FDI and Child Labour

Abstract

Concomitant to the disparate empirical evidences on how FDI impinges on the incidence of child labour in the developing world, the first part of this chapter in terms of two simple general equilibrium models with child labour and the informal sector aims at explaining why growth led by foreign capital has failed to lessen the gravity of the problem in some cases, while in general the incidence has declined in the developing countries in the liberalized regime. The latter part of it concentrates in identifying the different channels through which economic liberalization like increased FDI inflow can affect the child labour problem by constructing a three-sector HT-type general equilibrium model with endogenous skill formation. The analysis finds that reduction in poverty is not a necessary condition for improvement of the problem of child labour. FDI can mitigate the gravity of problem and improve welfare of each poor working family that supplies child labour even by raising the return to education and lowering the earning opportunities of children.

Appendices

Appendices

8.1.1 Appendix 8.1: (Model 1) Derivation of Child Labour Supply Function

Maximizing Eq. (8.2) with respect to C ₁, C ₃ and l _C subject to the budget constraint (8.3), the following first-order conditions are obtained:

$$ \left(\frac{\left(\alpha U\right)}{\left({P}_1{C}_1\right)}\right)=\left(\frac{\left(\beta U\right)}{\left({P}_3{C}_3\right)}\right)=\left(\frac{\left(\gamma U\right)}{\left( n-{l}_C\right){W}_C}\right) $$

(8.A.1)

From (8.A.1) we get the following expressions:

$$ {C}_1=\left\{\frac{\alpha \left( n-{l}_C\right){W}_C}{\left(\gamma {P}_1\right)}\right\} $$

(8.A.2)

$$ {C}_3=\left\{\frac{\beta \left( n-{l}_C\right){W}_C}{\left(\gamma {P}_3\right)}\right\} $$

(8.A.3)

Substitution of the values of C ₁ and C ₃ into the budget constraint and further simplifications give us the following child labour supply function of each poor working household:

$$ {l}_C=\left\{\left(\alpha +\beta \right) n-\gamma \left(\frac{W}{W_C}\right)\right\} $$

(8.4)

8.1.2 Appendix 8.2: (Model 1) Effects on X ₁ and X ₂

Totally differentiating Eqs. (8.6), (8.7) and (8.8) and writing in a matrix notation, we obtain

$$ \left[\begin{array}{rrr}\hfill {\theta}_{L1}& \hfill {\theta}_{C1}& \hfill {\theta}_{K1}\\ {}\hfill {\theta}_{L2}& \hfill {\theta}_{C2}& \hfill {\theta}_{K2}\\ {}\hfill 0& \hfill 0& \hfill {\theta}_{K3}\end{array}\right]\left[\begin{array}{c}\hfill \widehat{W}\hfill \\ {}\hfill {\widehat{W}}_C\hfill \\ {}\hfill \widehat{r}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 0\hfill \\ {}\hfill {\widehat{P}}_2\hfill \\ {}\hfill -{\theta}_{23}{\widehat{P}}_2\hfill \end{array}\right] $$

(8.A.4)

where θ _ji  = distributive share of the jth input in the ith sector and ‘⋀’ = proportional change.

Solving (8.A.4) by Cramer’s rule and using (8.1.1), we find the following expressions:

$$ \begin{array}{l}\widehat{W}=\left(\frac{{\widehat{P}}_2}{\left|\theta \right|}\right)\left[\left({\theta}_{K1}{\theta}_{C2}-{\theta}_{C1}{\theta}_{K2}\right){\theta}_{23}-{\theta}_{C1}{\theta}_{K3}\right]\\ {}\kern2.5em \left(-\right)\kern4em \left(-\right)\end{array} $$

(8.A.5)

$$ \begin{array}{l}{\widehat{W}}_C=\left(\frac{{\widehat{P}}_2}{\left|\theta \right|}\right)\left[{\theta}_{L1}{\theta}_{K3}+{\theta}_{23}\left({\theta}_{L1}{\theta}_{K2}-{\theta}_{K1}{\theta}_{L2}\right)\right]\\ {}\kern3em \left(-\right)\kern9.5em \left(+\right)\ \end{array} $$

(8.A.6)

$$ \widehat{r}=\left(\frac{\theta_{23}{\widehat{P}}_2}{\left|\theta \right|}\right)\left({\theta}_{C1}{\theta}_{L2}-{\theta}_{L1}{\theta}_{C2}\right)=-\left(\frac{\theta_{23}}{\theta_{K3}}\right){\widehat{P}}_2 $$

(8.A.7)

$$ \left|\theta \right|={\theta}_{K3}\left({\theta}_{L1}{\theta}_{C2}-{\theta}_{L2}{\theta}_{C1}\right)<0. $$

(8.A.8)

where |θ| is the determinant to the coefficient matrix given by (8.A.4).

Again differentiating (8.9), (8.10) and (8.11.1) and arranging in a matrix notation, one finds

$$ \left[\begin{array}{rrr}{\lambda}_{L1}& \hfill {\lambda}_{L2}& \hfill {\lambda}_{L3}\\ {}\hfill {\lambda}_{K1}& \hfill {\lambda}_{K2}& \hfill {\lambda}_{K3}\\ {}\hfill {\lambda}_{C1}& \hfill {\lambda}_{C2}& \hfill \frac{\lambda_{L3}}{1-{\lambda}_{L3}}\end{array}\right]\left[\begin{array}{l}{\widehat{X}}_1\\ {}{\widehat{X}}_2\\ {}{\widehat{X}}_3\end{array}\right]=\left[\begin{array}{l} -{A}_1{\widehat{P}}_2\\ {} \widehat{K}-{A}_2{\widehat{P}}_2\\ {} -{A}_3{\widehat{P}}_2\end{array}\right] $$

(8.A.9)

where

$$ \begin{array}{l}{A}_1=\left(\frac{1}{\left|\theta \right|}\right)\big[\left({\lambda}_{L1}{S}_{L L}^1+{\lambda}_{L2}{S}_{L L}^2\right)\left\{{\theta}_{K1}{\theta}_{23}{\theta}_{C2}-{\theta}_{C1}\left({\theta}_{K3}+{\theta}_{23}{\theta}_{K2}\right)\right\}\\ {}\kern3em \left(-\right)\kern4em \left(-\right)\kern7em \left(-\right)\\ \qquad{}+\left({\lambda}_{L1}{S}_{L C}^1+{\lambda}_{L2}{S}_{L C}^2\right)\left\{{\theta}_{L1}\left({\theta}_{K3}+{\theta}_{23}{\theta}_{K2}\right)-{\theta}_{K1}{\theta}_{23}{\theta}_{L2}\right\}\\ \quad{}\kern5.5em \left(+\right)\kern10.5em \left(+\right)\kern0.75em \\ \qquad{}+\left({\lambda}_{L1}{S}_{L K}^1+{\lambda}_{L2}{S}_{L K}^2+{\lambda}_{L3}{S}_{L K}^3\right){\theta}_{23}\left({\theta}_{C1}{\theta}_{L2}-{\theta}_{L1}{\theta}_{C2}\right)\big]<0\\ {}\kern6.7em \left(+\right)\kern12em \left(+\right)\end{array} $$

(8.A.10.1)

$$ \begin{array}{c}{A}_2=\left(\frac{1}{\left|\theta \right|}\right)\left[\left({\lambda}_{K1}{S}_{K L}^1+{\lambda}_{K2}{S}_{K L}^2\right)\left\{{\theta}_{K1}{\theta}_{23}{\theta}_{C2}-{\theta}_{C1}\left({\theta}_{K3}+{\theta}_{23}{\theta}_{K2}\right)\right\}\right.\\ {}\kern-5.8em \left(-\right)\kern4em \left(+\right)\kern8em \left(-\right)\\ {}\ +\left({\lambda}_{K1}{S}_{K C}^1+{\lambda}_{K2}{S}_{K C}^2\right)\left\{{\theta}_{L1}\left({\theta}_{K3}+{\theta}_{23}{\theta}_{K2}\right)-{\theta}_{K1}{\theta}_{23}{\theta}_{L2}\right\}\\ {}\kern.5em \left(+\right)\kern11em \left(+\right)\\ \ \qquad\left.+\left({\lambda}_{K1}{S}_{K K}^1+{\lambda}_{K2}{S}_{K K}^2+{\lambda}_{K3}{S}_{K K}^3\right){\theta}_{23}\left({\theta}_{C1}{\theta}_{L2}-{\theta}_{L1}{\theta}_{C2}\right)\right]=?\\ {}\kern3.5em \left(-\right)\kern10em \left(+\right)\end{array} $$

(8.A.10.2)

$$ \begin{array}{c}{A}_3=\left(\frac{1}{\left|\theta \right|}\right)\left[\left({\lambda}_{C1}{S}_{C L}^1+{\lambda}_{C2}{S}_{C L}^2+\frac{\gamma W}{l_C{W}_C}\right)\left\{{\theta}_{K1}{\theta}_{23}{\theta}_{C2}-{\theta}_{C1}\left({\theta}_{K3}+{\theta}_{23}{\theta}_{K2}\right)\right\}\right.\\ {}\kern0.5em \left(-\right)\kern5em \left(+\right)\kern9em \left(-\right)\\ \ {}+\left({\lambda}_{C1}{S}_{C C}^1+{\lambda}_{C2}{S}_{C C}^2-\frac{\gamma W}{l_C{W}_C}\right)\left\{{\theta}_{L1}\left({\theta}_{K3}+{\theta}_{23}{\theta}_{K2}\right)-{\theta}_{K1}{\theta}_{23}{\theta}_{L2}\right\}\\ {}\kern4em \left(-\right)\kern12em \left(+\right)\\ \ \kern-7pt\left.+\left({\lambda}_{C1}{S}_{C K}^1+{\lambda}_{C2}{S}_{C K}^2+\frac{\lambda_{L3}}{1-{\lambda}_{L3}}{S}_{L K}^3\right){\theta}_{23}\left({\theta}_{C1}{\theta}_{L2}-{\theta}_{L1}{\theta}_{C2}\right)\right]=?\\ {}\kern6em \left(+\right)\kern8em \left(+\right)\end{array} $$

(8.A.10.3)

$$ \left|\lambda \right|=\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right){\left|\lambda \right|}_{L K}^{12}+{\lambda}_{K3}{\left|\lambda \right|}_{CL}^{12}-{\lambda}_{L3}{\left|\lambda \right|}_{CK}^{12} $$

(8.A.10.4)

where |λ| is the determinant of the coefficient matrix and

$$ \left.\begin{array}{cc}\hfill {\left|\lambda \right|}_{L K}^{12}=\left({\lambda}_{L1}{\lambda}_{K2}-{\lambda}_{L2}{\lambda}_{K1}\right)>0;\hfill & \hfill {\left|\lambda \right|}_{C L}^{12}=\left({\lambda}_{C1}{\lambda}_{L2}-{\lambda}_{C2}{\lambda}_{L1}\right)>0;\hfill \\ {}\hfill {\left|\lambda \right|}_{C K}^{12}=\left({\lambda}_{C1}{\lambda}_{K2}-{\lambda}_{C2}{\lambda}_{K1}\right)>0\hfill & \hfill \hfill \end{array}\right\} $$

(8.A.11)

(The signs are found by using (8.1.1).)

