Pollution control, worker productivity, and wage inequality

Abstract

Poor environmental quality can reduce worker productivity, but how this effect is associated with skilled-unskilled wage inequality is still unclear. This paper studies how stricter pollution control impacts wage inequality in the presence of such a worker productivity effect by establishing general equilibrium models with two urban sectors and by conducting empirical analysis with country-level panel data. The models show that when the non-polluting sector is under perfect competition, wage inequality is closely related to the worker productivity effect, while when it is under monopolistic competition, productivity gains of skilled workers from stricter pollution control exhibit complete pass-through. Empirical analysis suggests that the worker productivity effect of pollution control is unskill-biased, which helps mitigate inequality.

Appendices

Appendix A: Proofs

Proof of Proposition 1

Writing Eqs. (7), (8), (13), and (14) in the matrix form, we have:

$$\begin{aligned} \left( {\begin{array}{*{20}{c}} {{\theta _{SX}}}&{}0&{}{{\theta _{KX}}}&{}0\\ 0&{}{{\theta _{UY}}}&{}{{\theta _{KY}}}&{}{{\theta _{EY}}}\\ {{\lambda _{KX}}{\sigma _{SKX}}}&{}{{\lambda _{KY}}{\sigma _{UKY}}}&{}{ - {A_1}}&{}0\\ 0&{}{{\sigma _{UEY}}}&{}0&{}{ - {\sigma _{UEY}}} \end{array}} \right) \left( {\begin{array}{*{20}{c}} {{{{\hat{w}}}_S}}\\ {{{{\hat{w}}}_U}}\\ {{\hat{r}}}\\ {{\hat{\rho }} } \end{array}} \right) = - \left( {\begin{array}{*{20}{c}} {{\varepsilon _S}{\theta _{SX}}}\\ {{\varepsilon _U}{\theta _{UY}}}\\ 0\\ { - 1} \end{array}} \right) {\hat{E}}, \end{aligned}$$

where ${A_1} = {\lambda _{KX}}{\sigma _{SKX}} + {\lambda _{KY}}{\sigma _{UKY}}$.

By solving the above equation, we can obtain:

$$\begin{aligned}{} & {} \frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} = \frac{{{\theta _{SX}}{\sigma _{UEY}}\left[ {\left( {{\theta _{EY}} + {\theta _{UY}}} \right) {\lambda _{KX}}{\sigma _{SKX}} + {\lambda _{KY}}{\sigma _{UKY}}} \right] }}{{{\Delta _1}}}\left[ {{f_1}\left( {{\varepsilon _U}} \right) - {\varepsilon _S}} \right] , \\{} & {} \frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} = \frac{{{\theta _{KY}}{\theta _{SX}}{\lambda _{KX}}{\sigma _{SKX}}{\sigma _{UEY}}}}{{{\Delta _1}}}\left[ {{\varepsilon _S} - {f_2}\left( {{\varepsilon _U}} \right) } \right] , \end{aligned}$$

where $\Delta _1 = \sigma _{UEY}[ {\left( {{\theta _{EY}} + {\theta _{UY}}} \right) {\lambda _{KX}}{\sigma _{SKX}} + {\theta _{SX}}{\lambda _{KY}}{\sigma _{UKY}}} ]$, $f_1( {{\varepsilon _U}} ) = {B_1}( {\varepsilon _U}{\theta _{UY}}{\sigma _{UEY}} - {\theta _{EY}} )$, $f_2( \varepsilon _U ) = B_2 ( \varepsilon _U\theta _{UY}\sigma _{UEY} - \theta _{EY} )$, ${B_1} = \frac{{{\theta _{KX}}{\lambda _{KY}}{\sigma _{UKY}}}}{{{\theta _{SX}}{\sigma _{UEY}}\left[ {\left( {{\theta _{EY}} + {\theta _{UY}}} \right) {\lambda _{KX}}{\sigma _{SKX}} + {\lambda _{KY}}{\sigma _{UKY}}} \right] }}$, and ${B_2} = \frac{{{\lambda _{KX}}{\sigma _{SKX}} + {\theta _{SX}}{\lambda _{KY}}{\sigma _{UKY}}}}{{{\theta _{KY}}{\theta _{SX}}{\lambda _{KX}}{\sigma _{SKX}}{\sigma _{UEY}}}}$.

