FDI in Healthcare

Abstract

The healthcare sector has historically been publicly funded in developing countries due to commitments of governments to provide universal access to health services at low cost. A major part of the empirical literature on FDI in healthcare is analytical in nature, with an apparent polarization of views for and against FDI. However, empirical evidence on the likely impact of FDI in health service is virtually non-existent. In this chapter, we formally investigate the effects of FDI in the healthcare sector on the welfare and human capital formation of a developing economy in terms of a three-sector, full-employment general equilibrium model with a non-traded sector that produces healthcare services the consumption of which directly raises the efficiency of the workers. The results of the analysis indicate that although FDI of the type which is specific to the healthcare sector raises the human capital endowment of the economy, it may adversely affect social welfare. On the contrary, FDI of the other type which is used in all the sectors of the economy is likely to be welfare improving under reasonable conditions. These results question the desirability of allowing entry of foreign capital in the healthcare sector that generates externalities.

Appendices

Appendices

9.1.1 Appendix 9.1: Derivations of Certain Useful Expressions

Totally differentiating Eqs. (9.3) and (9.7), one gets, respectively,

$$\left.\begin{array}{l}\widehat{R}=\left(\frac{{\widehat{P}}_G}{\theta_{N G}}\right)\hfill \\ {}{\widehat{X}}_G=\widehat{N}\hfill \end{array}\right\}$$

(9.A.1)

Differentiating Eqs. (9.5) and (9.6) and using (9.A.1) yield

$${\lambda}_{L1}{\widehat{X}}_1+{\lambda}_{L2}{\widehat{X}}_2=-\left({\lambda}_{L G}-{\varepsilon}_h\right)\widehat{N}$$

(9.A.2)

$${\lambda}_{K1}{\widehat{X}}_1+{\lambda}_{K2}{\widehat{X}}_2=\widehat{K}-{\lambda}_{LG}\widehat{N}$$

(9.A.3)

where ε _h  = (dh/dX _G )(X _G /h) > 0 is the elasticity of the efficiency function of labour with respect to the output of sector G, i.e. X _G .

It is assumed that the healthcare sector (sector G) is a net supplier of labour in efficiency unit, implying that λ _LG  < ε _h which means (λ _LG  − ε _h ) < 0.

Solving (9.A.2) and (9.A.3) by Cramer’s rule yields

$$\left.\begin{array}{llll}{\widehat{X}}_1&=\left(1/\left|\lambda \right|\right)\left[\widehat{N}\left\{{\lambda}_{K G}{\lambda}_{L2}-{\lambda}_{K2}\left({\lambda}_{L G}-{\varepsilon}_h\right)\right\}-{\lambda}_{L2}\widehat{K}\right] \ \ \mathrm{and}\\ {\widehat{X}}_2&=\left(1/\left|\lambda \right|\right)\left[{\lambda}_{L1}\widehat{K}+\left\{{\lambda}_{K1}\left({\lambda}_{L G}-{\varepsilon}_h\right)-{\lambda}_{K G}{\lambda}_{L1}\right\}\widehat{N}\right]\end{array}\right\}$$

(9.A.4)

where

$$\left|\lambda \right|=\left({\lambda}_{L1}{\lambda}_{K2}-{\lambda}_{L2}{\lambda}_{K1}\right)>0$$

(9.A.5)

(This is because sector 2 is more intensive in the use of capital of type K vis-à-vis sector 1 with respect to labour.)

Using (9.A.5) from (9.A.4), we find that

$$\left.\begin{array}{lll}&{\widehat{X}}_1<0; \ \mathrm{and} \ {\widehat{X}}_2>0 \ \mathrm{when} \ \widehat{K}>0; \ \mathrm{and} \\ &{\widehat{X}}_1>0; \ \mathrm{and} \ {\widehat{X}}_2<0 \ \mathrm{when} \ \widehat{N}>0 \end{array}\right\}$$

(9.A.4.1)

9.1.2 Appendix 9.2: Derivation for Change in Welfare

Differentiation of Eq. (9.14.1) gives

$$\begin{array}{llll}[b] d{D}_1+{P}_2^{*} d{D}_2+{P}_G^{*} d{D}_G&= d{X}_1+{P}_2^{*} d{X}_2+{P}_G d{X}_G+{X}_G d{P}_G+ t{P}_2 dM\\ &\quad - rd{K}_F- RdN-{D}_G d{P}_G^{*}\end{array}$$

(9.A.6)

(Note that z is a policy parameter, which here does not change. So we have dz = 0.)

