Youth Dependency, Technological Progress, and Economic Development

Abstract

In this chapter, assuming that R&D activities need more resources in order to sufficiently improve the quality of intermediate goods to be combined with higher labor abilities in production, we show that a negative relationship between total-factor-productivity (TFP) growth and youth dependency, demonstrated by Kögel (J Dev Econ 76(1):147–173, 2005) using cross-country data, does not necessarily obtain over the development process, especially in earlier stages of economic development.

Appendix 1

1.1 1.1 Derivations of (11.4–11.6)

The problem of individuals is:

$$ Max\;{u}_t= \ln {c}_t^1+\rho \ln {c}_{t+1}^2+\varepsilon \ln {n}_t+\beta \eta \ln \left({e}_t+\theta \right)+ \ln \mu $$

(11.23)

$$ \mathrm{s}.\mathrm{t}.\kern0.36em {m}_t\left(1- z{n}_t\right)-{c}_t^1-\left({c}_{t+1}^2/{r}_{t+1}\right)-{n}_t{e}_t=0\kern0.24em \mathrm{and} $$

(11.24)

$$ {c}_t^1\ge {\overline{c}}^1. $$

(11.25)

Letting λ _t and γ _t be the Lagrange multipliers attached to (11.24) and (11.25), respectively, we obtain the first-order conditions for utility maximization as:

$$ 1/{c}_t^1-{\lambda}_t+{\gamma}_t=0 $$

(11.26a)

$$ \rho /{c}_{t+1}^2-{\lambda}_t/{r}_{t+1}=0 $$

(11.26b)

$$ \varepsilon /{n}_t-{\lambda}_t\left({m}_t z+{e}_t\right)=0 $$

(11.26c)

$$ \beta \eta /\left({e}_t+\theta \right)-{\lambda}_t{n}_t=0. $$

(11.26d)

We have two cases, (i) ${\gamma}_t=0$ and ${c}_t^1\ge {\overline{c}}^1$ and (ii) ${\gamma}_t\ge 0$ and ${c}_t^1={\overline{c}}^1$. We examine these two cases in turn.

Case (i)

From (11.26a–11.26d) and letting ${\gamma}_t=0$, we obtain

$$ 1/{\lambda}_t={m}_t/\left(1+\rho +\varepsilon \right)={c}_t^1. $$

(11.27)

Inserting (11.27) into (11.26b), it follows that

$$ {c}_{t+1}^2/{r}_{t+1}=\left[\rho /\left(1+\rho +\varepsilon \right)\right]{m}_t={a}_{t+1}. $$

(11.4b)

Assuming interior solutions and inserting (11.27) into (11.26c, 11.26d), we have

$$ \frac{\varepsilon}{1+\rho +\varepsilon}{m}_t={n}_t\left({m}_t z+{e}_t\right)\kern0.24em \mathrm{and} $$

(11.28a)

$$ \frac{\beta \eta}{1+\rho \varepsilon}{m}_t={n}_t\left({e}_t+\theta \right). $$

(11.28b)

Eliminating e _t from (11.28a, 11.28b), we obtain

$$ {n}_t=\frac{\varepsilon -\beta \eta}{1+\rho +\varepsilon}\frac{m_t}{m_t z-\theta}. $$

(11.6c)

Using (11.6) and (11.28a) we obtain

$$ {e}_t=\frac{\beta \eta z{m}_t-\varepsilon \theta}{\varepsilon -\beta \eta}. $$

(11.5b)

On the other hand, if $\beta \eta z{m}_t-\varepsilon \theta \le 0$ or if ${m}_t\le \varepsilon \theta /\beta \eta z\left[\equiv \tilde{m}\right]$, we have a corner solution for education, i.e., ${e}_t=0$. In this case, by setting ${e}_t=0$ in (11.28a), we obtain

$$ {n}_t=\frac{\varepsilon}{1+\rho +\varepsilon}\frac{1}{z}. $$

(11.6b)

Case (ii)

Setting ${c}_t^1={\overline{c}}^1$ in (11.26a) we have

$$ {\gamma}_t={\lambda}_t-1/{\overline{c}}^1. $$

(11.29a)

From (11.26a, 11.26b) and (11.24) with ${c}_t^1={\overline{c}}^1$, we obtain

$$ 1/{\lambda}_t=\left({m}_t-{\overline{c}}^1\right)/\left(\rho +\varepsilon \right). $$

(11.29b)

From (11.29a, 11.29b), it follows that

$$ {\gamma}_t=\left[\left(1+\rho +\varepsilon \right){\overline{c}}^1-{m}_t\right]/\left[\left({m}_t-{\overline{c}}^1\right){\overline{c}}^1\right]>0, $$

(11.29c)

where

$$ d{\gamma}_t/ d{m}_t<0\kern0.24em \mathrm{and} $$

(11.29d)

$$ {\gamma}_t=0\kern0.24em \mathrm{when}\kern0.24em {m}_t=\left(1+\rho +\varepsilon \right){\overline{c}}^1\left[\equiv \overline{m}\right]. $$

(11.29e)

Therefore, we have

$$ {c}_t^1={\overline{c}}^1\kern0.24em \mathrm{if}\kern0.24em {\overline{c}}^1\le {m}_t<\overline{m}\left[=\left(1+\rho +\varepsilon \right){\overline{c}}^1\right]. $$

(11.30)

Inserting (11.29b) into (11.26b), we have

$$ \frac{\rho}{\rho +\varepsilon}\left({m}_t-{\overline{c}}^1\right)={c}_{t+1}^2/{r}_{t+1}={a}_{t+1}. $$

