Land Acquisition and Land Diversion for Mining Towards Industrial Growth: Interest Conflict and Negotiation Game for Sustainable Development

Abstract

Acquisition of community land for mining activities leads to conflict of interest between miners and traditional communities. This can be resolved by two types of institutions: either the community and miner negotiate and reach the Nash solution in game theoretic framework; or settlement is made by intervention of the social planner. For both cases the long-term impacts of mine reserve depletion and change of land-use pattern are analytically compared and sustainable development is discussed.

Appendix

1.1 Appendix 1

The Hamiltonian for this problem is:

$$ H = \left( {p - \Psi } \right)Ye^{ - \delta t} - C\left( {Y,R} \right)e^{ - \delta t} - \gamma R\left( {A_{c} } \right)e^{ - \delta t} + \lambda_{1} \left[ {f\left( {A_{c} ,x} \right) - Y\left( {A_{c} } \right)} \right] + \lambda_{2} \left[ {f\left( {A_{c} ,x} \right)} \right] $$

Differentiating H with respect to Y, R, x and $A_{c}$ we get:

$$ pe^{ - \delta t} - C_{Y} e^{ - \delta t} - \Psi e^{ - \delta t} - \lambda_{1} = 0 $$

Solving this, we get, $\lambda_{1} = \left( {p - C_{Y} - \Psi } \right)e^{ - \delta t}$

$$ \begin{array}{l} \dot{\lambda }_{1} = \left( {C_{R} + \gamma } \right)e^{ - \delta t} \\ \dot{\lambda }_{2} = - f_{x} \left( {\lambda_{1} + \lambda_{2} } \right) \\ \end{array} $$

Again differentiating H with respect to A _c we obtain:

$$ \left[ {\left( {p - C_{Y} - \Psi } \right)Y_{{A_{c} }} - \left( {C_{R} + \gamma } \right)R_{{A_{c} }} } \right]e^{ - \delta t} + f_{{A_{c} }} \left( {\lambda_{1} + \lambda_{2} } \right) - \lambda_{1} Y_{{A_{c} }} = 0 $$

Plugging the value of $\lambda_{1}$ in the above expression we get:

$$ \left[ {\left( {p - C_{Y} - \Psi } \right)Y_{{A_{c} }} - \left({C_{R} + \gamma } \right)R_{{A_{c} }} } \right]e^{ - \delta t} +f_{{A_{c} }} \left( {\lambda_{1} + \lambda_{2} } \right) - \left({p - C_{Y} - \Psi } \right)e^{ - \delta t} Y_{{A_{c} }} = 0 $$

$$ \begin{array}{cl}\left[ { - \left( {C_{R} + \gamma } \right)R_{{A_{c} }} } \right]e^{ - \delta t} + f_{{A_{c} }} \left( {\lambda_{1} + \lambda_{2} } \right) & = 0 \\ \left[ { - \left( {C_{R} + \gamma } \right)R_{{A_{c} }} } \right]e^{ - \delta t} - \frac{{f_{{A_{c} }} }}{{f_{x} }}\dot{\lambda }_{2} &= 0 \\ \end{array} $$

$$ \dot{\lambda }_{2} = - \left[ {C_{R} + \gamma } \right]R_{{A_{c} }} \frac{{f_{x} }}{{f_{{A_{c} }} }}e^{ - \delta t} $$

1.2 Appendix 2

Now differentiating Eq. (6) with respect to time we get the time path of land acquisition chosen by miner:

$$ \begin{array}{lllll} \left[ { - \delta \left[ {\left( {p - C_{Y} - \Psi } \right)f_{{A_{c} }} - \left( {C_{R} + \gamma } \right)R_{{A_{c} }} } \right] + \left[ {\left( {p - C_{Y} - \Psi } \right)f_{{A_{c} A_{c} }} - \left( {C_{R} + \gamma } \right)R_{{A_{c} A_{c} }} } \right]\dot{A}_{c} } \right]e^{ - \delta t} + f_{{A_{c} A_{c} }} \dot{A}_{c} \lambda_{2} + f_{{A_{c} }} \dot{\lambda }_{2} & = 0 \\ - \delta \left[ {\left( {p - C_{Y} - \Psi } \right)f_{{A_{c} }} - \left( {1 + f_{x} } \right)\left( {C_{R} + \gamma } \right)R_{{A_{c} }} } \right] + \left( {\frac{{f_{{A_{c} A_{c} }} R_{{A_{c} }} }}{{f_{{A_{c} }} }} - R_{{A_{c} A_{c} }} } \right)\left( {C_{R} + \gamma } \right)\dot{A}_{c} & = 0 \\ \\ \end{array} $$

$$ \dot{A}_{c}^{{\rm{M}}} = \frac{{\delta \left[ {\left( {p - C_{Y} - \Psi } \right)f_{{A_{c} }} - \left( {1 + f_{x} } \right)\left( {C_{R} + \gamma } \right)R_{{A_{c} }} } \right]}}{{\left( {\frac{{f_{{A_{c} A_{c} }} R_{{A_{c} }} }}{{f_{{A_{c} }} }} - R_{{A_{c} A_{c} }} } \right)\left( {C_{R} + \gamma } \right)}} $$