We also note that X ₁ = F ¹(L ₁, K ₁) and X ₂ = F ²(L ₂, K ₂) are the two production functions in sectors 1 and 2, respectively. Besides, we have fixed-coefficient technology in sector G where a _LG units of labour (in efficiency unit), a _KG units of capital of type K and a _NG units of capital of type N together produce one unit output. The full-employment conditions for the three inputs are L ₁ + L ₂ + L _G  = h(X _G )L; K ₁ + K ₂ + K _G  = K _D  + K _F  = K;  and N _G  = N.

Hence, differentiating Eq. (9.11) and the production functions yields

$$\begin{array}{llll} d Y&= d{X}_1+{P}_2^{*} d{X}_2+{P}_G d{X}_G+{X}_G d{P}_G- rd{K}_F- RdN+ t{P}_2 d M\\ &={F}_L^1 d{L}_1+{F}_K^1 d{K}_1+{P}_2^{*}{F}_L^2 d{L}_2+{P}_2^{*}{F}_K^2 d{K}_2+{P}_G d{X}_G\\ & \ \quad +{X}_G d{P}_G- rd{K}_F- RdN+ t{P}_2 d M\end{array}$$

(Note that P _G X _G  = W*dL _G  + rdK _G  + RdN _G ).

So,

$$\begin{array}{llll} d Y&= Wd{L}_1+{W}^{*}\left( d{L}_2+ d{L}_G\right)+ r\left( d{K}_1+ d{K}_2+ d{K}_G\right)+ Rd{N}_G\\ &\quad +{X}_G d{P}_G- rd{K}_F- RdN+ t{P}_2 dM\\ &=-\left({W}^{*}- W\right) d{L}_1+{X}_G d{P}_G+ t{P}_2 dM+{W}^{*}{h}^{\prime}\left(\cdot \right) Ld{X}_G \end{array}$$

(9.A.7)

(Note that (dL ₂ + dL _G  = h′(⋅)LdX _G  − dL ₁) and (dK ₁ + dK ₂ + dK _G  = dK _F ) as dK _D  = 0 and dN _G  = dN).

Differentiating Eq. (9.15) and using (9.A.7) we find

$$dM=\left(\frac{\partial {D}_2}{\partial Y}\right)\left[-\left({W}^{*}\!-\! W\right) d{L}_1\!+\!{X}_G d{P}_G+ t{P}_2 dM+{W}^{*}{h}^{\prime}\left(\cdot \right) Ld{X}_G\right]- d{X}_2$$

(Note that P ^*₂ does not change.)

On simplification we get

$$\begin{array}{llll} &d M\left(\frac{1+ t\left(1- m\right)}{1+ t}\right)=\left(\frac{m}{P_2^{*}}\right)\left[-\left({W}^{*}- W\right) d{L}_1+{X}_G d{P}_G+ t{P}_2 dM\right.\\ &\qquad \qquad \qquad \qquad \qquad \ \left.+ {W}^{*}{h}^{\prime}\left(\cdot \right) Ld{X}_G\right]- d{X}_2\\&d M\!=\! v\left[\left(\frac{m}{P_2^{*}}\right)\!\left\{-\left({W}^{*}\!-\! W\right)\! d{L}_1\!+\!{X}_G d{P}_G\!+\! t{P}_2 dM+{W}^{*}{h}^{\prime}\!\left(\cdot \right)\! Ld{X}_G\right\}\!-\! d{X}_2\right]\end{array}$$

(9.A.8)

Use of (9.A.8) in (9.A.7) and simplification yield

$$dY= v\left[-\left({W}^{*}- W\right) d{L}_1+{X}_G d{P}_G- t{P}_2 d{X}_2+{W}^{*}{h}^{\prime}\left(\cdot \right) Ld{X}_G\right]$$

(9.A.9)

where v = [(1 + t)/{1 + t(1 − m)}] and m = P ^*₂ (∂D ₂/∂Y) is the marginal propensity to consume commodity 2.

