Skill formation, environmental pollution, and wage inequality

Abstract

This paper studies the effects of government-led skill formation on environmental quality and the skilled–unskilled wage gap through the three-sector general equilibrium approach. The process of skill formation is divided into two periods. In the basic model, an increase in the subsidy on skill formation can improve the environment in both periods, but may widen or narrow down the wage gap, depending on the relative capital intensities of the urban sectors, and on the environmental dependence of the rural sector. In the presence of urban unemployment, the main results of the basic model still more or less hold. This paper suggests that, although the subsidy on skill formation does not necessarily reduce wage inequality, it does serve as an effective policy to improve environmental quality.

Appendices

Appendix 1: Proof of Proposition 1

The determinant of the coefficient matrix of Eq. (15), which is denoted as $\Delta_{1}$, cannot be judged by direct calculation. However, the adjustment processes in Appendix E suggest that stability requires $\Delta_{1}$ to be negative.

We obtain the following results by using the Cramer’s rule to solve Eq. (15):

$$ \begin{aligned} \frac{{\hat{w}_{\text{S}} }}{{\hat{s}}} & = - \frac{{e_{s} \theta_{{{\text{K}}X}} \theta_{{{\text{T}}Z}} \theta_{{{\text{U}}Y}} \lambda_{{{\text{K}}Y}}^{2} }}{{\Delta_{1} }} > 0, \\ \frac{{\hat{w}_{\text{U}} }}{{\hat{s}}} & = - \frac{{e_{s} \theta_{{{\text{K}}Y}} \theta_{{{\text{S}}X}} \theta_{{{\text{T}}Z}} \lambda_{{{\text{K}}Y}}^{2} }}{{\Delta_{1} }} > 0, \\ \frac{{\hat{E}}}{{\hat{s}}} & = - \frac{{e_{s} \theta_{{{\text{T}}Z}} \theta_{{{\text{U}}Y}} \lambda_{{{\text{K}}Y}} \lambda_{Y} \left( {\lambda_{{{\text{K}}X}} \sigma_{X} + \theta_{{{\text{S}}X}} \theta_{{{\text{T}}Z}} \theta_{{{\text{U}}Y}} \lambda_{{{\text{K}}Y}} \sigma_{Y} } \right)}}{{\Delta_{1} }} > 0. \\ \end{aligned} $$

The first part of Proposition 1 can be verified from $\frac{{\hat{E}}}{{\hat{s}}} > 0$. As for the second part, we can calculate wage inequality as follows:

$$ \frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} = - \frac{{e_{s} \theta_{{{\text{T}}Z}} \lambda_{{{\text{K}}Y}}^{2} \left( {\theta_{{{\text{K}}X}} - \theta_{{{\text{K}}Y}} } \right)}}{{\Delta_{1} }}, $$

which implies that if $\theta_{{{\text{K}}X}} > \theta_{{{\text{K}}Y}}$, then $\frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} > 0$; and if $\theta_{{{\text{K}}X}} < \theta_{{{\text{K}}Y}}$, then $\frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} < 0$.

Appendix 2: Proof of Proposition 2

With the help of adjustment processes similar to Appendix E, we can know that stability requires the determinant of the coefficient matrix of Eq. (16) (denoted as $\Delta_{2}$) to be negative.

We have the following results by using the Cramer’s rule to solve Eq. (16):

