Environmental amenities as a renewable resource: management and conflicts

Abstract

The major assumption made in this paper, is that the environment offers at large two distinct services each of different kind. First, the environmental resources may serve as inputs to the production of conventional goods. An example is the exploitation of an oil source from which, one firm extracts the oil which in turn is used as a fossil fuel for an industry. In the worst case, the use of the natural resources for industrial purposes will negatively affect the environment, e.g. the air quality over an industrial area. Nevertheless, saving abatement costs, production cost decreases due to possibility to pollute. Therefore, this first environmental service is evaluated positively by the economic agents (firms and consumers). The second service provided is the environment itself which offers amenities (i.e. clean air, blue coasts, natural creeks, clean rivers and lakes etc.) The crucial difference between the uses of the above services is how environmental quality affected and how much is the environmental degradation. From the pure economic point of view, the uses of the environmental services are consumptive and non-consumptive. Conversely in natural resources means, the environmental stock may be used as a raw material for the industrial production of conventional goods providing simultaneously a positive externality. Hence, the main purpose of this paper is twofold. First it considers the management at which the social planer has to steer emissions in an optimal way, meaning that both environmental quality and stock of pollutants remain optimal. Second it considers the conflict between the representative polluting producer and the representative environmental quality enjoyer which actually abates. In both cases we explore the complex limit cycle equilibrium, but additionally in the second case the analytical expressions of the crucial variables of the model are computed.

Appendices

Appendix 1: Proof of proposition 4

With the specifications, given in subsection entitled “Periodic solutions”, one can compute

$$ G^{\prime } \left( X \right) = R\left( {1 - 2X} \right),\quad G^{\prime \prime } \left( X \right) = - 2R,\quad \phi_{u} \left( {u, \nu } \right) = \gamma u^{\gamma - 1} ,\quad \phi_{\nu } \left( {u, \nu } \right) = u^{\gamma } ,\quad C^{\prime } \left( u \right) = C,\quad A^{\prime } \left( \nu \right) = \nu^{\xi - 2} ,\quad D^{\prime } \left( X \right) = D,\quad \upsilon^{\prime } \left( X \right) = \upsilon $$

$$ \frac{{\partial H_{1} }}{\partial u} = 0 \Leftrightarrow \left( {1 - \lambda } \right)\phi_{u} \left( {u,\nu } \right) = C^{\prime}\left( u \right) \Leftrightarrow \left( {1 - \lambda } \right)\gamma u^{\gamma - 1} \nu = C $$

(45)

$$ \frac{{\partial H_{2} }}{\partial \nu } = 0 \Leftrightarrow A^{{\prime }} \left( \nu \right) = \mu \phi_{\nu } \left( {u,\nu } \right) \Leftrightarrow \mu u^{\gamma } = \nu^{\xi - 2} $$

(46)

Combining (45) and (46) the optimal strategies take the following forms

$$ u^{*} = \mu_{{}}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} \left[ {\frac{C}{{\gamma \left( {1 - \lambda } \right)}}} \right]^{{{{\left( {\xi - 2} \right)} \mathord{\left/ {\vphantom {{\left( {\xi - 2} \right)} {\left[ {1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)} \right]}}}} $$

(47)

$$ \nu^{*} = \mu_{{}}^{{{{\left( {\gamma - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\gamma - 1} \right)} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} \left[ {\frac{C}{{\gamma \left( {1 - \lambda } \right)}}} \right]^{{{\gamma \mathord{\left/ {\vphantom {\gamma {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} $$

(48)

and the optimal environmental damage becomes

$$ \phi \left( {u^{*} ,\nu^{*} } \right) = \mu^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} \left[ {\frac{C}{{\gamma \left( {1 - \lambda } \right)}}} \right]^{{{{\gamma \left( {\xi - 1} \right)} \mathord{\left/ {\vphantom {{\gamma \left( {\xi - 1} \right)} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} $$

(49)

with the following partial derivatives

$$ \begin{aligned} \frac{\partial \phi }{\partial \lambda } & = \frac{{\mu^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} \left[ {\frac{C}{{\gamma \left( {1 - \lambda } \right)}}} \right]^{{{{\gamma \left( {\xi - 1} \right)} \mathord{\left/ {\vphantom {{\gamma \left( {\xi - 1} \right)} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} }}{{\left( {1 - \lambda } \right)}}\frac{{\gamma \left( {\xi - 1} \right)}}{{1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)}} \\ & = \frac{{\phi \left( {u^{*} ,\nu^{*} } \right)}}{{\left( {1 - \lambda } \right)}}\frac{{\gamma \left( {\xi - 1} \right)}}{{1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)}} \\ \end{aligned} $$

