FDI and Relative Wage Inequality

Abstract

An important manifestation of inequality in the labour market is the inequality between the wages of skilled and unskilled labour. Empirical evidences suggest that the relative wage inequality has worsened in many of the developing countries after trade liberalization which is contrary to the predictions of the standard Heckscher–Ohlin (HO) model with Stolper–Samuelson theorem at its core. This calls for theoretical explanations. This chapter is devoted to offer such explanations in terms of a few simple general equilibrium models taking adequately into consideration some of the salient features of these countries like labour market imperfections, rural urban migration and presence and type of non-traded goods.

Appendices

Appendices

5.1.1 Appendix 5.1: Derivation of the Expression for Change in Relative Wage Inequality

Totally differentiating Eqs. (5.31), (5.32), (5.33.1), (5.34), (5.36), (5.37), (5.38), (5.39) and (5.40); keeping all parameters, except K, unchanged; and arranging in a matrix notation, one gets

$$\left(\begin{array}{llllll} {\theta}_{L1} & {\theta}_{N1} & 0 & 0 & 0 & 0 \\ {} {\theta}_{L2} & 0 & {\theta}_{K2} & 0 & -1 & 0 \\ {} {\theta}_{L3} & 0 & {\theta}_{K3} & 0 & {\theta}_{23} & 0 \\ {} 0 & 0 & {\theta}_{K4} & {\theta}_{S4} & 0 & 0 \\ {} -{A}_1 & {A}_2 & {A}_3 & 0 & 0 & {\lambda}_L \\ {} {A}_4 & 0 & -{A}_5 & {A}_6 & 0 & {\lambda}_K \end{array}\right)\;\left(\begin{array}{c} \widehat{W} \\ {} \widehat{R} \\ {} \widehat{r} \\ {} {\widehat{W}}_{\mathrm{S}} \\ {} {\widehat{P}}_2 \\ {} {\widehat{X}}_3 \end{array}\right)=\left(\begin{array}{c} 0 \\ {} 0 \\ {} 0 \\ {} 0 \\ {} 0 \\ {} \widehat{K} \end{array}\right) $$

(5.A.1)

where

$$ \left.\begin{array}{l}{A}_1=\left({\lambda}_{L1}{S}_{L N}^1+{\lambda}_{L2}{S}_{L K}^2+{\lambda}_{L3}{S}_{L K}^3+{\lambda}_{L1}{S}_{NL}^1\right)>0\hfill \\[4pt] {}{A}_2={\lambda}_{L1}\left({S}_{L N}^1+{S}_{NL}^1\right)>0;{A}_3=\left({\lambda}_{L2}{S}_{L K}^2+{\lambda}_{L3}{S}_{L K}^3\right)>0\hfill \\[4pt] {}{A}_4=\left({\lambda}_{K2}{S}_{K L}^2+{\lambda}_{K3}{S}_{K L}^3\right)>0\hfill \\[4pt] {}{A}_5=\left({\lambda}_{K2}{S}_{K L}^2+{\lambda}_{K3}{S}_{K L}^3+{\lambda}_{K4}{S}_{\it KS}^4+{\lambda}_{K4}{S}_{S K}^4\right)>0\hfill \\[4pt] {}{A}_6={\lambda}_{K4}\left({S}_{K S}^4+{S}_{S K}^4\right)>0\hfill \\[4pt] {}{\lambda}_L=\left({\lambda}_{L2}+{\lambda}_{L3}\right)>0;{\lambda}_K=\left({\lambda}_{K2}+{\lambda}_{K3}\right)>0\hfill \end{array}\right\} $$

(5.A.2)

The determinant to the coefficient matrix in (5.A.1) is given by

$$ \begin{aligned}\Delta & ={\theta}_{L1}{\theta}_{S4}{A}_2{\lambda}_K\left({\theta}_{23}{\theta}_{K2}+{\theta}_{K3}\right)+{\theta}_{N1}\left[{\theta}_{S4}\left({\theta}_{23}{\theta}_{K2}+{\theta}_{K3}\right)\left({A}_1{\lambda}_K+{A}_4{\lambda}_L\right) \right. \\ {}& \quad \left. +\left({\theta}_{23}{\theta}_{L2}+{\theta}_{L3}\right)\left\{{\theta}_{K4}{A}_6{\lambda}_L+{\theta}_{S4}\left({A}_3{\lambda}_K+{A}_5{\lambda}_L\right)\right\}\right]>0\end{aligned} $$

(5.A.3)

As commodity 2 is internationally non-traded, its market must clear domestically through adjustments in its price, P ₂. The stability condition in the market for commodity 2 requires that

(d(X ^D₂  − X ₂)/dP ₂) < 0. This implies around equilibrium, initially, X ^D₂  = X ₂. Thus, $\left(\left({\widehat{X}}_2^D/{\widehat{P}}_2\right)-\left({\widehat{X}}_2/{\widehat{P}}_2\right)\right)<0$. This requires that Δ > 0. In this case, of course, the stability condition is automatically satisfied. This is because from (5.A.2) and (5.A.3), it follows that ∆ is unconditionally positive.