Wage inequality can be expressed by:

$$\begin{aligned} \frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} = \frac{{A_1 {\theta _{SX}}{\sigma _{UEY}}}}{{{\Delta _1}}}\left[ {{f_3}\left( {{\varepsilon _U}} \right) - {\varepsilon _S}} \right] , \end{aligned}$$

where ${f_3}\left( {{\varepsilon _U}} \right) = {B_3}\left( {{\varepsilon _U}{\theta _{UY}}{\sigma _{UEY}} - {\theta _{EY}}} \right)$, and ${B_3} = \frac{1}{{{\theta _{SX}}{\sigma _{UEY}}}}$. We can find that ${B_1}< {B_3} < {B_2}$.

The implications of stricter pollution control on the skilled wage rate, the unskilled wage rate, and wage inequality are summarized as follows:

• 1. When ${\varepsilon _U} < \frac{{{\theta _{EY}}}}{{{\theta _{UY}}{\sigma _{UEY}}}}$, we have $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} < 0$, $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} > 0$, and $\frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$.

• 2. When ${\varepsilon _U} > \frac{{{\theta _{EY}}}}{{{\theta _{UY}}{\sigma _{UEY}}}}$, we have:

  • (a) If ${\varepsilon _S} < {f_1}\left( {{\varepsilon _U}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} > 0$, $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$, and $\frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} > 0$.

  • (b) If ${f_1}\left( {{\varepsilon _U}} \right)< {\varepsilon _S} < {f_3}\left( {{\varepsilon _U}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} < 0$, $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$, and $\frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} > 0$.

  • (c) If ${f_3}\left( {{\varepsilon _U}} \right)< {\varepsilon _S} < {f_2}\left( {{\varepsilon _U}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} < 0$, $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$, and $\frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$.

  • (d) If ${\varepsilon _S} > {f_2}\left( {{\varepsilon _U}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} < 0$, $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} > 0$, and $\frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$.

$\square$

Proof of Proposition 2

Stricter pollution control reduces the production cost of the skilled sector, which is simply because $u'\left( E \right) > 0$. ${\varepsilon _S}$ measures the productivity effect of pollution control on skilled workers. It does not enter Eqs. (26) and (28), and it also does not enter the other equations in the economic system (i.e., Eqs. (4), (10), (12), and (22)).

Since Eqs. (4), (10), (12), (22), (26), and (28) pin down ${w_S}$, ${w_U}$, r, $\rho$, n, and Y, these variables will not be affected by ${\varepsilon _S}$. When ${w_S}$, ${w_U}$, r, $\rho$, n, and Y are solved from these equations, we can solve p from Eq. (23), which shows that p will be negatively affected by ${\varepsilon _S}$ originating from stricter pollution control (i.e., a reduction in E). From Eqs. (23) and (24), we have ${\hat{x}} = {\hat{r}} - {{\hat{w}}_S} - {\varepsilon _S}{\hat{E}}$. Hence, x will be positively affected by ${\varepsilon _S}$. Since new firms will enter the skilled sector, the results show that the productivity effect on skilled workers will decrease the relative price but increase output of the non-polluting good. $\square$

Proof of Proposition 3

Writing Eqs. (8), (14), (28), and (29) in the matrix form, we have:

$$\begin{aligned} \left( {\begin{array}{*{20}{c}} {{A_2}}&{}{ - {\alpha _U}}&{}{ - {\alpha _K}}&{}{ - {\alpha _E}}\\ 0&{}{{\theta _{UY}}}&{}{{\theta _{KY}}}&{}{{\theta _{EY}}}\\ {{\lambda _{KX}}}&{}{{\lambda _{KY}}{\sigma _{UKY}}}&{}{ - {A_3}}&{}0\\ 0&{}{{\sigma _{UEY}}}&{}0&{}{ - {\sigma _{UEY}}} \end{array}} \right) \left( {\begin{array}{*{20}{c}} {{{{\hat{w}}}_S}}\\ {{{{\hat{w}}}_U}}\\ {{\hat{r}}}\\ {{\hat{\rho }} } \end{array}} \right) = - \left( {\begin{array}{*{20}{c}} { - {\alpha _E}}\\ {{\varepsilon _U}{\theta _{UY}}}\\ 0\\ { - 1} \end{array}} \right) {\hat{E}}, \end{aligned}$$

where ${A_2} = {\alpha _E} + {\alpha _K} + {\alpha _U}$, and ${A_3} = {\lambda _{KX}} + {\lambda _{KY}}{\sigma _{UKY}}$.