From (8.A.10.4) and (8.A.11), it follows that

$$ \left|\lambda \right|>0\kern0.5em \mathrm{if}\kern0.5em \left(\frac{\lambda_{K3}}{\lambda_{L3}}\right)\ge \left(\frac{{\left|\lambda \right|}_{CK}^{12}}{{\left|\lambda \right|}_{CL}^{12}}\right);\kern0.5em \mathrm{or},\kern0.5em \mathrm{if}\left(\frac{1}{1-{\lambda}_{L3}}\right)\ge \left(\frac{{\left|\lambda \right|}_{CK}^{12}}{{\left|\lambda \right|}_{L K}^{12}}\right) $$

(8.A.12)

Solving (8.A.9) by Cramer’s rule, the following expressions are found:

$$ \begin{array}{lll}{\widehat{X}}_2&=\left(\frac{{\widehat{P}}_2}{\left|\lambda \right|}\right)\left[{A}_3\left({\lambda}_{L1}{\lambda}_{K3}-{\lambda}_{L3}{\lambda}_{K1}\right)+{A}_2{\lambda}_{L3}\left({\lambda}_{C1}-\frac{\lambda_{L1}}{1-{\lambda}_{L3}}\right)\right.\\ &\quad \left. + {A}_1\left(\frac{\lambda_{K1}{\lambda}_{L3}}{1-{\lambda}_{L3}}-{\lambda}_{K3}{\lambda}_{C1}\right)\right] +\left(\frac{\lambda_{L3}\widehat{K}}{\left|\lambda \right|}\right)\left(\frac{\lambda_{L1}}{1-{\lambda}_{L3}}-{\lambda}_{C1}\right)\end{array} $$

(8.A.13)

$$ \begin{array}{lll}{\widehat{X}}_3&=\left(\frac{{\widehat{P}}_2}{\left|\lambda \right|}\right)\big[{A}_3\left({\lambda}_{L2}{\lambda}_{K1}-{\lambda}_{L1}{\lambda}_{K2}\right)+{A}_2\left({\lambda}_{L1}{\lambda}_{C2}-{\lambda}_{L2}{\lambda}_{C1}\right)\\ & \ \quad + {A}_1\left({\lambda}_{K2}{\lambda}_{C1}-{\lambda}_{K1}{\lambda}_{C2}\right)\big]+\left(\frac{\widehat{K}}{\left|\lambda \right|}\right)\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\end{array} $$

(8.A.14)

8.1.3 Appendix 8.3: Derivation of Stability Condition in the Market for the Non-traded Input (in Model 1)

For Walrasian static stability in the market for the non-traded input (good 2), we need

$$ \left(\frac{{\widehat{X}}_3}{{\widehat{P}}_2}\right)-\left(\frac{{\widehat{X}}_2}{{\widehat{P}}_2}\right)<0 $$

(8.A.15)

Using (8.A.13) and (8.A.14) from (8.A.15), we can write

$$ \begin{array}{lll}&\left[\left(\frac{{\widehat{X}}_3}{{\widehat{P}}_2}\right)-\left(\frac{{\widehat{X}}_2}{{\widehat{P}}_2}\right)\right]=\left(\frac{1}{\left|\lambda \right|}\right)\left[{A}_3\left({\lambda}_{L2}{\lambda}_{K1}-{\lambda}_{L1}{\lambda}_{K2}\right)+{A}_2\left({\lambda}_{L1}{\lambda}_{C2}-{\lambda}_{L2}{\lambda}_{C1}\right)\right.\\ &\left.\qquad\qquad\qquad\qquad\quad +{A}_1\left({\lambda}_{K2}{\lambda}_{C1}-{\lambda}_{K1}{\lambda}_{C2}\right)\right]\\ & \qquad -\left(\frac{1}{\left|\lambda \right|}\right)\left[{A}_3\left({\lambda}_{L1}{\lambda}_{K3}-{\lambda}_{L3}{\lambda}_{K1}\right)+{A}_2{\lambda}_{L3}\left({\lambda}_{C1}-\frac{\lambda_{L1}}{1-{\lambda}_{L3}}\right)\right.\\ & \qquad\qquad \left. +{A}_1\left(\frac{\lambda_{K1}{\lambda}_{L3}}{1-{\lambda}_{L3}}-{\lambda}_{K3}{\lambda}_{C1}\right)\right]<0\end{array} $$

or

$$ \begin{array}{lll}&\left(\frac{A_3}{\left|\lambda \right|}\right)\left[{\lambda}_{K1}\left({\lambda}_{L2}+{\lambda}_{L3}\right)-{\lambda}_{L1}\left({\lambda}_{K2}+{\lambda}_{K3}\right)\right]+\left(\frac{A_2}{\left|\lambda \right|}\right)\left[{\lambda}_{L1}\left({\lambda}_{C2}+\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\right.\\ &\qquad\left.-{\lambda}_{C1}\left({\lambda}_{L2}+{\lambda}_{L3}\right)\right]\\ &+\left(\frac{A_1}{\left|\lambda \right|}\right)\left[{\lambda}_{C1}\left({\lambda}_{K2}+{\lambda}_{K3}\right)-{\lambda}_{K1}\left({\lambda}_{C2}+\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\right]= D<0\end{array} $$

(8.A.15.1)

Thus, for static stability in the market for good 2, we require that

$$ D<0 $$

8.1.4 Appendix 8.4: Effect of FDI on P ₂ (in Model 1)

Differentiating Eq. (8.12) we obtain

$$ {\widehat{X}}_3={\widehat{X}}_2 $$

(8.A.16)

Using Eq. (8.A.13) and (8.A.14) from Eq. (8.A.16), it follows that

$$ \begin{array}{lll}&\left(\frac{{\widehat{P}}_2}{\left|\lambda \right|}\right)\left[{A}_3\left({\lambda}_{L2}{\lambda}_{K1}-{\lambda}_{L1}{\lambda}_{K2}\right)+{A}_2\left({\lambda}_{L1}{\lambda}_{C2}-{\lambda}_{L2}{\lambda}_{C1}\right)\right.\\ &\left.\quad+{A}_1\left({\lambda}_{K2}{\lambda}_{C1}-{\lambda}_{K1}{\lambda}_{C2}\right)\right]+\left(\frac{\widehat{K}}{\left|\lambda \right|}\right)\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\\ &=\left(\frac{{\widehat{P}}_2}{\left|\lambda \right|}\right)\left[{A}_3\left({\lambda}_{L1}{\lambda}_{K3}-{\lambda}_{L3}{\lambda}_{K1}\right)+{A}_2{\lambda}_{L3}\left({\lambda}_{C1}-\frac{\lambda_{L1}}{1-{\lambda}_{L3}}\right)\right.\\ &\left.\quad +{A}_1\left(\frac{\lambda_{K1}{\lambda}_{L3}}{1-{\lambda}_{L3}}-{\lambda}_{K3}{\lambda}_{C1}\right)\right]+\left(\frac{\lambda_{L3}\widehat{K}}{\left|\lambda \right|}\right)\left(\frac{\lambda_{L1}}{1-{\lambda}_{L3}}-{\lambda}_{C1}\right)\end{array} $$

Collecting terms, simplifying and using (8.A.15.1), one finds

$$ \begin{array}{l}\left(\dfrac{{\widehat{P}}_2}{\widehat{K}}\right)=\left(\dfrac{1}{D\left|\lambda \right|}\right)\left[\dfrac{\lambda_{L1}{\lambda}_{C2}-{\lambda}_{C1}{\lambda}_{L2}}{\left({\lambda}_{L1}+{\lambda}_{L2}\right)}\right]>0\\ {}\kern4.5em \left(-\right)\left(+\right)\kern3em \left(-\right)\ \end{array} $$

(8.A.17)

So, an inflow of foreign capital raises the price of the non-traded input.