Solving (5.A.1) by Cramer’s rule, the following expressions are obtained:

$$ \widehat{W}=\left({\theta}_{N1}{\theta}_{S4}{\lambda}_L{\tilde{\theta}}_{K3}\right)\widehat{K}/\Delta $$

(5.A.4)

$$ {\widehat{W}}_{\mathrm{S}}=\left({\theta}_{N1}{\theta}_{K4}{\lambda}_L{\theta}_{L3}\right)\widehat{K}/\Delta $$

(5.A.5)

$$ \widehat{r}=-\left({\theta}_{N1}{\theta}_{S4}{\lambda}_L{\theta}_{L3}\right)\widehat{K}/\Delta $$

(5.A.6)

$$ {\widehat{X}}_3={\theta}_{S4}\left[{\tilde{\theta}}_{K3}\left({\theta}_{L1}{A}_2+{\theta}_{N1}{A}_1\right)+\left({\theta}_{N1}{\tilde{\theta}}_{L3}\right){A}_3\right]\widehat{K}/\Delta \kern0.5em \mathrm{and} $$

(5.A.7)

$$ {\widehat{P}}_2=\left({\theta}_{N1}{\theta}_{S4}{\lambda}_L\right)\left({\theta}_{L2}{\theta}_{K3}-{\theta}_{L3}{\theta}_{K2}\right)\widehat{K}/\Delta $$

(5.A.8)

where ${\tilde{\theta}}_{L3}=\left({\theta}_{L2}{\theta}_{23}+{\theta}_{L3}\right)$ and ${\tilde{\theta}}_{K3}=\left({\theta}_{K2}{\theta}_{23}+{\theta}_{K3}\right)$. Differentiating (5.41), using (5.A.4), (5.A.6) and (5.A.7), and simplifying, one can derive the following expression:

$$\begin{aligned}{\widehat{W}}_A &=\left(\frac{W\widehat{W}}{W_A}\right)\left[1{+}\left(\alpha -1\right){\lambda}_{L3}\right]{+}\frac{W}{W_A}\left(\alpha -1\right){\lambda}_{L3}{\theta}_{S4}\frac{\widehat{K}}{\Delta}\left[{\lambda}_{L1}{\tilde{\theta}}_{K3}\right({S}_{L N}^1+{S}_{N L}^1\Big)\\ & \quad +{\theta}_{N1}\left({S}_{L K}^2-{S}_{L K}^3\right){\lambda}_{L2}\Big]\end{aligned}$$

(5.A.9)

Using (5.A.2) and (5.A.3) from (5.A.4), (5.A.5), (5.A.6), (5.A.7), (5.A.8) and (5.A.9), the following results are obtained.

When $\widehat{K}>0$, (1) $\widehat{W}>0$; (2) ${\widehat{W}}_{\mathrm{S}}>0$; (3) $\widehat{r}<0$; (4) ${\widehat{X}}_3>0$; (5) ${\widehat{W}}_{\mathrm{A}}>0$ if S ² _LK  ≥ S ³ _LK ; and (6) ${\widehat{P}}_2>0$ (as θ _L2 θ _K3 > θ _L3 θ _K2, i.e. sector 3 is more capital-intensive relative to sector 2 with respect to unskilled labour in value sense).

Subtracting (5.A.9) from (5.A.5), using (5.A.4) and after a little manipulation, one obtains the following expression:

$$ \begin{aligned} {}& \left({\widehat{W}}_{\mathrm{S}}-{\widehat{W}}_{\mathrm{A}}\right)=\left(\frac{\widehat{K}}{\Delta}\right)\left[\left({\theta}_{N1}{\lambda}_L\right)\left({\theta}_{K4}{\tilde{\theta}}_{L3}-{\theta}_{S4}{\tilde{\theta}}_{K3}\right) \right. \\ {}& \quad \left. \left. -\left\{\frac{W{\lambda}_{L3}{\theta}_{S4}}{W_A}\left(\alpha -1\right)\right\}{\lambda}_{L1}{\tilde{\theta}}_{K3}\left({S}_{L N}^1+{S}_{N L}^1\right)+{\theta}_{N1}\left({S}_{L K}^2-{S}_{L K}^3\right){\lambda}_{L2}\right\}\right] \end{aligned} $$

(5.42)

5.1.2 Appendix 5.2: Derivations of Certain Useful Expressions

Total differentials of (5.44), (5.45), (5.46) and (5.47) holding the parameters unaffected yield the following expressions, respectively:

$$ {\theta}_{L1}\widehat{W}+{\theta}_{N1}\widehat{R}=0 $$

(5.A.10)

$$ {\theta}_{L2}\widehat{W}+{\theta}_{N2}\widehat{R}-{\widehat{P}}_2=0 $$

(5.A.11)

$$ {\theta}_{L3}\widehat{W}+{\theta}_{K3}\widehat{r}=0 $$

(5.A.12)

$$ {\theta}_{S4}{\widehat{W}}_S+{\theta}_{K4}\widehat{r}=0 $$

(5.A.13)