By solving the above equation, we can obtain:

$$\begin{aligned}{} & {} \frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} = {\theta _{UY}}\left[ {{g_1}\left( {{\sigma _{UEY}}} \right) - {\varepsilon _U}} \right] , \\{} & {} \frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} = {\theta _{UY}}\left[ {{g_2}\left( {{\sigma _{UEY}}} \right) - {\varepsilon _U}} \right] , \end{aligned}$$

where $g_1( \sigma _{UEY} ) = \frac{\alpha _E[ ( \theta _{EY} + \theta _{UY} )\lambda _{KX} + \lambda _{KY}\sigma _{UKY} ]}{\theta _{UY}\Delta _2}\sigma _{UEY} + C_1$, ${g_2}\left( {{\sigma _{UEY}}} \right) = - \frac{{{\alpha _E}{\theta _{KY}}{\lambda _{KX}}}}{{{\theta _{UY}}{\Delta _2}}}{\sigma _{UEY}} + {C_2}$, $\Delta _2 = \sigma _{UEY}[ ( \alpha _E + \alpha _U )\lambda _{KX} + A_2 \lambda _{KY}\sigma _{UKY} ]$, $C_1 = \frac{1}{\theta _{UY}\Delta _2}\{ ( \alpha _U\theta _{EY} - \alpha _E\theta _{UY} )\lambda _{KX} + [ ( {\alpha _K} + {\alpha _U} )\theta _{EY} - \alpha _E( \theta _{KY} + \theta _{UY} ) ]\lambda _{KY}\sigma _{UKY} \}$, and $C_2 = \frac{1}{\theta _{UY}\Delta _2}\{ [ ( \alpha _E + \alpha _U )\theta _{EY} + \alpha _E\theta _{KY} ]\lambda _{KX} + A_2 \theta _{EY}\lambda _{KY}\sigma _{UKY} \}$.

Wage inequality can be expressed by:

$$\begin{aligned} \frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}}&= {\theta _{UY}}\left[ {{g_1}\left( {{\sigma _{UEY}}} \right) - {g_2}\left( {{\sigma _{UEY}}} \right) } \right] \nonumber \\ {}&= \frac{{{\alpha _E} A_3}}{{{\Delta _2}}}\left( {{\sigma _{UEY}} - 1} \right) . \end{aligned}$$

(A.1)

The implications of stricter pollution control on the skilled wage rate, the unskilled wage rate, and wage inequality are summarized as follows:

• 1. When ${\sigma _{UEY}} > 1$, we have $\frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} > 0$. Since ${g_1}\left( {{\sigma _{UEY}}} \right) > {g_2}\left( {{\sigma _{UEY}}} \right)$, we also have:

  • (a) If ${\varepsilon _U} < {g_2}\left( {{\sigma _{UEY}}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} > 0$, and $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} > 0$.

  • (b) If ${g_2}\left( {{\sigma _{UEY}}} \right)< {\varepsilon _U} < {g_1}\left( {{\sigma _{UEY}}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} > 0$, and $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$.

  • (c) If ${\varepsilon _U} > {g_1}\left( {{\sigma _{UEY}}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} < 0$, and $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$.

• 2. When ${\sigma _{UEY}} < 1$, we have $\frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$. Since ${g_1}\left( {{\sigma _{UEY}}} \right) < {g_2}\left( {{\sigma _{UEY}}} \right)$, we also have:

  • (a) If ${\varepsilon _U} < {g_1}\left( {{\sigma _{UEY}}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} > 0$, and $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} > 0$.

  • (b) If ${g_1}\left( {{\sigma _{UEY}}} \right)< {\varepsilon _U} < {g_2}\left( {{\sigma _{UEY}}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} < 0$, and $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} > 0$.