(50)

$$ \begin{aligned} \frac{\partial \phi }{\partial \mu } & = \frac{{\mu^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} \left[ {\frac{C}{{\gamma \left( {1 - \lambda } \right)}}} \right]^{{{{\gamma \left( {\xi - 1} \right)} \mathord{\left/ {\vphantom {{\gamma \left( {\xi - 1} \right)} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \left( {1 - \gamma } \right)\left( {1 - \xi } \right)} \right]}}}} }}{{\lambda_{2} }}\frac{ - 1}{{1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)}} \\ & = \frac{{\phi \left( {u^{*} ,\nu^{*} } \right)}}{\mu }\frac{ - 1}{{1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)}} \\ \end{aligned} $$

(51)

Both derivatives (50), (51) are negatives due to the assumptions on the parameters $\gamma , \xi \in \left( {0,1} \right)$ and on the signs of the functions derivates, that is $\phi_{u} > 0, \phi_{\nu } > 0, \upsilon^{{\prime }} \left( x \right) > 0, D^{{\prime }} \left( x \right) > 0$, which ensures the positive sign of the adjoints $\lambda , \mu$.

Bifurcation condition $w = \frac{\det \left( J \right)}{{{\text{tr }}\left( J \right)}}$ now becomes

$\rho_{1} \rho_{2} \left[ {\rho_{1} + \rho_{2} - 2G^{{\prime }} \left( X \right)} \right] = \lambda \rho_{1} G^{{{\prime \prime }}} \left( X \right)\frac{\partial \phi }{\partial \lambda } + \mu \rho_{2} G^{{{\prime \prime }}} \left( X \right)\frac{\partial \phi }{\partial \mu }$, which after substituting the values from (50), (51) and making the rest of algebraic manipulations, finally yields (at the steady states)

$$ \frac{{\phi \left( {u_{\infty } ,\nu_{\infty } } \right)G^{{{\prime \prime }}} \left( X \right)}}{{1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)}}\left[ {\rho_{1} \gamma \left( {1 - \xi } \right)\frac{D}{{D + G^{{\prime }} \left( X \right) - \rho_{1} }} - \rho_{2} } \right] - \rho_{1} \rho_{2} \left[ {\rho_{1} + \rho_{2} - 2G^{{\prime }} \left( X \right)} \right] = 0 $$

(52)

Where we have set $\frac{\lambda }{1 - \lambda } = \frac{D}{{\rho_{1} - G^{{\prime }} \left( X \right) - D}}$ stemming from the adjoint equation $\dot{\lambda } = \lambda \left( {\rho_{1} - G^{{\prime }} \left( X \right)} \right) - D^{{\prime }} \left( X \right)$, which at the steady states reduces into $\lambda = D^{\prime } \left( X \right)/D^{\prime } \left( X \right)\left( {\rho_{1} - G^{\prime } \left( X \right)} \right).\left( {\rho_{1} - G^{\prime } \left( X \right)} \right)$.

Condition w > 0 after substitution the values from (50), (51) becomes

$$ w = \rho_{1} \rho_{2} - \left[ {G^{\prime } \left( X \right)} \right]^{2} + \frac{{\phi \left( {u,\nu } \right)G^{\prime \prime } \left( X \right)}}{{1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)}}\left[ {\gamma \left( {1 - \xi } \right)\frac{ - D}{{G^{\prime}\left( X \right) + D - \rho_{1} }} + 1} \right] > 0 $$

(53)

The division of (52) by ρ ₁ yields

$$ \frac{{\phi \left( {u_{\infty } ,\nu_{\infty } } \right)G^{{{\prime \prime }}} \left( X \right)}}{{1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)}}\left[ {\gamma \left( {1 - \xi } \right)\frac{D}{{D + G^{{\prime }} \left( X \right) - \rho_{1} }} - \frac{{\rho_{2} }}{{\rho_{1} }}} \right] - \rho_{2} \left[ {\rho_{1} + \rho_{2} - 2G^{{\prime }} \left( X \right)} \right] = 0 $$

(54)

The sum (53) + (54) must be positive, thus after simplifications and taking into account, at the steady state, that ϕ(u _∞, ν _∞) = G(X), we have:$G\left( X \right)G^{{{\prime \prime }}} \left( X \right)\frac{{\rho_{1} - \rho_{2} }}{{\rho_{1} \left[ {1 + \left( {1 - \xi } \right)\left( {1 - \gamma } \right)} \right]}} > \left[ {\rho_{2} - G^{{\prime }} \left( X \right)} \right]^{2}$ and the result ρ ₂ > ρ ₁ follows from the strict concavity of the logistic growth G″ < 0.

