Time zone difference, skill formation and corrupt informal sector: the role of virtual trade

Abstract

This paper uses competitive general equilibrium model of trade for small open economy with informal sector to check the possible effects of virtual trade. We show that skilled labors and educational capital owners benefit from virtual trade. The service sector expands while the formal and informal sector contract along with the number of people engaged in corruption-related intermediation. Following this, we also check the effect of a fall in the extent of cost of corruption. Results show an increase in unskilled wage and outflow of educational capital thus hurting the skill-intensive sector. We proceed further to club the effects of both virtual trade and fall in intermediation cost, and explore the consequences. Though, both skilled and unskilled labors benefit, the effect on output and intermediators, however, is ambiguous. We then modify the basic model to endogenize the cost of corruption, include punishment aspect of intermediators, etc. In this case, owing to time zone difference exploitation, we experience an increase in wage of both types of labor, an expansion of the service sector and contraction of the informal sector. Interestingly, the cost of intermediation rises while the number of intermediators falls in the extended model.

Appendices

Appendices

1.1 Appendix A

1.1.1 Determination of unskilled-labor wage in the formal sector

The wage in sector $Y$ is determined through the utility maximization problem of labor unions. Union’s utility function is given as:

$$U = U\left( {w_{Y} , N,p_{Y} } \right).$$

Here, $w_{Y}$ is the wage rate, $N$ is employment of unskilled labor in the union sector or in sector $Y$, and $p_{Y}$ is the relative price of $Y$. $U$ is increasing and quasi-concave in $N$ and $w_{Y}$. $\frac{\delta u}{{\delta w_{Y} }}, \frac{\delta u}{\delta N}, \frac{{\delta^{2} u}}{{\delta w_{Y} \delta N}} > 0; \; \frac{\delta u}{\delta p} < 0,$ and $\frac{{\delta^{2} u}}{{\delta w_{Y}^{2} }}, \frac{{\delta^{2} u}}{{\delta N^{2} }} < 0$.

The problem is to maximize the utility function subject to the labor demand curve, $N = Yc_{w} = N\left( {w_{Y} ,r} \right);$ where $Y$ is output, $r$ is rent, $\frac{\delta N}{{\delta w_{Y} }} < 0$ and $c_{w}$ is the partial derivative, with respect to $w_{Y}$, of the representative firm’s unit cost function, $c\left( {w_{Y} ,r} \right)$. The First Order Condition for equilibrium is:

$$\frac{\delta u}{{\delta w_{Y} }} + \frac{\delta u}{\delta N}\frac{\delta N}{{\delta w_{Y} }} = 0 \Rightarrow \frac{{\delta u/\delta w_{Y} }}{\delta u/\delta N} = - \frac{\delta N}{{\delta w_{Y} }}.$$

$\Rightarrow MRS_{{w_{Y} ,N}} =$ slope of the labor demand curve.

The above equality yields the equilibrium for labor union where wage rate is determined. In our paper, the unionized wage is denoted as $\bar{w}$. Note that, here firms are price takers and the Second Order Condition for maximization is also satisfied.

1.2 Appendix B

1.2.1 Effect of rise in $\delta$

Writing Eqs. (1)–(5) in relative change form we have,

$$\theta_{SX} \hat{w}_{S} + \theta_{KX} \hat{r} + \theta_{SX} \hat{a}_{SX} + \theta_{KX} \hat{a}_{KX} = \delta \hat{P}_{X} + \hat{\delta }\delta .$$

Since $\hat{P}_{X} = 0$ and $a_{SX}$ and $a_{KX}$ are assumed to be fixed, we have

$$\theta_{SX} \hat{w}_{S} + \theta_{KX} \hat{r} = \hat{\delta }\delta .$$

(B.1)

Similarly, from Eq. (2), $0 + \theta_{KY} \hat{r} + \theta_{LY} \hat{a}_{LY} + \theta_{KY} \hat{a}_{KY} = 0.$

From Envelope Theorem, $\theta_{LY} \hat{a}_{LY} + \theta_{KY} \hat{a}_{KY} = 0$; therefore we have,

$$\theta_{KY} \hat{r} = 0.$$

(B.2)

