{{{\left( {1 - e^{{ - nR}} } \right)}}}{{{\left( {e^{{ - nR}} - e^{{ - nT}} } \right)}}}\).Obviously, this is so when the interest rate is different to the birth rate. If

*r*=*n*, the social security is actuarially fair because the present value of social security benefits is equal to the present value of contributions. In other words, net benefits from the pension system are equal to zero, and therefore, changes in the legal retirement age are neutral.Notice the different effect of changes in

*n*or*r*on the optimal decision. While a variation in the interest rate would affect the discount present value of the entire lifetime income, a change in the population growth rate would only affect either the working period or the retirement period depending on the defined parameter of the social security.The actuarially fair system in the case of

*r*=*n*would be completely equivalent to private savings. Therefore, in our context of certain lifetimes and perfect capital markets, individuals would only have to replace private savings with public savings if the contribution rate is not set at their desired level. See Crawford and Lilien (1981) for an extensive analysis of the effect of a pension system on the retirement decision when the three commonly maintained assumptions—perfect capital markets, actuarial fairness and certain lifetimes—are relaxed.When the interest rate is higher than the birth rate

*r*>*n*, the single-peakness property of the indirect utility function*U*(*c*,*R*) cannot be guaranteed for any value of the interest and birth rates.This means that the pension system creates intergenerational redistribution.

These optimal retirement ages have been calculated for

*T*=60 and*v*=0.008513. The constant parameters are*p*=0.908 in the defined pension scheme and*τ*=0.3 in the defined contribution scheme.For determined parameters we have observed how reductions in the birth rate lead to higher optimal legal retirement ages under the defined contribution scheme. For instance, calculated for

*T*=60,*τ*=0.3 and*v*=0.0206, we have obtained \(R^{*}_{\tau } = 40.00\) with*r*=0.06 and*n*=0.02, and \(R^{*}_{\tau } = 40.19\) with*r*=0.06 and*n*=0.019.

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## Acknowledgements

The authors thank Ignacio Ortuño Ortin for all his constructive suggestion. We also owe thanks for helpful comments to Georges Casamatta. Valuable comments from Subir Chattopaday, Ramón Faulí, Francisco Marhuenda, Slomo Webber and two anonymous referees on a previous version of this paper are also gratefully acknowledged.

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*Responsible editor*: Alessandro Cigno

## Appendix

### Appendix

### Proof proposition 1

1. We have to prove that ∂*R*
_{
p
}*/∂*n* is negative. It can easily be checked that ∂^{2}
*U*/∂*R*
^{2} is negative-evaluated at *R*=*R*
_{
p
}*. Therefore, the sign of ∂*R*
_{
p
}*/∂*n* coincides with the sign of ∂ (∂*U*/∂*R*)/∂*n*,

The constant consumption *c*
_{
p
} is as follows:

First, ∂*c*
_{
p
}/∂*R* and ∂*c*
_{
p
}/∂*n* are strictly positive: an increase in *R* augments the working period and reduces the contribution rate, both effects implying higher consumption; an increase in *n* reduces the dependency ratio and the contribution rate, and this effect also implies higher consumption.

Secondly, ∂ (∂*c*
_{
p
}/∂*R*)/∂*n* can be reduced to

It is easy to check that

and therefore, ∂ (∂*c*
_{
p
}/∂*R*)/∂*n*<0.

2. We have to prove that \({\partial R^{*}_{\tau } } \mathord{\left/ {\vphantom {{\partial R^{*}_{\tau } } {\partial n}}} \right. \kern-\nulldelimiterspace} {\partial n} > 0\). It is easy to check that Eq. 2 is strictly concave; therefore, the sign of \({\partial R^{*}_{\tau } } \mathord{\left/ {\vphantom {{\partial R^{*}_{\tau } } {\partial n}}} \right. \kern-\nulldelimiterspace} {\partial n}\) also coincides with the sign of Eq. 10.

