Can urban pollution shrink rural districts?

Abstract

This paper discusses how the externality of environmental damage affects the equilibrium properties of a simple overlapping generations model with multiple regions. Simulation results indicate that the environmental policy of the government decreases capital accumulation. When the government imposes an environmental tax on the urban sector, the urban-to-rural population ratio decreases, whereas the total fertility rate increases.

Appendix A: Comparative statics of (25)

Applying the implicit function theorem to (25) obtains

$$\begin{aligned} \frac{d\hat{k}}{d \tau }=\frac{\Phi }{\Psi }, \end{aligned}$$

(26)

where

$$\begin{aligned} \Phi&= \gamma \left( 1- \sigma \theta \hat{k}^{\delta } \right) \left[ \theta ^{1-\alpha } \hat{k}^{\alpha + \left( 1- \alpha \right) \delta } \left( - \left( 1- \alpha \right) -b \left( 1- \gamma \right) \theta ^{-1-\alpha }\right. \right. \nonumber \\&+\left. \left. \left( 2\!-\! \alpha \right) b \left( 1\!-\! \gamma \right) \tau \frac{\partial \theta }{\partial \tau } \right) \!+\! \hat{k}^{\alpha (1\!-\! \delta )}\theta ^{- \alpha } \left( \left( 1\!-\! \alpha \right) \frac{\partial \theta }{\partial \tau } b \left( 1\!-\! \gamma \right) \tau \!+\! \theta b \left( 1- \gamma \right) \right) \right] \nonumber \\&+\left[ \left( \left( 1\!-\! \tau \right) \left( 1\!-\! \alpha \right) \!-\!b \left( 1\!-\! \gamma \right) \tau \theta \right) \theta ^{1\!-\!\alpha } \hat{k}^{\alpha \!+\! \left( 1- \alpha \right) \delta } + \theta ^{1-\alpha } b \left( 1\!-\! \gamma \right) \tau \hat{k}^{\alpha \left( 1\!-\! \delta \right) } \right] \sigma \frac{\partial \theta }{\partial \tau } \hat{k}^{\delta },\nonumber \\ \end{aligned}$$

(27)

and

$$\begin{aligned} \Psi&= \frac{\left[ \gamma \left( 1- \sigma \theta \hat{k}^{\delta } \right) \right] ^{2}}{M} -\gamma \left( 1- \sigma \theta \hat{k}^{\delta } \right) \Bigl [ \left( \left( 1- \tau \right) \left( 1- \alpha \right) -b \left( 1- \gamma \right) \tau \theta \right) \theta ^{1- \alpha }\nonumber \\&\left( \alpha + (1- \alpha ) \delta \right) \hat{k}^{\alpha + (1- \alpha ) \delta -1} +\alpha (1- \delta ) \theta ^{1- \alpha } b \left( 1- \gamma \right) \tau \hat{k}^{\alpha (1- \delta ) -1}\Bigr ]\nonumber \\&-\left[ \left( \left( 1\!-\! \tau \right) \left( 1\!-\! \alpha \right) \!-\!b \left( 1 \!-\! \gamma \right) \tau \theta \right) \theta ^{1-\alpha } \hat{k}^{\alpha + \left( 1- \alpha \right) \delta } \!+\! \theta ^{1-\alpha } b \left( 1\!-\! \gamma \right) \tau \hat{k}^{\alpha \left( 1\!-\! \delta \right) } \right] \sigma \theta \delta \hat{k}^{\delta -1}.\nonumber \\ \end{aligned}$$

(28)

The sign of (26) is ambiguous. Thus, we determine the sign of (26). From the numerator of (26), $\Phi$, as $\left( 1- \tau \right) \left( 1- \alpha \right) -b \left( 1- \gamma \right) \tau \theta >0$, if $\alpha$ is less than $\tau$, $\Phi$ is negative. On the other hand, on denominator of (26), $\Psi$, if

$$\begin{aligned}&\frac{\left[ \gamma \left( 1- \sigma \theta \hat{k}^{\delta } \right) \right] ^{2}}{M}>\gamma \left( 1- \sigma \theta \hat{k}^{\delta } \right) \Bigl [ \left( \left( 1- \tau \right) \left( 1- \alpha \right) -b \left( 1- \gamma \right) \tau \theta \right) \theta ^{1- \alpha }\nonumber \\&\quad \left( \alpha + (1- \alpha ) \delta \right) \hat{k}^{\alpha + (1- \alpha ) \delta -1} +\alpha (1- \delta ) \theta ^{1- \alpha } b \left( 1- \gamma \right) \tau \hat{k}^{\alpha (1- \delta ) -1}\Bigr ]\nonumber \\&\quad +\left[ \left( \left( 1\!-\! \tau \right) \left( 1\!-\! \alpha \right) \!-\!b \left( 1\!-\! \gamma \right) \tau \theta \right) \theta ^{1-\alpha } \hat{k}^{\alpha + \left( 1\!-\! \alpha \right) \delta } \!+\! \theta ^{1-\alpha } b \left( 1\!-\! \gamma \right) \tau \hat{k}^{\alpha \left( 1\!-\! \delta \right) } \right] \sigma \theta \delta \hat{k}^{\delta -1},\nonumber \\ \end{aligned}$$

(29)

$\Psi$ is positive.

Government policy affects the steady-state capital either directly or indirectly. In the indirect effect, government policy influences capital by improving environmental quality. We can demonstrate the effects of tax on disposable income in (27), $\tau$, and the effects of tax on disposable income through changing government expenditure in (27), $\alpha$. When the effect of the latter exceeds that of the former, $\Phi$ is negative.

The direct efect by which such a policy affects saving behavior in both the urban and the rural areas is shown in (29). The left-han side of (29) shows how tax affects savings in the urban area. The right-hand side of (29) shows the effects of tax on savings in the rural area. When the effect of the former is weaker than that of the latter, the denominator of (26), $\Psi$ is positive. Therefore, if these conditions are satisfied, $\frac{d\hat{k}}{d \tau }$, is negative.

We subsequently examine the ratio of the urban to the total population, $\hat{\phi }$, and the total fertility rate, $\hat{m}$, in a steady state. The effect of $\tau$ on $\hat{\phi }$ is negative as $\frac{d \hat{\phi }}{d \tau }= \frac{d \theta }{d \tau } \hat{k}^{\delta }+ \theta \delta \hat{k}^{\delta -1}\frac{d \hat{k}}{d\tau }<0$ from (20). A higher tax contributes to the environmental quality and the wage rates in the rural area. This tax promotes immigration from urban to rural. Furthermore, the effect of $\tau$ on $\hat{m}$ is positive by $\frac{d \hat{m}}{d \tau } = \frac{- \gamma Z \sigma \left( \hat{k}^{\delta }\frac{d \theta }{d \tau }+\delta \hat{k}^{\delta -1}\frac{d \hat{k}}{d \tau }\right) }{Z^{2}}>0$ from (22). The effect of $\tau$ on $\hat{k}$ is negative. Thus, urban wages decrease, whereas rural wages increase. Therefore, given the increasing wages and the reduced cost of raising children, an increased tax rate enhances the fertility rate in the rural area.