Life Expectancy, Money, and Growth

Abstract

In this chapter, assuming two assets of money and the claims on physical capital, we examine the effect of an increase in life expectancy on portfolio choices of individuals between these two assets. Money is introduced based on the money-in-the-utility-function approach. With capital externalities as an engine of growth, changes in portfolios of individuals affect the economic growth rate.

This chapter is the revised and expanded version of Yakita (2006).

Appendix 1

1.1 1.1 Uniqueness of Balanced-Growth Equilibrium

The left- and right-hand sides of (2.17) are

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

(2.23)

and

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

(2.24)

as depicted in Fig. 2.1. φ(π) is a quadratic function of π and has two intersections with the horizontal axis at $-1$ and $\left\{\left[1-\eta \frac{1}{1+\rho}\right]/\left[\left(1+ r\right)\left(1-\frac{1}{1+\rho}\right)\right]\right\}-1$, while χ(π) is a line and intersects the axis of the abscissa at ${\left\{\left(1+ r\right)\left[1-\left(1-\eta \right)\frac{1}{1+\rho}\frac{\mu}{1+\mu}\right]\right\}}^{-1}-1$. Since we can easily see that $\left\{\left[1-\eta \frac{1}{1+\rho}\right]/\left[\left(1+ r\right)\left(1-\frac{1}{1+\rho}\right)\right]\right\}-1>{\left\{\left(1+ r\right)\left[1-\left(1-\eta \right)\frac{1}{1+\rho}\frac{\mu}{1+\mu}\right]\right\}}^{-1}-1$ for $p,\eta \in \left(0,1\right)$, Eq. (2.17) has two real roots, π _ℓ and π _h , such that $-1<{\pi}_{\ell }$ $<{\left\{\left(1+ r\right)\left[1-\left(1-\eta \right)\frac{1}{1+\rho}\frac{\mu}{1+\mu}\right]\right\}}^{-1}-1$ $<{\pi}_h$ (see Fig. 2.1). With condition (2.18), however, the smaller root, π _ℓ , is not a solution to our model.

Therefore, solving (2.17) for π, we obtain

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

(2.25)

where

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

(2.26)

from which it follows that

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

(2.27)

This implies that $\frac{\partial \chi}{\partial \pi}-\frac{\partial \varphi}{\partial \pi}<0$. Thus, we have $sign\left(\frac{d\pi}{d p}\right)= sign\left(\frac{\partial \chi}{\partial p}-\frac{\partial \varphi}{\partial p}\right)$.

1.2 1.2 Increase in the Growth Rate of Money Supply

Differentiating χ(π; μ), we have

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

from which it follows that

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

(2.28)

Since we can easily see that

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

(2.29)

while taking (2.18) into account, it follows that $\pi -\left\{{\left(1+ r\right)}^{-1}{\left[1-\left(1-\eta \right)\frac{1}{1+\rho}\right]}^{-1}-1\right\}>0$. The curve φ(π) does not depend on μ, so the upward shift of χ(π; μ) increases the inflation rate.