Similarly from (3), (4) and (5), we have

$$\theta_{LZ} \hat{w} + \theta_{KZ} \hat{r} = 0,$$

(B.3)

$$\hat{w} + \hat{L}_{N} = \hat{Z},$$

(B.4)

$$\theta_{ES} \hat{R} + \theta_{{LS^{\prime}}} \hat{w} = \left( {1 - \theta_{SS'} } \right)\hat{w}_{S} .$$

(B.5)

From (B.2), $\hat{r} = 0.$ Therefore from (B.1), $\hat{w}_{S} = \frac{1}{{\theta_{SX} }}\delta \hat{\delta } > 0.$ Putting the value of $\hat{r}$ in (B.3), $\hat{w} = 0$. Substituting $\hat{w}$ and $\hat{w}_{S}$ in (B.5), we get $\hat{R} = \frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{ES} \theta_{SX} }}\delta \hat{\delta } > 0$. Since, $\hat{w} = 0$, from (B.4), $\hat{L}_{N} = \hat{Z}$.

Here the input coefficients of $X$ is fixed; the input coefficients of $Y$ and $Z$ are variable but as there is no change in the factor prices there will not be any change in the input coefficients. The amount of unskilled labor required to produce one unit of skilled labor is one (i.e., $a_{{LS^{\prime}}} = 1)$ and the amount of educational capital required is assumed as fixed. Therefore, by Envelope condition, the per unit skilled labor requirement for training remains unchanged. Thus, even though there are changes in $w_{S}$ and $R$ there will not be any change in the input coefficients of $S$ production. However, rise in $R$ will make domestic $R$ higher than the international $R$, leading to inflow of educational capital. Inflow of educational capital will initiate inter-sectoral movement of factors and the size of different sectors will change. From (9)

$$\hat{S}' = \hat{E}.$$

(B.6)

From Eq. (6), $\lambda_{SX} \hat{X} = \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)\hat{S}'$

$$\Rightarrow \hat{X} = \frac{{\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\hat{E}.$$

(B.7)

From Eq. (7),

$$\lambda_{KY} \hat{Y} + \lambda_{KZ} \hat{Z} = - \frac{{\lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\hat{E}.$$

(B.8)

From (10)

$$\hat{L}_{S} = \hat{S}'.$$

(B.9)

From (8), $\lambda_{LY} \hat{Y} + \lambda_{LZ} \hat{Z} + \lambda_{LN} \hat{L}_{N} + \lambda_{{LS^{\prime}}} \hat{L}_{S} = 0.$

Since $\hat{L}_{N} = \hat{Z}$ and $\hat{L}_{S} = \hat{S}'$; $\lambda_{LY} \hat{Y} + \lambda_{LZ} \hat{Z} + \lambda_{LN} \hat{Z} + \lambda_{{LS^{\prime}}} \hat{S}' = 0.$

Using (B.6),

$$\lambda_{LY} \hat{Y} + \left( {\lambda_{LZ} + \lambda_{LN} } \right)\hat{Z} = - \lambda_{{LS^{\prime}}} \hat{E}.$$

(B.10)

Writing (B.8) and (B.10) in matrix form,

$$\left( {\begin{array}{*{20}c} {\lambda_{KY} } & {\lambda_{KZ} } \\ {\lambda_{LY} } & {\left( {\lambda_{LZ} + \lambda_{LN} } \right)} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\hat{Y}} \\ {\hat{Z}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { - \frac{{\lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\hat{E}} \\ { - \lambda_{{LS^{\prime}}} \hat{E}} \\ \end{array} } \right).$$

Now, $\left| \lambda \right| = \lambda_{KY} \left( {\lambda_{LZ} + \lambda_{LN} } \right) - \lambda_{LY} \lambda_{KZ} > 0$; since $Y$ is $K$ intensive and $Z$ is $L$ intensive.

