Endogenous Altruism and Impact of Child Labour Ban and Education Subsidy on Child Labour

Abstract

This paper builds an overlapping generations household economy model where child labour is present. Child schooling is determined by parental altruism. The degree of parental altruism is determined by the level of schooling of the parent. A more educated parent has a greater willingness to invest in the human capital formation of the child. These differences in the preferences of parents towards their offspring’s schooling have significant effects on the long-run dynamics of schooling. The dynamics of schooling exhibit the possibility of the existence of a child labour trap. If the economy is trapped in an inefficient equilibrium, increasing the child wage and the adult unskilled wage can help the economy get rid of the child labour trap. In this paper, we also study the efficacy of child labour ban and education subsidy in enhancing schooling and reducing child labour. We find that education subsidy is always likely to increase child schooling and reduce child labour. But banning child labour will increase schooling if the adult wage exceeds the sum of schooling cost and subsistence consumption expenditure. Once the economy reaches the advanced stage, banning child labour is desirable to take the stable equilibrium to full schooling equilibrium, but before that, banning child labour is not desirable.

Appendices

Appendix

The Lagrangian function is.

Z = ln(c_t-$\underline{\mathrm c}$) + s_t-1ln (bs_t) + λ [{$\underline{\mathrm W}$ + δbs_t-1 +$\underline{\mathrm W}$ φ (1-s_t)—p_cc_{t -}ρs_t] + θ(c_t-$\underline{\mathrm c}$).

where λ is the Lagrange multiplier. The decision variables of the household are c_t and s_t._. The first order conditions for maximization of utility are given by:

$$\frac{\mathrm\delta\mathrm Z}{\mathrm\delta{\mathrm{c}}_\mathrm{t}}=\frac1{{\mathrm{c}}_\mathrm{t}-\underline{\mathrm c}}-{\mathrm\lambda\mathrm p}_\mathrm{c}+\theta=0$$

(6)

$$\frac{\delta\mathrm Z}{\mathrm\delta s_{\mathrm t}}=\frac{{\mathrm s}_{\mathrm t-1}}{{\mathrm s}_{\mathrm t}}-\lambda\left(\underline{\mathrm W}\mathrm\varphi+\rho\right)=0$$

(7)

  

$$\theta\geq0,\theta\left(\mathrm{c_t}-\underline{\mathrm c}\right)=0$$

(8)

From (6) and budget constraint $\underline{\mathrm W}$ + δbs_t-1 + $\underline{\mathrm W}$ φ (1-s_t) = p_cc_{t +}ρs_t, we get

$$\frac1{\underline{\mathrm W}+\underline{\mathrm W}\mathrm\varphi(1-{\mathrm{s}}_\mathrm{t})-{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}-\rho{\mathrm{s}}_\mathrm{t}}=\lambda$$

(9)

From (7) and (9) we get,

$${\mathrm{s}}_\mathrm{t}=\frac{{\mathrm{s}}_{\mathrm{t}-1}\left[\underline{\mathrm W}+\mathrm\delta\mathrm b{\mathrm{s}}_{\mathrm{t}-1}+\underline{\mathrm W}\mathrm\varphi-{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\right]}{\left(\underline{\mathrm W}\mathrm\varphi+\rho\right)(1+{\mathrm{s}}_{\mathrm{t}-1})}$$

(10)

$$\frac{{\mathrm{ds}}_\mathrm{t}}{\mathrm d\mathrm\varphi}=\frac{{\underline{\mathrm W}\mathrm{s}}_{\mathrm{t}-1}}{(1+{\mathrm{s}}_{\mathrm{t}-1})}\left[\frac{\rho-\mathrm\delta\mathrm b{\mathrm{s}}_{\mathrm{t}-1}-\underline{\mathrm W}+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}}{\left(\underline{\mathrm W}\mathrm\varphi+\rho\right)^2}\right]$$