Appendix 2

Proof that the bifurcation condition $w = \frac{\det \left( J \right)}{{{\text{tr}} \left( J \right)}}$ (56) can be written as:

$$ \rho_{1} \rho_{2} \left[ {\rho_{1} + \rho_{2} - 2G^{{\prime }} \left( X \right)} \right] = \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right]\rho_{1} + \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right]\rho_{2} $$

(55)

Until now we have:

$$ {\text{tr}} \left( J \right) = \rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right) $$

(56)

$$ \begin{aligned} \det \left( J \right) & = G^{{\prime }} \left( X \right)\left( {\rho_{1} - G^{{\prime }} \left( X \right)} \right)\left( {\rho_{2} - G^{{\prime }} \left( X \right)} \right) - \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left( {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right) \\ & \quad \left( {\rho_{2} - G^{{\prime }} \left( X \right)} \right) - \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left( {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right)\left( {\rho_{1} - G^{{\prime }} \left( X \right)} \right) \\ \end{aligned} $$

(57)

$$ w = \rho_{1} \rho_{2} - \left[ {G^{{\prime }} \left( X \right)} \right]^{2} - \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right] - \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right] $$

(58)

First we multiply (56) by (58), and

$$ \begin{aligned} \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right] \cdot \left[ {\rho_{1} \rho_{2} - \left[ {G^{{\prime }} \left( X \right)} \right]^{2} - \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right] - \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right]} \right] \\ = \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right]\left[ {\rho_{1} \rho_{2} - \left[ {G^{{\prime }} \left( X \right)} \right]^{2} } \right] - \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right]\frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right] - \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right]\frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right] \\ \end{aligned} $$

(59)

Equating (57) = (59)

$$ \begin{array}{l} G^{{\prime }} \left( X \right)\left( {\rho_{1} - G^{{\prime }} \left( X \right)} \right)\left( {\rho_{2} - G^{{\prime }} \left( X \right)} \right) - \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left( {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right)\left( {\rho_{2} - G^{{\prime }} \left( X \right)} \right) \hfill \\ \quad - \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left( {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right)\left( {\rho_{1} - G^{{\prime }} \left( X \right)} \right) \hfill \\ = \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right]\left[ {\rho_{1} \rho_{2} - \left[ {G^{{\prime }} \left( X \right)} \right]^{2} } \right] - \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right]\frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right] \hfill \\ \quad - \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right]\frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right] \hfill \\ \Leftrightarrow G^{{\prime }} \left( X \right)\left[ {\rho_{1} - G^{{\prime }} \left( X \right)} \right]\left[ {\rho_{2} - G^{{\prime }} \left( X \right)} \right] = \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right]\left[ {\rho_{1} \rho_{2} - \left[ {G^{{\prime }} \left( X \right)} \right]^{2} } \right] \hfill \\ \quad - \rho_{1} \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right] - \rho_{2} \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right] \hfill \\ \Leftrightarrow \left[ {\rho_{1} + \rho_{2} - G^{{\prime }} \left( X \right)} \right]\left[ {\rho_{1} \rho_{2} - \left[ {G^{{\prime }} \left( X \right)} \right]^{2} } \right] - G^{{\prime }} \left( X \right)\left[ {\rho_{1} - G^{{\prime }} \left( X \right)} \right]\left[ {\rho_{2} - G^{{\prime }} \left( X \right)} \right] \hfill \\ = \rho_{1} \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right] + \rho_{2} \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right] \hfill \\ \Leftrightarrow \left( {\rho_{1} + \rho_{2} } \right)\rho_{1} \rho_{2} - \left( {\rho_{1} + \rho_{2} } \right)\left[ {G^{{\prime }} \left( X \right)} \right]^{2} - \rho_{1} \rho_{2} G^{{\prime }} \left( X \right) + \left[ {G^{{\prime }} \left( X \right)} \right]^{3} \hfill \\ \quad - \rho_{1} \rho_{2} G^{{\prime }} \left( X \right) + \rho_{1} \left[ {G^{{\prime }} \left( X \right)} \right]^{2} + \rho_{2} \left[ {G^{{\prime }} \left( X \right)} \right]^{2} - \left[ {G^{{\prime }} \left( X \right)} \right]^{3} \hfill \\ = \rho_{1} \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right] + \rho_{2} \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right] \hfill \\ \Leftrightarrow \rho_{1} \rho_{2} \left( {\rho_{1} + \rho_{2} - 2G^{{\prime }} \left( X \right)} \right) = \rho_{1} \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \lambda }\left[ {\lambda G^{{{\prime \prime }}} \left( X \right) + D^{{{\prime \prime }}} \left( X \right)} \right] + \rho_{2} \frac{{\partial \phi \left( {u,\nu } \right)}}{\partial \mu }\left[ {\mu G^{{{\prime \prime }}} \left( X \right) + \upsilon^{{{\prime \prime }}} \left( X \right)} \right] \hfill \\ \end{array} $$

which is the condition for cyclical strategies mentioned in the main text.