Using Cramer’s Rule, we solve $\hat{Y}$ as follows:

$$\hat{Y} = \frac{{\hat{E}}}{{\left| \lambda \right|\lambda_{SX} }}\left\{ \lambda_{{LS^{\prime}}} \lambda_{KZ} \lambda_{SX} - \lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)\left( {\lambda_{LZ} + \lambda_{LN} } \right)\right\} .$$

In the above equation we have

$$\begin{aligned} & \lambda_{{LS^{\prime}}} \lambda_{KZ} \lambda_{SX} - \lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)\left( {\lambda_{LZ} + \lambda_{LN} } \right) \hfill \\ &\quad = \frac{1}{SKL}\left\{ {a_{{LS^{\prime}}} S^{\prime}a_{KZ} Za_{SX} X + a_{KX} Xa_{{ss^{\prime}}} S^{\prime}L_{N} + a_{KX} Xa_{{ss^{\prime}}} S^{\prime}a_{LZ} Z - \left( {a_{KX} XS^{\prime}a_{LZ} Z + a_{KX} XS^{\prime}L_{N} } \right)} \right\} \hfill \\ & \quad = \frac{{2XS^{\prime}}}{SKL}\{ a_{KZ} Z - a_{LZ} Z\left( {1 - a_{{ss^{\prime}}} } \right) - L_{N} \left( {1 - a_{{ss^{\prime}}} } \right) \hfill \\ & \quad = \frac{{2XS^{\prime}}}{SKL}\left\{ {a_{KZ} Z - \left( {a_{LZ} Z + L_{N} } \right)\left( {1 - a_{{ss^{\prime}}} } \right)} \right\}. \hfill \\ \end{aligned}$$

Here if $a_{KZ}$ and $a_{{ss^{\prime}}}$ are very low, then the given expression is negative. This implies $Y$ is negative. The reverse is true when the values of $a_{KZ}$ and $a_{{ss^{\prime}}}$ are large.

Similarly, $\hat{Z} = \frac{{\hat{E}}}{{\left| \lambda \right|\lambda_{SX} }}\left\{ {\lambda_{LY} \lambda_{KX} \lambda_{{S^{\prime}S}} - \lambda_{LY} \lambda_{KX} \lambda_{{SS^{\prime}}} - \lambda_{KY} \lambda_{{LS^{\prime}}} \lambda_{SX} } \right\}.$

Here, $\lambda_{LY} \lambda_{KX} \lambda_{{S^{\prime}S}} = \frac{{a_{LY} Y}}{L}\frac{2X}{K}\frac{{S^{\prime}}}{S}$; and $\lambda_{KY} \lambda_{{LS^{\prime}}} \lambda_{SX} = \frac{{a_{KY} Y}}{K}\frac{{L_{S} }}{L}\frac{2X}{S} = \frac{{a_{KY} Y}}{K}\frac{2X}{S}\frac{S'}{L}$. Since, $Y$ is capital intensive $\lambda_{LY} \lambda_{KX} \lambda_{{S^{\prime}S}} < \lambda_{KY} \lambda_{{LS^{\prime}}} \lambda_{SX}$. This implies $\hat{Z} < 0.$

Now, as $\hat{Z} = \hat{L}_{N}$,

$$\hat{L}_{N} = \frac{{\hat{E}}}{{\left| \lambda \right|\lambda_{SX} }}\left\{ {\lambda_{LY} \lambda_{KX} \lambda_{{S^{\prime}S}} - \lambda_{LY} \lambda_{KX} \lambda_{{SS^{\prime}}} - \lambda_{KY} \lambda_{{LS^{\prime}}} \lambda_{SX} } \right\} < 0.$$

1.3 Appendix C

1.3.1 The effect of a fall in $\alpha$

The relative change form of (2) is $\theta_{KY} \hat{r} = 0$

$$\Rightarrow \hat{r} = 0.$$

(C.1)

From (1), $\theta_{SX} \hat{w}_{S} + \theta_{KX} \hat{r} = 0$. Using (C.1), $\hat{w}_{S} = 0$

With a fall in $\alpha$, Eq. (3) gives, $\theta_{LZ} \hat{w} + \theta_{KZ} \hat{r} = - \alpha \hat{\alpha }\varvec{ } \Rightarrow \hat{w} = - \frac{{\alpha \hat{\alpha }}}{{\theta_{LZ} }}$

From Eqs. (4) and (5), we have, respectively,

$$\hat{w} + \hat{L}_{N} = \hat{Z} + \hat{\alpha }.$$

(C.2)