Using (5.54), Eq. (5.53) may be rewritten as follows:

$$ {a}_{K3}{X}_3+\left(\frac{a_{K4} S}{a_{S4}}\right)= K $$

(5.A.14)

Differentiating totally Eqs. (5.51), (5.52) and (5.A.14) and allowing only parameter K to change, one gets, respectively,

$$ {B}_1\widehat{W}-{B}_2\widehat{R}+{\lambda}_{N1}{\widehat{X}}_1+{\lambda}_{N2}{\widehat{X}}_2=0 $$

(5.A.15)

$$ -{B}_3\widehat{W}+{B}_4\widehat{R}+{B}_5\widehat{r}+{\lambda}_{L1}{\widehat{X}}_1+{\lambda}_{L2}{\widehat{X}}_2+{\lambda}_{L3}{\widehat{X}}_3=0 $$

(5.A.16)

$$ {B}_6\widehat{W}-{B}_7\widehat{r}+{B}_8{\widehat{W}}_S+{\lambda}_{K3}{\widehat{X}}_3=\widehat{K} $$

(5.A.17)

Also differentiating (5.49) and (5.50), one may obtain

$$ {B}_9\widehat{W}+{B}_{10}\widehat{R}+{B}_{11}\widehat{r}+{B}_{12}\widehat{r}+{B}_{13}{\widehat{X}}_3+{E}_P{\widehat{P}}_2-{\widehat{X}}_2+{B}_{13}{\widehat{X}}_3=0 $$

(5.A.18)

where

$$ \left.\begin{array}{l}{B}_1={B}_2=\left({\lambda}_{N1}{S}_{N L}^1+{\lambda}_{N2}{S}_{N L}^2\right)>0\hfill \\[4pt] {}{B}_3=\left({\lambda}_{L1}{S}_{L N}^1+{\lambda}_{L2}{S}_{L N}^2+{\lambda}_{L3}{S}_{L K}^3\right)>0\hfill \\[4pt] {}{B}_4=\left({\lambda}_{L1}{S}_{L N}^1+{\lambda}_{L2}{S}_{L N}^2\right)>0;{B}_5=\left({\lambda}_{L3}{S}_{L K}^3\right)>0;{B}_6=\left({\lambda}_{K3}{S}_{K L}^3\right)>0\hfill \\[4pt] {}{B}_7=\left({\lambda}_{K3}{S}_{K L}^3+{\lambda}_{K4}\left({S}_{K S}^4+{S}_{S K}^4\right)\right)>0;{B}_8={\lambda}_{K4}\left({S}_{K S}^4+{S}_{S K}^4\right)\Big)>0\hfill \\[4pt] {}{B}_9=\left({E}_Y W/ Y\right)\left[ L+\right(\alpha -1\left){a}_{L3}{X}_3\right(1-{S}_{L K}^3\left)\right];{B}_{10}=\Big({E}_Y RN/ Y\Big)>0\hfill \\[4pt] {}{B}_{11}=\left({E}_Y/ Y\right)\left[ r{K}_D+\big(\alpha {-}1\big) W{a}_{L3}{X}_3{S}_{L K}^3\right]{>}0;{B}_{12}{=}\Big({E}_Y{W}_S S/ Y\Big){>}0\;\mathrm{and}\hfill \\[4pt] {}{B}_{13}=\left[\left({E}_Y/ Y\right)\right(\alpha -1\left) W{a}_{L3}{X}_3\right]>0\hfill \end{array}\right\} $$

(5.A.19)

Arranging (5.A.10), (5.A.11), (5.A.12), (5.A.13) and (5.A.15), (5.A.16), (5.A.17) and (5.A.18) in a matrix notation, we get the following:

$$\left(\begin{array}{llllllll} {\theta}_{L1} & {\theta}_{N1} & 0 & 0 & 0 & 0 & 0 & 0 \\ {} {\theta}_{L2} & {\theta}_{N2} & 0 & 0 & -1 & 0 & 0 & 0 \\ {} {\theta}_{L3} & 0 & {\theta}_{K3} & 0 & 0 & 0 & 0 & 0 \\ {} 0 & 0 & {\theta}_{K4} & {\theta}_{S4} & 0 & 0 & 0 & 0 \\ {} {B}_1 & -{B}_2 & 0 & 0 & 0 & {\lambda}_{N1} & {\lambda}_{N2} & 0 \\ {} -{B}_3 & {B}_4 & {B}_5 & 0 & 0 & {\lambda}_{L1} & {\lambda}_{L2} & {\lambda}_{L3} \\ {} {B}_6 & 0 & -{B}_7 & {B}_8 & 0 & 0 & 0 & {\lambda}_{K3} \\ {} {B}_9 & {B}_{10} & {B}_{11} & {B}_{12} & {E}_P & 0 & -1 & {B}_{13} \end{array}\right)\;\left(\begin{array}{c} \widehat{W} \\ {} \widehat{R} \\ {} \widehat{r} \\ {} {\widehat{W}}_S \\ {} {\widehat{P}}_2 \\ {} {\widehat{X}}_1 \\ {} {\widehat{X}}_2 \\ {} {\widehat{X}}_3 \end{array}\right)=\left(\begin{array}{c} 0 \\ {} 0 \\ {} 0 \\ {} 0 \\ {} 0 \\ {} 0 \\ {} \widehat{K} \\ {} 0 \end{array}\right)$$

