Water Conservation Versus Soil Salinity Control

Abstract

This paper tackles the increasingly significant problem of irrigation-induced soil salinity within a groundwater management model. Irrigation can result not only in heavier salt concentrations but also in the removal of salt from the soil through return flows. Given these contradictory observations, we are interested in the effects on soil salt concentration if irrigation efficiency is improved. We develop a model of salt concentration patterns in both soil and groundwater. We introduce a negative externality to the production process by assuming that soil degradation due to higher soil salinity affects total factor productivity. Within this framework, we show that in the presence of this externality, increasing irrigation efficiency can lead to higher or lower soil salt concentration, depending on the social cost of transferring salt from one reservoir to another.

Appendices

Appendix 1: Proof of proposition 1

By construction, the steady state $\{w^{\ast };S^{\ast };C^{\theta \ast };\mu ^{\ast }\}$ satisfies:

$$ \left\{ \begin{array}{l} w^{\ast }=\frac{R}{e} \\ C^{\theta ^{\ast }}(S^{\ast })=C^{S^{\ast }}(S^{\ast })=\frac{M}{ S^{\ast}+\theta } \\ \mu ^{\ast }(S^{\ast })=-\frac{A^{\prime }\left( \frac{M}{S^{\ast }+\theta } \right) F(R)eS^{\ast }\theta }{\rho eS^{\ast }\theta +R(S^{\ast }+\theta )}\\ \phi (S^{\ast })=\rho \left[ A\left( \frac{M}{S^{\ast }+\theta }\right)\cdot F^{\prime }(R)-\frac{c(S^{\ast })}{e}\right] +c^{\prime }(S^{\ast })\frac{R}{ e}\notag\\ {\kern35pt}+\frac{A^{\prime }\left( \frac{M}{S^{\ast }+\theta }\right) \cdot F(R)\cdot M}{Z(S^{\ast })\cdot (S^{\ast }+\theta )}\left[ \rho S^{\ast }e+R\right] =0. \end{array} \right. $$

We only have to verify that $\phi (S)=0$ admits at least one solution. First, given assumptions 1 and 2, we observe that

$$\begin{array}{@{}rcl@{}} \displaystyle\lim\limits_{S\rightarrow 0}\phi (S) &=&\rho \left( A\left( \frac{M}{ \theta }\right) F^{\prime }(R)-\frac{1}{e}\displaystyle\lim\limits_{S\rightarrow0}c(S)\right) +\frac{R}{e}\displaystyle\lim\limits_{S\rightarrow 0}\notag\\ &&c^{\prime }(S)+\frac{A^{\prime }\left( \frac{M}{\theta }\right) \cdot F(R)\cdot M}{\theta ^{2}}=-\infty \notag\\ \phi (\overline{S}) &\,=\,&\rho A( \overline{M}) {} \cdot {} F^{\prime }(R)\,+\, \frac{A^{\prime }( \overline{M}) {} \cdot {} F(R) {} \cdot {} \overline{M}}{Z(\overline{S})}( \rho \overline{S}e\,+\,R)\notag \end{array} $$

with $\overline {M}=\frac {M}{\overline {S}+\theta }$ defined above.

We can also show that the function $\phi (\cdot )$ is monotonic under assumptions 1 and 2:

$$\begin{array}{@{}rcl@{}} \phi ^{\prime }(S)&=&\rho \left[ -\frac{M}{(S+\theta )^{2}}A^{\prime }\left( \frac{M}{S+\theta }\right) \cdot F^{\prime }(R)-\frac{c^{\prime }(S)}{e} \right]\notag\\ &&+c^{\prime \prime }(S)\frac{R}{e} \notag\\ &+&\frac{u\cdot[ \rho S^{\ast }e+R] -\frac{A^{\prime }\left( \frac{M}{S^{\ast }+\theta }\right) \cdot F(R)\cdot M}{Z\cdot (S^{\ast }+\theta )}\rho e}{[ \rho S^{\ast }e+R] ^{2}}>0\notag \end{array} $$

with $u{}={}\frac {-\frac {M}{(S+\theta )^{2}}\cdot A^{\prime \prime }\left ({}\frac {M}{S+\theta }{}\right ) \cdot F^{\prime }(R)\cdot Z(S)\cdot (S+\theta )-A^{\prime }\left ({}\frac {M}{S+\theta }{}\right ) \cdot F^{\prime }(R)\cdot M\cdot \lbrack Z(S)+(\rho e\theta +R)(S+\theta )]}{Z^{2}(S+\theta )^{2}}{}>{}0$, given we assume that $A^{\prime \prime }(\cdot ) \leq 0$. We therefore obtain that the function $\phi (\cdot )$ monotonically increases from $-\infty$ to $\phi (\overline {S})$ for values of S between 0 and $\overline {S}$. Thus, a sufficient condition for the existence of a unique steady state is that $\phi (\overline {S})$ is positive.

