Low level equilibrium trap, unemployment, efficiency of education system, child labour and human capital formation

Abstract

This paper builds an overlapping generations household economy model to examine the impact of adult unemployment on the human capital formation of a child and on child labour, as viewed through the lens of the adult’s expectations of future employability. The model indicates that the higher the adult unemployment rate in the skilled sector, the lesser is the time allocated by an unskilled adult towards schooling of her child. We also find that an increase in the unskilled adult’s wage may or may not decrease child labour in the presence of unemployment. The model predicts that an increase in child wage increases schooling and human capital growth rate only if the adults in the unskilled sector earn less than subsistence consumption expenditure. As the responsiveness of skilled wage to human capital increases, schooling and human capital growth rates increase. The model dynamics bring out the importance of education efficiency and parental human capital in human capital formation of the child. In the case of an inefficient education system, generations will be trapped into low level equilibrium. Only in the presence of an efficient education system, steady growth of human capital is possible. Suitable policies that may be framed to escape the child labour trap are discussed as well.

Appendix

The optimization problem of the household, headed by an unskilled adult is to maximize

$$\begin{aligned} \hbox {Z}= & {} {\upbeta }_{1}\ln (\hbox {c}_{\mathrm {t}}- \underline{\mathrm {c}}) +{\upbeta }_{1}\ln [\hbox {f}\updelta (\hbox {bs}_{\mathrm {t}}\hbox {h}_{\mathrm {t}}+\underline{\mathrm {h}} ) + (1-\hbox {f}) \hbox {A}]\\&+\uplambda [\hbox {A} + \hbox {A}\upvarphi (1-\hbox {s}_{\mathrm {t }}) - \hbox {p}_{\mathrm {c }}\hbox {c}_{\mathrm {t}}]+\uptheta (\hbox {c}_{\mathrm {t}}- \underline{\mathrm {c}}), \end{aligned}$$

where $\uplambda$ is the Lagrange multiplier. The decision variables of the household are $\hbox {c}_{\mathrm {t }}$ and $\hbox {s}_{\mathrm {t}}$. The first order conditions for maximization of utility are given by:

$$\begin{aligned}&\displaystyle \frac{{\partial \hbox {Z}}}{{\partial }\mathrm {c}_{\mathrm {t}}}= \frac{{\upbeta }_{{1}}}{\mathrm {c}_{\mathrm {t}}\mathrm {-}\underline{\mathrm {c}}} -\uplambda \hbox {p}_{\mathrm {c }}+\uptheta =0, \end{aligned}$$

(A.1)

$$\begin{aligned}&\displaystyle \frac{{\partial \hbox {Z}}}{{\partial }\mathrm {s}_{\mathrm {t}}}= \frac{{\upbeta }_{{2}}{\mathrm {f}\updelta \hbox {b}}\mathrm {h}_{\mathrm {t}}}{{\mathrm {f}\updelta (\hbox {b}}\mathrm {s}_{\mathrm {t}}\mathrm {h}_{\mathrm {t}}\mathrm {+}\underline{\mathrm {h}}{) + (1-\hbox {f}) \,\hbox {A}}}- \uplambda \hbox {A}\upvarphi =0, \end{aligned}$$

(A.2)

$$\begin{aligned}&\displaystyle \uptheta \ge 0, \uptheta (\hbox {c}_{\mathrm {t}}- \underline{\mathrm {c}})= 0, \end{aligned}$$

(A.3)

From budget constraint $\hbox {A}+ \hbox {A}\upvarphi (1-\hbox {s}_{\mathrm {t}}) = \hbox {p}_{\mathrm {c }}\hbox {c}_{\mathrm {t}}$, we get

$$\begin{aligned} \hbox {c}_{\mathrm {t}} = \frac{\mathrm {A}}{\mathrm {p}_{\mathrm {c}}} + \frac{{\mathrm {A}\upvarphi (1-}\mathrm {s}_{\mathrm {t}}{) }}{\mathrm {p}_{\mathrm {c}}}, \end{aligned}$$

(A.4)