(5.A.20)

where

$${{\begin{aligned}\Omega &= -\left[\left(\left|\theta \right|\left|\lambda \right|{E}_P\right)-\right\{\left({B}_9{\theta}_{N1}{\theta}_{K3}{\theta}_{S4}\left|\lambda \right|\right)-\left({B}_{10}{\theta}_{L1}{\theta}_{K3}{\theta}_{S4}\left|\lambda \right|\right)-\left({B}_{11}{\theta}_{N1}{\theta}_{L3}{\theta}_{S4}\left|\lambda \right|\right)\\ {}& \quad +\left({B}_{12}{\theta}_{N1}{\theta}_{L3}{\theta}_{K4}\left|\lambda \right|\right)\\ {}& \quad -{B}_{13}\left({\lambda}_{L2}{\lambda}_{N1}{-}{\lambda}_{N2}{\lambda}_{L1}\right){\theta}_{N1}\left({\theta}_{K3}{\theta}_{S4}{B}_6{+}{\theta}_{L3}{\theta}_{K4}{B}_8{+}{\theta}_{L3}{\theta}_{S4}{B}_7\right)\\ {}& \quad -\left({\theta}_{N1}{\theta}_{K3}{\theta}_{S4}\right)\left({\lambda}_{N1}{\lambda}_{K3}{B}_3+{\lambda}_{N1}{\lambda}_{L3}{B}_6+{\lambda}_{L1}{\lambda}_{K3}{B}_1\right)\\ {}& \quad -\left({\lambda}_{K3}{\theta}_{K3}{\theta}_{S4}{\theta}_{L1}\right)\left({B}_2{\lambda}_{L1}+{B}_4{\lambda}_{N1}\right)-\left({\lambda}_{N1}{\theta}_{N1}{\theta}_{L3}{\theta}_{S4}\right)\left({B}_5{\lambda}_{K3}+{B}_7{\lambda}_{L3}\right)\\ {}& \quad -\left({\lambda}_{N1}{\lambda}_{L3}{\theta}_{N1}{\theta}_{L3}{\theta}_{K4}{B}_8\right)\left\}\right]\end{aligned}}}$$

(5.A.21)

$$ \left|\theta \right|={\theta}_{K3}{\theta}_{S4}\left({\theta}_{L1}{\theta}_{N2}-{\theta}_{N1}{\theta}_{L2}\right)\kern1em \textit{and} $$

(5.A.22)

$$ \left|\lambda \right|={\lambda}_{K3}\left({\lambda}_{N1}{\lambda}_{L2}-{\lambda}_{N2}{\lambda}_{L1}\right) $$

(5.A.23)

So, we always have

$$ \left|\theta \right|\left|\lambda \right|<0 $$

(5.A.24)

Using the stability condition in the market for commodity 2 (see Appendix 5.5), it can be shown that Ω < 0.

5.1.3 Appendix 5.3: Effects on Factor and Non-traded Commodity Prices

Solving (5.A.20) by Cramer’s rule, the following expressions are obtained:

$$ \widehat{W}=-\left({\theta}_{N1}{\theta}_{K3}{\theta}_{S4}\left|{\lambda}^{*}\right|\widehat{K}/\varOmega \right) $$

(5.A.25)

$$ {\widehat{W}}_{\mathrm{S}}=-\left({\theta}_{N1}{\theta}_{L3}{\theta}_{K4}\left|{\lambda}^{*}\right|\widehat{K}/\varOmega \right) $$

(5.A.26)

$$ \widehat{r}=\left({\theta}_{N1}{\theta}_{L3}{\theta}_{S4}\left|{\lambda}^{*}\right|\widehat{K}/\varOmega \right) $$

(5.A.27)

$$ {\widehat{P}}_2=\left({\theta}_{K3}{\theta}_{S4}\left|{\lambda}^{*}\right|\widehat{K}/\varOmega \right)\left({\theta}_{L1}{\theta}_{N2}-{\theta}_{N1}{\theta}_{L2}\right) $$

(5.A.28)

where

$$ \left|{\lambda}^{*}\right|=\left[\left({\lambda}_{L2}{\lambda}_{N1}-{\lambda}_{N2}{\lambda}_{L1}\right){B}_{13}+{\lambda}_{N1}{\lambda}_{L3}\right] $$

(5.A.29)

${\widehat{X}}_3$ also can be solved in the same manner. The final expression for ${\widehat{X}}_3$ has been derived in Appendix 5.6. Using the stability condition in the market for commodity 2, it can be shown that ${\widehat{X}}_3>0$ when $\widehat{K}>0$.