$$ \begin{aligned} \frac{{\hat{w}_{\text{S}} }}{{\hat{s}}} & = - \frac{{e_{s} \theta_{{{\text{K}}X}} \theta_{{{\text{U}}Y}} \lambda_{{{\text{K}}Y}} \left( {\varepsilon \lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} - \theta_{{{\text{T}}Z}} C} \right)}}{{\Delta_{2} }}, \\ \frac{{\hat{w}_{\text{U}} }}{{\hat{s}}} & = - \frac{{e_{s} \theta_{{{\text{K}}Y}} \theta_{{{\text{S}}X}} \lambda_{{{\text{K}}Y}} \left( {\varepsilon \lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} - \theta_{{{\text{T}}Z}} C} \right)}}{{\Delta_{2} }}, \\ \frac{{\hat{E}}}{{\hat{s}}} & = - \frac{{e_{s} \lambda_{Y} \left\{ {\theta_{{{\text{T}}Z}} \left[ {\theta_{{{\text{U}}Y}} \lambda_{{{\text{K}}X}} \sigma_{X} + \theta_{{{\text{S}}X}} \lambda_{{{\text{K}}Y}} \left( {\theta_{{{\text{U}}Y}} + \theta_{{{\text{K}}Y}} \lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Y}} \lambda_{L} } \right)\sigma_{Y} } \right] + \theta_{{{\text{K}}Y}} \theta_{{{\text{S}}X}} \lambda_{{{\text{K}}X}} \lambda_{{{\text{K}}Y}} \lambda_{{{\text{U}}Z}} \lambda_{L} \sigma_{Z} } \right\}}}{{\Delta_{2} }} > 0. \\ \end{aligned} $$

We immediately obtain the first part of Proposition 2. As for the second part and the third part, wage inequality can be expressed as follows:

$$ \frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} = - \frac{{e_{s} \lambda_{{{\text{K}}Y}} \left( {\theta_{{{\text{K}}X}} - \theta_{{{\text{K}}Y}} } \right)\left( {\varepsilon \lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} - \theta_{{{\text{T}}Z}} C} \right)}}{{\Delta_{2} }}, $$

which implies that when $\varepsilon > \frac{{\theta_{{{\text{T}}Z}} C}}{{\lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} }}$, if $\theta_{{{\text{K}}X}} > \theta_{{{\text{K}}Y}}$, then $\frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} > 0$, and if $\theta_{{{\text{K}}X}} < \theta_{{{\text{K}}Y}}$, then $\frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} < 0$; and when $\varepsilon < \frac{{\theta_{{{\text{T}}Z}} C}}{{\lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} }}$, if $\theta_{{{\text{K}}X}} < \theta_{{{\text{K}}Y}}$, then $\frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} > 0$, and if $\theta_{{{\text{K}}X}} > \theta_{{{\text{K}}Y}}$, then $\frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} < 0$.

Appendix 3: Proof of Proposition 3

By calculation and simplification, the determinant of the coefficient matrix of Eq. (20), which we denote as $\Delta_{3}$, is negative:

$$ \Delta_{3} = - \theta_{{{\text{S}}X}} \theta_{{{\text{K}}Y}} \lambda_{{{\text{K}}Y}}^{2} \left( {e_{{w_{\text{U}} }} \theta_{{{\text{T}}Z}} + \theta_{{{\text{T}}Z}} \lambda_{{{\text{U}}Y}} + \lambda_{{{\text{U}}Z}} \sigma_{Z} } \right) < 0. $$

We get the following results by using the Cramer’s rule to solve Eq. (20):

$$ \begin{aligned} \frac{{\hat{w}_{\text{S}} }}{{\hat{s}}} & = 0, \\ \frac{{\hat{w}_{\text{U}} }}{{\hat{s}}} & = - \frac{{e_{s} \theta_{{{\text{K}}Y}} \theta_{{{\text{S}}X}} \theta_{{{\text{T}}Z}} \lambda_{{{\text{K}}Y}}^{2} }}{{\Delta_{3} }} > 0, \\ \frac{{\hat{E}}}{{\hat{s}}} & = 0. \\ \end{aligned} $$

The first part of Proposition 3 is straightforward. As for the second part and the third part, note that:

$$ \begin{aligned} \frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} & = \frac{{e_{s} \theta_{{{\text{K}}Y}} \theta_{{{\text{S}}X}} \theta_{{{\text{T}}Z}} \lambda_{{{\text{K}}Y}}^{2} }}{{\Delta_{3} }} < 0, \\ \frac{{\hat{\mu }}}{{\hat{s}}} & = - \frac{1 + \mu }{\mu } \cdot \frac{{\hat{w}_{\text{U}} }}{{\hat{s}}} < 0, \\ \end{aligned} $$

where the second equation is derived from Eq. (17).