8.1.5 Appendix 8.5: Effect of FDI on Factor Prices, P ₂ and X ₃ (in Model 1)

Using (8.A.17) from (8.A.5), (8.A.6) and (8.A.7), one obtains the following expressions:

$$ \begin{array}{l}\left(\dfrac{\widehat{W}}{\widehat{K}}\right)\!=\!\left(\dfrac{1}{D\left|\lambda \right|\left|\theta \right|}\right)\left[\dfrac{\lambda_{L1}{\lambda}_{C2}-{\lambda}_{C1}{\lambda}_{L2}}{\left({\lambda}_{L1}+{\lambda}_{L2}\right)}\right]\left[\left({\theta}_{K1}{\theta}_{C2}-{\theta}_{C1}{\theta}_{K2}\right){\theta}_{23}-{\theta}_{C1}{\theta}_{K3}\right]>0\\ {}\kern4em \left(-\right)\left(+\right)\left(-\right)\kern3em \left(-\right)\kern6em \left(-\right)\end{array} $$

(8.A.5.1)

$$ \begin{array}{l}\left(\dfrac{{\widehat{W}}_C}{\widehat{K}}\right)\!=\!\left(\dfrac{1}{D\left|\lambda \right|\left|\theta \right|}\right)\left[\dfrac{\lambda_{L1}{\lambda}_{C2}-{\lambda}_{C1}{\lambda}_{L2}}{\left({\lambda}_{L1}+{\lambda}_{L2}\right)}\right]\left[{\theta}_{L1}{\theta}_{K3}\!+\!{\theta}_{23}\left({\theta}_{L1}{\theta}_{K2}\!-\!{\theta}_{K1}{\theta}_{L2}\right)\right]<0\\ {}\kern4em \left(-\right)\left(+\right)\left(-\right)\kern4em \left(-\right)\kern11.8em \left(+\right)\end{array} $$

(8.A.6.1)

$$ \begin{array}{l}\left(\dfrac{\widehat{r}}{\widehat{K}}\right)\!=\!\left(\dfrac{\theta_{23}{\widehat{P}}_2}{\left|\theta \right|\widehat{K}}\right)\left({\theta}_{C1}{\theta}_{L2}\!-\!{\theta}_{L1}{\theta}_{C2}\right)\!=\!-\left(\!\dfrac{\theta_{23}}{\theta_{K3}}\!\right)\left(\!\dfrac{1}{D\left|\lambda \right|}\!\right)\left[\!\dfrac{\lambda_{L1}{\lambda}_{C2}-{\lambda}_{C1}{\lambda}_{L2}}{\left({\lambda}_{L1}+{\lambda}_{L2}\right)}\!\right]\!<\!0.\\ {}\kern20.1em (\hbox{--})(\hbox{+})\kern3.8em \left(-\right)\end{array} $$

(8.A.7.1)

Subtraction of (8.A.6.1) from (8.A.5.1) shows that

$$ \begin{array}{l}\left(\frac{\widehat{W}}{\widehat{K}}-\frac{{\widehat{W}}_C}{\widehat{K}}\right)>0\\ {}\left(+\right)\kern1em \left(-\right)\end{array} $$

(8.A.18)

Therefore, from (8.A.5.1), (8.A.6.1), (8.A.7.1) and (8.A.18), one finds that an inflow of foreign capital raises the adult wage rate and lowers both the return to capital and the child wage rate. The (W _C /W) ratio decreases as a consequence.

Using (8.A.17) and (8.A.10.1), (8.A.10.2), (8.A.10.3) and (8.A.10.4) and simplifying from (8.A.14), we find

$$ \begin{array}{l}\left(\dfrac{{\widehat{X}}_3}{\widehat{K}}\right)=\left[\dfrac{\lambda_{L3}\left({\lambda}_{L1}+{\lambda}_{L2}\right){\left|\lambda \right|}_{C L}^{12}}{\left(1-{\lambda}_{L3}\right) D{\left(\left|\lambda \right|\right)}^2}\right]\left[\dfrac{\left({\lambda}_{K1}+{\lambda}_{K2}\right)}{\left({\lambda}_{L1}+{\lambda}_{L2}\right)}-\dfrac{\lambda_{K3}}{\lambda_{L3}}\right]\left[{\lambda}_{L1}{A}_3-{\lambda}_{C1}{A}_1\right]\\ {}\kern9.3em \left(-\right)\kern9.9em \left(-\right)\kern7em \left(-\right)\end{array} $$

(8.A.14.1)

> 0 if A ₃ ≥ 0

Note that if technologies of production are of fixed-coefficient type, S ^k _ji  = 0. From (8.A.10.1) and (8.A.10.3) it then follows that A ₁ = 0 and A ₃ > 0. From (8.A.14.1) it is, therefore, evident that

$$ \left(\frac{{\widehat{X}}_3}{\widehat{K}}\right)>0 $$

So, the formal sector (sector 3) expands following inflows of foreign capital under the sufficient condition that A ₃ ≥ 0. Sector 3 unambiguously expands if the technologies of production are of fixed-coefficient type.

8.1.6 Appendix 8.6: (Model 2) Derivations for Obtaining the Expressions for Changes in Output Composition

Total differential of (8.23) yields

$$ \Sigma {a}_{Li} d{X}_i=-\Sigma {X}_i d{a}_{Li} $$

or

$$ \begin{array}{lll}\Sigma \left(\frac{X_i{a}_{L i}}{L}\right){\widehat{X}}_i&=-\left(\frac{X_1}{L}\right)\left\{\left(\frac{\partial {a}_{L1}}{\partial W}\right) d W+\left(\frac{\partial {a}_{L1}}{\partial {W}_C}\right) d{W}_C+\left(\frac{\partial {a}_{L1}}{\partial R}\right) d R\right\}\\ &\quad-\left(\frac{X_2}{L}\right)\left\{\left(\frac{\partial {a}_{L2}}{\partial W}\right) d W+\left(\frac{\partial {a}_{L2}}{\partial {W}_C}\right) d{W}_C\right\}\\ &\quad-\left(\frac{X_3}{L}\right)\left\{\left(\frac{\partial {a}_{L3}}{\partial R}\right) d R\right\}\end{array} $$

(8.A.19)

Using the result that $\widehat{r}=0$ (see (8.26)) from the above expression, we can write

$$ {\lambda}_{L1}{\widehat{X}}_1+{\lambda}_{L2}{\widehat{X}}_2+{\lambda}_{L3}{\widehat{X}}_3\!=\!-\left({\lambda}_{L1}{S}_{L L}^1+{\lambda}_{L2}{S}_{L L}^2\right)\widehat{W}-\left({\lambda}_{L1}{S}_{L C}^1+{\lambda}_{L2}{S}_{L C}^2\right){\widehat{W}}_C $$

(8.A.20)

Substituting the values of $\widehat{W}$ and ${\widehat{W}}_C$ from (8.27) and (8.28) into (8.A.20) and simplifying, we get the following:

$$ {\lambda}_{L1}{\widehat{X}}_1+{\lambda}_{L2}{\widehat{X}}_2+{\lambda}_{L3}{\widehat{X}}_3=-{A}_1{\widehat{P}}_2 $$

(8.A.21)

where ${A}_1=\left[\left({\lambda}_{L1}{S}_{L C}^1+{\lambda}_{L2}{S}_{L C}^2\right)\left({\theta}_{L1}+{\theta}_{C1}\right)+{\lambda}_{L1}{S}_{L K}^1{\theta}_{C1}\right]\left(\frac{1}{\left|\theta \right|}\right)$

(Note that S ¹ _LL  + S ¹ _LC  + S ¹ _LK  = 0 ⇒ S ¹ _LL  = − (S ¹ _LC  + S ¹ _LK ) and S ² _LL  + S ² _LC  = 0)

Now differentiating (8.24) we get

$$ {\lambda}_{K1}{\widehat{X}}_1+{\lambda}_{K3}{\widehat{X}}_3=\widehat{K}-{\lambda}_{K1}\left({S}_{K L}^1\widehat{W}-{S}_{K C}^1{\widehat{W}}_C\right) $$

Substituting the values of $\widehat{W}$ and ${\widehat{W}}_C$ from (8.27) and (8.28) into the above expression and simplifying, one gets the following:

$$ {\lambda}_{K1}{\widehat{X}}_1+{\lambda}_{K3}{\widehat{X}}_3=\widehat{K}+\left(\frac{\lambda_{K1}}{\left|\theta \right|}\right)\left[{S}_{K L}^1{\theta}_{C1}-{S}_{K C}^1{\theta}_{L1}\right]{\widehat{P}}_2 $$

or

$$ {\lambda}_{K1}{\widehat{X}}_1+{\lambda}_{K3}{\widehat{X}}_3=\widehat{K}+{A}_2{\widehat{P}}_2 $$

(8.A.22)

where ${A}_2=\left(\frac{\lambda_{K1}}{\left|\theta \right|}\right)\left[{S}_{K L}^1{\theta}_{C1}-{S}_{K C}^1{\theta}_{L1}\right]$

Similarly differentiating Eq. (8.25.1) we get

$$ \begin{array}{lll} {\lambda}_{C1}{\widehat{X}}_1+{\lambda}_{C2}{\widehat{X}}_2+\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right){\widehat{X}}_3&=-\left\{{\lambda}_{C1}{S}_{C L}^1+{\lambda}_{C2}{S}_{C L}^2+{G}_2\right\}\widehat{W}\\ &\quad-\left\{{\lambda}_{C1}{S}_{C C}^1+{\lambda}_{C2}{S}_{C C}^2-{G}_1\right\}{\widehat{W}}_C-{G}_3\widehat{E} \end{array} $$

(8.A.23)

Substituting the values of $\widehat{W}$ and ${\widehat{W}}_C$ into (8.A.23) and simplifying, we get the following:

$$ \begin{array}{c}{\lambda}_{C1}{\widehat{X}}_1+{\lambda}_{C2}{\widehat{X}}_2+\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right){\widehat{X}}_3=\frac{{\widehat{P}}_2}{\left|\theta \right|}\left[{\lambda}_{C1}{S}_{C L}^1\left({\theta}_{C1}+{\theta}_{L1}\right)+{\lambda}_{C2}{S}_{C L}^2\left({\theta}_{C1}+{\theta}_{L1}\right)\right.\\ \qquad\qquad\qquad\qquad\qquad {}\left.\kern1em +{\lambda}_{C1}{S}_{C K}^1{\theta}_{L1}+{G}_2{\theta}_{C1}+{G}_1{\theta}_{L1}\right]-{G}_3\widehat{E}\end{array} $$

This is rewritten as follows:

$$ {\lambda}_{C1}{\widehat{X}}_1+{\lambda}_{C2}{\widehat{X}}_2+\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right){\widehat{X}}_3={A}_3{\widehat{P}}_2-{G}_3\widehat{E} $$

(8.A.24)

where

$$ \begin{array}{lll} {G}_1&=\frac{\gamma {W}_C\left\{ W+ n B(E)\right\}}{l_C{\left\{{W}_C- B(E)\right\}}^2}>0;\kern0.5em {G}_2=\frac{\gamma W}{l_C\left\{{W}_C- B(E)\right\}}>0;\\ {G}_3&=\frac{\gamma {B}^{\prime } E\left\{ W+ n{W}_C\right\}}{l_C{\left\{{W}_C- B(E)\right\}}^2}>0 \end{array}$$

and

$$ \begin{array}{lll} {A}_3&=\frac{1}{\left|\theta \right|}\left[{\lambda}_{C1}{S}_{C L}^1\left({\theta}_{C1}+{\theta}_{L1}\right)+{\lambda}_{C2}{S}_{C L}^2\left({\theta}_{C1}+{\theta}_{L1}\right)+{\lambda}_{C1}{S}_{C K}^1{\theta}_{L1}\right.\\ &\left.\quad +{G}_2{\theta}_{C1}+{G}_1{\theta}_{L1}\right] \end{array} $$

