Mining-Induced Desiccation of Water Bodies and Consequent Impact on Traditional Economic Livelihood: An Analytical Framework

Abstract

Lowering of water table by rampant mining creates severe threat to traditional economy. An analytical framework is designed for planning social optimal mine extraction path and social welfare path. If society sets a maxi–min rule of sacrificing some traditional output to increase mine production, the minimum amount of water required for the society is determined here. Two alternative tax measures are proposed and compared—on rate of mine-resource depletion; and on mining-induced loss of traditional output.

Appendix

A2.1: The waste dump is assumed to have conical shape. If its height is h, basal radius is r and slope is $\theta$, (i.e. sub vertical angle is $90^\circ \, - \,\theta$),

Then its volume ($V_{W}$) is:

$V_{W} = \frac{1}{3}\pi r^{2} h = \frac{1}{3}\pi r^{2} .r.\tan \theta = \frac{1}{3}A_{W} \Bigg( {\sqrt {\frac{{A_{W} }}{\pi }} } \Bigg)\tan \theta = \frac{1}{3\sqrt \pi }\Bigg( {A_{W} } \Bigg)^{\frac{3}{2}} \tan \theta$ Here $A_{W}$ is the area covered by waste dump.

The volume of the waste dump can be expressed as the ratio of its mass and average density, i.e. $V_{W} \, = \,\frac{W}{{\rho_{W} }}$

Therefore

$$ A_{W} = \left[ {\frac{3\sqrt \pi }{\tan \theta }\left( {V_{W} } \right)} \right]^{\frac{2}{3}} = \left(\frac{3\sqrt \pi W}{{\tan \theta \rho_{W} }}\right)^{\frac{2}{3}} = \left(\frac{{3\sqrt \pi k_{WY} }}{{\tan \theta \rho_{W} }}\right)^{\frac{2}{3}} Y^{\frac{2}{3}} $$

(A2.1)

A2.2: The materials dug out from the mine comprise the desired material (Y) and the waste material (W). Volume of a material is equal to its mass divided by its average density. Therefore, the volume of desired material and waste material are $\frac{{Y_{{}} }}{{\rho_{Y} }}$ and $\frac{{W_{{}} }}{{\rho_{W} }}$ respectively. Therefore the total volume of the excavation:

$$ V_{E} = \Bigg( {\frac{{Y_{{}} }}{{\rho_{Y} }} + \frac{{W_{{}} }}{{\rho_{W} }}} \Bigg) = \Bigg( {\frac{1}{{\rho_{Y} }} + \frac{{k_{WY} }}{{\rho_{W} }}} \Bigg)Y $$

A3: The shadow cost path of desiccation can be obtained by $\dot{\lambda }_{\eta } + \lambda_{\eta } = \varphi - \sigma \Upomega_{\eta } e^{ - \delta t}$

Here, the integrating factor is $\int e^{dt} = e^{t}$. Multiplying both sides by $e^{t}$ we get:

$$ \lambda_{\eta }^{\prime } e^{t} \, + \,\lambda_{\eta } e^{t} \, = \,\varphi e^{t} \, - \,\sigma \Upomega_{\eta } e^{ - \delta t + t} \;or,\;\left( {\lambda_{\eta } e^{t} } \right)^{\prime } \, = \,\varphi e^{t} \, - \,\sigma \Upomega_{\eta } e^{{\left( {1 - \delta } \right)t}} $$

Integrating both sides we get:

$$ \begin{aligned} \lambda_{\eta } e^{t}\;=\;& \,\varphi \int {e^{t} } dt\, - \,\int {\sigma \Upomega_{\eta } e^{{\left( {1 - \delta } \right)t}} } dt \\ \;=\;& \varphi e^{t} \, - \,\sigma \left( {1\, - \,\delta } \right)\Upomega_{\eta } e^{{\left( {1 - \delta } \right)t}} \, + \,K \\ \end{aligned} $$

At $t\, = \,0$, $\lambda_{\eta } \, = \,\lambda_{{\eta_{0} }}$ and thus, $K\, = \,\sigma \left( {1\, - \,\delta } \right)\Upomega_{{\eta_{{}} }} \, - \,\varphi \, - \,\lambda_{{\eta_{0} }}$. Plugging this value into the above equation, we get:

The shadow cost path of desiccation can be obtained by $\dot{\lambda }_{\eta } \, + \,\lambda_{\eta } \, = \,\varphi \, - \,\sigma \Upomega_{\eta } e^{ - \delta t}$

Here, the integrating factor is $\int e^{dt} \, = \,e^{t}$. Multiplying both sides by $e^{t}$ we get:

$$ \lambda^{\prime }_{\eta } e^{t} + \lambda_{\eta } e^{t} = \varphi e^{t} - \sigma \Upomega_{\eta } e^{ - \delta t + t} \;or,\left( {\lambda_{\eta } e^{t} } \right)^{\prime } = \varphi e^{t} - \sigma \Upomega_{\eta } e^{{\left( {1 - \delta } \right)t}} $$

Integrating both sides we get:

$$ \begin{aligned} \lambda_{\eta } e^{t} = & \varphi \int {e^{t} } dt\, - \,\int {\sigma \Upomega_{\eta } e^{{\left( {1 - \delta } \right)t}} } dt \\ = & \varphi e^{t} \, - \,\sigma \left( {1\, - \,\delta } \right)\Upomega_{\eta } e^{{\left( {1 - \delta } \right)t}} \, + \,K \\ \end{aligned} $$

At $t\, = \,0$, $\lambda_{\eta } \, = \,\lambda_{{\eta_{0} }}$ and thus, $K\, = \,\sigma \left( {1\, - \,\delta } \right)\Upomega_{{\eta_{{}} }} \, - \,\varphi \, - \,\lambda_{{\eta_{0} }}$. Plugging this value into the above equation, we get:

$$ \lambda_{\eta } \, = \,\varphi \, - \,e^{ - \delta t} \sigma \left( {1\, - \,\delta } \right)\Upomega_{\eta } \, - \,\left( {\varphi - \left( {1\, - \,\delta } \right)\sigma \Upomega_{\eta } \, + \,\lambda_{{\eta_{0} }} } \right)e^{ - t} \, = \,\left( {1\, - \,e^{ - t} } \right)\varphi \, - \,\sigma \left( {1\, - \,\delta } \right)\Upomega_{\eta } \left( {e^{{^{ - \delta t} }} \, - \,e^{ - t} } \right)\, - \,\lambda_{{\eta_{0} }} e^{ - t} $$