  • (c) If ${\varepsilon _U} > {g_2}\left( {{\sigma _{UEY}}} \right)$, then $\frac{{{{{\hat{w}}}_S}}}{{{\hat{E}}}} < 0$, and $\frac{{{{{\hat{w}}}_U}}}{{{\hat{E}}}} < 0$.

$\square$

Appendix B: Extension

In this appendix, we extend the model presented in Sect. 3 by assuming the production function of the non-polluting skilled firm is $x=F(ul,k)$, where u is skilled worker productivity, $u^\prime (E)<0$, l is skilled labor input, k is capital input, and F(.) satisfies strict quasi-concavity and linear homogeneity. In other words, we maintain consistency with the basic model in terms of the production function. We also need to assume that there is no fixed cost, and the number of firms is given. Otherwise, maintaining free entry would contradict the assumption of constant returns to scale. In this extension, we find that both Propositions 2 and 3 may still hold.

Specifically, Eq. (18) now becomes:

$$\begin{aligned} p\left( {1 - \frac{1}{\sigma }} \right) = {a_{SX}}{w_S} + {a_{KX}}r. \end{aligned}$$

The price p is still endogenously determined by Eq. (19), but the profit $\Pi$ is not zero, which is calculated as:

$$\begin{aligned} \mathrm{{\Pi }} = \left[ {p - \left( {{a_{SX}}{w_S} + {a_{KX}}r} \right) } \right] X = \frac{\left( {{a_{SX}}{w_S} + {a_{KX}}r} \right) X}{ {\sigma - 1} }. \end{aligned}$$

Equations (4), (9)–(12) still hold, with $X=nx$. By solving for the new economic system, we have the matrix as follows:

$$\begin{aligned} \left( {\begin{array}{*{20}{c}} B&{}{ - {\alpha _U}}&{}{ - {\beta _K}}&{}{ - {\alpha _E}}\\ 0&{}{{\theta _{UY}}}&{}{{\theta _{KY}}}&{}{{\theta _{EY}}}\\ {{\lambda _{KX}}{\sigma _{SKX}}}&{}{{\lambda _{KY}}{\sigma _{UKY}}}&{}{ - {A_1}}&{}0\\ 0&{}{{\sigma _{UEY}}}&{}0&{}{ - {\sigma _{UEY}}} \end{array}} \right) \left( {\begin{array}{*{20}{c}} {{{{\hat{w}}}_S}}\\ {{{{\hat{w}}}_U}}\\ {{\hat{r}}}\\ {{\hat{\rho }} } \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{\alpha _E}}\\ { - {\varepsilon _U}{\theta _{UY}}}\\ 0\\ 1 \end{array}} \right) {\hat{E}}, \end{aligned}$$

(B.1)

where $B = \left( {1 - {\alpha _P}} \right) \left( {{\theta _{SX}} + {\theta _{KX}}{\sigma _{SKX}}} \right) - {\alpha _S}$, ${\beta _K} = {\alpha _K} + \left( {1 - {\alpha _P}} \right) \left( {{\sigma _{SKX}} - 1} \right) {\theta _{KX}}$, and $\alpha _P$ is the ratio of profits to total income. The determinant of the coefficient matrix can be calculated as ${\Delta _3} = {\sigma _{UEY}}\left[ {B{\lambda _{KY}}{\sigma _{UKY}} + \left( {{\alpha _E} + {\alpha _U}} \right) {\lambda _{KX}}{\sigma _{SKX}}} \right]$. Wage inequality can be expressed by:

$$\begin{aligned} \frac{{{{{\hat{w}}}_S} - {{{\hat{w}}}_U}}}{{{\hat{E}}}} = \frac{{{\alpha _E}{A_1}}}{{{\Delta _3}}}\left( {{\sigma _{UEY}} - 1} \right) , \end{aligned}$$

which is almost the same as Eq. (A.1) presented in Proof of Proposition 3.

Therefore, we can conclude that Proposition 3 can likely hold when skilled firms adopt constant returns to scale technologies but compete imperfectly. Moreover, since $\varepsilon _S$ does not enter Eq. (B.1), pollution control does not impact the skilled wage rate via the skilled worker productivity channel, and thus, Proposition 2 can also likely hold.