(8.A.21), (8.A.22) and (8.A.24) can be written in a matrix notation as follows:

$$ \left[\begin{array}{lll}{\lambda}_{L1}\hfill & {\lambda}_{L2}\hfill & {\lambda}_{L3}\hfill \\ {}{\lambda}_{K1}\hfill & 0\hfill & {\lambda}_{K3}\hfill \\ {}{\lambda}_{C1}\hfill & {\lambda}_{C2}\hfill & \left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\widehat{X}}_1\hfill \\ {}\hfill {\widehat{X}}_2\hfill \\ {}\hfill {\widehat{X}}_3\hfill \end{array}\right]=\left[\begin{array}{c} -{A}_1{\widehat{P}}_2 \\ {}\left(\widehat{K}+{A}_2{\widehat{P}}_2\right)\\ {}\left({A}_3{\widehat{P}}_2-{G}_3\widehat{E}\right)\end{array}\right] $$

(8.A.25)

Solving (8.A.25) by Cramer’s rule and simplifying, one gets

$$ {\widehat{X}}_1=\left(\frac{1}{\left|\lambda \right|}\right)\left[{\lambda}_{L3}\left({\lambda}_{C2}-\frac{\lambda_{L2}}{1-{\lambda}_{L3}}\right)\widehat{K}+{Z}_1{\widehat{P}}_2-{\lambda}_{L2}{\lambda}_{K3}{G}_3\widehat{E}\right] $$

(8.29)

$$ {\widehat{X}}_2=\left(\frac{1}{\left|\lambda \right|}\right)\left[{\lambda}_{L3}\left(\frac{\lambda_{L1}}{1-{\lambda}_{L3}}-{\lambda}_{C1}\right)\widehat{K}+{Z}_2{\widehat{P}}_2+\left({\lambda}_{L1}{\lambda}_{K3}-{\lambda}_{K1}{\lambda}_{L3}\right){G}_3\widehat{E}\right] $$

(8.30)

and

$$ {\widehat{X}}_3=\left(\frac{1}{\left|\lambda \right|}\right)\left[\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\widehat{K}-{Z}_3{\widehat{P}}_2+{\lambda}_{L2}{\lambda}_{K1}{G}_3\widehat{E}\right] $$

(8.31)

where

$$ \begin{array}{lll}{Z}_1&=\left[{\lambda}_{K3}\left({A}_1{\lambda}_{C2}+{A}_3{\lambda}_{L2}\right)+{\lambda}_{L3}{A}_2\left({\lambda}_{C2}-\frac{\lambda_{L2}}{1-{\lambda}_{L3}}\right)\right]\\ {Z}_2&=\left[{A}_2{\lambda}_{L3}\left(\frac{\lambda_{L1}}{1-{\lambda}_{L3}}-{\lambda}_{C1}\right)-{A}_3\left({\lambda}_{L1}{\lambda}_{K3}-{\lambda}_{K1}{\lambda}_{L3}\right)\right.\\ &\left.\quad+{A}_1\left\{\frac{\lambda_{K1}{\lambda}_{L3}}{\left(1-{\lambda}_{L3}\right)}-{\lambda}_{K3}{\lambda}_{C1}\right\}\right]\\ \left|\lambda \right|&={\lambda}_{K3}\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)+\left(\frac{\lambda_{K1}{\lambda}_{L3}}{1-{\lambda}_{L3}}\right)\left({\lambda}_{C2}\left(1-{\lambda}_{L3}\right)-{\lambda}_{L2}\right)\\ &=\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\left(\frac{\lambda_{K3}-{\lambda}_{L3}}{1-{\lambda}_{L3}}\right)\end{array} $$

(8.32.1)

(see Appendix 8.A.10)

and

$$ {Z}_3=\left[-{A}_2\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)+{A}_3{\lambda}_{L2}{\lambda}_{K1}+{A}_1{\lambda}_{K1}{\lambda}_{C2}\right] $$

8.1.7 Appendix 8.7: (Model 2) Derivation of the Stability Condition in the Market for the Non-traded Final Commodity

Differentiating the demand function for commodity 2 given by Eq. (8.20), we can derive

$$ {\widehat{X}}_2^D={E}_{P2}{\widehat{P}}_2+{E}_Y\left(\frac{W^{*} L{\lambda}_{L3}}{Y}\right){\widehat{X}}_3-\left(\frac{T}{Y}\right){E}_Y\widehat{T} $$

(8.A.26)

Substituting ${\widehat{X}}_3$ from (8.31) and simplifying, one obtains the following:

$$ \begin{array}{lll} {\widehat{X}}_2^D & = {E}_{P2}{\widehat{P}}_2+\left(\frac{E_Y}{\left|\lambda \right|}\right)\left(\frac{W^{*} L{\lambda}_{L3}}{Y}\right)\left\{\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\right.\widehat{K}\\ &\quad \left. -{Z}_3{\widehat{P}}_2+{\lambda}_{L2}{\lambda}_{K1}{G}_3\widehat{E}\right\}-\left(\frac{T}{Y}\right){E}_Y\widehat{T} \end{array} $$

(8.A.27)

Allowing only P ₂ to change, while keeping all parameters unchanged, we find that

$$ \frac{{\widehat{X}}_2^D}{{\widehat{P}}_2}={E}_{P2}-\left(\frac{E_Y}{\left|\lambda \right|}\right)\left(\frac{W^{*} L{\lambda}_{L3}}{Y}\right){Z}_3 $$

(8.A.28)

Similarly, from Eq. (8.30) one obtains the following:

$$ \frac{{\widehat{X}}_2}{{\widehat{P}}_2}=\left(\frac{Z_2}{\left|\lambda \right|}\right) $$

(8.A.29)

For static stability we require that

$$ \left(\frac{{\widehat{X}}_2^D}{{\widehat{P}}_2}-\frac{{\widehat{X}}_2}{{\widehat{P}}_2}\right)=\left[{E}_{P2}-\left(\frac{E_Y}{\left|\lambda \right|}\right)\left(\frac{W^{*} L{\lambda}_{L3}}{Y}\right){Z}_3-\left(\frac{Z_2}{\left|\lambda \right|}\right)\right]=\Delta <0 $$

(8.33)

8.1.8 Appendix 8.8: (Model 2) Derivation of Eq. (8.34.1)

In equilibrium in the market for commodity 2, we have ${\widehat{X}}_2^D={\widehat{X}}_2$. Now using (8.A.27) and (8.30), we can write

$$ \begin{array}{lll}{}&{E}_{P2}{\widehat{P}}_2+\left(\frac{E_Y}{\left|\lambda \right|}\right)\left(\frac{W^{*} L{\lambda}_{L3}}{Y}\right)\left\{\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\widehat{K}-{Z}_3{\widehat{P}}_2+{\lambda}_{L2}{\lambda}_{K1}{G}_3\widehat{E}\right\}\\ &\quad -\left(\frac{T}{Y}\right){E}_Y\widehat{T}\\ {}&=\left(\frac{1}{\left|\lambda \right|}\right)\left[{\lambda}_{L3}\left(\frac{\lambda_{L1}}{1-{\lambda}_{L3}}-{\lambda}_{C1}\right)\widehat{K}+{Z}_2{\widehat{P}}_2+\left({\lambda}_{L1}{\lambda}_{K3}-{\lambda}_{K1}{\lambda}_{L3}\right){G}_3\widehat{E}\right]\end{array} $$

or

$$ \begin{array}{lll}&\left[{E}_{P2}-\left(\frac{E_Y}{\left|\lambda \right|}\right)\left(\frac{W^{*} L{\lambda}_{L3}}{Y}\right){Z}_3-\frac{Z_2}{\left|\lambda \right|}\right]{\widehat{P}}_2\\ &\quad =\left(\frac{1}{\left|\lambda \right|}\right)\left[{\lambda}_{L3}\left(\frac{\lambda_{L1}}{1-{\lambda}_{L3}}-{\lambda}_{C1}\right)-\left(\frac{E_Y{W}^{*} L{\lambda}_{L3}}{Y}\right)\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\right]\widehat{K}\\ &\qquad +\!\left(\frac{1}{\left|\lambda \right|}\right)\left[\left({\lambda}_{L1}{\lambda}_{K3}-{\lambda}_{K1}{\lambda}_{L3}\right)\!-\!\left(\!\frac{E_Y{W}^{*} L{\lambda}_{L3}}{Y}\!\right){\lambda}_{L2}{\lambda}_{K1}\right]{G}_3\widehat{E}\!+\!\left(\frac{T}{Y}\right){E}_Y\widehat{T}\end{array} $$

Using (8.33) the above expression may be rewritten as follows:

$$ {\widehat{P}}_2={Q}_1\widehat{K}+{Q}_2\widehat{E}+{Q}_3\widehat{T} $$

(8.34.1)

where

$$ \left.\begin{array}{lll}{Q}_1&\!=\!\left(\!\frac{1}{\left|\lambda \right|}\!\right)\left(\!\frac{1}{\Delta}\!\right)\left[\!{\lambda}_{L3}\left(\frac{\lambda_{L1}}{1\!-\!{\lambda}_{L3}}-{\lambda}_{C1}\right)\!-\!\left(\!\frac{E_Y{W}^{*} L{\lambda}_{L3}}{Y}\right)\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\!\right)\!\right]\!;\\ {Q}_2&=\left(\frac{G_3}{\left|\lambda \right|}\right)\left(\frac{1}{\Delta}\right)\left[\left({\lambda}_{L1}{\lambda}_{K3}-{\lambda}_{K1}{\lambda}_{L3}\right)-\left(\frac{E_Y{W}^{*} L{\lambda}_{L3}}{Y}\right){\lambda}_{L2}{\lambda}_{K1}\right];\kern1em \mathrm{and}\\ {Q}_3&=\left(\left(\frac{E_Y}{\Delta}\right)\left(\frac{T}{Y}\right)\right)<0\end{array}\right\} $$