If we remind that the elasticities $\varepsilon _{F/R}= \frac {R\cdot F^{\prime }(R)}{F(R)}>0$ and $\varepsilon _{F/\overline {M}}=\frac { \overline {M}\cdot A^{\prime }(\overline {M})}{A(\overline {M})}>0$, then the function $\phi (\overline {S})$ is positive if

$$\begin{array}{@{}rcl@{}} &&\rho \cdot \frac{A( {\normalsize \overline{M}}) }{A^{\prime }( {\normalsize \overline{M}}) \cdot {\normalsize \overline{M}}} \cdot \frac{F^{\prime }(R)\cdot R}{F(R)}>\frac{R\cdot \left( \rho \overline{S }e+R\right) }{Z(\overline{S})}\notag \\ &&\frac{\varepsilon _{F/R}}{\varepsilon _{A/\overline{M}}}>\frac{R\left( \rho \overline{S}e+R\right) }{\rho \cdot Z(\overline{S})}.\notag \end{array} $$

This concludes the proof.

Appendix 2: Proof of proposition 2

As usual, focusing on the local stability of the dynamic system, we can derive the following Jacobian matrix:

$$\begin{array}{@{}rcl@{}} J=\left( \begin{array}{llll} \frac{\partial \dot{w}}{\partial w} & \frac{\partial \dot{w}}{\partial C^{\theta }} & \frac{\partial \dot{w}}{\partial S} & \frac{\partial \dot{w}}{ \partial \mu } \\ \frac{\partial \dot{C^{\theta }}}{\partial w} & \frac{\partial \dot{ C^{\theta }}}{\partial C^{\theta }} & \frac{\partial \dot{C^{\theta }}}{ \partial S} & \frac{\partial \dot{C^{\theta }}}{\partial \mu } \\ \frac{\partial \dot{S}}{\partial w} & \frac{\partial \dot{S}}{\partial C^{\theta }} & \frac{\partial \dot{S}}{\partial S} & \frac{\partial \dot{S}}{ \partial \mu } \\ \frac{\partial \dot{\mu}}{\partial w} & \frac{\partial \dot{\mu}}{\partial C^{\theta }} & \frac{\partial \dot{\mu}}{\partial S} & \frac{\partial \dot{ \mu}}{\partial \mu } \end{array}\right) _{(w^{\ast },C^{\theta ^{\ast }},S^{\ast },\mu ^{\ast })}\notag\\ = \left( \begin{array}{llll} \frac{\partial \dot{w}}{\partial w} & \frac{\partial \dot{w}}{\partial C^{\theta }} & \frac{\partial \dot{w}}{\partial S} & \frac{\partial \dot{w}}{ \partial \mu } \\ \frac{\partial \dot{C^{\theta }}}{\partial w} & \frac{\partial \dot{ C^{\theta }}}{\partial C^{\theta }} & \frac{\partial \dot{C^{\theta }}}{ \partial S} & 0 \\ -e & 0 & 0 & 0 \\ \frac{\partial \dot{\mu}}{\partial w} & \frac{\partial \dot{\mu}}{\partial C^{\theta }} & \frac{\partial \dot{\mu}}{\partial S} & \frac{\partial \dot{ \mu}}{\partial \mu } \end{array} \right) _{(w^{\ast },C^{\theta ^{\ast }},S^{\ast },\mu ^{\ast })}\notag \end{array} $$