From (5.A.25), (5.A.26), (5.A.27) and (5.A.28), the following results can be found:

• 1. If sector 1 is more land-capital-intensive vis-à-vis sector 2 with respect to unskilled labour (i.e. |λ|, |λ * | > 0; |θ| < 0) when $\widehat{K}>0$, (i) $\widehat{W}>0$, (ii) ${\widehat{W}}_S>0$, (iii) $\widehat{r}<0$ and (iv) ${\widehat{P}}_2>0$.

• 2. If sector 1 is more labour-intensive (but not sufficiently labour-intensive) than sector 2 (i.e. |λ| < 0;  |λ * |,  |θ| > 0) when $\widehat{K}>0$, (i) $\widehat{W}>0$, (ii) ${\widehat{W}}_S>0$, (iii) $\widehat{r}<0$ and (iv) ${\widehat{P}}_2<0$.

• 3. If sector 1 is sufficiently labour-intensive (i.e. |λ|, |λ * | < 0;  |θ| > 0) when $\widehat{K}>0$, (i) $\widehat{W}<0$, (ii) ${\widehat{W}}_S<0$, (iii) $\widehat{r}>0$ and (iv) ${\widehat{P}}_2>0$.

5.1.4 Appendix 5.4: Derivation of Expression for Relative Wage Inequality

Differentiating (5.41), one gets:

$$ \begin{aligned} {\widehat{W}}_{\mathrm{A}}&=\left(\frac{W\widehat{W}}{W_{\mathrm{A}}}\right)\left[1+\left(\alpha -1\right){\lambda}_{L3}\left(1-{S}_{L K}^3\right)\right]+ W\left(\alpha -1\right)\left(\frac{\lambda_{L3}{S}_{L K}^3}{W_A}\right)\widehat{r}\\ & \quad + W\left(\alpha -1\right)\left(\frac{\lambda_{L3}}{W_A}\right){\widehat{X}}_3 \end{aligned}$$

Using (5.A.25) and (5.A.27), the above expression may be simplified to

$$\begin{aligned}{\widehat{W}}_{\mathrm{A}}&=\left(\frac{W\widehat{K}}{\varOmega {W}_A}\right)\left[\left({\theta}_{N1}{\theta}_{S4}\left|\lambda *\right|\right)\left\{\left(\alpha -1\right){\lambda}_{L3}\left({S}_{L K}^3-{\theta}_{K3}\right)-{\theta}_{K3}\right\} \right. \\ {}& \quad +\left(\frac{W}{W_{\mathrm{A}}}\left(\alpha -1\right){\lambda}_{L3}\right){\widehat{X}}_3\end{aligned}$$

(5.A.30)

From (5.A.30), it is evident that:

• 1. When $\widehat{K}>0$, ${\widehat{W}}_A>0$ if (i) (|λ| > 0 and λ _L3 > 0 ⇒ |λ * | > 0)/(|λ| < 0 and λ _L3 > 0 such that |λ * | > 0) and (ii) θ _K3 ≥ S ³ _LK . Note that when sector 2 (sector 1) is labour-intensive (land-capital-intensive) (i.e. |λ| > 0) and the proportion of unskilled labour employed in sector 3 (i.e. λ _L3) is not sufficiently small, this allocative share rises following an inflow of foreign capital under the sufficient condition that θ _K3 ≥ S ³ _LK . The average unskilled wage, W _A, also rises in this situation.

• 2. When $\widehat{K}>0$, ${\widehat{W}}_{\mathrm{A}}<0$ if (i) |λ| < 0 and λ _L3 ≅ 0 so that |λ * | < 0.

Subtracting (5.A.30) from (5.A.26), using (5.A.25) and (5.A.27) and simplifying, we get

$$ \begin{aligned}\left({\widehat{W}}_{\mathrm{S}}-{\widehat{W}}_{\mathrm{A}}\right)&=-\left(\frac{\theta_{N1}\widehat{K}}{\Omega}\right)\left[\left({\lambda}_{L2}{\lambda}_{N1}-{\lambda}_{N2}{\lambda}_{L1}\right){B}_{13}+{\lambda}_{N1}{\lambda}_{L3}\right]\left[\left({\theta}_{K4}-{\theta}_{K3}\right) \right. \\ {}& \quad \left. -\left(\alpha -1\right){\lambda}_{L3}\left({\theta}_{S4}{\theta}_{K3}-{S}_{L K}^3\right)\right]-\frac{W\left(\alpha -1\right){\lambda}_{L3}{\widehat{X}}_3}{W_{\mathrm{A}}}\end{aligned}$$

