Groundwater Management in a Food Security Context

Abstract

This article analyzes the sustainability of market-based instruments such as tradable permits for the management of a renewable aquifer used for irrigated agriculture. In our dynamic hydro-economic model, a water agency aims at satisfying a food security constraint within a tradable permit scheme in the presence of myopic heterogeneous agents. We identify analytically the viability kernel that defines the states of the resource yielding inter-temporal feasible paths able to satisfy the set of constraints over time and the associated set of viable quota policies. We then illustrate the theoretical results of the paper with numerical simulations based on the Western La Mancha aquifer.

Appendix

1.1 Optimality Conditions

The Lagrangian corresponding to the problem (4) with $w_{i} \ge 0$ and $q_{i} \ge 0$ is

$$\begin{aligned} L=\pi _{i}\left( H,w_{i}\right) -mq_{i}+\lambda _{i}\left( q_{i}-w_{i}\right) \end{aligned}$$

The Kuhn-Tucker conditions are

$$\begin{aligned} L_{w_{i}}^{\prime }= & {} \frac{\partial \pi _{i}\left( H,w_{i}\right) }{ \partial w_{i}}-\lambda _{i} \le 0; ({=}0 \text{ if } w_{i}>0),\\ L_{q_{i}}^{\prime }= & {} -m+\lambda _{i} \le 0; ({=}0 \text{ if } q_{i}>0),\\ \lambda _{i}\ge & {} 0\text {, }\lambda _{i}\left( q_{i}-w_{i}\right) =0. \end{aligned}$$

If $\lambda _{i}=0$, the constraint does not bind then $w_{i}^{*}$ is solution of $\partial \pi _{i}\left( H,w_{i}\right) /\partial w_{i}=0$ and $q_{i}=0$. Otherwise $\lambda _{i}>0$ implies a positive quota price $m>0$. It shows that the constraint binds $q_{i}=w_{i}$ and $\frac{\partial \pi _{i}\left( H,q_{i}\right) }{\partial q_{i}}=m$ which gives Eq.  (5).

1.2 Proof of the Viability Kernel

Proof

Consider the dynamics

$$\begin{aligned} H(t+1)=H(t)+\frac{1-\mu }{AS}\left( Q_{R}-Q(t)\right) , \end{aligned}$$

with $Q_{R}=\frac{R}{1-\mu }$.

We first show that $Q_{FS}\le Q_{R}$ implies $Viab=\left[ H_{\lim }, S_L \right]$.

Assume that $H_0 \ge H_{\lim }$, choose $Q(t)=Q_{FS}$ then

$$\begin{aligned} H(t)+\frac{1-\mu }{AS}\left( Q_{R}-Q(t)\right) =H(t)+\frac{1-\mu }{AS}\left( Q_{R}-Q_{FS}\right) \ge H{(t)} \ge H_{\lim }. \end{aligned}$$

Hence $\left[ H_{\lim }, S_L \right]$ is viable and $Viab=\left[ H_{\lim }, S_L \right]$.

Now if $Q_{FS}>Q_R$, we show by forward induction that $\forall H_0 \ge H_{\lim }$

$$\begin{aligned} H(1)= & {} H(0)- \left( \frac{1-\mu }{AS}\right) \left( Q_{FS}-Q_{R})\right) , \\ H(2)= & {} H(0)-2 \left( \frac{1-\mu }{AS}\right) \left( Q_{FS}-Q_{R})\right) , \\ H(t)= & {} H(0)-t \left( \frac{1-\mu }{AS}\right) \left( Q_{FS}-Q_{R})\right) . \end{aligned}$$

Hence $\exists t^{*}$ such that $H(t^{*})< H_{\lim }$, it implies that $Viab=\emptyset$.

1.3 Dynamic Constraint $Q_D$

The constraint on the state variable $H_{\lim } \le H(t+1)$ implies

$$\begin{aligned} H_{\lim }\le & {} H(t)+\frac{R}{AS}-\frac{(1-\mu )}{AS}Q(t), \\\Longleftrightarrow & {} Q(t)\le Q_{R}-\frac{AS}{(1-\mu )}H_{\lim } +\frac{AS}{ (1-\mu )}H(t). \end{aligned}$$

By denoting $Q_D= Q_{R}-\frac{AS}{(1-\mu )}H_{\lim } +\frac{AS}{(1-\mu )}H(t)$, it gives $Q(t)\le Q_D(H(t))$. We look at the conditions depending on the sign of $Q_M-Q_D$ under which this dynamic constraint is binding and reduces the viability kernel. By definition, $Q_D(H_{\lim })=Q_R$ and since Q is bounded by $Q_R$, it implies that for $H=H_{\lim }$

$$\begin{aligned} Q_M(H_{\lim })< Q_D(H_{\lim }). \end{aligned}$$

It yields

$$\begin{aligned} \overline{W}-Q_R<\frac{\beta }{p_{y}}(c_{0}- c_{1}H_{\lim }). \end{aligned}$$

The expression of $Q_M-Q_D$ is given by

$$\begin{aligned} Q_M-Q_D=\overline{W}-Q_R-\frac{\beta c_{0}}{p_{y}} +\frac{c_{1}\beta }{p_{y}} H-\frac{AS}{1-\mu }(H-H_{\lim }). \end{aligned}$$

Using the two previous expressions gives

$$\begin{aligned} Q_M-Q_D< & {} \frac{\beta }{p_{y}}(c_{0}- c_{1}H_{\lim }) -\frac{\beta c_{0}}{ p_{y}} +\frac{c_{1}\beta }{p_{y}}H-\frac{AS}{1-\mu }(H-H_{\lim }), \nonumber \\ Q_M-Q_D< & {} \frac{c_{1}\beta }{p_{y}}(H-H_{\lim })-\frac{AS}{1-\mu }(H-H_{\lim }). \end{aligned}$$

When $H>H_{\lim }$ the condition ensuring $Q_M-Q_D>0$ is

$$\begin{aligned} \frac{c_1(1-\mu )(p_y/\beta )}{AS} > 1, \end{aligned}$$

and corresponds to Eq.  (29) in the text.