## Abstract

We examine the effect of an increase in life expectancy on portfolio choices of individuals and, thereby, on economic growth in a simple endogenous growth model populated by overlapping generations, in which money is introduced based on the money-in-the-utility-function approach. It is shown that an increase in longevity raises the balanced growth rate and lowers the inflation rate, offsetting the Tobin effect, if spillovers from accumulated capital to labor productivity sufficiently raise wage income and real savings, and, if not, it may retard economic growth and aggravate inflation. Under plausible conditions, the former will be the case.

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## Notes

van der Ploeg and Alogoskoufis (1994) suggested that the birth of new generations and the absence of intergenerational bequests may bring about an adverse relationship between inflation and economic growth in a continuous time overlapping generations model. This relationship holds for our model, too.

We assume in this study that the retirement age is given institutionally.

We assume that an individual does not hold real balances at the end of his second period.

For the interpretation, see, for example, Wigger (1999).

We assume here that 0<

*μ*<1, for example, on an annual base, but it is not necessarily needed for our result.This is a version of the AK model, and therefore, there is no transition.

*π*is a jump variable.A greater

*η*and/or a higher*r*tend to make the left-hand side smaller, while higher*r*is associated with higher*ω*/*a*under the assumption that 1>(*af*′/*f*)[1-(*af*″/*f*′)]. The condition is satisfied when*f*(*a*)=*Aa*^{α}(0<*A*;0<*α*<1).We have (

*∂ɛ*/*∂π*)−(*∂ϕ*/*∂π*)>0 (see Appendix). It should be noted that*π*may not be monotonic in*λ*when the growth rate of money supply,*μ*, is constant and relatively high enough for Eq. (19) to be satisfied. If the inflation rate reaches*π*=*η*[1/(1+*r*)+(1−*η*)]+(1−*η*)[*μ*/(*ω*/*a*)]-1, either from above or from below, an increase in life expectancy no longer affects the inflation rate and, thus, economic growth.The negative relation between growth and inflation is often documented in cross-country data, e.g., Gomme (1993). Roubini and Sala-i-Martin (1992) argued that the negative correlation is likely to be spurious as both are caused by policies of financial repression. The negative correlation obtained in our study is conditional on the life expectancy variable.

A smaller

*η*makes the inflation rate slightly lower and the growth rate slightly higher in each case. We have similar numerical results in the case of a CES production function.For a survey of different approaches and the results, see, for example, Orphanides and Solow (1990). Based on the cash-and-credit-goods approach with a cash-in-advance constraint a la Lucas and Stokey (1983) and Batina and Ihori (2000), we can show that an increase in life expectancy leads to higher growth and lower inflation.

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## Acknowledgements

The author thanks Kazutoshi Miyazawa, the referees and the editor of this journal, and the seminar participants at Chukyo University for their helpful comments and suggestions. The usual disclaimer applies.

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*Responsible editor*: *Alessandro Cigno*

## Appendices

### Uniqueness of balanced growth equilibrium

From Eq. (18),

and

as depicted in Fig. 1. *ϕ*(*π*) is a quadratic function of *π* and has two intersections with the horizontal axis at −1 and \({\left\{ {{{\left[ {1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right]}} \mathord{\left/ {\vphantom {{{\left[ {1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right]}} {{\left[ {{\left( {1 + r} \right)}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {{\left( {1 + r} \right)}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)}} \right]}}} \right\}} - 1\), while *ɛ*(*π*) is a line and intersects the abscissa at \({\left\{ {{\left( {1 + r} \right)}{\left[ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}\frac{\mu }{{1 + \mu }}} \right]}} \right\}}^{{ - 1}} - 1\). Since we can easily see that \({\left\{ {{{\left[ {1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right]}} \mathord{\left/ {\vphantom {{{\left[ {1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right]}} {{\left[ {{\left( {1 + r} \right)}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {{\left( {1 + r} \right)}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)}} \right]}}} \right\}} - 1 > {\left\{ {{\left( {1 + r} \right)}{\left[ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}\frac{\mu }{{1 + \mu }}} \right]}} \right\}}^{{ - 1}} - 1\) for *λ*, *η* ∈ (0,1), Eq. (18) has two real roots, \(\pi _{{\ell }} \) and *π*
_{
h
}, such that \( - 1 < \pi _{{\ell }} < {\left\{ {{\left( {1 + r} \right)}{\left[ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}\frac{\mu }{{1 + \mu }}} \right]}} \right\}}^{{ - 1}} - 1 < \pi _{h} \) (see Fig. 1). With condition (19), however, the smaller root, \(\pi _{{\ell }} \), is not a solution to our model. Therefore, solving Eq. (18) for *π*, we obtain

where

from which it follows that

This implies that \(\frac{{\partial \varepsilon }}{{\partial \pi }} - \frac{{\partial \phi }}{{\partial \pi }} > 0\). Thus, we have \(sign{\left( {\frac{{d\pi }}{{d\lambda }}} \right)} = sign{\left( {\frac{{\partial \phi }}{{\partial \lambda }} - \frac{{\partial \varepsilon }}{{\partial \lambda }}} \right)}.\)

### Increase in the growth rate of money supply

Differentiating *ɛ*(*π*; *μ*), we have

from which it follows that

Since we can easily see that

while taking Eq. (19) into account, it follows that

The curve *ϕ*(*π*) does not depend on *μ*, so the upward shift of *ɛ*(*π*, *μ*) increases the inflation rate.

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### Cite this article

Yakita, A. Life expectancy, money, and growth.
*J Popul Econ* **19**, 579–592 (2006). https://doi.org/10.1007/s00148-005-0017-z

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DOI: https://doi.org/10.1007/s00148-005-0017-z

### Keywords

- Economic growth
- Life expectancy
- Money

### JEL Classification

- D91
- E31
- J11