(8.34.2)

8.1.9 Appendix 8.9: (Model 2) Derivation of Eqs. (8.35.1) and (8.35.2)

Differentiating Eq. (8.16) we get

$$ {\widehat{L}}_C={G}_1{\widehat{W}}_C-{G}_2\widehat{W}-{G}_3\widehat{E}-\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right){\widehat{X}}_3 $$

(8.A.30)

Substitution of $\widehat{W},{\widehat{W}}_C$ and ${\widehat{X}}_3$ from (8.27), (8.28) and (8.31) into (8.A.30) yields

$$ \begin{array}{lll}{\widehat{L}}_C=&\left(\frac{G_1}{\left|\theta \right|}\right){\theta}_{L1}{\widehat{P}}_2+\left(\frac{G_2}{\left|\theta \right|}\right){\theta}_{C1}{\widehat{P}}_2-{G}_3\widehat{E}\\ {}&-\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\left(\frac{1}{\left|\lambda \right|}\right)\left[\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\widehat{K}-{Z}_3{\widehat{P}}_2+{\lambda}_{L2}{\lambda}_{K1}{G}_3\widehat{E}\right]\end{array} $$

(8.35.1)

Rearranging terms we write

$$ \begin{array}{l}{\widehat{L}}_C=\left[\left(\frac{G_1}{\left|\theta \right|}\right){\theta}_{L1}+\left(\frac{G_2}{\left|\theta \right|}\right){\theta}_{C1}+\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\left(\frac{1}{\left|\lambda \right|}\right){Z}_3\right]{\widehat{P}}_2-\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\left(\frac{1}{\left|\lambda \right|}\right)\\ \qquad \quad \left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\widehat{K} -\left\{1+\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\left(\frac{1}{\left|\lambda \right|}\right){\lambda}_{L2}{\lambda}_{K1}\right\}{G}_3\widehat{E}\end{array} $$

Now substituting ${\widehat{P}}_2$ from (8.34.1) into the above expression, using (8.32.1) and simplifying, one finally gets

$$ \begin{array}{c}{\widehat{L}}_C=\left[\left({I}_{W C}+{I}_W+{I}_L\right){Q}_1-\left(\frac{\lambda_{L3}}{\lambda_{K3}-{\lambda}_{L3}}\right)\right]\widehat{K}+\left[\left({I}_{W C}+{I}_W+{I}_L\right){Q}_2\right.\\ {}\kern1em \left.-\left\{1+\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\left(\frac{\lambda_{L2}{\lambda}_{K1}}{\left|\lambda \right|}\right)\right\}{G}_3\right]\widehat{E}+\left[\left({I}_{W C}+{I}_W+{I}_L\right){Q}_3\right]\widehat{T}\end{array} $$

(8.35.2)

where

$$ {I}_{W C}=\left(\frac{G_1}{\left|\theta \right|}\right){\theta}_{L1};\kern1em {I}_W=\left(\frac{G_2}{\left|\theta \right|}\right){\theta}_{C1};\kern1em {I}_L=\left(\frac{\lambda_{L3}}{1-{\lambda}_{L3}}\right)\left(\frac{1}{\left|\lambda \right|}\right){Z}_3 $$

(8.36)

8.1.10 Appendix 8.10: (Model 2) Effects on Child Labour Incidence Under Alternate Factor Intensity Conditions

In this model, sector 3 has been assumed to be more capital-intensive (in physical sense) vis-à-vis sector 1 with respect to adult labour. This implies the following:

$$ \frac{\lambda_{K3}}{\lambda_{L3}}>\frac{\lambda_{K1}}{\lambda_{L1}}\Rightarrow \frac{\lambda_{K3}}{\lambda_{K1}}>\frac{\lambda_{L3}}{\lambda_{L1}}\Rightarrow \frac{1}{\lambda_{K1}}>\frac{\left({\lambda}_{L3}+{\lambda}_{L1}\right)}{\lambda_{L1}}\Rightarrow {\lambda}_{L1}>{\lambda}_{K1}\left({\lambda}_{L3}+{\lambda}_{L1}\right) $$

(8.A.31)

and

$$ \frac{\lambda_{L1}}{\lambda_{L3}}>\frac{\lambda_{K1}}{\lambda_{K3}}\Rightarrow \frac{\left({\lambda}_{L1}+{\lambda}_{L3}\right)}{\lambda_{L3}}>\frac{1}{\lambda_{K3}}\Rightarrow {\lambda}_{K3}>\frac{\lambda_{L3}}{\left({\lambda}_{L1}+{\lambda}_{L3}\right)}\Rightarrow {\lambda}_{K3}>{\lambda}_{L3} $$

(8.A.32)

(Note that (λ _L1 + λ _L3) < 1)

The expression for |λ| is as follows:

$$ \left|\lambda \right|={\lambda}_{K3}\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)+\left(\frac{\lambda_{K1}{\lambda}_{L3}}{1-{\lambda}_{L3}}\right)\left({\lambda}_{C2}\left(1-{\lambda}_{L3}\right)-{\lambda}_{L2}\right) $$

(8.A.33)

Substitution of (λ _L1 + λ _L2) in place of (1 − λ _L3) in (8.A.33) and simplification yield

$$ \left|\lambda \right|=\left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)\left(\frac{\lambda_{K3}-{\lambda}_{L3}}{1-{\lambda}_{L3}}\right)$$

(8.32.1)

From the stability condition in the market for commodity 2, we have

$$ \left[{E}_{P2}-\left(\frac{E_Y}{\left|\lambda \right|}\right)\left(\frac{W^{*} L{\lambda}_{L3}}{Y}\right){Z}_3-\left(\frac{Z_2}{\left|\lambda \right|}\right)\right]=\Delta <0$$

(8.33)

As both sectors 1 and 2 use adult labour and child labour as inputs, these together form a HOSS given the rental to capital, R. So, these sectors can be classified in terms of relative factor intensities.

8.1.10.1 Case I

We first consider the case where the export sector is more child labour-intensive vis-à-vis the non-traded sector with respect to adult labour. In other words,

$$ \left(\frac{a_{C1}}{a_{L1}}\right)>\left(\frac{a_{C2}}{a_{L2}}\right)\Rightarrow \left(\frac{\lambda_{C1}}{\lambda_{L1}}\right)>\left(\frac{\lambda_{C2}}{\lambda_{L2}}\right)\Rightarrow \left({\lambda}_{L2}{\lambda}_{C1}-{\lambda}_{L1}{\lambda}_{C2}\right)>0$$

(8.A.34)

In this case, we also have

$$ \left({\theta}_{C1}{\theta}_{L2}>{\theta}_{C2}{\theta}_{L1}\right)\Rightarrow \left|\theta \right|=\left({\theta}_{L1}{\theta}_{C2}-{\theta}_{C1}{\theta}_{L2}\right)<0 $$

(8.A.35)

$$ \left.\begin{array}{rr}\hfill \mathrm{From}\kern0.5em \left(8.\mathrm{A}.34\right)\kern0.5em \mathrm{we}\kern0.5em \mathrm{find}\kern0.5em \mathrm{that}& \hfill \frac{\lambda_{L2}}{\lambda_{L1}}>\frac{\lambda_{C2}}{\lambda_{C1}}\Rightarrow \frac{\left(1-{\lambda}_{L3}\right)}{\lambda_{L1}}>\frac{1}{\lambda_{C1}}\Rightarrow {\lambda}_{C1}>\frac{\lambda_{L1}}{\left(1-{\lambda}_{L3}\right)}\\[5pt] {}\hfill \mathrm{Alternatively}& \hfill \frac{\lambda_{C1}}{\lambda_{C2}}>\frac{\lambda_{L1}}{\lambda_{L2}}\Rightarrow \frac{1}{\lambda_{C2}}>\frac{\left(1-{\lambda}_{L3}\right)}{\lambda_{L2}}\Rightarrow \frac{\lambda_{L2}}{\left(1-{\lambda}_{L3}\right)}>{\lambda}_{C2}\end{array}\right\}$$

(8.A.36)

Assuming a _K1 to be technologically given and using (8.33), (8.A.31), (8.A.32) and (8.A.34), (8.A.35) and (8.A.36) from (8.32.1), (8.32.2) and (8.34.2), it is easy to check that

$$ \begin{array}{ll} {}\left(\mathrm{i}\right)\hfill & \left|\lambda \right|>0;\hfill \\ {}\left(\mathrm{i}\mathrm{i}\right)\hfill & {A}_1<0;\hfill \\ {}\left(\mathrm{i}\mathrm{i}\mathrm{i}\right)\hfill & {A}_2=0;\hfill \\ {}\left(\mathrm{i}\mathrm{v}\right)\hfill & {A}_3<0;\hfill \\ {}\left(\mathrm{v}\right)\hfill & {Q}_1>0;\hfill \\ {}\left(\mathrm{v}\mathrm{i}\right)\hfill & {Q}_2>0\kern1em \mathrm{i}\mathrm{f}\kern1em \left[{\lambda}_{L1}{\lambda}_{K3}\le \left(\frac{E_Y{W}^{*} L{\lambda}_{L3}}{Y}\right){\lambda}_{L2}{\lambda}_{K1}\right]\hfill \\ {}\left(\mathrm{v}\mathrm{i}\mathrm{i}\right)\hfill & {Z}_1<0;\hfill \\ {}\left(\mathrm{v}\mathrm{i}\mathrm{i}\mathrm{i}\right)\hfill & {Z}_3<0;\hfill \\ {}\left(\mathrm{i}\mathrm{x}\right)\hfill & {I}_{W C},{I}_W,{I}_L<0.\hfill \end{array}$$

(8.A.37)

With the help of (8.A.37) from (8.29 and 8.34.1), we can get the following results:

$$ \left.\begin{array}{lll} {}\left(\mathrm{a}\right)\hfill & {\widehat{X}}_1<0\hfill & \mathrm{when}\;\widehat{K}>0\hfill \\ {}\left(\mathrm{b}\right)\hfill & {\widehat{X}}_1<0\hfill & \mathrm{when}\;\widehat{E}>0\hfill \\ {}\left(\mathrm{c}\right)\hfill & {\widehat{X}}_1<\left(>\right)0\hfill & \mathrm{when}\;{\widehat{P}}_2>\left(<\right)0\hfill \\ {}\left(\mathrm{d}\right)\hfill & {\widehat{P}}_2>0\hfill & \mathrm{when}\;\widehat{K}>0\hfill \\ {}\left(\mathrm{e}\right)\hfill & {\widehat{P}}_2>0\hfill & \mathrm{when}\;\widehat{E}>0\kern1em \mathrm{if}\;\left[{\lambda}_{L1}{\lambda}_{K3}\le \left(\frac{E_Y{W}^{*} L{\lambda}_{L3}}{Y}\right){\lambda}_{L2}{\lambda}_{K1}\right]\hfill \\ {}\left(\mathrm{f}\right)\hfill & {\widehat{P}}_2<0\hfill & \mathrm{when}\;\widehat{T}>0\hfill \end{array}\right\} $$

(8.A.38)

Using (8.A.34), (8.A.36) and (8.A.37) from (8.30) and (8.31), we find that X ₂ decreases (increases) while X ₃ increases (increases) following an inflow of foreign capital (an education subsidy policy). Besides, an increase in P ₂ raises X ₃. We use these results while explaining Proposition 8.3 verbally.