And

$$\theta_{ES} \hat{R} + \theta_{{LS^{\prime}}} \hat{w} = \left( {1 - \theta_{{SS^{\prime}}} } \right)\hat{w}_{S} \Rightarrow \hat{R} = \frac{{\theta_{{LS^{\prime}}} }}{{\theta_{ES} }}\frac{{\alpha \hat{\alpha }}}{{\theta_{LZ} }}.$$

Since wage of unskilled labor is changing there will be substitution between factors in $Z$. The elasticity of substitution between $L$ and $K$ in $Z$ is $\sigma_{Z} = \frac{{\hat{a}_{LZ} - \hat{a}_{KZ} }}{{\hat{r} - \hat{w}}}.$

As $\hat{r} = 0$, $\hat{a}_{KZ} = \hat{a}_{LZ} + \sigma_{Z} \hat{w}.$

From the Envelope theorem, $\theta_{LZ} \hat{a}_{LZ} + \theta_{KZ} \hat{a}_{KZ} = 0 \Rightarrow \hat{a}_{LZ} = - \frac{{\theta_{KZ} }}{{\theta_{LZ} }}\hat{a}_{KZ} .$

Therefore, $\hat{a}_{KZ} = - \frac{{\theta_{KZ} }}{{\theta_{LZ} }}\hat{a}_{KZ} + \sigma_{Z} \hat{w} \Rightarrow \hat{a}_{KZ} = \sigma_{Z} \theta_{LZ} \hat{w}$

$$\Rightarrow \hat{a}_{KZ} = - \sigma_{Z} \alpha \hat{\alpha }.$$

(C.3)

Now,

$$\hat{a}_{LZ} = \frac{{\theta_{KZ} }}{{\theta_{LZ} }}\sigma_{Z} \hat{\alpha }\alpha < 0.$$

(C.4)

From Eqs. (6)–(10) we have, respectively,

$$\lambda_{SX} \hat{X} = \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)\hat{S}',$$

(C.5)

$$\lambda_{KX} \hat{X} + \lambda_{KY} \hat{Y} + \lambda_{KZ} \hat{Z} + \lambda_{KZ} \hat{a}_{KZ} = 0,$$

(C.6)

$$\lambda_{LY} \hat{Y} + \lambda_{LZ} \hat{Z} + \lambda_{LZ} \hat{a}_{LZ} + \lambda_{LN} \hat{L}_{N} + \lambda_{LS '} \hat{L}_{S} = 0,$$

(C.7)

$$\hat{S}' = \hat{E},$$

(C.8)

$$\hat{L}_{S} = \hat{S}'.$$

(C.9)

Putting (C.8) in (C.5)

$$\hat{X} = \frac{{\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\hat{E} < 0.$$

(C.10)

Using (C.10) in (C.6)

$$\lambda_{KY} \hat{Y} + \lambda_{KZ} \hat{Z} = - \lambda_{KZ} \hat{a}_{KZ} - \frac{{\lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\hat{E}.$$

(C.11)

Using (C.2) and (C.9) in (C.7)

$$\lambda_{LY} \hat{Y} + \left( {\lambda_{LZ} + \lambda_{LN} } \right)\hat{Z} = - \lambda_{LZ} \hat{a}_{LZ} - \lambda_{LS '} \hat{E} - \lambda_{LN} \left( {\hat{\alpha } + \frac{{\hat{\alpha }\alpha }}{{\theta_{LZ} }}} \right).$$

(C.12)

Writing (C.11) and (C.12) in matrix form:

$$\left( {\begin{array}{*{20}c} {\lambda_{KY} } & {\lambda_{KZ} } \\ {\lambda_{LY} } & {\left( {\lambda_{LZ} + \lambda_{LN} } \right)} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\hat{Y}} \\ {\hat{Z}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\left\{ { - \lambda_{KZ} \hat{a}_{KZ} - \frac{{\lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\hat{E}} \right\}} \\ {\left\{ { - \lambda_{LZ} \hat{a}_{LZ} - \lambda_{{LS^{\prime}}} \hat{E} - \lambda_{LN} \left( {\hat{\alpha } + \frac{{\hat{\alpha }\alpha }}{{\theta_{LZ} }}} \right)} \right\}} \\ \end{array} } \right),$$

$$\left| \lambda \right| = \lambda_{KY} \left( {\lambda_{LZ} + \lambda_{LN} } \right) - \lambda_{LY} \lambda_{KZ} > 0.$$