1.3 Appendix 3

The present value Hamiltonian for this problem with the single control variable L is:

$$ H = w.g\left( L \right)e^{ - \theta t} + \Psi .p.Y\left( {A_{c} } \right)e^{ - \theta t} + \gamma R\left( {A_{c} } \right)e^{ - \theta t} - \xi_{1} A_{c} $$

(14)

Differentiating H with respect to $A_{c}$ and L we get:

$$ H_{{A_{c} }} = \left( {\Psi .p.Y_{{A_{c} }} + \gamma R_{{A_{c} }} } \right)e^{ - \theta t} - \xi_{1} = 0\,\,H_{L} = - \dot{\xi }_{1} = w.g_{L} .e^{ - \theta t} $$

The optimal path is found from the necessary conditions by substituting the equation of motion $\dot{L} = - A{}_{c}$ into (14) and taking the time derivative of the resulting equation.

$$ \begin{array}{c} \left( {\Psi Y_{{A_{c} }} \left( { - \dot{L}} \right) - \gamma R_{{A_{c} }} \left( { - \dot{L}} \right)} \right)e^{ - \theta t} - \xi_{1} = 0 \\ \left( {\Psi Y_{{A_{c} L}} \left( { - \dot{L}} \right)\ddot{L} + \gamma R_{{A_{c} L}} \left( { - \dot{L}} \right)\ddot{L} - \theta \left( { - w.g_{{A_{c} }} + \Psi Y_{{A_{c} }} \left( { - \dot{L}} \right) + \gamma R_{{A_{c} }} \left( { - \dot{L}} \right)} \right)} \right) - w.g_{L} = 0 \\ \end{array} $$

$$ \left( {\Psi Y_{{A_{c} L}} + \gamma R_{{A_{c} L}} } \right)\ddot{L} = \theta \left( {\Psi Y_{{A_{c} }} + \gamma R_{{A_{c} }} } \right) - w.g_{L} $$

As defined in (13), $\ddot{L} = - \dot{A}_{c}$, i.e., rate of change of available land for traditional production is the change of acquired land over time.

$$ \left( {\Psi Y_{{A_{c} L}} + \gamma R_{{A_{c} L}} } \right)\dot{A}_{c} = w.g_{L} - \theta \left( {\Psi Y_{{A_{c} }} + \gamma R_{{A_{c} }} } \right) $$

1.4 Appendix 4

The Hamiltonian function with the single control variable $Y\left( {A_{c} } \right)$ and two state variables R and xis:

$$ \tilde{H} = e^{ - \delta t} \left[ {P.Y\left( {A_{c} } \right) - C\left( {Y\left( {A_{c} } \right),R\left( {A_{c} } \right)} \right) - \left[ {\Psi .p.Y\left( {A_{c} } \right) + \gamma R\left( {A_{c} } \right)} \right]\left( {1 - \rho .} \right) + g\left( {\bar{L} - A_{c} } \right)} \right] + \mu_{1} \left[ {f\left( {x,A_{c} } \right) - Y} \right] + \mu_{2} .f\left( {x,A_{c} } \right) $$

Differentiating it with respect to $A_{c}$ and Y we get:

$$ \begin{array}{r} \tilde{H}_{{A_{c} }} = e^{ - \delta t} \left[ {\left[ {P - C_{Y} - \Psi .p\left( {1 - \rho } \right)} \right]Y_{{A_{c} }} - \left[ {C_{R} + \gamma \left( {1 - \rho } \right)} \right]R_{{A_{c} }} - g} \right] - \mu_{1} Y_{{A_{c} }} + \left( {\mu_{1} + \mu_{2} } \right).f_{{A_{c} }} = 0 \\ \tilde{H}_{Y} = e^{ - \delta t} \left[ {\left( {p - C_{Y} - \Psi .p\left( {1 - \rho } \right)} \right)} \right] - \mu_{1} = 0 \\ Or,\,\mu_{1} = \left[ {p - C_{Y} - \Psi .p.\left( {1 - \rho } \right)} \right]e^{ - \delta t} \\ \end{array} $$