Differentiating Eq. (9.13) one obtains

$$\left(\frac{1}{V_1}\right) dV= d{D}_1+{P}_2^{*} d{D}_2+{P}_G^{*} d{D}_G$$

(9.A.10)

Differentiation of Eq. (9.9), use of (9.A.6) and (9.A.9) in (9.A.10) and simplification give

$$\begin{array}{llll} \left(\frac{1}{V_1}\right) dV&=- v\left[\left({W}^{*}- W\right){a}_{L1} d{X}_1+ t{P}_2 d{X}_2\right]+ v{X}_G d{P}_G\\& \ \quad + v{W}^{*}{h}^{\prime } Ld{X}_G- s{P}_G d{D}_G\end{array}$$

(9.A.11)

(Note that dL ₁ = a _L1 dX ₁.)

Differentiation of Eq. (9.8) yields

$${\widehat{D}}_G=\left(\frac{E_{P_G^{*}}^G}{P_G^{*}}\right) d{P}_G^{*}+\left(\frac{E_Y^G}{Y}\right) d Y$$

(9.A.12)

Using (9.A.4) and (9.A.9), Eq. (9.A.12) can be rewritten as follows:

$$\begin{array}{llll}{\widehat{D}}_G&={E}_{P_G^{*}}^G{\widehat{P}}_G^{*}+\left(\frac{v{ E}_Y^G{P}_G{X}_G}{Y}\right){\widehat{P}}_G\\ & \ \quad -\left(\frac{v{ E}_Y^G}{Y\left|\lambda \right|}\right)\left[\left({W}^{*}- W\right){a}_{L1}{X}_1\left\{{\lambda}_{K G}{\lambda}_{L2}-{\lambda}_{K2}\left({\lambda}_{L G}-{\varepsilon}_h\right)\right\}\right.\\ & \ \quad \left. + t{P}_2{X}_2\left\{{\lambda}_{K1}\left({\lambda}_{L G}-{\varepsilon}_h\right)-{\lambda}_{K G}{\lambda}_{L1}\right\}-{W}^{*}{h}^{\prime}\left(\cdot \right) L{X}_G\left|\lambda \right|\right]\widehat{N}\\ & \ \quad -\left[\left(\frac{v{ E}_Y^G{\lambda}_{L1}{X}_2}{Y\left|\lambda \right|}\right)\left\{\left({W}^{*}- W\right){a}_{L2}- t{P}_2\right\}\right]\widehat{K}\end{array}$$

(9.A.13)

Now, differentiating Eq. (9.9) and equation ((P ^* _G  = (1 − s)P _G ) and simplifying, we obtain

$${\widehat{P}}_G=\left(1- s\right){\widehat{P}}_G^{*}- s{\widehat{D}}_G$$

(9.A.14)

Rearranging terms in (9.A.13) and using (9.A.14) yield

$${H}_1{\widehat{D}}_G={H}_2{\widehat{P}}_G^{*}-{H}_3\widehat{N}+{H}_4\widehat{K}$$

(9.A.15)

where

$$\left.\begin{array}{llll}{H}_1&=\left[1+\left(\frac{sv{ E}_Y^G{P}_G{X}_G}{Y}\right)\right]>0; \\ {H}_2&=\left[{E}_{P_G^{*}}^G+\left(1- s\right)\left(\frac{v{ E}_Y^G{P}_G{X}_G}{Y}\right)\right]; \\ {H}_3&=\left(\frac{v{ E}_Y^G}{Y\left|\lambda \right|}\right)\left[\left({W}^{*}- W\right){a}_{L1}{X}_1\left\{{\lambda}_{K G}{\lambda}_{L2}-{\lambda}_{K2}\left({\lambda}_{L G}-{\varepsilon}_h\right)\right\}\right. \\ &\left.\left. \ \quad + t{P}_2{X}_2\left\{{\lambda}_{K1}\left({\lambda}_{L G}-{\varepsilon}_h\right)-{\lambda}_{K G}{\lambda}_{L1}\right\}-{W}^{*}{h}^{\prime}\left(\cdot \right) L{X}_G\left|\lambda \right|\right\}\right]; \ \mathrm{and} \\ {H}_4&=\left(\frac{v{ E}_Y^G{\lambda}_{L1}{X}_2}{Y\left|\lambda \right|}\right)\left[\left({W}^{*}- W\right){a}_{L2}- t{P}_2\right] \end{array}\right\}$$