From (A.1) and (A.2) and using (A.4) we get,

$$\begin{aligned}&\displaystyle \hbox {s}_{\mathrm{t}}= \frac{{\upbeta }_{\mathrm {2 }}{\mathrm {f}\updelta \hbox {b}}\mathrm {h}_{\mathrm {t}}\left( {\mathrm {A}+\mathrm {A}\upvarphi -}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \mathrm {-}{\upbeta }_{\mathrm {1 }}{\mathrm {A}\upvarphi [\hbox {f}\updelta }\underline{\mathrm {h}}\mathrm {+A}\left( \mathrm {1-f} \right) \mathrm {]}}{{\mathrm {A}\upvarphi \mathrm {f}\updelta \mathrm {b}}\mathrm {h}_{\mathrm {t}}\mathrm {(}{\upbeta }_{\mathrm {1 }}{+ }{\upbeta }_{\mathrm {2 }}{)}} , \end{aligned}$$

(A.5)

$$\begin{aligned}&\displaystyle \frac{{\partial }\mathrm {s}_{\mathrm {t}}}{{\partial \updelta }}=\frac{{\upbeta }_{\mathrm {1 }}\mathrm {A}^{\mathrm {2 }}{\upvarphi (1-\mathrm {f})}}{{\mathrm {A}\upvarphi \hbox {f}}{\updelta }^{{2}}\mathrm {b}\mathrm {h}_{\mathrm {t}}{(}{\upbeta }_{\mathrm {1 }}\mathrm {+ }{\upbeta }_{\mathrm {2 }}{)}} , \end{aligned}$$

(A.6)

$$\begin{aligned}&\displaystyle \frac{{\partial }\mathrm {s}_{\mathrm {t}}}{\mathrm { \partial \upvarphi }} = \frac{{\upbeta }_{\mathrm {2 }}\left( \mathrm {-A+}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) }{\mathrm {A}\mathrm {\upvarphi }^{{2}}\mathrm {(}{\upbeta }_{\mathrm {1 }}\mathrm {+ }{\upbeta }_{\mathrm {2 }}\mathrm {)}}, \end{aligned}$$

(A.7)

$$\begin{aligned}&\displaystyle \frac{{\partial }\mathrm {s}_{\mathrm {t}}}{{\partial }{\upbeta }_{{1}}}= \frac{{{-\upbeta }}_{\mathrm {2 }}{\mathrm {f}\updelta \hbox {b}}\mathrm {h}_{\mathrm {t}}\left( {\mathrm {A}+\mathrm {A}\upvarphi -}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \mathrm {-}{\upbeta }_{\mathrm {2 }}{\mathrm {A}\upvarphi \hbox {f}\updelta }\underline{\mathrm {h}}\mathrm {-}{\upbeta }_{\mathrm {2 }}\mathrm {A}^{\mathrm {2 }}{\upvarphi (1-\mathrm {f})}}{{\mathrm {A}\upvarphi \mathrm {f}\updelta \mathrm {b}}\mathrm {h}_{\mathrm {t}}{{(}{\upbeta }_{\mathrm {1 }}\mathrm {+}{\upbeta }_{\mathrm {2 }}\mathrm {)}}^{{2}}}, \end{aligned}$$

(A.8)

$$\begin{aligned}&\displaystyle \frac{{\partial }\mathrm {s}_{\mathrm {t}}}{{\partial \hbox {A}}}= \frac{{\upbeta }_{\mathrm {2 }}{\mathrm {f}\updelta \hbox {b}}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}}\mathrm {h}_{\mathrm {t}}\mathrm {-}{\upbeta }_{\mathrm {1 }}\mathrm {A}^{\mathrm {2 }}{\upvarphi (1-\mathrm {f})}}{\mathrm {A}^{{2}}{\upvarphi \mathrm {f}\updelta \hbox {b}}\mathrm {h}_{\mathrm {t}}{(}{\upbeta }_{\mathrm {1 }}\mathrm {+ }{\upbeta }_{\mathrm {2 }}{)}}, \end{aligned}$$

(A.9)

Let the growth rate of human capital $\frac{\mathrm {h}_{\mathrm {t+1}}\mathrm {-}\mathrm {h}_{\mathrm {t}}}{\mathrm {h}_{\mathrm {t}}}$ be denoted by $\Psi$. Then,

$$\begin{aligned}&\displaystyle \Psi = \frac{\mathrm {h}_{\mathrm {t+1}}\mathrm {-}\mathrm {h}_{\mathrm {t}}}{\mathrm {h}_{\mathrm {t}}}= \frac{{\upbeta }_{\mathrm {2 }}{\mathrm {f}\updelta \mathrm {b}}\left( \mathrm {A+A\upvarphi -}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \mathrm {-}\frac{{\upbeta }_{{1}}\mathrm {A\upvarphi [f\updelta }\underline{\mathrm {h}}\mathrm {+A}\left( \mathrm {1-f} \right) \mathrm {]}}{\mathrm {h}_{\mathrm {t}}}}{\mathrm {A\upvarphi f\updelta (}{\upbeta }_{\mathrm {1 }}\mathrm {+ }{\upbeta }_{\mathrm {2 }}\mathrm {)}}\mathrm {+ }\frac{\underline{\mathrm {h}}}{\mathrm {h}_{\mathrm {t}}}-1,\qquad \end{aligned}$$