(11)

$$\frac{\mathrm{ds}^\ast}{\mathrm d\mathrm\rho}=\frac{\mathrm\delta\mathrm b+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}-\underline{\mathrm W}-\underline{\mathrm W}\mathrm\varphi}{{(\underline{\mathrm W}\mathrm\varphi+\rho-\mathrm\delta\mathrm b)}^2}$$

(12)

$$\frac{\mathrm{ds}^\ast}{\mathrm{d}\underline{\mathrm W}}=\frac{\rho+\mathrm\rho\mathrm\varphi+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\mathrm\varphi-\mathrm\delta\mathrm b}{{(\underline{\mathrm W}\mathrm\varphi+\rho-\mathrm\delta\mathrm b)}^2}$$

(13)

 

$$\frac{\mathrm{ds}^\ast}{\mathrm d\mathrm\delta}=\frac{\mathrm{b}(\underline{\mathrm W}-\rho-{\mathrm{p}}_\mathrm{c}\underline{\mathrm c})}{{(\underline{\mathrm W}\mathrm\varphi+\rho-\mathrm\delta\mathrm b)}^2}$$

(14)

$$\frac{\mathrm{ds}^\ast}{\mathrm{db}}=\frac{\mathrm\delta(\underline{\mathrm W}-\rho-{\mathrm{p}}_\mathrm{c}\underline{\mathrm c})}{{(\underline{\mathrm W}\mathrm\varphi+\rho-\mathrm\delta\mathrm b)}^2}$$

(15)

 

Dynamics of schooling

Differentiating s_t with respect to ${{\mathrm{s}}}_{{\mathrm{t}}-1}$ we get

$$\frac{{{\mathrm{ds}}}_{{\mathrm{t}}}}{{{\mathrm{ds}}}_{{\mathrm{t}}-1}}={{\mathrm{s}}}_{{\mathrm{t}}}^{\mathrm{^{\prime}}}=\frac{\mathrm{\delta b}{{\mathrm{s}}}_{{\mathrm{t}}-1}^{2}+ 2\mathrm{\delta b}{{\mathrm{s}}}_{{\mathrm{t}}-1}+\underline{\mathrm W}+\underline{\mathrm W}\mathrm{\varphi }-{{\mathrm{p}}}_{{\mathrm{c}}}\underline{\mathrm c}}{\left(\underline{\mathrm W}\mathrm{\varphi }+\uprho \right){\left(1+{\mathrm{ s}}_{{\mathrm{t}}-1}\right)}^{2}}$$

$${{\mathrm{s}}}_{{\mathrm{t}}}^{\mathrm{^{\prime}}} > 0\mathrm{ if} {{\mathrm{s}}}_{{\mathrm{t}}-1} > \sqrt{\frac{1}{\mathrm{\delta b}}\{\left(\mathrm{\delta b}+{{\mathrm{p}}}_{{\mathrm{c}}}\underline{\mathrm c}\right)-\underline{\mathrm W}\left(1+\mathrm{\varphi }\right)\}}-1$$

When $\left(\mathrm\delta\mathrm b+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\right)$ < $\underline{\mathrm W}(1+\mathrm\varphi)$, ${{\mathrm{s}}}_{{\mathrm{t}}}^{\mathrm{^{\prime}}}$ >0 for all ${{\mathrm{s}}}_{{\mathrm{t}}-1}$

But, if $\left(\mathrm\delta\mathrm b+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\right)$ > $\underline{\mathrm W}(1+\mathrm\varphi)$, then for ${{\mathrm{s}}}_{{\mathrm{t}}-1}$ > $\sqrt{\frac1{\mathrm\delta\mathrm b}\{\left(\mathrm\delta\mathrm b+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\right)-\underline{\mathrm W}\left(1+\mathrm\varphi\right)\}}$ – 1, ${{\mathrm{s}}}_{{\mathrm{t}}}^{\mathrm{^{\prime}}}$ >0.