(9.A.16)

Using (9.A.14) in (9.A.15) and simplifying, we obtain

$${\widehat{D}}_G=\left(\frac{{\overline{H}}_2}{{\overline{H}}_1}\right){\widehat{P}}_G-\left(\frac{H_3}{{\overline{H}}_1}\right)\widehat{N}+\left(\frac{H_4}{{\overline{H}}_1}\right)\widehat{K}$$

(9.A.17)

where

$$\left.\begin{array}{lll}&{\overline{H}}_1=\left({H}_1-\frac{s{ H}_2}{1- s}\right) \ \mathrm{and} \\ &{\overline{H}}_2=\frac{H_2}{1- s} \end{array}\right\}$$

(9.A.17.1)

For the sake of analytical simplicity, let us assume that D _G is a negative function of P _G , i.e. $\left({\widehat{D}}_G/{\widehat{P}}_G\right)<0$ (from (9.A.14) we find that there is a one-to-one correspondence between P ^* _G and P _G ). This means (from (9.A.17) that

$$\left(\frac{{\widehat{D}}_G}{{\widehat{P}}_G}\right)=\left(\frac{{\overline{H}}_2}{{\overline{H}}_1}\right)<0$$

(9.A.17.2)

Using (9.A.16) and (9.A.17.1) and simplifying from (9.A.17.2), we get

$$\begin{array}{lr}\left(\frac{{\overline{H}}_2}{{\overline{H}}_1}\right)&=\left[\frac{\left\{{H}_2/\left(1- s\right)\right\}}{\left\{\left(\left(1- s\right){H}_1- s{H}_2\right)/\left(1- s\right)\right\}}\right]=\left[\frac{H_2}{\left(1- s\right)- s{E}_{P_G^{*}}^G}\right]<0\\& {} \ \left(-\right)\qquad\qquad\end{array}$$

(9.A.17.3)

It follows from (9.A.17.3) that

$${H}_2<0$$

(9.A.18)

Using Eqs. (9.A.18) and (9.A.16) from Eqs. (9.A.16) and (9.A.17.1), it is easy to check that

$$\left.\begin{array}{lll}&{\overline{H}}_1=\left({H}_1-\frac{s{H}_2}{1- s}\right)=\left(1-\frac{s{ E}_{P_G^{*}}^G}{1- s}\right)>0; \\ &{\overline{H}}_2=\frac{H_2}{1- s}<0; \\ &{H}_2<0; \\ &{H}_2<0 \ \mathrm{i}\mathrm{f} \ \left(\mathrm{i}\right) \ \left({W}^{*}- W\right){a}_{L2}\ge t{P}_2 \ \mathrm{and} \ \left(\mathrm{i}\mathrm{i}\right) \ {\varepsilon}_h \ \mathrm{i}\mathrm{s} \ \mathrm{low}; \\ &{H}_4>0 \ \mathrm{i}\mathrm{ff} \ \left({W}^{*}- W\right){a}_{L2}> t{P}_2 \end{array}\right\}$$

(9.A.19)

From (9.A.1) one gets

$$\left(\frac{{\widehat{X}}_G}{{\widehat{P}}_G}\right)=0$$

(9.A.20)

Now, Walrasian stability in the market for the non-traded healthcare sector requires that

$$\left(\frac{{\widehat{D}}_G}{{\widehat{P}}_G}\right)-\left(\frac{{\widehat{X}}_G}{{\widehat{P}}_G}\right)<0$$

(9.A.21)

This is automatically satisfied as $\left(\left({\widehat{D}}_G/{\widehat{P}}_G\right)<0\kern0.5em \mathrm{and}\kern0.5em \left({\widehat{X}}_G/{\widehat{P}}_G\right)=0\right)$ (see Eqs. (9.A.17.2) and (9.A.20)).