Therefore,

$$ \begin{aligned}\hat{Y} &= \frac{1}{\left| \lambda \right|}\left[ \vphantom{{+ \left\{ \lambda_{KZ} \lambda_{{LS^{\prime}}} - \frac{{\lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\left( {\lambda_{LZ} + \lambda_{LN} } \right)\right\} \hat{E}}}{\left\{ {\lambda_{KZ} \left( {\lambda_{LZ} + \lambda_{LN} } \right)\sigma_{Z} \hat{\alpha }\alpha + \lambda_{KZ} \lambda_{LZ} \frac{{\theta_{KZ} }}{{\theta_{LZ} }}\sigma_{Z} \hat{\alpha }\alpha + \lambda_{KZ} \lambda_{LN} \left( {\hat{\alpha } + \frac{{\hat{\alpha }\alpha }}{{\theta_{LZ} }}} \right)} \right\} }\right.\\&\quad\left.{+ \left\{ \lambda_{KZ} \lambda_{{LS^{\prime}}} - \frac{{\lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\left( {\lambda_{LZ} + \lambda_{LN} } \right)\right\} \hat{E}} \right].\end{aligned}$$

In the above equation, $\left\{ {\lambda_{KZ} \left( {\lambda_{LZ} + \lambda_{LN} } \right)\sigma_{Z} \hat{\alpha }\alpha + \lambda_{KZ} \lambda_{LZ} \frac{{\theta_{KZ} }}{{\theta_{LZ} }}\sigma_{Z} \hat{\alpha }\alpha + \lambda_{KZ} \lambda_{LN} \left( {\hat{\alpha } + \frac{{\hat{\alpha }\alpha }}{{\theta_{LZ} }}} \right)} \right\} < 0$, as $\hat{\alpha } < 0$; and $\{ \lambda_{KZ} \lambda_{{LS^{\prime}}} - \frac{{\lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}\left( {\lambda_{LZ} + \lambda_{LN} } \right)\}$ can be both positive or negative as mentioned in Appendix B. If it is positive then with $\hat{E} < 0$, $\hat{Y} < 0;$ and if it is negative, then the value of $\hat{Y}$ depends on both factor prices and the values of $\hat{\alpha }$ and $\hat{E}$.

$$ \begin{aligned}\hat{Z} &= \left( - \right)\frac{1}{\left| \lambda \right|}\left[ {\lambda_{KY} \lambda_{LZ} \frac{{\theta_{KZ} }}{{\theta_{LZ} }}\sigma_{Z} \hat{\alpha }\alpha + \lambda_{KY} \lambda_{LN} \left( {\hat{\alpha } + \frac{{\hat{\alpha }\alpha }}{{\theta_{LZ} }}} \right) + \lambda_{LY} \lambda_{KZ} \sigma_{Z} \hat{\alpha }\alpha }\right. \\&\quad\left.{+ \hat{E}\left\{ {\lambda_{KY} \lambda_{{LS^{\prime}}} - \lambda_{LY} \lambda_{KX} \frac{{\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}} \right\}} \right] > 0 .\end{aligned}$$

Here, as $\hat{\alpha } < 0,\left\{ {\lambda_{KY} \lambda_{LZ} \frac{{\theta_{KZ} }}{{\theta_{LZ} }}\sigma_{Z} \hat{\alpha }\alpha + \lambda_{KY} \lambda_{LN} \left( {\hat{\alpha } + \frac{{\hat{\alpha }\alpha }}{{\theta_{LZ} }}} \right) + \lambda_{LY} \lambda_{KZ} \sigma_{Z} \hat{\alpha }\alpha } \right\} < 0$. The expression$\lambda_{KY} \lambda_{{LS^{\prime}}} - \lambda_{LY} \lambda_{KX} \frac{{\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }} = \frac{1}{{\lambda_{SX} }}\left\{ {\lambda_{KY} \lambda_{{LS^{\prime}}} \lambda_{SX} - \lambda_{LY} \lambda_{KX} \left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)} \right\} = \frac{2XY}{{SKL\lambda_{SX} }}\left\{ {a_{KY} S^{\prime} - a_{LY} S^{\prime}\left( {1 - a_{{SS^{\prime}}} } \right)} \right\} > 0$, since, $a_{KY} > a_{LY}$; and $a_{{SS^{\prime}}}$ being less than 1, $S^{\prime} > S^{\prime}\left( {1 - a_{{SS^{\prime}}} } \right)$.