Finally, using (8.A.32) and (8.A.37) from Eq. (8.35.2), the following results trivially follow:

• (R.1) ${\widehat{L}}_C<0$ when $\widehat{K}>0$.

• (R.2) ${\widehat{L}}_C<0$ when $\widehat{E}>0$ if Q ₂ ≥ 0 i.e. if [λ _L1 λ _K3 ≤ (E _Y W*Lλ _L3/Y)λ _L2 λ _K1].

• (R.3) ${\widehat{L}}_C>0$ when $\widehat{T}>0$.

Note that, in (R.2), [λ _L1 λ _K3 ≤ (E _Y W*Lλ _L3/Y)λ _L2 λ _K1] is only a sufficient condition for L _C to fall following an education subsidy policy.

8.1.10.2 Case II

Let us now consider the case where sector 2 is more child labour-intensive relative to sector 1 with respect to adult labour. This implies the case where

$$ \left(\frac{a_{C1}}{a_{L1}}\right)<\left(\frac{a_{C2}}{a_{L2}}\right)\Rightarrow \left(\frac{\lambda_{C1}}{\lambda_{L1}}\right)<\left(\frac{\lambda_{C2}}{\lambda_{L2}}\right) $$

(8.A.39)

In this case, we find that

$$ \left.\begin{array}{l} {}\left({\theta}_{C1}{\theta}_{L2}<{\theta}_{C2}{\theta}_{L1}\right)\Rightarrow \left|\theta \right|=\left({\theta}_{L1}{\theta}_{C2}-{\theta}_{C1}{\theta}_{L2}\right)>0\\ {}\frac{\lambda_{L2}}{\lambda_{L1}}<\frac{\lambda_{C2}}{\lambda_{C1}}\Rightarrow \frac{\left(1-{\lambda}_{L3}\right)}{\lambda_{L1}}<\frac{1}{\lambda_{C1}}\Rightarrow {\lambda}_{C1}<\frac{\lambda_{L1}}{\left(1-{\lambda}_{L3}\right)}\\ {}\frac{\lambda_{C1}}{\lambda_{C2}}<\frac{\lambda_{L1}}{\lambda_{L2}}\Rightarrow \frac{1}{\lambda_{C2}}<\frac{\left(1-{\lambda}_{L3}\right)}{\lambda_{L2}}\Rightarrow {\lambda}_{C2}>\frac{\lambda_{L2}}{\left(1-{\lambda}_{L3}\right)}\\ {}\left|\lambda \right|<0\\ {}{A}_1>0\\ {}{A}_2=0\;\left(\mathrm{assuming}\kern0.5em {a}_{K1}\;\mathrm{to}\kern0.5em \mathrm{be}\kern0.5em \mathrm{given}\kern0.5em \mathrm{technologically}\right)\\ {}{A}_3>0\\ {}{Q}_1>0\\ {}{Q}_2<0\;\mathrm{if}\kern1em \left[{\lambda}_{L1}{\lambda}_{K3}\le \left(\frac{E_Y{W}^{*} L{\lambda}_{L3}}{Y}\right){\lambda}_{L2}{\lambda}_{K1}\right]\\ {}{Z}_1>0\\ {}{Z}_3>0\\ {}{I}_{W C},{I}_W>0;{I}_L<0\end{array}\right\} $$

(8.A.40)

Using (8.A.40) from (8.29) and (8.34.1), the following results follow:

$$ \left.\begin{array}{lll}\left(\mathrm{a}\right)\hfill & {\widehat{X}}_1<0\hfill & \mathrm{when}\;\widehat{K}>0\hfill \\ {}\left(\mathrm{b}\right)\hfill & {\widehat{X}}_1>0\hfill & \mathrm{when}\;\widehat{E}>0\hfill \\ {}\left(\mathrm{c}\right)\hfill & {\widehat{X}}_1<\left(>\right)0\hfill & \mathrm{when}\;{\widehat{P}}_2>\left(<\right)0\hfill \\ {}\left(\mathrm{d}\right)\hfill & {\widehat{P}}_2>0\hfill & \mathrm{when}\;\widehat{K}>0\hfill \\ {}\left(\mathrm{e}\right)\hfill & {\widehat{P}}_2<0\hfill & \mathrm{when}\;\widehat{E}>0\kern1em \mathrm{if}\;\left[{\lambda}_{L1}{\lambda}_{K3}\le \left(\frac{E_Y{W}^{*} L{\lambda}_{L3}}{Y}\right){\lambda}_{L2}{\lambda}_{K1}\right]\hfill \\ {}\left(\mathrm{f}\right)\hfill & {\widehat{P}}_2<0\hfill & \mathrm{when}\;\widehat{T}>0\hfill \end{array}\right\} $$

(8.A.41)

Using (8.A.39) and (8.A.40) from (8.30) and (8.31), we find that X ₂ decreases (increases) while X ₃ increases (decreases) following an inflow of foreign capital (an education subsidy policy). Besides, an increase in P ₂ raises X ₃. These results are useful in explaining proposition 8.4 intuitively.

Using (8.A.32) and (8.A.41) from (8.35.2), it is easy to derive the following results:

• (R.4) $\begin{array}{lll}{\widehat{L}}_C>0\kern0.5em \mathrm{when}\kern0.5em \widehat{K}>0 \kern0.5em \mathrm{if}\kern0.5em \left[\left({I}_{W C}+{I}_L\right){Q}_1\ge \left(\frac{\lambda_{L3}}{\lambda_{K3}-{\lambda}_{L3}}\right)\right]\\ {}\kern7em \mathrm{or}\kern0.5em \mathrm{if}\kern0.5em \left[\left({I}_W+{I}_L\right){Q}_1\ge \left(\frac{\lambda_{L3}}{\lambda_{K3}-{\lambda}_{L3}}\right)\right].\end{array}$

• (R.5) $\begin{array}{lll}{\widehat{L}}_C<0\kern0.75em \mathrm{when}\kern0.5em \widehat{E}>0\kern0.5em \mathrm{i}\mathrm{f}\kern1em \left(\mathrm{i}\right)\kern0.5em \left({\lambda}_{L1}{\lambda}_{K3}\le \frac{\lambda_{L3}{\lambda}_{L2}{\lambda}_{K1}{E}_Y{W}^{*} L}{Y}\right)\\ {}\kern7em \mathrm{and},\kern0.5em \left(\mathrm{i}\mathrm{i}\right)\kern0.5em \left[\left({I}_{W C}+{I}_W+{I}_L\right){Q}_2\le \frac{\lambda_{L3}{\lambda}_{L2}{\lambda}_{K1}{G}_3}{\left(1-{\lambda}_{L3}\right)\left|\lambda \right|}\right].\end{array}$

• (R.6) $\begin{array}{lll}{\widehat{L}}_C<0\kern0.5em \mathrm{when}\kern0.5em \widehat{T}>0\kern1em \mathrm{if}\kern0.5em \left({I}_{W C}+{I}_L\right){Q}_3\le 0\\ {}\kern7em \mathrm{or},\mathrm{if}\kern0.5em \left({I}_W+{I}_L\right){Q}_3\le 0.\end{array}$

8.1.11 Appendix 8.11: (Model 3) Derivations for Obtaining Expressions for Effects of FDI on Child Wage, Skilled and Unskilled Wages and Inequality Thereof

Totally differentiating Eqs. (8.42), (8.43.1) and (8.44) and using envelope conditions, the following expressions are obtained:

$$ {\theta}_{L1}\widehat{W}+{\theta}_{C1}{\widehat{W}}_C+{\theta}_{K1}\widehat{R}=0 $$

(8.A.42)

$$ {\theta}_{L2}{E}_W\widehat{W}+{\theta}_{K2}\widehat{R}=0 $$

(8.A.43)

$$ {\theta}_{S3}{\widehat{W}}_{\mathrm{S}}+{\theta}_{K3}\widehat{R}=0 $$

(8.A.44)

Totally differentiating Eqs. (8.45.1), (8.46), (8.47) and (8.49), collecting terms and simplifying, we get the following expressions:

$$ {\overline{S}}_{L L}\widehat{W}+{\lambda}_{L1}{S}_{L C}^1{\widehat{W}}_C+{\overline{S}}_{L K}\widehat{R}+{\lambda}_{L1}{\widehat{X}}_1+{\lambda}_{L2}{\widehat{X}}_2=0 $$

(8.A.45)

$$ {\overline{S}}_{K L}\widehat{W}+{A}_2\widehat{R}+{A}_1{\widehat{W}}_S+{\lambda}_{K1}{\widehat{X}}_1+{\lambda}_{K2}{\widehat{X}}_2=\widehat{K} $$

(8.A.46)

$$ \left({S}_{C L}^1+ E\right)\widehat{W}+\left({S}_{C C}^1- F\right){\widehat{W}}_C+ G{\widehat{W}}_{\mathrm{S}}+{\widehat{X}}_1=0 $$

(8.A.47)

(Note that we have used ${\widehat{X}}_3=-{S}_{S S}^3{\widehat{W}}_{\mathrm{S}}-{S}_{S R}^3\widehat{R}$ from (8.47)).