At equilibrium in the market for sector G, we have

$${\widehat{D}}_G={\widehat{X}}_G$$

(9.A.22)

Using (9.A.17) and (9.A.1) and collecting terms from (9.A.22), one gets

$${H}_5{\widehat{P}}_G={H}_6\widehat{N}-{H}_7\widehat{K}$$

(9.A.23)

where

$$\left.\begin{array}{llll}&{H}_5=\left(\frac{{\overline{H}}_2}{{\overline{H}}_1}\right)<0; \\ &{H}_6=\left(\frac{{\overline{H}}_1+{H}_3}{{\overline{H}}_1}\right)>0 \ \mathrm{i}\mathrm{f} \ \ \left(\mathrm{i}\right) \ \left({W}^{*}- W\right){a}_{L2}\ge t{P}_2 \ \ \mathrm{and (ii) } \ \varepsilon_h \ \mathrm{is low; and} \\ &{H}_7=\frac{H_4}{{\overline{H}}_1}>\left(=\right)< \ \ \mathrm{according} \ \ \mathrm{to} \ \ \left({W}^{*}- W\right){a}_{L2}>\left(=\right)< t{P}_2 \end{array}\right\}$$

(9.A.24)

From (9.A.23) we find that

$$\left.\begin{array}{lll}&\left(\frac{{\widehat{P}}_G}{\widehat{K}}\right)=\left(-\right)\frac{H_7}{H_5}>0 \ \ \mathrm{i}\mathrm{ff} \ \ \left({W}^{*}- W\right){a}_{L2}> t{P}_2 \ \ \mathrm{and} \\ &\left(\frac{{\widehat{P}}_G}{\widehat{N}}\right)=\frac{H_6}{H_5}<0 \ \mathrm{i}\mathrm{f} \ \ \left(\mathrm{i}\right) \ \left({W}^{*}- W\right){a}_{L2}\ge t{P}_2 \ \ \mathrm{and} \\ & \quad\qquad\qquad \qquad\left(\mathrm{i}\mathrm{i}\right) \ {\varepsilon}_h \ \ \mathrm{i}\mathrm{s} \ \ \mathrm{low} \end{array}\right\}$$

(9.A.25)

From (9.A.1) and (9.A.22), we have

$$\left(\frac{{\widehat{X}}_G}{\widehat{K}}\right),\left(\frac{{\widehat{D}}_G}{\widehat{K}}\right)=0\kern0.5em \mathrm{and}$$

(9.A.26.1)

$$\left(\frac{{\widehat{X}}_G}{\widehat{N}}\right),\left(\frac{{\widehat{D}}_G}{\widehat{N}}\right)=1>0$$

(9.A.26.2)

9.1.2.1 Welfare Consequence of Capital of K Type

From (9.A.11) after using (9.A.26.1), one can write

$$\left(\frac{1}{V_1}\right)\frac{d V}{ d K}=- v\left[\left({W}^{*}- W\right){a}_{L1}\left(\frac{d{X}_1}{ d K}\right)+ t{P}_2\left(\frac{d{X}_2}{ d K}\right)\right]+ v{X}_G\left(\frac{d{P}_G}{ d K}\right)$$

(9.A.27)

The first term in (9.A.27) is the following:

$${T}_{1 K}=- v\left({W}^{*}- W\right){a}_{L1}\left(\frac{d{ X}_1}{ d K}\right)$$

Using (9.A.4) and simplifying, we obtain

$${T}_{1 K}=\left[\frac{v\left({W}^{*}- W\right){a}_{L1}{X}_1{\lambda}_{L2}}{\left|\lambda \right| K}\right]$$

(9.A.28)

Now the second term in (9.A.27) is

$${T}_{2 K}=- vt{P}_2\left(\frac{d{ X}_2}{ d K}\right)$$

Using (9.A.4) and simplifying, one gets

$${T}_{2 K}=-\left(\frac{vt{ P}_2{X}_2{\lambda}_{L1}}{\left|\lambda \right| K}\right)$$

(9.A.29)

Finally, the third term is given by

$${T}_{3 K}= v{X}_G\left(\frac{d{ P}_G}{ d K}\right)$$

Using (9.A.25), (9.A.24), (9.A.19) and (9.A.16) and simplifying, the above term can be reduced to the following:

$${T}_{3 K}=-\left[\left(\frac{v{ E}_Y^G{\lambda}_{L1}{X}_2}{Y\left|\lambda \right|{H}_2 K}\right)\left(1- s\right) v{P}_G{X}_G\left\{\left({W}^{*}- W\right){a}_{L2}- t{P}_2\right\}\right]$$