Therefore, as $\hat{E} < 0$, $\hat{Z} > 0$.

Substituting $\hat{Z}$ in (C.2),

$$\begin{aligned} \hat{L}_{N} = & \frac{1}{\left| \lambda \right|}\left[ {\hat{E}\left\{ {\lambda_{LY} \lambda_{KX} \frac{{\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }} - \lambda_{KY} \lambda_{{LS^{\prime}}} } \right\} - \left( {\lambda_{KY} \lambda_{LZ} \frac{{\theta_{KZ} }}{{\theta_{LZ} }}\sigma_{Z} \hat{\alpha }\alpha + \lambda_{LY} \lambda_{KZ} \sigma_{Z} \hat{\alpha }\alpha } \right)} \right] \hfill \\ & + \left[ {\left\{ {\left( - \right)\frac{1}{\left| \lambda \right|}\lambda_{KY} \lambda_{LN} \left( {\hat{\alpha } + \frac{{\hat{\alpha }\alpha }}{{\theta_{LZ} }}} \right)} \right\} + \left( {\hat{\alpha } + \frac{{\hat{\alpha }\alpha }}{{\theta_{LZ} }}} \right)} \right]. \hfill \\ \end{aligned}$$

Here the first expression separated by square brackets is positive. However, the second expression is negative. Therefore the effect on $L_{N}$ depends on the magnitudes of the two expressions.

For the combined effect of a rise in $\delta$ and a fall in $\alpha$, from Eqs. (1) to (5) we get, respectively: $\theta_{SX} \hat{w}_{S} + \theta_{KX} \hat{r} = \delta \hat{\delta }; \quad \theta_{KY} \hat{r} = 0; \quad \theta_{LZ} \hat{w} + \theta_{KZ} \hat{r} = - \alpha \hat{\alpha };\quad \hat{w} + \hat{L}_{N} = \hat{Z} + \hat{\alpha };$ and $\theta_{ES} \hat{R} + \theta_{{LS^{\prime}}} \hat{w} = \left( {1 - \theta_{{SS^{\prime}}} } \right)\hat{w}_{S} .$ Using these, the change in factor prices are: $\hat{r} = 0; \quad \hat{w}_{S} = \frac{1}{{\theta_{SX} }}\delta \hat{\delta } > 0; \quad \hat{w} = - \frac{1}{{\theta_{LZ} }}\hat{\alpha }\alpha > 0;$ and $\hat{R} = \frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)\delta \hat{\delta }}}{{\theta_{ES} \theta_{SX} }} + \frac{{\theta_{{LS^{\prime}}} }}{{\theta_{ES} \theta_{LZ} }}\hat{\alpha }\alpha .$

Therefore, effect on $\hat{R}$ depends on

$$\left| {\frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)\delta \hat{\delta }}}{{\theta_{ES} \theta_{SX} }}} \right|{ \; \gtreqless \; }\left| {\frac{{\theta_{{LS^{\prime}}} }}{{\theta_{ES} \theta_{LZ} }}\hat{\alpha }\alpha } \right|.$$

The calculations for change in input coefficients and output of different sectors are same as in case of fall in $\alpha$ alone. However, whether the effects are positive or negative depend on the direction of flow of educational capital.