Appendix 4: Proof of Proposition 4

The adjustment processes similar to Appendix E suggest that stability requires the determinant of the coefficient matrix of Eq. (21) (denoted as $\Delta_{4}$) to be negative.

We obtain the following results by using the Cramer’s rule to solve Eq. (21):

$$ \begin{aligned} \frac{{\hat{w}_{\text{S}} }}{{\hat{s}}} & = 0, \\ \frac{{\hat{w}_{\text{U}} }}{{\hat{s}}} & = - \frac{{e_{s} \theta_{{{\text{K}}Y}} \theta_{{{\text{S}}X}} \lambda_{{{\text{K}}Y}} \left( {\varepsilon \lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} - \theta_{{{\text{T}}Z}} C} \right)}}{{\Delta_{4} }}, \\ \frac{{\hat{E}}}{{\hat{s}}} & = - \frac{{\theta_{{{\text{S}}X}} \theta_{{{\text{K}}Y}} \lambda_{{{\text{K}}X}} \lambda_{Y} \lambda_{{{\text{K}}Y}} \lambda_{L} e_{s} \left( {\theta_{{{\text{T}}Z}} \lambda_{{{\text{U}}Y}} + \lambda_{{{\text{U}}Z}} \sigma_{Z} } \right)}}{{\Delta_{4} }} > 0. \\ \end{aligned} $$

We immediately have the first part of Proposition 4. As for the second part, note that:

$$ \begin{aligned} \frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} & = \frac{{e_{s} \theta_{{{\text{K}}Y}} \theta_{{{\text{S}}X}} \lambda_{{{\text{K}}Y}} \left( {\varepsilon \lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} - \theta_{{{\text{T}}Z}} C} \right)}}{{\Delta_{4} }}, \\ \frac{{\hat{\mu }}}{{\hat{s}}} & = - \frac{1 + \mu }{\mu } \cdot \frac{{\hat{w}_{\text{U}} }}{{\hat{s}}} = \frac{1 + \mu }{\mu } \cdot \frac{{e_{s} \theta_{{{\text{K}}Y}} \theta_{{{\text{S}}X}} \lambda_{{{\text{K}}Y}} \left( {\varepsilon \lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} - \theta_{{{\text{T}}Z}} C} \right)}}{{\Delta_{4} }}, \\ \end{aligned} $$

which implies that if $\varepsilon > \frac{{\theta_{{{\text{T}}Z}} C}}{{\lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} }}$, then $\frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} < 0$ and $\frac{{\hat{\mu }}}{{\hat{s}}} < 0$; and if $\varepsilon < \frac{{\theta_{{{\text{T}}Z}} C}}{{\lambda_{{{\text{K}}X}} \lambda_{{{\text{U}}Z}} \lambda_{L} \lambda_{Y} \sigma_{Z} }}$, then $\frac{{\hat{w}_{\text{S}} - \hat{w}_{\text{U}} }}{{\hat{s}}} > 0$ and $\frac{{\hat{\mu }}}{{\hat{s}}} > 0$.

Appendix 5: The adjustment processes

The methodology used here is in line with Beladi et al. (2008). According to Eqs. (3), (7), and (9)–(14), we can specify the adjustment processes as follows:

$$ \dot{X} = d_{1} \left( {p_{X} - a_{{{\text{S}}X}} w_{\text{S}} - a_{{{\text{K}}X}} r} \right), $$

(22)

$$ \dot{Y} = d_{2} \left( {p_{Y} - a_{{{\text{U}}Y}} w_{\text{U}} - a_{{{\text{K}}Y}} r} \right), $$

(23)

$$ \dot{Z} = d_{3} \left( {1 - a_{{{\text{U}}Z}} w_{\text{U}} - a_{{{\text{T}}Z}} \tau } \right), $$

(24)

$$ \dot{E} = d_{4} \left( {\bar{E} - \lambda Y - E} \right), $$

(25)

$$ \dot{w}_{\text{S}} = d_{5} \left( {a_{{{\text{S}}X}} X - \bar{L}_{\text{S}} } \right), $$