where

$$ \left.\begin{array}{l}\kern2pt{\overline{S}}_{L L}=\left[{\lambda}_{L2}^{*}\left\{\left({E}_W-1\right)+{S}_{L L}^2\right\}+\left({\lambda}_{L1}{S}_{L L}^1\right)\right]<0;{\overline{S}}_{L K}={\lambda}_{L2}^{*}{S}_{L K}^2>0\\[4pt] {}{\overline{S}}_{K K}=\left({\lambda}_{K2}{S}_{K K}^2+{\lambda}_{K3}{S}_{K K}^3\right)<0;{\overline{S}}_{K L}={\lambda}_{K2}{S}_{K L}^2>0;\\[4pt] \quad \qquad {A}_1={\lambda}_{K3}\left({S}_{S K}^3+{S}_{K S}^3\right)>0\\[4pt] \kern8pt{A}_2=\left({\overline{S}}_{K K}-{\lambda}_{K3}{S}_{S K}^3\right)<0;{\lambda}_{L2}^{*}=\frac{W^{*}}{W}{\lambda}_{L2}>0\\[4pt] \quad\kern2pt A=\frac{W_S. W}{\left(1+\beta \right){L}_C{\left({W}_S- W\right)}^2}>0; B=\frac{\beta}{\left(1+\beta \right){L}_C{W}_C}>0\\[4pt] \quad\kern.8pt E=\left(- nA\left( L+ S\right)+ BLW\right); F= B\left( L W+ S{W}_S\right)>0;\\[4pt] \qquad\qquad G=\left[ nA\left(L+ S\right)+ BS{W}_S\right]>0\end{array}\right\} $$

(8.A.48)

S ^k _ji  = the degree of substitution between factors jand i in the kth sector, j, i = L, S, L _C , K and k = 1, 2, 3. S ^k _ji  > 0 for j ≠ i; S ^k _jj  < 0; and λ _ji  = proportion of the jth input employed in the ith sector.

Arranging (8.A.42), (8.A.43), (8.A.44), (8.A.45), (8.A.46), (8.A.47) in the matrix notation, we get the following:

$$ \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} {\theta}_{L1} & {\theta}_{C1} & {\theta}_{K1} & 0 & 0 & 0 \\ {} {\theta}_{L2}{E}_W & 0 & {\theta}_{K2} & 0 & 0 & 0 \\ {} 0 & 0 & {\theta}_{K3} & {\theta}_{S3} & 0 & 0 \\ {} {\overline{S}}_{L L} & {\lambda}_{L1}{S}_{L C}^1 & {\overline{S}}_{L K} & 0 & {\lambda}_{L1} & {\lambda}_{L2}^{*} \\ {} {\overline{S}}_{K L} & 0 & {A}_2 & {A}_1 & {\lambda}_{K1} & {\lambda}_{K2} \\ {} \left({S}_{C L}^1+ E\right) & \left({S}_{C C}^1- F\right) & 0 & G & 1 & 0 \end{array}\right]\left[\begin{array}{c} \widehat{W} \\ {} {\widehat{W}}_C \\ {} \widehat{R} \\ {} {\widehat{W}}_S \\ {} {\widehat{X}}_1 \\ {} {\widehat{X}}_2 \end{array}\right]=\left[\begin{array}{c} 0 \\ {} 0 \\ {} 0 \\ {} 0 \\ {} \widehat{K} \\ {} 0 \end{array}\right] $$

(8.A.49)

Solving (8.A.49) by Cramer’s rule the following expressions are obtained:

$$ \widehat{W}=-\left(\frac{\theta_{S3}{\theta}_{C1}{\theta}_{K2}{\lambda}_{L2}^{*}}{\Delta}\right)\widehat{K} $$

(8.A.50)

$$ {\widehat{W}}_C=\left(\frac{\theta_{S3}\left|\theta \right|{\lambda}_{L2}^{*}}{\Delta}\right)\widehat{K} $$

(8.A.51)

$$ \widehat{R}=\left(\frac{\theta_{S3}{\theta}_{C1}{E}_W{\theta}_{L2}{\lambda}_{L2}^{*}}{\Delta}\right)\widehat{K} $$

(8.A.52)

$$ {\widehat{W}}_S=-\left(\frac{\theta_{K3}{\theta}_{C1}{E}_W{\theta}_{L2}{\lambda}_{L2}^{*}}{\Delta}\right)\widehat{K} $$

(8.A.53)

$$ \left({\widehat{W}}_S-\widehat{W}\right)=-\left(\frac{\theta_{C1}{\lambda}_{L2}^{*}\left({\theta}_{L2}{E}_W{\theta}_{K3}-{\theta}_{S3}{\theta}_{K2}\right)}{\Delta}\right)\widehat{K} $$

(8.A.54)

$$ \begin{array}{l}{\widehat{X}}_2=\left(\frac{\widehat{K}}{\Delta}\right)\left[-{\theta}_{S3}{\lambda}_{L1}\left|\theta \right|\left\{{S}_{L C}^1-\left({S}_{C C}^1- F\right)\right\}-{\theta}_{L2}{\theta}_{C1}{E}_W\left({\theta}_{K3}{\lambda}_{L1} G+{\theta}_{S3}{\overline{S}}_{L K}\right)\right.\\ {} \ \qquad \left.+{\theta}_{C1}{\theta}_{K2}{\theta}_{S3}\left\{{\overline{S}}_{L L}-{\lambda}_{L1}\left({S}_{C L}^1+ E\right)\right\}\right]\end{array} $$

(8.A.55)

where

$$ \begin{array}{lll} \Delta &=-{\theta}_{K3}{\theta}_{C1}{E}_W{\theta}_{L2}\left\{BS{W}_S\left|\lambda \right|+{A}_1{\lambda}_{L2}^{*}\right\}-{\theta}_{S3}{\theta}_{C1}{\theta}_{L2}{E}_W\left({\overline{S}}_{L K}{\lambda}_{K2}-{A}_2{\lambda}_{L2}^{*}\right)\\ & \ \quad+{\theta}_{S3}\left|\theta \right|\left\{\left({S}_{C C}^1- F\right)\left|\lambda \right|-{\lambda}_{K2}{\lambda}_{L1}{S}_{L C}^1\right\}-{\theta}_{S3}{\theta}_{C1}{\theta}_{K2}\left\{\left({S}_{C L}^1+ BLW\right)\right.\\ &\qquad\left.\left|\lambda \right|-\left({\lambda}_{K2}{\overline{S}}_{L L}-{\lambda}_{L2}^{*}{\overline{S}}_{K L}\right)\right\}\\ & \ \quad+ nA\left( L+ S\right){\theta}_{C1}\left|\lambda \right|\left({\theta}_{K2}-{\theta}_{K3}\right)\\\end{array} $$

(8.A.56)

$$ \left.\begin{array}{c}{\left|\lambda \right|}_{L K}=\left({\lambda}_{L1}{\lambda}_{K2}-{\lambda}_{K1}{\lambda}_{L2}^{*}\right)>0\kern1em \mathrm{and}\\ {}{\left|\theta \right|}_{L K}=\left({\theta}_{L1}{\theta}_{K2}-{\theta}_{K1}{E}_W{\theta}_{L2}\right)>0\end{array}\right\} $$

(8.A.57)

(Note that |λ|, |θ| > 0 as sector 2 is more capital-intensive than sector 1 with respect to adult unskilled labour.)

Using (8.A.48) and (8.A.57) from (8.A.56), it follows that

$$ \Delta <0\kern1em \mathrm{if}\kern0.5em {\theta}_{K3}{E}_W{\theta}_{L2}>{\theta}_{K2}{\theta}_{S3} $$

(8.A.58)

However, θ _K3 E _W θ _L2 > θ _K2 θ _S3 is only a sufficient condition for Δ to be negative.

Using (8.A.48), (8.A.56) and (8.A.57) from (8.A.50), (8.A.51), (8.A.52), (8.A.53), (8.A.54) and (8.A.55), we can obtain the following results:

$$ \left.\begin{array}{ll} {} \left(\mathrm{i}\right)\hfill & \widehat{W}>0\;\mathrm{when}\;\widehat{K}>0\hfill \\ {}\left(\mathrm{i}\mathrm{i}\right)\hfill & {\widehat{W}}_C<0\;\mathrm{when}\;\widehat{K}>0\hfill \\ {}\left(\mathrm{i}\mathrm{i}\mathrm{i}\right)\hfill & \widehat{R}<0\;\mathrm{when}\;\widehat{K}>0\hfill \\ {}\left(\mathrm{i}\mathrm{v}\right)\hfill & {\widehat{W}}_S>0\;\mathrm{when}\;\widehat{K}>0\hfill \\ {}\left(\mathrm{v}\right)\hfill & \left({\widehat{W}}_S-\widehat{W}\right)>0\;\mathrm{when}\;\widehat{K}>0\kern1em \mathrm{i}\mathrm{ff}\;{\theta}_{K3}{E}_W{\theta}_{L2}>{\theta}_{K2}{\theta}_{S3}\hfill \\ {}\left(\mathrm{v}\mathrm{i}\right)\hfill & {\widehat{X}}_2>0\;\mathrm{when}\;\widehat{K}>0\hfill \end{array}\right\} $$

(8.A.59)

8.1.12 Appendix 8.12: (Model 3) Effects of FDI on Incidence on Child Labour

We use Eq. (8.51) to examine the impact of foreign capital inflow on the incidence of child labour in the economy. Totally differentiating Eq. (8.51) we get

$$ {\widehat{L}}_C=- nA\left( L+ S\right)\left({\widehat{W}}_{\mathrm{S}}-\widehat{W}\right)- LBW\widehat{W}- SB{W}_{\mathrm{S}}{\widehat{W}}_{\mathrm{S}}+ B{\widehat{W}}_C\left({W}_{\mathrm{S}} S+ LW\right) $$

(8.A.60)

Using (8.A.50), (8.A.51), (8.A.52), (8.A.53) and (8.A.54), the expression (8.A.60) may be rewritten as follows:

$$ \begin{array}{lll}{\widehat{L}}_C&=\left(\frac{1}{\Delta}\right)\left[ nA\left( L+ S\right){\theta}_{C1}{\lambda}_{L2}^{*}\left({\theta}_{K3}{E}_W{\theta}_{L2}-{\theta}_{\mathrm{S}3}{\theta}_{K2}\right)+ LBW{\theta}_{\mathrm{S}3}{\theta}_{C1}{\theta}_{K2}{\lambda}_{L2}^{*}\right.\\ &\left.\quad + SB{W}_{\mathrm{S}}{\theta}_{K3}{\theta}_{C1}{\theta}_{L2}{\lambda}_{L2}^{*}+ B\left( S{W}_{\mathrm{S}}+ LW\right){\theta}_{\mathrm{S}3}\left|\theta \right|{\lambda}_{\mathrm{L}2}^{*}\right]\widehat{K}\end{array} $$

(8.A.61)

From (8.A.61) we find that

${\widehat{L}}_C<0$ when $\widehat{K}>0$ if θ _K3 E _W θ _L2 > θ _S3 θ _K2

So, the incidence of child labour decreases following inflows of foreign capital under the sufficient condition: θ _K3 E _W θ _L2 > θ _S3 θ _K2. This implies that sector 3 is capital-intensive relative to sector 2. However, this result may hold under other sufficient conditions as well.

