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Life expectancy, money, and growth

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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Fig. 1

Notes

  1. 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.

  2. Some authors assume a linkage between life expectancy and spending on health care. See, for example, Ehrlich and Chuma (1990), Philipson and Becker (1998), and Kalemli-Ozcan (2002).

  3. The fertility decisions can be endogenized. If parents derive direct utility from the number of their children, the fertility decisions of parents and the effect of declining mortality on them are similar to those in Zhang et al. (2001) and Yakita (2001).

  4. It has been shown that with imperfect annuity markets and accidental bequests, the relation between mortality decline and economic growth can be nonmonotonic. See, for example, Zhang et al. (2003) and Miyazawa (2003).

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

  6. This is a two-period-lived-agent version of the utility function of Mino and Shibata (1995). Drazen (1981), considering instead the case in which money provides utility in both periods of life, showed that the effect of inflation on capital may depend on the seigniorage transfer policy.

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

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

  9. We assume here that 0<μ<1, for example, on an annual base, but it is not necessarily needed for our result.

  10. We can see that condition Eq. (7) is satisfied when condition Eq. (19) holds. Even when the interest rate is negative, individuals are willing to hold real balances as long as the nominal interest rate is positive.

  11. This is a version of the AK model, and therefore, there is no transition. π is a jump variable.

  12. 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).

  13. 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.

  14. 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.

  15. 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.

  16. 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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Correspondence to Akira Yakita.

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Responsible editor: Alessandro Cigno

Appendices

Uniqueness of balanced growth equilibrium

From Eq. (18),

$$\phi {\left( \pi \right)} \equiv {\left( {1 + \pi } \right)}\frac{\omega }{a}{\left[ {{\left( {1 + r} \right)}{\left( {1 + \pi } \right)}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)} - {\left( {1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)}} \right]}$$

and

$$\varepsilon {\left( \pi \right)} \equiv {\left( {1 + \mu } \right)}{\left\{ {{\left[ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}\frac{\mu }{{1 + \mu }}} \right]}{\left( {1 + r} \right)}{\left( {1 + \pi } \right)} - 1} \right\}}$$

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

$$\pi = \frac{{\frac{\omega }{a}{\left( {1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)} + {\left( {1 + \mu } \right)}{\left( {1 + r} \right)}{\left\{ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}\frac{\mu }{{1 + \mu }}} \right\}} + D^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{2\frac{\omega }{a}{\left( {1 + r} \right)}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)}}} - 1$$

where

$$D = {\left[ {\frac{\omega }{a}{\left( {1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)} + {\left( {1 + \mu } \right)}{\left( {1 + r} \right)}{\left\{ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}\frac{\mu }{{1 + \mu }}} \right\}}} \right]}^{2} - 4\frac{\omega }{a}{\left( {1 + \mu } \right)}{\left( {1 + r} \right)}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)}$$

from which it follows that

$${\left( {1 + \pi } \right)}{\left( {1 + r} \right)}\frac{{2\omega }}{a}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)} - {\left[ {\frac{\omega }{a}{\left( {1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)} + {\left( {1 + \mu } \right)}{\left( {1 + r} \right)}{\left\{ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}\frac{\mu }{{1 + \mu }}} \right\}}} \right]} > 0$$

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

$$\frac{{\partial \varepsilon }}{{\partial \mu }} = {\left( {1 + r} \right)}{\left( {1 + \pi } \right)}{\left\{ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right\}} - 1$$

from which it follows that

$$sign{\left( {\frac{{\partial \varepsilon }}{{\partial \mu }}} \right)} = sign{\left[ {\pi - {\left\{ {{\left( {1 + r} \right)}^{{ - 1}} {\left[ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right]}^{{ - 1}} - 1} \right\}}} \right]}$$

Since we can easily see that

$$\frac{{1 - \eta \frac{{1 + \rho }}{{2 + \rho - \lambda }}}}{{{\left( {1 + r} \right)}{\left( {1 - \frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right)}}} - 1 > \frac{1}{{{\left( {1 + r} \right)}{\left[ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right]}}} - 1$$

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

$$\pi - {\left\{ {{\left( {1 + r} \right)}^{{ - 1}} {\left[ {1 - {\left( {1 - \eta } \right)}\frac{{1 + \rho }}{{2 + \rho - \lambda }}} \right]}^{{ - 1}} - 1} \right\}} > 0$$

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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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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Keywords

  • Economic growth
  • Life expectancy
  • Money

JEL Classification

  • D91
  • E31
  • J11