To find the properties of this matrix, we shall apply a method developed by Dockner [10] to investigate in a simple way the stability properties of a linearized four-dimensional dynamic system. Using this method, the eigenvalues of the system can easily be computed according to the following simple formula:

$$p_{1,2,3,4}=\frac{\rho }{2}\pm \sqrt{\left( \frac{\rho }{2}\right) ^{2}- \frac{\Omega }{2}\pm \frac{1}{2}\sqrt{\Omega ^{2}-4detJ}} $$

with

$$\begin{array}{rll}\Omega &=&\left|\begin{array}{ll}\frac{\partial \dot{w}}{\partial w} & \frac{\partial \dot{w}}{\partial S} \\[2pt]\frac{\partial \dot{S}}{\partial w} & \frac{\partial \dot{S}}{\partial S}\end{array}\right| +\left|\begin{array}{ll}\frac{\partial \dot{C^{\theta }}}{\partial C^{\theta }} & \frac{\partial\dot{C^{\theta }}}{\partial \mu } \\[2pt]\frac{\partial \dot{\mu}}{\partial C^{\theta }} & \frac{\partial \dot{\mu}}{\partial \mu }\end{array}\right| +2\left|\begin{array}{ll}\frac{\partial \dot{w}}{\partial C^{\theta }} & \frac{\partial \dot{w}}{\partial \mu } \\[2pt]\frac{\partial \dot{S}}{\partial C^{\theta }} & \frac{\partial \dot{S}}{\partial \mu }\end{array}\right|\\[12pt] &=& \left|\begin{array}{ll}\frac{\partial \dot{w}}{\partial w} & \frac{\partial \dot{w}}{\partial S} \\[2pt]-e & 0\end{array}\right| +\left|\begin{array}{ll}\frac{\partial \dot{C^{\theta }}}{\partial C^{\theta }} & 0 \\[2pt]\frac{\partial \dot{\mu}}{\partial C^{\theta }} & \frac{\partial \dot{\mu}}{\partial \mu }\end{array}\right| +2\left|\begin{array}{ll}\frac{\partial \dot{w}}{\partial C^{\theta }} & \frac{\partial \dot{w}}{\partial \mu } \\[2pt]0 & 0\end{array}\right|.\end{array} $$

According to this method, the equilibrium of the system is saddle path if $\det (J)>0$ and $\Omega <0$.

Let use compute the required derivative.

$$\begin{array}{@{}rcl@{}} \frac{\partial \dot{w}}{\partial C^{\theta }} &=&\frac{1}{eA(\cdot )F^{\prime \prime }(\cdot )} \left\{ \frac{R(\theta +S^{\ast })}{e\theta S^{\ast }}\left[A^{\prime}(\cdot )F^{\prime}(\cdot)-\frac{\mu^{\ast}}{e\theta }\left(\frac{\theta}{S^{\ast }}+1-e\right)\right]+\rho A^{\prime}(\cdot )F^{\prime}(\cdot)\right.\notag \\ &&\left.+ \frac{\mu^{\ast}}{e}\left[\frac{\rho}{\theta}\left(\frac{\theta}{ S^{\ast }}+1-e\right)+\frac{R}{(S^{\ast})^{2}}\right]+A^{\prime\prime}(\cdot )F(\cdot)\frac{C^{\theta^{\ast } }}{\theta}\right\} \notag\\ &=&\frac{1}{eA(\cdot )F^{\prime \prime }(\cdot )} \left\{A^{\prime}(\cdot )F^{\prime}\cdot\frac{Z(S^{\ast})}{e\theta S^{\ast}}+A^{\prime\prime}(\cdot )F(\cdot)\frac{C^{\theta^{\ast } }}{\theta}+A^{\prime}(\cdot )F(\cdot)\cdot \frac{R(\theta +S^{\ast })}{e^{2}\theta^{2} S^{\ast }}\cdot \left(\frac{ \theta}{S^{\ast }}+1-e\right)\right.\notag \\ &&\left.-A^{\prime}(\cdot )F(\cdot)\cdot\frac{\theta S^{\ast}}{Z(S^{\ast})} \left[\frac{\rho}{\theta}\left(\frac{\theta}{S^{\ast }}+1-e\right)+\frac{R}{ (S^{\ast})^{2}}\right] \right\}\lessgtr 0 \notag\\ \frac{\partial \dot{w}}{\partial S} &=&\frac{1}{eA(\cdot )F^{\prime \prime }(\cdot )} \left\{ - \frac{R\cdot M}{eS^{\ast }\theta(\theta +S^{\ast })} \left[A^{\prime}(\cdot )F^{ \prime}(\cdot)-\frac{\mu^{\ast}}{e\theta}\left( \frac{\theta}{S^{\ast}}+1-e\right)\right] \right. \notag\\ &&\left.+ \frac{\rho}{e}\left(-c^{\prime}(S^{\ast})+\frac{ \mu^{\ast}C^{\theta^{\ast}}}{\theta S^{\ast }}\right) +c^{\prime\prime}(S^{\ast})\frac{R}{e}+ \frac{\mu^{\ast}C^{\theta^{\ast}}R}{ e\theta (S^{\ast })^{2}}\right\}<0 \notag\\ \frac{\partial \dot{w}}{\partial \mu} &=&\frac{C^{\theta^{\ast } }R}{ e^{2}\theta^{2}A(\cdot )F^{\prime \prime}(\cdot)}<0 \notag\\ \frac{\partial \dot{C}^{\theta }}{\partial C^{\theta }} &=&-\frac{R(\theta +S^{\ast })}{e\theta S^{\ast }} <0\:\:;\:\: \frac{\partial \dot{C}^{\theta } }{\partial S}= -\frac{R\cdot C^{\theta^{\ast }}}{eS^{\ast }\theta}<0 \notag\\ \frac{\partial \dot{\mu}}{\partial C^{\theta }}&=& A^{\prime \prime }(\cdot )F(\cdot )<0\:\:;\:\:\frac{\partial \dot{\mu}}{\partial S } = - \frac{ R\mu^{\ast }}{e(S^{\ast })^{2}}<0 \:\:;\:\: \frac{\partial \dot{\mu}}{ \partial \mu } =\rho +\frac{R}{e}\left( \frac{1}{S^{\ast }}+\frac{1}{\theta } \right)= \frac{Z(S^{\ast })}{e\theta S^{\ast }}>0.\notag\\ \end{array} $$