(26)

$$ \dot{w}_{\text{U}} = d_{6} \left[ {a_{{{\text{U}}Y}} Y + a_{{{\text{U}}Z}} Z - L\left( {w_{\text{U}} ,w_{\text{S}} ,s} \right)} \right], $$

(27)

$$ \dot{r} = d_{7} \left( {a_{{{\text{K}}X}} X + a_{{{\text{K}}Y}} Y - \bar{K}} \right), $$

(28)

$$ \dot{\tau } = d_{8} \left( {a_{{{\text{T}}Z}} Z - \bar{T}} \right), $$

(29)

where the dot notation over a variable represents the time derivative (e.g., $\dot{X} = \frac{{{\text{d}}X}}{{{\text{d}}t}}$), $d_{i} > 0$$\left( {i = 1,2, \ldots ,8} \right)$ stands for the speed of adjustments, and the mathematic expressions in the brackets of the right-hand sides of Eqs. (22)–(29) denote the excess demand functions.¹⁷ Taking linear approximations of Eqs. (22)–(29) around the equilibrium values of the variables, and denoting the Jacobian as J, we have:

$$ J = Q_{1} \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & { - \theta_{{{\text{S}}X}} } & 0 & { - \theta_{{{\text{K}}X}} } & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \theta_{{{\text{U}}Y}} } & { - \theta_{{{\text{K}}Y}} } & 0 \\ 0 & 0 & 0 & \varepsilon & 0 & { - \theta_{{{\text{U}}Z}} } & 0 & { - \theta_{{{\text{T}}Z}} } \\ 0 & { - \lambda_{Y} } & 0 & { - 1} & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & { - \theta_{{{\text{K}}X}} \sigma_{X} } & 0 & {\theta_{{{\text{K}}X}} \sigma_{X} } & 0 \\ 0 & {\lambda_{{{\text{U}}Y}} } & {\lambda_{{{\text{U}}Z}} } & 0 & {e_{{w_{\text{S}} }} } & { - C_{1} } & {\lambda_{{{\text{U}}Y}} \theta_{{{\text{K}}Y}} \sigma_{Y} } & {\lambda_{{{\text{U}}Z}} \theta_{{{\text{T}}Z}} \sigma_{Y} } \\ {\lambda_{{{\text{K}}X}} } & {\lambda_{{{\text{K}}Y}} } & 0 & 0 & {\lambda_{{{\text{K}}X}} \theta_{{{\text{S}}X}} \sigma_{X} } & {\lambda_{{{\text{K}}Y}} \theta_{{{\text{U}}Y}} \sigma_{Y} } & { - C_{2} } & 0 \\ 0 & 0 & 1 & 0 & 0 & {\theta_{{{\text{U}}Z}} \sigma_{Z} } & 0 & { - \theta_{{{\text{U}}Z}} \sigma_{Z} } \\ \end{array} } \right)Q_{2} , $$

where $C_{1} = \lambda_{{{\text{U}}Y}} \theta_{{{\text{K}}Y}} \sigma_{Y} + \lambda_{{{\text{U}}Z}} \theta_{{{\text{T}}Z}} \sigma_{Z} + e_{{W_{\text{U}} }} > 0$, $C_{2} = \lambda_{{{\text{K}}X}} \theta_{{{\text{S}}X}} \sigma_{X} + \lambda_{{{\text{K}}Y}} \theta_{{{\text{U}}Y}} \sigma_{Y} > 0$, $Q_{1} = {\text{diag}}(d_{1} p_{X} ,d_{2} p_{Y} ,d_{3} ,d_{4} E,d_{5} \bar{L}_{\text{S}} ,d_{6} L,d_{7} \bar{K},d_{8} \bar{T})$, and $Q_{2} = {\text{diag}}(1/X,1/Y,1/Z,1/E,1/w_{\text{S}} ,1/w_{\text{U}} ,1/r,1/\tau )$.

