Demographic change, human capital accumulation, and sectoral employment

Abstract

Many studies examine the relationship between aging and economic growth; however, only a few theoretical studies find a possible non-linear relationship. Thus, this study theoretically investigates the impact of population aging on economies. We construct a three-period overlapping generations model with two sectors: non-education and education. We assume that learning-by-doing effects compound as the share of employment increases and improves productivity. Both adults and old agents consume non-education goods and services, while only adults demand education services for their children to gain human capital. Our results demonstrate that whether an increase in life expectancy positively or negatively influences income growth per capita depends on the productivity of the non-education and education sectors.

Appendices

Appendix 1: Proof of Proposition 1

Equation (16) is transformed into

$$\begin{aligned} K_{t+1}=\frac{\beta q}{1+\gamma +\beta q}h_tA_t(1-\alpha )k^{\alpha }_tN_t \end{aligned}$$

(38)

by substituting Eqs. (2) and (11). We modify Eq. (38) as follows.

$$\begin{aligned}&\frac{K_{t+1}}{A_{t+1}h_{t+1}l^MN_t}\frac{A_{t+1}h_{t+1}l^MN_{t+1}}{A_{t}h_{t}l^MN_t}=\frac{\beta q}{1+\gamma +\beta q}(1-\alpha )k^{\alpha }_t\frac{N_t}{l^Mh_tN_t}. \end{aligned}$$

Thus, the above equation can be written as

$$\begin{aligned} k_{t+1}{\phi }_Ag^h_tl^Mn(q)=\frac{\beta q}{1+\gamma +\beta q}(1-\alpha )k_t^{\alpha }. \end{aligned}$$

(39)

If there is BGP, $\displaystyle k^*=\left[ \frac{\beta q}{1+\gamma +\beta q}\frac{1-\alpha }{{\phi }_Ag^{h*}l^M n(q)}\right] ^{\frac{1}{1-\alpha }}$ is satisfied; thus, it is unique. We define Eq. (39) as

$$\begin{aligned} \Delta k_t=\Omega \frac{k^\alpha _t}{g^h_{t}}-k_t, \end{aligned}$$

(40)

where

$$\begin{aligned} \Delta k_t\equiv k_{t+1}-k_t \end{aligned}$$

and

$$\begin{aligned} \Omega \equiv \frac{\beta q}{1+\gamma +\beta q}\frac{1-\alpha }{{\phi }_A\nu n}. \end{aligned}$$

When $\Delta k_t=0$, Eq. (40) is re-written as

$$\begin{aligned} g^h_t=\Omega k^{\alpha -1}_t. \end{aligned}$$

(41)

Equation (41) is described in Fig. 3. The horizon is reached in $g^{h*}$. Thus, the physical capital accumulation monotonously converges to the intersection point.

Fig. 3

Dynamics of the physical capital growth rate

Appendix 2: The derivation of income growth rate per capita: equation (26)

To find the per capita income growth rate, we define the outputs per the effective labor in the education sector as

$$\begin{aligned} y^M_t=\frac{Y^M_t}{A_tH^M_{t}}. \end{aligned}$$

Thus, the aggregate production function is re-written as

$$\begin{aligned}&y^M_t=k^\alpha _t \end{aligned}$$

These equations, $k_t$ and $y^M_t$, are constant. Note that $\displaystyle k_t\equiv K_t/A_tH_t$.

Therefore, the output in the non-education sector is written as

$$\begin{aligned} Y^M_t=A_tk^\alpha _th_tl^MN_t. \end{aligned}$$

Next, the value of output in the education sector is written as

$$\begin{aligned} p_tE_t=(1-\alpha )A_tk^{\alpha }_th_tl^EN_t \end{aligned}$$

by applying Eqs. (2) to (4). Thus, Eq. (25) is transformed into

$$\begin{aligned}&y_t\equiv \frac{Y_t}{N_t}=A_th_t\xi , \end{aligned}$$

where $\xi \equiv k^\alpha _t[l^M +(1-\alpha )l^E]$ is constant. Then, income growth per capita g leads to

$$\begin{aligned} g\equiv \frac{y_{t+1}}{y_t}= \frac{A_{t+1}h_{t+1}}{A_{t}h_{t}}=\phi _A\left( {\phi }_B\right) ^{\frac{\sigma }{1-\sigma }}, \end{aligned}$$

which is identical to Eq. (26).