We calculate $\det (J)$ as follows:

$$\begin{array}{@{}rcl@{}} \det(J) &=& -e \left[\frac{\partial \dot{w}}{\partial C^{\theta }} \cdot \frac{\partial \dot{C}^{\theta }}{\partial S}\cdot\frac{\partial \dot{\mu}}{ \partial \mu } -\frac{\partial \dot{w}}{\partial S}\cdot \frac{\partial \dot{ C}^{\theta }}{\partial C^{\theta }}\cdot\frac{\partial \dot{\mu}}{\partial \mu } +\frac{\partial \dot{w}}{\partial \mu} \left(\frac{\partial \dot{C} ^{\theta }}{\partial C^{\theta }}\cdot\frac{\partial \dot{\mu}}{\partial S }- \frac{\partial \dot{C}^{\theta }}{\partial S}\cdot\frac{\partial \dot{\mu}}{ \partial C^{\theta }}\right) \right] \notag\\ &=& -e \left[ \underbrace{\frac{\partial \dot{C}^{\theta }}{\partial S}} _{<0}\cdot\left( \underbrace{\frac{\partial \dot{w}}{\partial C^{\theta }}} _{?} \cdot \underbrace{\frac{\partial \dot{\mu}}{\partial \mu }}_{>0}- \underbrace{\frac{\partial \dot{w}}{\partial \mu}}_{<0} \cdot\underbrace{ \frac{\partial \dot{\mu}}{\partial C^{\theta }}}_{<0}\right) +\underbrace{ \frac{\partial \dot{w}}{\partial \mu}}_{<0}\cdot\underbrace{\frac{\partial \dot{C}^{\theta }}{\partial C^{\theta }}}_{<0}\cdot\underbrace{\frac{ \partial \dot{\mu}}{\partial S }}_{<0} -\underbrace{\frac{\partial \dot{w}}{ \partial S}}_{<0}\cdot \underbrace{\frac{\partial \dot{C}^{\theta }}{ \partial C^{\theta }}}_{<0}\cdot\underbrace{\frac{\partial \dot{\mu}}{ \partial \mu }}_{>0} \right].\notag \end{array} $$

