An Efficient Pollution Control Instrument: The Case of Urban Wastewater Pollution

Abstract

Urban wastewater and agriculture activities are the main sources of urban water pollution and of eutrophying nutrients in many water ecosystems. Several EU directives have been adopted and affect the control of urban water pollution. The EU legislation requires the achievement of good ecological and chemical status in all water. This paper focuses on the use of economic instruments as a priority in the context of implementation. Our analysis only considers public wastewater utilities facing demand and capacity shocks. The proposed mechanism constitutes efficient means of moving towards sustainability by promoting full-cost pricing and considering external costs from wastewater services. Environmental damage associated with urban water pollution are internalized. The model also explicitly considers the investment needed to set-up wastewater system facilities. Our results indicate that savings in capacity could be achieved by adopting the proposed incentive-based mechanism that characterizes the optimal capacity selection rule.

Appendix

The Kuhn–Tucker necessary conditions for maximizing the program in Eq. 22 with respect to p, and k will be

$$ \begin{array}{rll}\label{eq34} \frac{\partial \underset{\omega }{E}\left( S^{\ast }\right) }{\partial k} \! = \! \int\limits_{\widetilde{\theta }}^{1}\left( \left. \frac{\partial \underset{\omega }{E}\left( S\right) }{\partial k}\right\vert _{\Psi _{\theta }=\Psi _{\theta }^{\ast }}\right) {\rm d}H\left( \theta \right) \! = \! -\int\limits_{\widetilde{\theta }}^{1}\Psi _{\theta }^{\ast }{\rm d}H\left( \theta \right)\\ \end{array} $$

(34)

$$ \begin{array}{rll} \dfrac{\partial \underset{\omega }{E}\left( \Pi \right) }{\partial k} &=&\int\limits_{\widetilde{\theta }}^{1}\left( k\frac{\partial \Psi _{\theta }^{\ast }}{\partial k}+\Psi _{\theta }^{\ast }\right) {\rm d}H\left( \theta \right)\\ &&+\,\left( p-c\right) \sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{ \theta }}^{1}\frac{\partial \Psi _{\theta }^{\ast }}{\partial k}\left( 1-\mathcal{F}_{\tau }\left( \omega _{\tau }^{\ast }\right) \right) {\rm d}H\left( \theta \right) \\ &&-\,\varphi \sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\omega } _{\tau }}^{\omega _{\tau }^{+}}\frac{\partial \left( X_{\tau }-\Lambda \right) }{\partial k}{\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right)\label{eq35} \end{array} $$

(35)

Combining the last two equations, and taking into account the break-even constraint, the optimal condition for a maximum k is giving by

$$ \begin{array}{rll} &&{\kern-6pt} \dfrac{\partial \underset{\omega }{E}\left( \mathcal{L} \right) }{\partial k} \\ &&{\kern3pt}=\Lambda \int\limits_{\widetilde{\theta }}^{1}\Psi _{\theta }^{\ast }{\rm d}H\left( \theta \right) \\ &&{\kern3pt}{\kern7pt} + \! \left( 1 \! + \! \lambda \right) \left\{ \left( p \! - \! c\right) \sum\limits_{\tau =1}^{\Gamma }\int\limits_{ \widetilde{\theta }}^{1}\frac{\partial \Psi _{\theta }^{\ast }}{\partial k} \left( 1 \!- \! \mathcal{F}_{\tau }\left( \omega _{\tau }^{\ast }\right) \right) {\rm d}H\left( \theta \right) \right\}\\ &&{\kern3pt}{\kern7pt} +\left( 1+\lambda \right) \!\left\{\! k\int\limits_{\widetilde{\theta }}^{1}\frac{\partial \Psi _{\theta }^{\ast }}{\partial k}{\rm d}H\left( \theta \right) \vphantom{\underset{a}{ \underbrace{\sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\omega } _{\tau }}^{\omega _{\tau }^{+}}\frac{\partial \left( X_{\tau }-\Lambda \right) }{\partial k}{\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right) }}}\right.\\ && \quad\qquad\qquad\left.-\varphi \underset{a}{ \underbrace{\sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\omega } _{\tau }}^{\omega _{\tau }^{+}}\frac{\partial \left( X_{\tau }-\Lambda \right) }{\partial k}{\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right) }}\! \right\} =0\\ \label{eq36} \end{array} $$

(36)