(A.10)

$$\begin{aligned}&\displaystyle \frac{{\partial \Psi }}{{\partial \upvarphi }}= \frac{{\upbeta }_{\mathrm {2 }}\mathrm {b}\left( \mathrm {-A+}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) }{\mathrm {A}\mathrm {\upvarphi }^{{2}}\mathrm {(}{\upbeta }_{\mathrm {1 }}\mathrm {+ }{\upbeta }_{\mathrm {2 }}{)}}, \end{aligned}$$

(A.11)

$$\begin{aligned}&\displaystyle \frac{{\partial \Psi }}{{\partial \hbox {A} }}= \frac{{\upbeta }_{\mathrm {2 }}\mathrm {f\updelta b}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}}{\mathrm {h}_{\mathrm {t}}\mathrm {- \upbeta }}_{{1}}\mathrm {A}^{{2}}{\upvarphi (1-\mathrm {f})}}{\mathrm {A}^{{2}}\mathrm {\upvarphi f\updelta }\mathrm {h}_{\mathrm {t}}\mathrm {(}{\upbeta }_{\mathrm {1 }}\mathrm {+ }{\upbeta }_{\mathrm {2 }}{)}}, \end{aligned}$$

(A.12)

Relationship between ${\hat{\mathbf{h}}}$ and ${\mathrm { \mathbf {h}}}_{\mathrm {\mathbf {0}}}$

$$\begin{aligned} {\hat{\hbox {h}}}\equiv & {} \frac{{\upbeta }_{{1}}{\mathrm {A}\upvarphi [\mathrm {f}\updelta }\underline{\mathrm {h}}\mathrm {+ A}\left( \mathrm {1-f} \right) \mathrm {]}}{\mathrm {f\updelta b[}{\upbeta }_{2}\left( \mathrm {A}-\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) {-\mathrm {A}\upvarphi }{\upbeta }_{{1}}{]}}.\\ {\mathrm { h}}_{{0}}\equiv & {} \frac{{\upbeta }_{{1}}{\mathrm {A}\upvarphi [\mathrm {f}\updelta }\underline{\mathrm {h}}\mathrm {+ A}\left( \mathrm {1-f} \right) \mathrm {]}}{{\mathrm {f}\updelta \mathrm {b}}{\upbeta }_{2}\left( {\mathrm {A}+\mathrm {A}\upvarphi }-\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) }. \end{aligned}$$

Here numerators of ${\hat{\hbox {h}}}$ and ${\mathrm {h}}_{{0}}$ are same.

Now denominator of ${\hat{\hbox {h}}}$—denominator of ${\mathrm { h}}_{{0}} = - ({\upbeta }_{{1}}+{\upbeta }_{2}) \,{\mathrm {f}\updelta \mathrm {b}}\,\hbox {A}\upvarphi < 0$. This implies that denominator of ${\hat{\hbox {h}}}<$ denominator of ${\mathrm { h}}_{{0}}$.

Therefore we can conclude that ${\hat{\hbox {h}}}$ is always greater than ${\mathrm { h}}_{{0}}$.