(5.A.31)

Using (5.A.29), equation (5.A.31) can be rewritten as follows:

$$ \begin{aligned} \left({\widehat{W}}_{\mathrm{S}}-{\widehat{W}}_{\mathrm{A}}\right)&=-\left(\frac{\theta_{N1}\widehat{K}}{\Omega}\right)\left|\lambda *\right|\left[\left({\theta}_{K4}-{\theta}_{K3}\right)-\left(\alpha -1\right){\lambda}_{L3}\left({\theta}_{S4}{\theta}_{K3}-{S}_{L K}^3\right)\right]\\ &\quad\,-\frac{W\left(\alpha -1\right){\lambda}_{L3}{\widehat{X}}_3}{W_{\mathrm{A}}}\end{aligned} $$

(5.55)

5.1.5 Appendix 5.5: Stability Condition of the Market for Commodity 2

As commodity 2 is internationally non-traded, its market must clear domestically through adjustments in its price, P ₂. The stability condition of the market for commodity 2 requires that (d(D ₂ − X ₂)/dP ₂) < 0. This implies around equilibrium, initially, D ₂ = X ₂. Thus,

$$ \left(\left({\widehat{D}}_2/{\widehat{P}}_2\right)-\left({\widehat{X}}_2/{\widehat{P}}_2\right)\right)<0 $$

(5.A.32)

Totally differentiating Eqs. (5.44), (5.45), (5.46) and (5.47) and solving, one can find out the following expressions:

$$ \left(\widehat{W}/{\widehat{P}}_2\right)=-\left({\theta}_{N1}{\theta}_{K3}{\theta}_{S4}/\left|\theta \right|\right) $$

(5.A.33)

$$ \left(\widehat{R}/{\widehat{P}}_2\right)=\left({\theta}_{L1}{\theta}_{K3}{\theta}_{S4}/\left|\theta \right|\right) $$

(5.A.34)

$$ \left(\widehat{r}/{\widehat{P}}_2\right)=\left({\theta}_{N1}{\theta}_{L3}{\theta}_{S4}/\left|\theta \right|\right)\kern0.5em \textit{and} $$

(5.A.35)

$$ \left({\widehat{W}}_{\mathrm{S}}/{\widehat{P}}_2\right)=-\left({\theta}_{N1}{\theta}_{L3}{\theta}_{K4}/\left|\theta \right|\right) $$

(5.A.36)

Then differentiating Eqs. (5.51), (5.52), (5.53) and (5.54), using (5.A.33), (5.A.34), (5.A.35) and (5.A.36), putting $\widehat{K}=0$ and solving by Cramer’s rule, the following expressions may be obtained:

$$ \begin{aligned}{\widehat{X}}_2&=\left(\frac{-{\widehat{P}}_2}{\left|\theta \right|\left|\lambda \right|}\right)\left[\left({\lambda}_{N1}{\lambda}_{K3}{B}_3+{\lambda}_{N1}{\lambda}_{L3}{B}_6+{\lambda}_{L1}{\lambda}_{K3}{B}_1\right)\left({\theta}_{N1}{\theta}_{K3}{\theta}_{S4}\right) \right. \\ & \quad + \left({\lambda}_{N1}{B}_4+{\lambda}_{L1}{B}_2\right){\lambda}_{K3}\left({\theta}_{K3}{\theta}_{S4}{\theta}_{L1}\right) +{\lambda}_{N1}\left({\lambda}_{K3}{B}_5+{\lambda}_{L3}{B}_7\right)\left({\theta}_{N1}{\theta}_{L3}{\theta}_{S4}\right)\\ & \quad \left. +{\lambda}_{N1}{\lambda}_{L3}{B}_8{\theta}_{N1}{\theta}_{L3}{\theta}_{K4}\right]\end{aligned} $$

(5.A.37)

$$ {\widehat{X}}_3=-\left(\frac{{\widehat{P}}_2}{\left|\lambda \right|\left|\theta \right|}\right)\left[\left({\lambda}_{N2}{\lambda}_{L1}-{\lambda}_{N1}{\lambda}_{L2}\right)\left({\theta}_{K3}{B}_6+{\theta}_{L3}{B}_7+{\theta}_{L3}{B}_8\right){\theta}_{N1}{\theta}_{S4}\right] $$

(5.A.38)

Differentiating Eqs. (5.48) and (5.49) and considering $\widehat{K}=0$, one can derive

$$ {\widehat{D}}_2={E}_P{\widehat{P}}_2+{B}_9\widehat{W}+{B}_{10}\widehat{R}+{B}_{11}\widehat{r}+{B}_{12}{\widehat{W}}_S+{B}_{13}{\widehat{X}}_3 $$

