Longevity, Education and Economic Growth

Abstract

This chapter examines the effect of population aging on development paths of an economy by distinguishing between human capital obtained through schooling before entering the labor force and the stock of common knowledge of contemporary workers about productivity improvements. Each individual determines his own human capital investment, i.e., schooling period, financing by borrowing. Therefore, spending on human capital formation reduces lifecycle savings and hence physical capital formation.

Appendix 1

1.1 1.1 Scale Effect

If we assume ${A}_t={K}_t/ a$ in output production, there will be a kind of scale effect due to human capital accumulation through schooling. Specifying the production function as ${Y}_t={K}_t^{\alpha}{\left({A}_t{L}_t\right)}^{1-\alpha}$ for exposition, the aggregate production function can be written as ${Y}_t={\left({h}_t/ a\right)}^{1-\alpha}{K}_t$. Then we have

$$ \frac{x_{t+1}}{x_t}={\left[\frac{\left(1-\eta \right)\frac{\rho}{1+\rho}\left(1-\alpha \right){\left( a/\theta \right)}^{\alpha -1}{\left(\eta \theta \frac{1-\alpha}{\alpha}\right)}^{\frac{\left(1-\alpha \right)\eta}{1-\eta}}}{1+\eta \frac{1-\alpha}{\alpha}}\right]}^{1-\eta}{x}_t^{\left(1-\alpha \right)\eta}, $$

corresponding to (9.14) in the text. If the right-hand side is greater than one, the growth rate of the capital-labor ratio is increasing in time and so is the per capita output growth rate

$$ 1+\widehat{\gamma}=\frac{Y_{t+1}}{Y_t}={\left(\frac{x_{t+1}}{x_t}\right)}^{\frac{1+\left(1-\alpha \right)\eta}{1-\eta}}. $$

Therefore, as can be seen from the above, the equilibrium-growth rate is not necessarily decreasing in η. In this case, the rate of change in educational expenditure is smaller than the equilibrium growth rate, i.e., $1+{\widehat{\gamma}}_e={\left(1+\widehat{\gamma}\right)}^{1/\left[1+\left(1-\alpha \right)\eta \right]}<1+\widehat{\gamma}$.

1.2 1.2 Education in the Family and Paid by Parents

Yakita (2010) considered both education in the family and in school. If both of them are available to parents, the human capital of an individual working in period $t+1$ with intergenerational transmission of human capital of parental generation can be produced as follows:

$$ {h}_{t+1}=\varepsilon {\left({e}_t+\theta {h}_t\right)}^{\delta}{\overline{h}}_t^{1-\delta}\kern1em 0<\delta \le 1;0<\varepsilon, \theta $$

(9.1′)

where h _t is the stock of human capital of an individual of the working generation in period t (which we call generation t), e _t is per child educational expenditure by his parent, and ${\overline{h}}_t$ is the average stock of human capital of generation t. In this case parents will choose ${e}_t=0$ when the wage rate is sufficiently low. Assuming the neoclassical constant-returns-to-scale production function, ${Y}_t={K}_t^{\alpha}{L}_t^{1-\alpha}$, we can show that there is a critical wage rate separating dynamics of the system into two phases. In this case the economy is likely to fall into the development trap, other things being equal, if the parameters are such that human capital investment tends to be discouraged, i.e., $e\approx 0$, that is, if (i) per child rearing time is less, (ii) the utility weight on human capital per child is smaller, (iii) the savings rate is greater, (iv) the scale parameter in human capital production is greater, and/or (v) the educational effect of the parent on children is greater.