Relationship between ${\hat{\mathbf{h}}}$ and ${\hbox {h}}^*$

$$\begin{aligned} {\hat{\hbox {h}}} -\hbox {h}^*= & {} \frac{\mathrm {A\upvarphi }}{\mathrm {f\updelta }}\left[ \left\{ \frac{\mathrm {f\updelta }\underline{\mathrm {h}}\mathrm {\upbeta }_{\mathrm {1}}\mathrm {+}\mathrm {\upbeta }_{\mathrm {1}}\mathrm {A}\left( \mathrm {1-f} \right) }{\mathrm {\upbeta }_{\mathrm {2}}\mathrm {b}\left( \mathrm {A-}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \mathrm {-A\upvarphi }\mathrm {\upbeta }_{\mathrm {1}}}\right\} \right. \\&\left. -\left\{ \frac{{{\upbeta }_{\mathrm {1}}\mathrm {A}\left( \mathrm {1-f} \right) {-\mathrm {f}\updelta } \underline{\mathrm {h}}{\upbeta }_{\mathrm {2}}}}{{\upbeta }_{\mathrm {2}}\mathrm {b}\left( {\mathrm {A}+\mathrm {A}\upvarphi } -\mathrm {p}_{\mathrm {c}} \underline{\mathrm {c}} \right) {-\mathrm {A}\upvarphi }\left( {\upbeta }_{\mathrm {1}}\mathrm {+}\mathrm {\upbeta }_{\mathrm {2}} \right) } \right\} \right] . \end{aligned}$$

In the above expression, for $\hbox {b}> 1$, the denominator of $\hbox {h}^*$ within the third bracket > denominator of ${\hat{\hbox {h}}}$ within the third bracket.

For $\hbox {b}=1$, the denominator of h* within the third bracket $=$ denominator of ${\hat{\hbox {h}}}$ within the third bracket.

The numerator of $\hbox {h}^{*}$ within the third bracket < the numerator of ${\hat{\hbox {h}}}$ within the third bracket.

Therefore for $\hbox {b}\ge 1$, $\hbox {h}^{*}< {\hat{\hbox {h}}}$.

However for $\hbox {b}< 1$, $\hbox {h}^{*}> {\hat{\hbox {h}}}$.

Relationship between ${\mathbf {h}}_{{\mathbf {N}}}$ and ${\hat{\mathbf{h}}}$

$$\begin{aligned} \mathrm {h}_{\mathrm {N}}- {\hat{\hbox {h}}}= & {} - \frac{{\mathrm {A}\upvarphi }}{{\mathrm {f}\updelta \upbeta }}\left[ \frac{\left( {\mathrm {f}\updelta }\underline{\mathrm {h}}{\upbeta }_{{2}} \mathrm {-}{\upbeta }_{{1}}\mathrm {A-}{\upbeta }_{{2}}\mathrm {Af} \right) \left\{ {\upbeta }_{{2}}\left( \mathrm {A-}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \mathrm {-A\upvarphi }{\upbeta }_{{1}} \right\} \mathrm {+}{\upbeta }_{{1}}{\upbeta }_{{2}}\left( {\mathrm {A}+\mathrm {A}\upvarphi -}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \left\{ {\hbox {f}\updelta }\underline{\mathrm {h}}\mathrm { +A}\left( \mathrm {1-f} \right) \right\} }{{\upbeta }_{{2}}\left( \mathrm {A+A\upvarphi -}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \left\{ {\upbeta }_{{2}}\left( \mathrm {A-}\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) {-\mathrm {A}\upvarphi }{\upbeta }_{{1}} \right\} }\right] \\= & {} - \frac{{{{\hbox {A}\upvarphi }}}}{{{{\hbox {f}\updelta \upbeta }}}}\left[ {\frac{{{{{\upbeta }}_2}{{\hbox {f}}}\left( {{{{\upbeta }}_1} + {{{\upbeta }}_2}} \right) \left( {{{\hbox {A}}} - {{{\hbox {p}}}_{{\mathrm{c}}}}\underline{{\hbox {c}}}} \right) \left\{ {\left( {{{\updelta }}\underline{{\hbox {h}}} - {{\hbox {A}}}} \right) - {{{\upbeta }}_1}{{\hbox {A}}}\left( {{{{\upbeta }}_1} + {{{\upbeta }}_2}} \right) {{\hbox {A}}}\left( {1 - {{\upvarphi }}} \right) } \right\} }}{{{{{\upbeta }}_2}\left( {{{\hbox {A}}} + {{\hbox {A}\upvarphi }} - {{{\hbox {p}}}_{{\mathrm{c}}}}\underline{{\hbox {c}}}} \right) \left\{ {{{{\upbeta }}_2}\left( {{{\hbox {A}}} - {{{\hbox {p}}}_{{\mathrm{c}}}}\underline{{\hbox {c}}}} \right) - {{\hbox {A}\upvarphi }}{{{\upbeta }}_1}} \right\} }}} \right] \end{aligned}$$