Let us denote the term between the round brackets by $D(S^{\ast })=\left . \left ( \frac {\partial \dot {w}}{\partial C^{\theta }}\cdot \frac {\partial \dot {\mu }}{\partial \mu }-\frac {\partial \dot {w}}{\partial \mu }\cdot \frac {\partial \dot {\mu }}{\partial C^{\theta }} \right ) \right \vert _{(w^{\ast },C^{\theta ^{\ast }},S^{\ast },\mu ^{\ast })}.$ Then, a sufficient condition for $\det (J)$ to be positive is to have $D(S^{\ast })\geq 0.$ Now, let us rewrite $D(S^{\ast })$ as follows:

$$\begin{array}{@{}rcl@{}} D(S^{\ast }) &=&\frac{1}{e^{2}A(\cdot )F^{\prime \prime }(\cdot )}\Bigg\{ \frac{Z(S^{\ast })}{(e\theta S^{\ast })^{2}}\cdot \frac{A^{\prime }(\cdot )}{ \theta S^{\ast }}\left( F^{\prime }(\cdot )\cdot Z(S^{\ast })+F(\cdot )\cdot \frac{R(\theta +S^{\ast })}{e\theta }\cdot \left( \frac{\theta }{S^{\ast }} +1-e\right) \right)\notag \\ && -A^{\prime }(\cdot )F(\cdot )\left[ \frac{\rho }{\theta }\left( \frac{\theta }{S^{\ast }}+1-e\right) +\frac{R}{(S^{\ast })^{2}}\right] + \frac{A^{\prime \prime }(\cdot )F^{{}}(\cdot )\cdot C^{\theta ^{\ast }}}{ \theta }( \rho eS^{\ast }+RS^{\ast }) \Bigg\}.\notag \end{array} $$

One can first remark that if the TFP is assumed constant (i.e., $A^{\prime }(\cdot )\rightarrow 0$), then $D(S^{\ast })=0$ and $\det (J)>0.$ Now, in the more general case, with $A^{\prime }(\cdot )<0$, we can compute the condition on the parameters to have a positive $D(S^{\ast })$ as follows:

$$\begin{array}{@{}rcl@{}} D(S^{\ast }) &\geq &0 \\ &\Leftrightarrow &\frac{\rho }{\theta }\left( \frac{\theta }{S^{\ast }} +1-e\right) +\frac{R}{(S^{\ast })^{2}}>\frac{Z(S^{\ast })}{e^{2}(\theta S^{\ast })^{3}}\left( \frac{F^{\prime }(\cdot )}{F(\cdot )}\cdot Z(S^{\ast })+\frac{R(\theta +S^{\ast })}{e\theta }\cdot \left( \frac{\theta }{S^{\ast } }+1-e\right) \right) \\ &&+\frac{A^{\prime \prime }(\cdot )\cdot C^{\theta ^{\ast }}}{A^{\prime }(\cdot )\theta }( \rho eS^{\ast }+RS^{\ast }) \\ &\Leftrightarrow &\frac{\rho }{\theta }\left( \frac{\theta }{S^{\ast }} +1-e\right) +\frac{R}{(S^{\ast })^{2}}>\frac{Z(S^{\ast })}{e^{2}(\theta S^{\ast })^{3}}\left( \frac{\varepsilon _{F/R}}{R}\cdot Z(S^{\ast })+\frac{ R(\theta +S^{\ast })}{e\theta }\cdot \left( \frac{\theta }{S^{\ast }} +1-e\right) \right) \\ &&+\frac{\varepsilon _{A^{\prime }/M}}{\theta (S^{\ast }+\theta )}\left( \rho eS^{\ast }+RS^{\ast }\right) A \end{array} $$

with $\varepsilon _{A^{\prime }/M}=\frac {MA^{\prime \prime }(\cdot )}{ A^{\prime }(\cdot )}>0$.

We therefore can deduce that $\det J>0$ under this condition.

Besides, in a similar way, we find easily that

$$ \Omega =e\underset{<0}{\underbrace{\frac{\partial \dot{w}}{\partial S}}}+ \underset{<0}{\underbrace{\frac{\partial \dot{C}^{\theta }}{\partial C^{\theta }}}}\cdot \underset{>0}{\underbrace{\frac{\partial \dot{\mu}}{ \partial \mu }}}<0. $$

We therefore obtain that under condition (A), $\det J>0$ and $\Omega <0$, the sufficient condition for real eigenvalues to exist and thus for local monotonicity to hold are fulfilled. This concludes the proof.