(9.A.30)

Using (9.A.28), (9.A.29) and (9.A.30) and simplifying from (9.A.27), one finally finds that

$$\left(\frac{1}{V_1}\right)\frac{dV}{dK}=\left(\frac{v{\lambda}_{L1}{X}_2}{\left|\lambda \right| K}\right)\left[\left({W}^{*}- W\right){a}_{L2}- t{P}_2\right]\left[1-\left(1- s\right)\left(\frac{v{ E}_Y^G{P}_G{X}_G}{Y{ H}_2}\right)\right]$$

(9.A.31)

From (9.A.31) we find that in the absence of both labour market distortion and tariff distortion, $\left(\frac{1}{V_1}\right)\frac{dV}{dK}=0$. Therefore, welfare does not change.

Using (9.A.18) from (9.A.31), it leads to

$$\left.\begin{array}{lll}&\left(\frac{1}{V_1}\right)\frac{dV}{dK}>0 \ \mathrm{iff} \ \left({W}^{*}- W\right){a}_{L2}> t{P}_2 \ \ \mathrm{and} \\ &\left(\frac{1}{V_1}\right)\frac{dV}{dK}<0 \ \mathrm{iff} \ \left({W}^{*}- W\right){a}_{L2}< t{P}_2 \end{array}\right\}$$

(9.A.32)

Besides, from (9.A.32) the following results also follow.

When W* = W, that is, there is no labour market distortion,

$$\left(\frac{1}{V_1}\right)\frac{dV}{dK}<0$$

(9.A.33)

On the other hand, when t = 0, that is, there is no tariff distortion,

$$\left(\frac{1}{V_1}\right)\frac{dV}{dK}>0$$

(9.A.34)

9.1.2.2 Welfare Consequence of Capital of Type N

From (9.A.11) one can obtain

$$\begin{array}{lllll}\left(\frac{1}{V_1}\right)\frac{d V}{ d N}&=- v\left[\left({W}^{*}- W\right){a}_{L1}\left(\frac{d{X}_1}{ d N}\right)+ t{P}_2\left(\frac{d{X}_2}{ d N}\right)\right]\\& \ \quad + v{X}_G\left(\frac{d{P}_G}{ d N}\right)+\left[ v{W}^{*}{h}^{\prime}\left(\cdot \right) L\left(\frac{d{X}_G}{ d N}\right)- s{P}_G\left(\frac{d{ D}_G}{ d N}\right)\right]\end{array}$$

(9.A.35)

In (9.A.35) the first term is

$${T}_{1 N}=- v\left[\left({W}^{*}- W\right){a}_{L1}\left(\frac{d{ X}_1}{ d N}\right)+ t{P}_2\left(\frac{d{ X}_2}{ d N}\right)\right]$$

Using (9.A.4) and simplifying, we obtain

$$\begin{array}{llll}{T}_{1 N}&=-\left(\frac{v}{\left|\lambda \right| N}\right)\left[{\lambda}_{K G}{\lambda}_{L1}{X}_2\Big\{\left({W}^{*}- W\right){a}_{L2}- t{P}_2\Big\}\right.\\ & \ \quad \left.-\left({\lambda}_{LG}-{\varepsilon}_h\right)\left(\frac{a_{L1}{X}_1{\lambda}_{K2}}{a_{L2}}\right)\left\{\left({W}^{*}- W\right){a}_{L2}- t{P}_2\left(\frac{a_{L2}{a}_{K1}}{a_{K2}{a}_{L1}}\right)\right\}\right]\end{array}$$

(9.A.36)

Note that (λ _LG − ε _h) < 0 and that (a _L2 a _K1/a _K2 a _L1) < 1 (since sector 1 is labour-intensive relative to sector 2 with respect to capital of type K).