If $\sqrt{\frac1{\mathrm\delta\mathrm b}\{\left(\mathrm\delta\mathrm b+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\right)-\underline{\mathrm W}\left(1+\mathrm\varphi\right)\}}$ – 1 < 0 for all ${{\mathrm{s}}}_{{\mathrm{t}}-1}, {{\mathrm{s}}}_{{\mathrm{t}}}^{\mathrm{^{\prime}}}$ >0.

But, if $\sqrt{\frac1{\mathrm\delta\mathrm b}\{\left(\mathrm\delta\mathrm b+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\right)-\underline{\mathrm W}\left(1+\mathrm\varphi\right)\}}$—1 > 0 then,

$${{\mathrm{s}}}_{{\mathrm{t}}}^{\mathrm{^{\prime}}}<0 {\mathrm{when}} {{\mathrm{s}}}_{{\mathrm{t}}-1}<\sqrt{\frac{1}{\mathrm{\delta b}}\{\left(\mathrm{\delta b}+{{\mathrm{p}}}_{{\mathrm{c}}}\underline{\mathrm c}\right)-\underline{\mathrm W}\left(1+\mathrm{\varphi }\right)\}}-1$$

$${{\mathrm{s}}}_{{\mathrm{t}}}^{\mathrm{^{\prime}}}<0 {\mathrm{when}} {{\mathrm{s}}}_{{\mathrm{t}}-1}<\sqrt{\frac{1}{\mathrm{\delta b}}\{\left(\mathrm{\delta b}+{{\mathrm{p}}}_{{\mathrm{c}}}\underline{\mathrm c}\right)-\underline{\mathrm W}\left(1+\mathrm{\varphi }\right)\}}-1$$

Now, $\sqrt{\frac1{\mathrm\delta\mathrm b}\{\left(\mathrm\delta\mathrm b+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\right)-\underline{\mathrm W}\left(1+\mathrm\varphi\right)\}}$ <1 implies ${\mathrm{p}}_\mathrm{c}\underline{\mathrm c}$< $\underline{\mathrm W}\left(1+\mathrm\varphi\right)$  

Note that when $\underline{\mathrm W}\left(1+\mathrm\varphi\right)$> ${\mathrm{p}}_\mathrm{c}\underline{\mathrm c}$, then $\underline{\mathrm W}\left(1+\mathrm\varphi\right)$> $\mathrm{\delta b}+$ ${\mathrm{p}}_\mathrm{c}\underline{\mathrm c}$  

Therefore, when $\underline{\mathrm W}\left(1+\mathrm\varphi\right)$> ${\mathrm{p}}_\mathrm{c}\underline{\mathrm c},$ slope of s_t i.e. ${{\mathrm{s}}}_{{\mathrm{t}}}^{\mathrm{^{\prime}}}$ is always positive.

$$\frac{\mathrm{d}^2{\mathrm{s}}_\mathrm{t}}{\mathrm{ds}_{\mathrm{t}-1}^2}=\frac{2\lbrack\mathrm\delta\mathrm b-\underline{\mathrm W}-\underline{\mathrm W}\mathrm\varphi+{\mathrm{p}}_\mathrm{c}\underline{\mathrm c}\rbrack}{\left(\underline{\mathrm W}\mathrm\varphi+\rho\right){(1+{\mathrm s}_{\mathrm{t}-1})}^3}$$

When $\mathrm{\delta b}$+${\mathrm{p}}_\mathrm{c}\underline{\mathrm c}$  > $\underline{\mathrm W}(1+\mathrm\varphi)$, $\frac{{{\mathrm{d}}}^{2}{{\mathrm{s}}}_{{\mathrm{t}}}}{{{\mathrm{ds}}}_{{\mathrm{t}}-1}^{2}}$ >0.

When $\mathrm{\delta b}$+${\mathrm{p}}_\mathrm{c}\underline{\mathrm c}$  < $\underline{\mathrm W}(1+\mathrm\varphi)$, $\frac{{{\mathrm{d}}}^{2}{{\mathrm{s}}}_{{\mathrm{t}}}}{{{\mathrm{ds}}}_{{\mathrm{t}}-1}^{2}}$ <0.