1.4 Appendix D

Writing (25)–(30) with ‘hat’ notation:

$$\theta_{SX} \hat{w}_{S} + \theta_{KX} \hat{r} = \hat{\delta }\delta ,$$

(D.1)

$$\theta_{KY} \hat{r} = 0,$$

(D.2)

$$\theta_{LZ} \hat{w} + \theta_{TZ} \hat{R} = - \alpha \hat{\alpha },$$

(D.3)

$$\hat{w}_{N} = \hat{w},$$

(D.4)

$$\theta_{KS '} \hat{r} + \theta_{LS '} \hat{w} = \left( {1 - \theta_{SS '} } \right)\hat{w}_{S,}$$

(D.5)

$$\hat{\alpha } = \hat{w}_{N} .$$

(D.6)

From (D.2), $\hat{r} = 0$. Using the value of $\hat{r}$, from (D.1), $\hat{w}_{S} = \frac{{\hat{\delta }\delta }}{{\theta_{SX} }} > 0$. With the value of $\hat{r}$ and $\hat{w}_{S}$ known, from (D.5), $\hat{w} = \frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{LS '} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }} > 0$. Using (D.4), $\hat{w}_{N} = \hat{w} = \frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{LS '} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }} > 0$. With the help of (D.6), we get, $\hat{\alpha } = \hat{w}_{N} = \hat{w} = \frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{LS '} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }} > 0$. Putting the value of $\hat{w}$ and $\hat{\alpha }$ in (D.3), $\hat{R} = - \frac{{\left( {\alpha + \theta_{LZ} } \right)}}{{\theta_{TZ} }}\hat{w} \Rightarrow \hat{R} = - \frac{{\left( {\alpha + \theta_{LZ} } \right)}}{{\theta_{TZ} }}\frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{LS '} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }} < 0$

The difference between the relative change in skilled wage and unskilled wage is

$$\hat{w}_{S} - \hat{w} = \frac{{\hat{\delta }\delta }}{{\theta_{SX} }}\left\{ {1 - \frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{{LS^{\prime}}} }}} \right\} = \frac{{\hat{\delta }\delta }}{{\theta_{SX} }}\frac{{(\theta_{{LS^{\prime}}} - 1 + {{\theta }}_{{{\text{ss}}^{\prime} }} )}}{{\theta_{{LS^{\prime}}} }} = - \frac{{\hat{\delta }\delta }}{{\theta_{SX} }}\frac{{\theta_{{KS^{\prime} }} }}{{\theta_{{LS^{\prime}}} }} < 0.$$

This implies there is a fall in the wage gap.

Similar to the previous section, there may be some changes in input usage in $Z$ due to change in $R$ and $w$. The elasticity of substitution, $\sigma_{Z}$, between $T$ and $L$ in $Z$ is:

$$\sigma_{Z} = \frac{{\hat{a}_{LZ} - \hat{a}_{TZ} }}{{\hat{R} - \hat{w}}} \Rightarrow \left( {\hat{R} - \hat{w}} \right)\sigma_{Z} + \hat{a}_{TZ} = \hat{a}_{LZ} .$$

(D.7)

From the envelope condition,

$$\hat{a}_{TZ} = \left( - \right)\frac{{\theta_{LZ} }}{{\theta_{TZ} }}\hat{a}_{LZ} .$$

(D.8)

Putting this in (D.7),

$$\hat{a}_{LZ} = - \left( {\alpha + 1} \right)\hat{w}\sigma_{Z} = - \sigma_{Z} \left( {\alpha + 1} \right)\frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{LS '} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }}.$$

(D.9)

Therefore, from (D.8),

$$\hat{a}_{TZ} = \frac{{\theta_{LZ} }}{{\theta_{TZ} }}\sigma_{Z} \left( {\alpha + 1} \right)\frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{LS '} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }}.$$

(D.10)

Now, because of change in $\delta ,\alpha$ and the input coefficients in $Z$, output of other sectors may also change. To calculate the change we express Eqs. (31) to (36) as follows:

$$\hat{X} = \frac{{\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right)}}{{\lambda_{SX} }}S',$$

(D.11)

$$\lambda_{KX} \hat{X} + \lambda_{KY} \hat{Y} + \lambda_{{KS^{\prime}}} \hat{S}' = 0,$$

(D.12)

$$\lambda_{LY} \hat{Y} + \lambda_{LZ} \hat{Z} + \lambda_{{LS^{\prime}}} \hat{L}_{S} + \lambda_{LN} \hat{L}_{N} = - \lambda_{LZ} \hat{a}_{LZ} ,$$