(5.A.39)

Using (5.A.33), (5.A.34), (5.A.35), (5.A.36), (5.A.38), equation (5.A.39) may be rewritten as follows:

$${{\begin{aligned} {\widehat{D}}_2&{=}\left({\widehat{P}}_2\right)\!\left[\!{E}_P{-}\!\left(\frac{1}{\left|\theta \right|}\right)\left\{{B}_9{\theta}_{N1}{\theta}_{K3}{\theta}_{S4}{-}{B}_{10}{\theta}_{K3}{\theta}_{S4}{\theta}_{L1}{-}{B}_{11}{\theta}_{N1}{\theta}_{L3}{\theta}_{S4}{+}{B}_{12}{\theta}_{N1}{\theta}_{L3}{\theta}_{K4} \right. \right. \\ {}& \quad \left. \left. -\left({B}_{13}/\left|\lambda \right|\right)\left({\lambda}_{N1}{\lambda}_{L2}-{\lambda}_{N2}{\lambda}_{L1}\right){\theta}_{N1}\left({\theta}_{K3}{\theta}_{S4}{B}_6+{\theta}_{L3}{\theta}_{S4}{B}_7+{\theta}_{L3}{\theta}_{S4}{B}_8\right)\right\}\vphantom{\left(\frac{1}{\left|\theta \right|}\right)}\right]\end{aligned} }}$$

(5.A.40)

Substituting the expressions for $\left({\widehat{D}}_2/{\widehat{P}}_2\right)$ and $\left({\widehat{X}}_2/{\widehat{P}}_2\right)$ from (5.A.40) and (5.A.37) in (5.A.32) and simplifying, one obtains

$${{\begin{aligned}{}&\left[{E}_P{-}\left(\frac{1}{\left|\lambda \right|\left|\theta \right|}\right)\right\{\left|\lambda \right|\left({B}_9{\theta}_{N1}{\theta}_{K3}{\theta}_{S4}{-}{B}_{10}{\theta}_{K3}{\theta}_{S4}{\theta}_{L1}{-}{B}_{11}{\theta}_{N1}{\theta}_{L3}{\theta}_{S4}{+}{B}_{12}{\theta}_{N1}{\theta}_{L3}{\theta}_{K4}\right)\\ & \quad-\left({B}_{13}{\theta}_{N1}\right)\left({\lambda}_{N1}{\lambda}_{L2}-{\lambda}_{N2}{\lambda}_{L1}\right)\left({\theta}_{K3}{\theta}_{S4}{B}_6+{\theta}_{L3}{\theta}_{S4}{B}_7+{\theta}_{L3}{\theta}_{S4}{B}_8\right)\\ & \quad-\left({\theta}_{N1}{\theta}_{K3}{\theta}_{S4}\right)\left({\lambda}_{N1}{\lambda}_{K3}{B}_3{+}{\lambda}_{N1}{\lambda}_{L3}{B}_6{+}{\lambda}_{L1}{\lambda}_{K3}{B}_1\right){-}\left({\lambda}_{K3}{\theta}_{K3}{\theta}_{S4}{\theta}_{L1}\right)\left({\lambda}_{N1}{B}_4{+}{\lambda}_{L1}{B}_2\right)\\ & \quad-\left({\lambda}_{N1}{\theta}_{N1}{\theta}_{L3}{\theta}_{S4}\right)\left({\lambda}_{K3}{B}_5+{\lambda}_{L3}{B}_7\right)-\left({\lambda}_{N1}{\lambda}_{L3}{B}_8{\theta}_{N1}{\theta}_{L3}{\theta}_{K4}\right)\left\}\right]<0\end{aligned} }}$$

(5.A.41)

Thus, the stability condition in the market for commodity 2 is given by (5.A.41).

Using (5.A.24) and (5.A.41) from (5.A.21), it now trivially follows that

$$ \Omega <0 $$

(5.A.42)

5.1.6 Appendix 5.6: Effect on X ₃

Solving (5.A.20), we can find the following expression:

$${{\begin{aligned}{\widehat{X}}_3&\,{=}\, -\left(\frac{\widehat{K}}{\Omega {\lambda}_{K3}}\right)\left[\left|\theta \right|\left|\lambda \right|{E}_P{-}\left\{{\theta}_{K3}{\theta}_{S4}\left({\theta}_{N1}{B}_9{-}{\theta}_{L1}{B}_{10}\right){+}{\theta}_{N1}{\theta}_{L3}\left({\theta}_{K4}{B}_{12}{-}{\theta}_{S4}{B}_{11}\right)\right\}\left|\lambda \right| \right. \\ {}& \,\quad \left.+{\theta}_{S4}{\lambda}_{K3}\left\{{\theta}_{N1}{\theta}_{K3}\left({\lambda}_{N1}{B}_3+{\lambda}_{L1}{B}_1\right)+{\theta}_{K3}{\theta}_{L1}\left({B}_2{\lambda}_{L1}+{B}_4{\lambda}_{N1}\right)+{\lambda}_{N1}{\theta}_{N1}{\theta}_{L3}{B}_5\right\}\right]\end{aligned} }}$$