For dynamic analysis we consider the case where $\hbox {A}+\hbox {A}\upvarphi -\hbox {p}_{\mathrm {c}}\underline{\hbox {c}}> 0$. Therefore, in the denominator, within the third bracket of the above expression, $\hbox {A}+\hbox {A}\upvarphi -\hbox {p}_{\mathrm {c}}\underline{\hbox {c}}> 0$. For dynamic analysis we also consider the case where $\upbeta _{{2}}(\hbox {A}- \hbox {p}_{\mathrm {c}}\underline{\hbox {c}})-A\upvarphi \upbeta _{{1}} > 0$. Since we assume $\hbox {A}> \updelta \underline{\mathrm {h}}$ ,the numerator within the third bracket of the above expression is negative. [$\hbox {If A}- \hbox {p}_{\mathrm {\mathrm{c}}}\underline{\hbox {c}}< 0$, then $\upbeta _{{2}}(\hbox {A}- \hbox {p}_{\mathrm {c}}\underline{\hbox {c}})-\hbox {A}\upvarphi \upbeta _{{1}} < 0$. Since we consider the case where $\upbeta _{{2}}(\hbox {A}- \hbox {p}_{\mathrm {c}}\underline{\hbox {c}})-\hbox {A}\upvarphi \upbeta _{{1}} > 0$, Thus for our dynamic analysis $\hbox {A}- \hbox {p}_{\mathrm {c}}\underline{\hbox {c}}> 0]$. Therefore $\hbox {h}_{\mathrm {N}}-{\hat{\hbox {h}}} > 0$ i.e. $\hbox {h}_{\mathrm {N}}> {\hat{\hbox {h}}}$.

Relationship between ${{\mathbf {h}}}_{{\mathbf {0}}}$ and $\underline{{\mathbf {h}}}$

$$\begin{aligned} \mathrm {h}_{{0}}> & {} \underline{\mathrm {h}}\\\Rightarrow & {} \frac{{\upbeta }_{{1}}{\mathrm {A}\upvarphi [\mathrm {f}\updelta }\underline{\mathrm {h}}\mathrm {+ A}\left( \mathrm {1-f} \right) \mathrm {]}}{{\mathrm {f}\updelta \mathrm {b}}{\upbeta }_{2}\left( {\mathrm {A}+\mathrm {A}\upvarphi }-\mathrm {p}_{\mathrm {c}} \underline{\mathrm {c}} \right) }> \underline{\mathrm {h}}\\\Rightarrow & {} {\upbeta }_{{1}}{\mathrm {A}\upvarphi \mathrm {f}\updelta }\underline{\mathrm {h}}\mathrm {+ }{\upbeta }_{{1}}\mathrm {A}^{{2}}\mathrm {\upvarphi }\left( \mathrm {1-f} \right) > [{\mathrm {f}\updelta \hbox {b}}{\upbeta }_{2}\left( {\mathrm {A}+\mathrm {A}\upvarphi }-\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) ]\underline{\mathrm {h}}\\\Rightarrow & {} [\mathrm {f\updelta b}{\upbeta }_{2}\left( {\mathrm {A}+\mathrm {A}\upvarphi }-\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \mathrm {-}{\upbeta }_{{1}}\mathrm {A\upvarphi f\updelta }]\underline{\mathrm {h}}< {\upbeta }_{{1}}\mathrm {A}^{{2}} \mathrm {\upvarphi }\left( \mathrm {1-f} \right) \\\Rightarrow & {} \underline{\mathrm {h}}< \frac{{\upbeta }_{{1}}\mathrm {A}^{{2}}{\upvarphi }\left( \mathrm {1-f} \right) }{{\mathrm {f}\updelta \mathrm {b}}{\upbeta }_{2}\left( {\mathrm {A}+\mathrm {A}\upvarphi }-\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \mathrm {-}{\upbeta }_{{1}}{\mathrm {A}\upvarphi \mathrm {f}\updelta }}. \end{aligned}$$

Therefore, $\underline{\mathrm {h}}<\frac{{\upbeta }_{{1}}\mathrm {A}^{{2}}{\upvarphi }\left( \mathrm {1-f} \right) }{{\mathrm {f}\updelta \mathrm {b}}{\upbeta }_{2}\left( {\mathrm {A}+\mathrm {A}\upvarphi }-\mathrm {p}_{\mathrm {c}}\underline{\mathrm {c}} \right) \mathrm {-}{\upbeta }_{{1}}\mathrm {A\upvarphi f\updelta }}$ is a necessary and sufficient condition for $\mathrm {h}_{{0}}> \underline{\mathrm {h}}$.