From (9.A.36) it now follows that

$${T}_{1 N}<0\kern1em \mathrm{if}\kern0.5em \left({W}^{*}- W\right){a}_{L2}\ge t{P}_2$$

(9.A.37)

We write the second term and the third term of (9.A.35) together as follows:

$${T}_{2 N+3 N}= v{X}_G\left(\frac{d{P}_G}{ d N}\right)+ v{W}^{*}{h}^{\prime}\left(\cdot \right) L\left(\frac{d{X}_G}{ d N}\right)- s{P}_G\left(\frac{d{ D}_G}{ d N}\right)$$

Using (9), (9.A.25), (9.A.24) and (9.A.26.2), we can write

$$\begin{array}{lll}{T}_{2 N+3 N}&= v\left[\left({X}_G\frac{P_G}{N}\right)\left(\frac{H_6}{H_5}\right)+{W}^{*}{h}^{\prime } L\left(\frac{d{X}_G}{ d N}\right)\right]-\left(\frac{z}{N}\right)\\ &= v\left[\left(\frac{P_G{X}_G}{N}\right)\left(\frac{{\overline{H}}_1}{{\overline{H}}_2}\right)+\left(\frac{P_G{X}_G}{N}\right)\left(\frac{H_3}{{\overline{H}}_2}\right)+\left(\frac{W^{*}{\varepsilon}_h hL}{N}\right)\right]-\left(\frac{z}{N}\right)\\ &= v\left[\left[\left(\frac{P_G{X}_G}{N}\right)\left(\frac{1- s- s{E}_{P_G^{*}}^G}{H_2}\right)-\left(\frac{P_G{X}_G}{{\overline{H}}_2 N}\right)\left(\frac{v{E}_Y^G{W}^{*}{\varepsilon}_h hL}{Y}\right)\right.\right.\\&\quad \left.\left.+\left(\frac{W^{*}{\varepsilon}_h hL}{N}\right)\right]\right. +\left(\frac{P_G{X}_G}{{\overline{H}}_2 N}\right)\left(\frac{v{E}_Y^G}{Y}\right)\left[\left({W}^{*}- W\right){a}_{L1}{X}_1\left\{{\lambda}_{K G}{\lambda}_{L2}\right.\right.\\ &\ \ \;\left.\left.\left.-{\lambda}_{K2}\big({\lambda}_{L G}-{\varepsilon}_h\big)\right\}+ t{P}_2{X}_2\left\{{\lambda}_{K1}\left({\lambda}_{L G}-{\varepsilon}_h\right)-{\lambda}_{K G}{\lambda}_{L1}\right\}\right]\right]-\left(\frac{z}{N}\right)\end{array}$$

(9.A.38)

Now

$$\begin{array}{llll}&\left(\frac{P_G{X}_G}{{\overline{H}}_2 N}\right)\left(\frac{v{ E}_Y^G}{Y}\right)\left[\left({W}^{*}- W\right){a}_{L1}{X}_1\left\{{\lambda}_{K G}{\lambda}_{L2}-{\lambda}_{K2}\left({\lambda}_{L G}-{\varepsilon}_h\right)\right\}\right.\\ & \ \quad\left. + t{P}_2{X}_2\left\{{\lambda}_{K1}\left({\lambda}_{L G}-{\varepsilon}_h\right)-{\lambda}_{K G}{\lambda}_{L1}\right\}\right]\ \\ &\quad=\left(\frac{P_G{X}_G}{{\overline{H}}_2 N}\right)\left(\frac{v{ E}_Y^G}{Y}\right)\left[{\lambda}_{K G}{\lambda}_{L1}{X}_2\left\{\left({W}^{*}- W\right){a}_{L2}- t{P}_2\right\}\right.\\ & \ \quad \quad \left. -\left({\lambda}_{L G}-{\varepsilon}_h\right)\left(\frac{a_{L1}{X}_1{\lambda}_{K2}}{a_{L2}}\right)\left\{\left({W}^{*}- W\right){a}_{L2}- t{P}_2\left(\frac{a_{L2}{a}_{K1}}{a_{K2}{a}_{L1}}\right)\right\}\right]\end{array}$$

(9.A.39)

$$<0\kern1em \mathrm{if}\kern0.5em \left({W}^{*}- W\right){a}_{L2}\ge t{P}_2$$

(9.A.39.1)

(Note that ${\overline{H}}_2<0$, (λ _LG  − ε _h ) < 0 and (a _L2 a _K1/a _K2 a _L1) < 1)