The constant consumption \(c_{\tau } \) is

Again, \({\partial c_{\tau } } \mathord{\left/ {\vphantom {{\partial c_{\tau } } {\partial R}}} \right. \kern-\nulldelimiterspace} {\partial R}\) and \({\partial c_{\tau } } \mathord{\left/ {\vphantom {{\partial c_{\tau } } {\partial n}}} \right. \kern-\nulldelimiterspace} {\partial n}\) are strictly positive: an increase in *R* augments the working period and raises pension benefits, both effects implying higher consumption; an increase in *n* reduces the dependency ratio and thus augments pension benefits, this effect also implying higher consumption.

\({\partial {\left( {{\partial c_{\tau } } \mathord{\left/ {\vphantom {{\partial c_{\tau } } {\partial R}}} \right. \kern-\nulldelimiterspace} {\partial R}} \right)}} \mathord{\left/ {\vphantom {{\partial {\left( {{\partial c_{\tau } } \mathord{\left/ {\vphantom {{\partial c_{\tau } } {\partial R}}} \right. \kern-\nulldelimiterspace} {\partial R}} \right)}} {\partial n}}} \right. \kern-\nulldelimiterspace} {\partial n}\) can be reduced to

It is easy to check that

Therefore, since ∂ (∂*c*/∂*R*)/∂*n*>0, the next step is to prove that the positive part of Eq. 10 is larger than the negative part.

Taking common factor and rearranging terms, Eq. 10 will be

Since *ρ*
_{
r
}(*c*)<1, Eq. 10 is positive if

Since we are evaluating at *n*=*r*, after some simplifications, the left-hand side (LHS) of Eq. 18 can be reduced to the following expression:

It can be shown numerically by means of a mathematical programme that LHS>1 for any *n*∈(0, 1) and for any *R*∈(0, *T*). Indeed, the mathematical programme runs results lower than 1 for values of *R* very close to 0. Therefore, we should impose a lower bound on *R*, although almost no restrictive, for instance *R*∈(*T*/100, *T*).

### Proof proposition 2

We have to prove that with strictly concave indirect utility functions, if *τ* (*n*, *p*, *R*
_{
p
}*)=*τ*, then \(R^{*}_{p} > R^{*}_{\tau } \).

Disregarding the scheme, the first-order condition of the maximization problem would be

Given that for *R*=*R*
_{
p
}*, *c*
_{
p
} will be equal to \(c_{\tau } \), we have to prove that at that point,

If this inequality holds, the single-peakness property of the utility function implies that \(R^{*}_{p} > R^{*}_{\tau } \). We calculate ∂*c*
_{
p
}/∂*R* and \({\partial c_{\tau } } \mathord{\left/ {\vphantom {{\partial c_{\tau } } {\partial R}}} \right. \kern-\nulldelimiterspace} {\partial R}\) to compare them:

and

Given that for *R*=*R*
_{
p
}*, *τ* (*n*, *p*, *R*
_{
p
}*)=*τ* and *p*(*n*, *τ*, *R*
_{
p
}*)=*p*, we only have to compare the second and the third addends. In the defined pension scheme

we know that *τ*=*τ* (*n*, *p*, *R*
_{
p
}*). Therefore, substituting Eq. 26 in Eqs. 24 and 25,that is, the constant contribution *τ* for its value, and comparing the two derivatives, we obtain that Eq. 21 holds if:

The inequality (Eq. 27) will be true for *r*>*n* since the dependency ratio is decreasing with respect to the population growth rate.

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### Cite this article

Lacomba, J.A., Lagos, F. Population aging and legal retirement age.
*J Popul Econ* **19**, 507–519 (2006). https://doi.org/10.1007/s00148-005-0044-9

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DOI: https://doi.org/10.1007/s00148-005-0044-9

### Keywords

- Social security
- Aging
- Legal retirement age

### JEL

- H55
- J26