Concerning p, we have

$$ \begin{array}{rll} \frac{\partial \underset{\omega }{E}\left( S^{\ast }\right) }{\partial p} &=&\sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\theta } }^{1}\left( \left. \frac{\partial \underset{\omega }{E}\left( S\right) }{ \partial p}\right\vert _{\Psi _{\theta }=\Psi _{\theta }^{\ast }}\right) {\rm d}H\left( \theta \right)\\ \frac{\partial \underset{\omega }{E}\left( S^{\ast }\right) }{\partial p} &=&\sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\theta } }^{1}\left\{ \int\limits_{\omega _{\tau }^{-}}^{\omega _{\tau }^{\ast }}-q_{\tau }^{\ast }\left( \theta \right) {\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right) \right\} {\rm d}H\left( \theta \right) \\ &&-\sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\theta } }^{1}\left\{ \Psi _{\theta }^{\ast }\left( 1-\mathcal{F}_{\tau }\left( \omega _{\tau }^{\ast }\right) \right) \right\} {\rm d}H\left( \theta \right)\label{eq37} \end{array} $$

(37)

$$ \begin{array}{rll} &&{\kern-6pt} \dfrac{\partial \underset{\omega }{E}\left( \Pi \right) }{\partial p}\\ &&= \sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\theta } }^{1}\!\left\{ \int\limits_{\omega _{\tau }^{-}}^{\omega _{\tau }^{\ast }}\!q_{\tau }^{\ast }\left( \theta \right) {\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right) \!+\!\Psi _{\theta }^{\ast }\left( 1\!-\!\mathcal{F}_{\tau }\left( \omega _{\tau }^{\ast }\right) \right) \right. \!+\!( p\!-\!c) \\ &&{\kern6pt} \times\!\left. \left( \int\limits_{\omega _{\tau }^{-}}^{\omega _{\tau }^{\ast }} \frac{\partial q_{\tau }^{\ast }\left( \theta \right) }{\partial p} {\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right) \!+\!\frac{\partial \Psi _{\theta }^{\ast }}{\partial p}\left( 1\!-\!\mathcal{F}_{\tau }\left( \omega _{\tau }^{\ast }\right) \right) \!\right) \!\!\right\}\! \mathit{{\rm d}H}\left( \theta \right) \\ &&{\kern6pt} +\,k\int\limits_{\widetilde{\theta }}^{1}\frac{\partial \Psi _{\theta }^{\ast }}{\partial p}{\rm d}H\left( \theta \right) -\varphi \sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\omega }_{\tau }}^{\omega _{\tau }^{+}} \frac{\partial \left( X_{\tau }-\Lambda \right) }{\partial p}{\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right)\\ \label{eq38} \end{array} $$

(38)

Combining the last two equations, and taking into account the break-even constraint, the optimal condition for a maximum p is giving by the following relation

$$ \begin{array}{rll} &&{\kern-6pt} \dfrac{\partial \underset{\omega }{E}\left( \mathcal{L} \right) }{\partial p} \\ &&= \left( 1+\lambda \right) \left\{ k\int\limits_{\widetilde{\theta }}^{1} \frac{\partial \Psi _{\theta }^{\ast }}{\partial p}{\rm d}H\left( \theta \right) \right\} + \lambda \sum\limits_{\tau =1}^{\Gamma }\\ &&{\kern6pt} \times\,\int\limits_{\widetilde{\theta } }^{1}\!\left\{ \int\limits_{\omega _{\tau }^{-}}^{\omega _{\tau }^{\ast }}q_{\tau }^{\ast }\left( \theta \right) {\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right) +\Psi _{\theta }^{\ast }\left( 1-\mathcal{F}_{\tau }\left( \omega _{\tau }^{\ast }\right) \right) \!\right\}\! {\rm d}H\left( \theta \right)\\ &&{\kern6pt} +\left( 1\!+\!\lambda \right) \!\left\{\! ( p\!-\!c) \sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\theta }}^{1}\!\left( \int\limits_{\omega _{\tau }^{-}}^{\omega _{\tau }^{\ast }}\frac{\partial q_{\tau }^{\ast }\left( \theta \right) }{\partial p}{\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right) \!\right)\! \mathit{{\rm d}H}\left( \theta \right) \!\!\right\}\\ &&{\kern6pt} +\left( 1\!+\!\lambda \right) \!\left\{ \!( p\!-\!c) \sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\theta }}^{1}\left( \frac{\partial \Psi _{\theta }^{\ast }}{\partial p}\left( 1-\mathcal{F}_{\tau }\left( \omega _{\tau }^{\ast }\right) \right) \!\right)\! {\rm d}H\left( \theta \right) \!\right\} \\ &&{\kern6pt} -\left( 1+\lambda \right) \left\{ \varphi \underset{b}{\underbrace{ \sum\limits_{\tau =1}^{\Gamma }\int\limits_{\widetilde{\omega }_{\tau }}^{\omega _{\tau }^{+}}\frac{\partial \left( X_{\tau }-\Lambda \right) }{ \partial p}{\rm d}\mathcal{F}_{\tau }\left( \omega _{\tau }\right) }}\right\} =0\label{eq39} \end{array} $$

(39)

Equations 36 and 39 can be arranged to give Eqs. 26 and 27 in the text.