Besides,

$$\begin{array}{llll}&\left[\left(\frac{P_G{X}_G}{N}\right)\left(\frac{1- s- s{E}_{P_G^{*}}^G}{H_2}\right)-\left(\frac{P_G{X}_G}{{\overline{H}}_2 N}\right)\left(\frac{v{E}_Y^G{W}^{*}{\varepsilon}_h hL}{Y}\right)+\left(\frac{W^{*}{\varepsilon}_h hL}{N}\right)\right]\\ &\quad =\left[\left(\frac{P_G{X}_G}{N}\right)\left(\frac{1- s- s{E}_{P_G^{*}}^G}{H_2}\right)+\left(\frac{W^{*}{\varepsilon}_h hL}{N}\right)\left\{1-\left(\frac{P_G{X}_G}{{\overline{H}}_2}\right)\left(\frac{v{E}_Y^G}{Y}\right)\right\}\right]\\ &\quad =\left[\left(\frac{P_G{X}_G}{N}\right)\left(\frac{1- s- s{E}_{P_G^{*}}^G}{H_2}\right)+\left(\frac{W^{*}{\varepsilon}_h hL{E}_{P_G^{*}}^G}{N{ H}_2}\right)\right]\\ &\quad =\left(\frac{1}{N{ H}_2}\right)\left[\left(1- s\right){P}_G{X}_G-\left( s{P}_G{X}_G-{W}^{*}{\varepsilon}_h hL\right){E}_{P_G^{*}}^G\right]\\ &\quad =\left(\frac{1}{N{ H}_2}\right)\left[\left(1- s\right){P}_G{X}_G-\left( z-{W}^{*}{\varepsilon}_h hL\right){E}_{P_G^{*}}^G\right] \end{array}$$

(9.A.40)

$$<0\kern1em \mathrm{if}\kern0.5em z\ge {W}^{*}{\varepsilon}_h hL$$

(9.A.40.1)

(Note that H ₂ < 0; ${E}_{P_G^{*}}^G<0$; and z = sP _G X _G .)

So, using (9.A.39), (9.A.39.1), (9.A.40) and (9.A.40.1) from (9.A.38), we find

$${T}_{2 N+3 N}<0\kern1em \mathrm{i}\mathrm{f}\kern0.5em \left(\mathrm{i}\right)\kern0.5em \left({W}^{*}- W\right){a}_{L2}\ge t{P}_2\kern0.5em \mathrm{and}\kern0.5em \left(\mathrm{i}\mathrm{i}\right)\kern0.5em z\ge {W}^{*}{\varepsilon}_h hL$$

(9.A.41)

It may be noted that the term T _2N + 3N measures the net effect of LEE and DVE (defined in the text) on social welfare.

Therefore, using (9.A.36), (9.A.37) and (9.A.38) and (9.A.41) from (9.A.35), it follows that

$$\left(\frac{1}{V_1}\right)\frac{dV}{dN}<0\kern1em \mathrm{i}\mathrm{f}\kern0.5em \left(\mathrm{i}\right)\kern0.5em \left({W}^{*}- W\right){a}_{L2}\ge t{P}_2\kern0.5em \mathrm{and}\kern0.5em \left(\mathrm{i}\mathrm{i}\right)\kern0.5em z\ge {W}^{*}{\varepsilon}_h hL$$

(9.A.42)

From Eqs. (9.A.38) and (9.A.40), it may be noted that the second sufficient condition (i.e. z ≥ W*ε _h hL) can be replaced by a few other alternative sufficient conditions.

From (9.A.40) it may be noted that the second sufficient condition (i.e. z ≥ W*ε _h hL) can be replaced by a few other alternative sufficient conditions.

From (9.A.42) one may note that in the absence of any tariff we have

$$\left(\frac{1}{V_1}\right)\frac{dV}{dN}<0\kern1em \mathrm{if}\kern0.5em z\ge {W}^{*}{\varepsilon}_h hL$$

(9.A.43)

9.1.2.3 Effect of $\widehat{N}>0$ on Aggregate Labour Endowment in Efficiency Unit

Differentiating Eq. (9.16) with respect to N gives

$$\frac{d C}{ d N}={h}^{\prime } L\frac{d{ X}_G}{ d N}$$

(9.A.44)

Using (9.A.1) Eq. (9.A.44) may be rewritten as follows:

$$\frac{dC}{dN}=\left(\frac{\varepsilon_h hL}{N}\right)>0$$

(9.A.45)

So an inflow of foreign capital of type N always improves the human capital stock of the economy.