(5.A.43)

It may be noted that

$$ \left.\begin{array}{l}{W}_S S={\theta}_{P4}{X}_4; W{a}_{L3}=\left({P}_3{\theta}_{L3}/\alpha \right);\hfill \\[4pt] r{K}_D=\left({\theta}_{K3}{P}_3{X}_3+{\theta}_{K4}{P}_4{X}_4- r{K}_F\right);\hfill \\[4pt] \left({\theta}_{N1} WL-{\theta}_{L1} RN\right)=\left[{P}_2{X}_2\right({\theta}_{N1}{\theta}_{L2}-{\theta}_{L1}{\theta}_{N2}\left)+{\theta}_{N1}\theta {}_{L3} P_3{X}_3\right];\hfill \\[4pt] \left({\theta}_{K4}{W}_S S-{\theta}_{S4} r{K}_D\right)={\theta}_{S4}\left(r{K}_F-{\theta}_{K3}{P}_3{X}_3\right)\hfill \end{array}\right\} $$

(5.A.44)

Inserting the values of B _i s from (5.A.19), using (5.A.44) and simplifying, it is easily found that

$$ \begin{aligned} {}& \left\{{\theta}_{K3}{\theta}_{S4}\left({\theta}_{N1}{B}_9-{\theta}_{L1}{B}_{10}\right)+{\theta}_{N1}{\theta}_{L3}\right({\theta}_{K4}{B}_{12}-{\theta}_{S4}{B}_{11}\left)\right\}\\ {}& \quad =\left(\frac{E_Y}{Y}\right)\left\{{\theta}_{N1}{\theta}_{S4} W\right(\alpha -1\left){a}_{L3}{X}_3\right({\theta}_{K3}-{S}_{L K}^3\left)+{\theta}_{N1}{\theta}_{S4}{\theta}_{L3} r{K}_F-\left|\theta \right|{P}_2{X}_2\right\}\end{aligned} $$

(5.A.45)

Using (5.A.45) and simplifying, from (5.A.43), the following expression can be easily derived:

$$ \begin{aligned}{\widehat{X}}_3&=-\left(\frac{\widehat{K}}{\lambda_{K3}\Omega}\right)\left[\left|\theta \right|\left|\lambda \right|\left({E}_P+\frac{E_Y{P}_2{X}_2}{Y}\right) \right. \\ {}& \,\quad -\left|\lambda \right|\left(\frac{E_Y{\theta}_{N1}{\theta}_{S4}{\theta}_{L3}}{Y}\right)\left\{\left(\frac{\alpha -1}{\alpha}\right){P}_3{X}_3\left({\theta}_{K3}-{S}_{L K}^3\right)+ r{K}_F\right\}\\ {}& \,\quad +{\theta}_{S4}{\lambda}_{K3}\left\{{\theta}_{N1}{\theta}_{K3}\left({\lambda}_{N1}{B}_3+{\lambda}_{L1}{B}_1\right)+{\theta}_{L1}{\theta}_{K3}\left({\lambda}_{L1}{B}_2+{\lambda}_{N1}{B}_4\right) \right. \\ {}& \quad \left. \left. +{\lambda}_{N1}{\theta}_{N1}{\theta}_{L3}{B}_5\right\}\right]\end{aligned} $$

(5.A.46)

Using (5.A.19) and (5.A.41) and comparing terms, we can check that the algebraic sign of the square-bracketed term in (5.A.46) is positive. As Ω < 0, from (5.A.46), it now follows that ${\widehat{X}}_3>0$ when $\widehat{K}>0$.

5.1.7 Appendix 5.7: Effect on X ₄

Differentiating Eq. (5.54), one gets

$$ {\widehat{X}}_4=-{\widehat{a}}_{S4}={S}_{S K}^4\left({\widehat{W}}_S-\widehat{r}\right) $$

(5.A.47)

Inserting the values of ${\widehat{W}}_S$ and $\widehat{r}$ from (5.A.26) and (5.A.27) in (5.A.47) and simplifying, the following expression is finally obtained:

$$ {\widehat{X}}_4=-\left(\frac{S_{S K}^4{\theta}_{N1}{\theta}_{L3}\left|\lambda *\right|\widehat{K}}{\Omega}\right) $$

(5.A.48)

From (5.A.48), the following results can be stated:

• 1. If sector 1 is land-capital-intensive or unskilled labour-intensive (but not sufficiently enough) (such that |λ* | > 0), ${\widehat{X}}_4>0$ when $\widehat{K}>0$.

• 2. If sector 1 is sufficiently unskilled labour-intensive (such that |λ* | < 0), ${\widehat{X}}_4<0$ when $\widehat{K}>0$.