(11.4a)

Assuming interior solutions for n _t and e _t , and using (11.26c, 11.26d) and (11.29b), we obtain

$$ {n}_t=\frac{\varepsilon -\beta \eta}{\rho +\varepsilon}\frac{m_t-{\overline{c}}^1}{m_t z-\theta}\kern0.24em \mathrm{and} $$

(11.31a)

$$ {e}_t=\frac{\beta \eta z{m}_t-\varepsilon \theta}{\varepsilon -\beta \eta}. $$

(11.31b)

As in the previous case, if ${m}_t\le \tilde{m}$, we have a corner solution for education, i.e., ${e}_t=0$. In this case, by setting ${e}_t=0$ in (11.26c) and using (11.29b), we obtain

$$ {n}_t=\frac{\varepsilon}{1+\rho +\varepsilon}\frac{1}{z}\frac{m_t-{\overline{c}}^1}{m_t}. $$

(11.6a)

In this chapter we assume that $\left(1+\rho +\varepsilon \right){\overline{c}}^1\left[\equiv \overline{m}\right]<\varepsilon \theta /\beta \eta z\left[=\tilde{m}\right]$. This assumption will be satisfied when per child rearing time, z, is sufficiently small.

1.2 1.2 Proof of Proposition 11.1

From (11.1) and (11.5) and making use of (11.21), we have $\frac{A_{t+2}}{A_{t+1}}=\widehat{\delta}{\left(\frac{m_{t+1} z-\theta}{m_{t+1}}\right)}^{\phi}/{\left(\frac{m_{t+1} z-\theta}{m_t z-\theta}\right)}^{\eta}$ where ${m}_{t+1}=\widehat{\delta}{\left(\frac{m_t z-\theta}{m_t}\right)}^{\phi}{m}_t$ from (11.20). Differentiating it with respect to m _t , we obtain

$$ \begin{array}{l}\frac{d}{d{ m}_t}\left(\frac{A_{t+2}}{A_{t+1}}\right)=\left(\frac{A_{t+2}}{A_{t+1}}\right)\frac{1}{m_{t+1} z-\theta}\\ {}\times \left\{\left[\frac{\theta \phi}{m_{t+1}}\left(1+\frac{\theta \phi}{m_t z-\theta}\right)-\frac{\eta z\theta \phi}{m_t z-\theta}\right]\frac{m_{t+1}}{m_t}+\eta z\left(\frac{m_{t+1} z-\theta}{m_t z-\theta}-\frac{m_{t+1}}{m_t}\right)\right\}.\end{array} $$

(11.32)

Since the coefficient of ${m}_{t+1}/{m}_t$ in the braces on the right-hand side of (11.32) approaches zero when m _t grows infinitely, the sign of $d\left({A}_{t+2}/{A}_{t+1}\right)/ d{m}_t$ depends on that of the last term in the braces. Thus, since $\frac{m_{t+1} z-\theta}{m_t z-\theta}-\frac{m_{t+1}}{m_t}=\frac{\theta \left({m}_{t+1}-{m}_t\right)}{m_t\left({m}_t z-\theta \right)}>0$, we can see that the TFP growth rate is increasing in m _t , i.e., $d\left({A}_{t+2}/{A}_{t+1}\right)/ d{m}_t>0$ for sufficiently great m _t . Therefore, when ${A}_{t+1}/{A}_t\left|{}_{m_t=\tilde{m}}\right.\ge {A}_{t+1}/{A}_t\left|{}_{t\to \infty}\right.$, the TFP growth rate must decline at the earlier stages just after the time when parents begin to spend on their children’s education. Since ${h}_{t+1}/{h}_t$ changes unsystematically from 1 to ${\left[1+\frac{\beta \eta z{m}_t-\varepsilon \theta}{\theta \left(\varepsilon -\beta \eta \right)}\right]}^{\eta}$ at the minimum level of per worker potential income that makes educational expenditures positive, ${A}_{t+1}/{A}_t$ will change correspondingly.

When ${A}_{t+1}/{A}_t\left|{}_{m_t=\tilde{m}}\right.<{A}_{t+1}/{A}_t\left|{}_{t\to \infty}\right.$, the TFP growth rate may not have the U-shaped dynamics, and may increase monotonically as per worker potential income grows. For exposition, assuming initially that ${A}_{t+1}/{A}_t=\left({m}_{t+1}/{m}_t\right)$ $=\widehat{\delta}{\left[\left(\varepsilon -\beta \eta \right) z/\varepsilon \right]}^{\phi}$ (>1) at ${m}_t=\tilde{m}$, we consider a marginal increase in m _t . The increased potential income induces parents to begin to invest in their children’s education, and the growth rate of per worker income becomes higher. Then we have

$$ \frac{d}{d{ m}_t}\left(\frac{A_{t+1}}{A_t}\right)\left|{}_{m_t=\tilde{m}}\right.=\left(\frac{m_{t+1}}{m_t}\right)\frac{\beta {\eta}^2 z}{\varepsilon -\beta \eta}\left(\frac{\beta \phi}{\varepsilon}-1\right). $$

(11.33)

Therefore, we can see that, if $\phi <\left(\varepsilon /\beta \right)$, the TFP growth rate declines when educational expenditures become positive. In this case the TFP growth rate will show the U-shape dynamics. However, when $\phi \ge \left(\varepsilon /\beta \right)$, the TFP growth rate will increase as per worker potential income grows.☐