(D.13)

$$\hat{L}_{S} = \hat{S} ' ,$$

(D.14)

$$\hat{Z} = - \hat{a}_{TZ} = \left( - \right)\frac{{\theta_{LZ} }}{{\theta_{TZ} }}\sigma_{Z} \left( {\alpha + 1} \right)\frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{LS '} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }}.$$

(D.15)

Putting the value of $\hat{X}$ in (D.12)

$$\lambda_{KY} \hat{Y} + \left\{ {\frac{{\lambda_{KX} }}{{\lambda_{SX} }}\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right) + \lambda_{{KS^{\prime}}} } \right\}\hat{S} '= 0.$$

(D.16)

Using Eqs. (31), (D.8) and (D.14), Equation (D.13) becomes

$$\lambda_{LY} \hat{Y} + \lambda_{{LS^{\prime}}} \hat{S}' = - \left\{ {\lambda_{LZ} + \left( {\lambda_{LZ} + \lambda_{LN} } \right)\frac{{\theta_{LZ} }}{{\theta_{TZ} }}} \right\}\hat{a}_{LZ} .$$

(D.17)

Writing (D.16) and (D.17) in matrix form:

$$\left( {\begin{array}{*{20}c} {\lambda_{KY} } & {\left\{ {\frac{{\lambda_{KX} }}{{\lambda_{SX} }}\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right) + \lambda_{{KS^{\prime}}} } \right\}} \\ {\lambda_{LY} } & {\lambda_{{LS^{\prime}}} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\hat{Y}} \\ {\hat{S}'} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 \\ { - \left\{ {\lambda_{LZ} + \left( {\lambda_{LZ} + \lambda_{LN} } \right)\frac{{\theta_{LZ} }}{{\theta_{TZ} }}} \right\}\hat{a}_{LZ} } \\ \end{array} } \right),$$

$$\left| {\lambda_{1} } \right| = \lambda_{KY} \lambda_{{LS^{\prime}}} - \lambda_{LY} \left\{ {\frac{{\lambda_{KX} }}{{\lambda_{SX} }}\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right) + \lambda_{{KS^{\prime}}} } \right\}.$$

Here, $\lambda_{KY} \lambda_{{LS^{\prime}}} - \lambda_{LY} \left\{ {\frac{{\lambda_{KX} }}{{\lambda_{SX} }}\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right) + \lambda_{{KS^{\prime}}} } \right\}$$= \frac{{YS^{\prime}2X}}{{\lambda_{SX} KLS}}\left[ {a_{KY} - a_{LY} \left\{ {a_{{KS^{\prime}}} + \left( {1 - a_{{SS^{\prime}}} } \right)} \right\}} \right] > 0;$ as $Y$ is $K$ intensive.

Using Cramer’s rule

$$\begin{aligned} \hat{Y} = & \left( - \right)\frac{1}{{\left| {\lambda_{1} } \right|}}\left\{ {\frac{{\lambda_{KX} }}{{\lambda_{SX} }}\left( {\lambda_{{S^{\prime}S}} - \lambda_{{SS^{\prime}}} } \right) + \lambda_{{KS^{\prime}}} } \right\}\left\{ {\lambda_{LZ} + \left( {\lambda_{LZ} + \lambda_{LN} } \right)\frac{{\theta_{LZ} }}{{\theta_{TZ} }}} \right\} \hfill \\ & \left\{ {\sigma_{Z} \left( {\alpha + 1} \right)\frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{{LS^{\prime}}} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }}} \right\} < 0. \hfill \\ \end{aligned}$$

And,

$$\hat{S}' = \frac{1}{{\left| {\lambda_{1} } \right|}}\lambda_{KY} \left\{ {\lambda_{LZ} + \left( {\lambda_{LZ} + \lambda_{LN} } \right)\frac{{\theta_{LZ} }}{{\theta_{TZ} }}} \right\}\left\{ {\sigma_{Z} \left( {\alpha + 1} \right)\frac{{\left( {1 - \theta_{{SS^{\prime}}} } \right)}}{{\theta_{{LS^{\prime}}} }}\frac{{\hat{\delta }\delta }}{{\theta_{SX} }}} \right\} > 0.$$