8.1.13 Appendix 8.13: (Model 3) Effects of FDI on Unemployment of Unskilled Labour

Differentiating (8.52) one gets

$$ \left(\frac{{\widehat{L}}_{\mathrm{U}}}{\widehat{K}}\right)=\left(\frac{{\widehat{X}}_2}{\widehat{K}}\right)-\left[{S}_{L K}^2{E}_W+\left(\frac{\lambda_{L2}+{\lambda}_{L U}}{\lambda_{L U}}\right)\left(1-{E}_W\right)\right]\left(\frac{\widehat{W}}{\widehat{K}}\right)+{S}_{L K}^2\left(\frac{\widehat{R}}{\widehat{K}}\right) $$

(8.A.62)

Using (8.A.50), (8.A.52) and (8.A.55) and simplifying from the above equation, we obtain

$$ \begin{array}{l}\left(\frac{{\widehat{L}}_{\mathrm{U}}}{\widehat{K}}\right)=\left(\frac{1}{\Delta}\right)\Big[\left[{\theta}_{\mathrm{S}3}\left\{{\theta}_{C1}{\theta}_{K2}\left({\overline{S}}_{L L}-{\lambda}_{L1}{S}_{C L}^1\right)-{\lambda}_{L1}\left|\theta \right|\left({S}_{L C}^1+{S}_{C L}^1+ F\right)\right\}\right.\\ \qquad\qquad\left.- nA\left( L+ S\right){\theta}_{C1}{\lambda}_{L1}\left({\theta}_{K3}{E}_W{\theta}_{L2}-{\theta}_{\mathrm{S}3}{\theta}_{K2}\right)\right]\end{array} $$

$$ \begin{array}{lll} &-\left(\frac{\theta_{C1}{\theta}_{K2}{\theta}_{\mathrm{S}3}}{\Delta}\right)\left[{\lambda}_{L1} B\left(\frac{\theta_{L2}{E}_W{\theta}_{K3} S{W}_{\mathrm{S}}}{\theta_{K2}{\theta}_{\mathrm{S}3}}+ WL\right)\right.\\ & \left.-{\lambda}_{L2}^{*}\left\{{E}_W{S}_{L K}^2+\left(\frac{\lambda_{L2}+{\lambda}_{L U}}{\lambda_{L U}}\right)\left(1-{E}_W\right)\right\}\right] \end{array} $$

(8.A.63)

$$ \begin{array}{lll} &\left(\frac{{\widehat{L}}_U}{\widehat{K}}\right)>0\kern0.5em \mathrm{if}\kern0.5em \left[\frac{\lambda_{L1}}{\lambda_{L2}^{*}} B\left(\frac{\theta_{L2}{E}_W{\theta}_{K3} S{W}_{\mathrm{S}}}{\theta_{K2}{\theta}_{\mathrm{S}3}}+ WL\right)\right.\\ & \left.\ge \left\{{E}_W{S}_{L K}^2+\left(\frac{\lambda_{L2}+{\lambda}_{L U}}{\lambda_{L U}}\right)\left(1-{E}_W\right)\right\}\right] \end{array} $$

(8.A.64)

8.1.14 Appendix 8.14: (Model 3) Effects of FDI on the Welfare of the Child Labour-Supplying Families

Differentiation of Eq. (8.37) yields

$$ dV={\widehat{C}}_1+\beta {\widehat{C}}_2 $$

(8.A.65)

Substituting the expression for l _C from (8.41) into (8.38) and (8.39) and simplifying, one gets

$$ {C}_1=\left[\frac{W\left({W}_{\mathrm{S}}- W\right)+ n{W}_C{W}_{\mathrm{S}}}{\left(1+\beta \right)\left({W}_{\mathrm{S}}- W\right)}\right]\kern1em \mathrm{and} $$

(8.A.66)

$${C}_2=\beta \left[\frac{W\left({W}_{\mathrm{S}}- W\right)+ n{W}_C{W}_{\mathrm{S}}}{\left(1+\beta \right){W}_C}\right] $$

(8.A.67)

Differentiating (8.A.66) and (8.A.67) we, respectively, find

$$\begin{array}{lll} &{\widehat{C}}_1=\\ & \left[\frac{\left({W}_S\!-\!W\right)\left[ W{\left({W}_{\mathrm{S}}- W\right)}^2\widehat{W}\!+\! n{W}_C{W}_{\mathrm{S}}\left\{\left({W}_{\mathrm{S}}\!-\! W\right){\widehat{W}}_C- W\left({\widehat{W}}_{\mathrm{S}}-\widehat{W}\right)\right\}\right]}{\left[ W\left({W}_{\mathrm{S}}- W\right)+ n{W}_C{W}_{\mathrm{S}}\right]}\right.\kern0.5em \mathrm{and} \end{array} $$

(8.A.68)

$$ {\widehat{C}}_2=\left[\frac{W_C\left[ n{W}_{\mathrm{S}}{\widehat{W}}_{\mathrm{S}}+\left(\frac{W}{W_C}\right)\left\{{W}_{\mathrm{S}}{\widehat{W}}_{\mathrm{S}}+\widehat{W}\left({W}_{\mathrm{S}}-2 W\right)-\left({W}_{\mathrm{S}}- W\right){\widehat{W}}_C\right\}\right]}{\left[ W\left({W}_{\mathrm{S}}- W\right)+ n{W}_C{W}_{\mathrm{S}}\right]}\right. $$

(8.A.69)

Substitution of the expressions for ${\widehat{C}}_1$ and ${\widehat{C}}_2$ into (8.A.65) and simplification produce

$$ \begin{array}{lll} dV&=\left[\frac{1}{\left[ W\left({W}_{\mathrm{S}}- W\right)+ n{W}_C{W}_{\mathrm{S}}\right]}\right]\left[ W\left({W}_{\mathrm{S}}\!-\! W\right)\widehat{W}\left\{{\left({W}_{\mathrm{S}}\!-\! W\right)}^2\!+\! n{W}_C{W}_{\mathrm{S}}+\beta \right\}\right.\\ &\quad+\beta W\left({W}_{\mathrm{S}}{\widehat{W}}_{\mathrm{S}}- W\widehat{W}\right)\\ &\left. \quad +\left({W}_{\mathrm{S}}\!-\! W\right){\widehat{W}}_C\left\{ n{W}_C{W}_{\mathrm{S}}\left({W}_{\mathrm{S}}\!-\! W\right)\!-\!\beta W\right\}\!+\! n{W}_C{W}_{\mathrm{S}}{\widehat{W}}_{\mathrm{S}}\left\{\beta\!-\! W\left({W}_{\mathrm{S}}\!{-}\! W\right)\right\}\!\right]\end{array} $$

or

$${\begin{array}{lll} \left(\frac{d V}{d K}\right)=\left[\frac{1}{\left[ W\left({W}_{\mathrm{S}}- W\right)+ n{W}_C{W}_{\mathrm{S}}\right]}\right]\left[\left({W}_{\mathrm{S}}- W\right)\left(\frac{d W}{d K}\right)\left\{{\left({W}_{\mathrm{S}}- W\right)}^2+ n{W}_C{W}_{\mathrm{S}}+\beta \right\}\right.\\ \qquad\qquad \ \ \qquad\left(+\right)\kern10.5em \left(+\right)\kern6.5em \left(+\right)\\ \qquad\qquad+\beta W\left\{\left(\frac{d{W}_{\mathrm{S}}}{d K}\right)-\left(\frac{d W}{ d K}\right)\right\}\\ \qquad\qquad\quad\quad\qquad\qquad\left(+\right)\\ \qquad\qquad\left.+\left({W}_{\mathrm{S}}\!-\! W\right)\left(\!\frac{d{W}_C}{ d K}\!\right)\left\{n{W}_{\mathrm{S}}\left({W}_{\mathrm{S}}\!-\! W\right)\!-\!\beta \frac{W}{W_C}\right\}\!+\! n{W}_C\left(\frac{d{W}_{\mathrm{S}}}{d K}\right)\left\{\beta\! -\! W\left({W}_{\mathrm{S}}- W\right)\right\}\right]\\ \qquad\qquad\qquad\qquad\qquad\left(-\right)\kern14.5em \left(+\right)\end{array}}$$

(8.A.70)

From (8.A.70) it follows that

$$ \left(\frac{dV}{dK}\right)>0\kern1em \mathrm{if}\kern0.5em \beta W\ge n{W}_C{W}_{\mathrm{S}}\left({W}_{\mathrm{S}}- W\right) $$

(8.A.71)

As n ≥ 1, W _S > W and W _C  < W, from (8.A.71), it follows that

$$ \left(\frac{dV}{dK}\right)>0\kern1em \mathrm{if}\kern0.5em \beta \ge n{W}_{\mathrm{S}}\left({W}_{\mathrm{S}}- W\right) $$

(8.A.72)

However, from (8.A.70) it is easily seen that β ≥ nW _S(W _S − W) is only a sufficient condition for (dV/dK) > 0. One can find out several other sufficient conditions under which (dV/dK) > 0.