Stability requires all eigenvalues of the coefficient matrix of the linearized system to have negative real parts (see, e.g., Dixit 1986, p. 110; Keen and Kotsogiannis 2004, p. 406). This implies that the trace is negative, and the sign of the determinant of the coefficient matrix takes the form of ${\text{sign}}\left| J \right| = \left( { - 1} \right)^{n}$, where n is the order of J (see Beladi et al. 2008, p. 901). For the matrix discussed above, first, since the diagonal elements are either zero or negative, the trace is always negative. Second, $n = 8$, and thus, stability requires $\left| J \right| > 0$, where $\left| J \right|$ is as follows:

$$ \left| J \right| = \varphi \left| {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & { - \theta_{{{\text{S}}X}} } & 0 & { - \theta_{{{\text{K}}X}} } & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \theta_{{{\text{U}}Y}} } & { - \theta_{{{\text{K}}Y}} } & 0 \\ 0 & 0 & 0 & \varepsilon & 0 & { - \theta_{{{\text{U}}Z}} } & 0 & { - \theta_{{{\text{T}}Z}} } \\ 0 & { - \lambda_{Y} } & 0 & { - 1} & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & { - \theta_{{{\text{K}}X}} \sigma_{X} } & 0 & {\theta_{{{\text{K}}X}} \sigma_{X} } & 0 \\ 0 & {\lambda_{{{\text{U}}Y}} } & {\lambda_{{{\text{U}}Z}} } & 0 & {e_{{w_{\text{S}} }} } & { - C_{1} } & {\lambda_{{{\text{U}}Y}} \theta_{{{\text{K}}Y}} \sigma_{Y} } & {\lambda_{{{\text{U}}Z}} \theta_{{{\text{T}}Z}} \sigma_{Y} } \\ {\lambda_{{{\text{K}}X}} } & {\lambda_{{{\text{K}}Y}} } & 0 & 0 & {\lambda_{{{\text{K}}X}} \theta_{{{\text{S}}X}} \sigma_{X} } & {\lambda_{{{\text{K}}Y}} \theta_{{{\text{U}}Y}} \sigma_{Y} } & { - C_{2} } & 0 \\ 0 & 0 & 1 & 0 & 0 & {\theta_{{{\text{U}}Z}} \sigma_{Z} } & 0 & { - \theta_{{{\text{U}}Z}} \sigma_{Z} } \\ \end{array} } \right|, $$

where $\varphi = \prod\nolimits_{i = 1}^{8} {d_{i} } \frac{{p_{X} p_{Y} \bar{L}_{\text{S}} L\bar{K}\bar{T}}}{{XYZw_{\text{S}} w_{\text{U}} r\tau }} > 0$. By adding a scalar multiple of one row to another row, and then interchanging the columns, we can obtain:

$$ \left| J \right| = - \frac{\varphi }{{\lambda_{{{\text{K}}Y}} }}\left| {\begin{array}{*{20}c} { - \theta_{{{\text{S}}X}} } & 0 & 0 & { - \theta_{{{\text{K}}X}} } & 0 \\ 0 & { - \theta_{{{\text{U}}Y}} } & 0 & { - \theta_{{{\text{K}}Y}} } & 0 \\ 0 & { - \theta_{{{\text{U}}Z}} } & { - \theta_{{{\text{T}}Z}} } & 0 & \varepsilon \\ {\lambda_{{{\text{K}}X}} \lambda_{Y} \sigma_{X} } & {\theta_{{{\text{U}}Y}} \lambda_{{{\text{K}}Y}} \lambda_{Y} \sigma_{Y} } & 0 & { - A_{1} } & { - \lambda_{{{\text{K}}Y}} } \\ {A_{2} } & { - A_{3} } & {\lambda_{{{\text{K}}Y}} \lambda_{{{\text{U}}Z}} \sigma_{Z} } & {A_{4} } & 0 \\ \end{array} } \right| = - \frac{\varphi }{{\lambda_{{{\text{K}}Y}} }}\Delta_{1} , $$

where $A_{i}$$\left( {i = 1,2,3,4} \right)$ is defined behind Eq. (15), and $\Delta_{1}$ is the determinant of the coefficient matrix of Eq. (15). Hence, stability in turn requires $\Delta_{1} < 0$.