1.5 Appendix 5

Differentiating $\tilde{H}$ with respect to $A_{c}$, we get:

$$ \tilde{H}_{{A_{c} }} = e^{ - \delta t} \left[ {\left[ { - \left[ {C_{R} + \gamma \left( {1 - \rho } \right)} \right]R_{{A_{c} }} - g} \right] + \left[ {\left( {p - C_{Y} + \Psi .p\left( {1 - \rho } \right)} \right)} \right]f_{{A_{c} }} } \right] + \mu_{2} .f_{{A_{c} }} = 0 $$

Again differentiating the above expression with respect to time we get:

$$ e^{ - \delta t} \left( {1 - \rho } \right)\left[ {\gamma R_{{A_{c} A_{c} }} + \Psi .pf_{{A_{c} A_{c} }} .} \right]\dot{A}_{c} + \delta \left[ {\left[ {C_{R} - \gamma \left( {1 - \rho } \right)} \right]R_{{A_{c} }} + g} \right] + \mu_{2} .f_{{A_{c} A_{c} }} \dot{A}_{c} + f_{{A_{c} }} \dot{\mu }_{2} = 0 $$

Plugging the value of $\dot{\mu }_{2} = - \left[ {\left[ {C_{R} + \gamma \left( {1 - \rho } \right)} \right]R_{{A_{c} }} + .g} \right]\frac{{f_{x} }}{{f_{{A_{c} }} }}e^{ - \delta t}$ and $\mu_{2}$ into it we get the land acquisition path under social planning:

$$ \begin{aligned} \dot{A}_{c}^{{{\text{SO}}}} = & \frac{{\left( {\delta - f_{x} } \right)\left[ {\left[ {C_{R} - \gamma \left( {1 - \rho } \right)} \right]R_{{A_{c} }} + g} \right]}}{{\left( {1 - \rho } \right)\left[ {\gamma R_{{A_{c} A_{c} }} + \Psi .p\left( {1 + \rho - \frac{1}{{.f_{{A_{c} }} }}} \right) + \left[ {\frac{1}{{f_{{A_{c} }} }}\left[ {\left[ {C_{R} + \gamma \left( {1 - \rho } \right)} \right]R_{{A_{c} }} - g} \right]\left( {p - C_{Y} } \right)} \right]} \right]}} \\ & = \frac{{f_{{A_{c} }} \left( {\delta - f_{x} } \right)\left[ {\left[ {C_{R} - \gamma \left( {1 - \rho } \right)} \right]R_{{A_{c} }} + g} \right]}}{{\left( {1 - \rho } \right)\left[ {\gamma R_{{A_{c} A_{c} }} + \Psi pf_{{A_{c} }} \left( {\left( {1 + \rho } \right) - 1} \right)\left[ {\left[ {\left[ {C_{R} + \gamma \left( {1 - \rho } \right)} \right]R_{{A_{c} }} - g} \right]\left( {p - C_{Y} } \right)} \right]} \right]}} \\ \end{aligned} $$

1.6 Appendix 6

If the shape of the excavated material is cubic with sides a, $V_{E}$ is the total volume of material excavated from the mine that includes the volumes of desired material and that of the waste material,

$$ A_{E} = a^{2} = \left( {a^{3} } \right)^{2/3} = \left( {V_{E} } \right)^{2/3} . $$

(46)

Expressing the volume as the ratio of mass and average density, the volume of the desired material and the waste materials are $\frac{Y}{{\rho_{Y} }}$ and $\frac{W}{{\rho_{W} }}$ respectively.

Therefore from (46) $A_{E} = (\frac{Y}{{\rho_{Y} }} + \frac{W}{{\rho_{W} }})^{\frac{2}{3}} = (\frac{1}{{\rho_{Y} }} + \frac{{k_{WY} }}{{\rho_{W} }})^{\frac{2}{3}} Y^{\frac{2}{3}}$, where $k_{WY}$ is the strip ratio.

If the area of an waste dump is having conical shape, let the height and radius of conical waste dump be h and r respectively, and its slope is θ. (i.e., subvertical angle is $90^{\circ} - \theta$).

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

$$ V_{W} = \frac{1}{3}\pi r^{2} h = \frac{1}{3}\pi r^{2} .r.\tan \theta = \frac{1}{3}A_{W} \left( {\sqrt {\frac{{A_{W} }}{\pi }} } \right)\tan \theta = \frac{1}{3\sqrt \pi }\left( {A_{W} } \right)^{\frac{3}{2}} \tan \theta $$

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