Using Tradable Water Permits in Irrigated Agriculture

Abstract

The present paper examines, both theoretically and empirically, the environmental and economic benefits of introducing a policy to optimally manage groundwater used in irrigated agriculture and the efficiency potential of allowing trading of water permits. We confirm the results appearing separately in the literature on optimal resource management and tradable permits, that both tradable water permits and non-tradable quotas provide the basic mechanism for sustainable water use and yield substantial economic benefits and that allowing trading of water permits improves efficiency. In order to derive the above results we develop a discrete time model to realistically describe farmer’s myopic behavior, while using a continuous time model to describe optimal management. We also incorporate into the analysis the economic effect of sea water intrusion in the aquifer and we estimate the cost of water overexploitation under myopic behavior. The empirical part of the paper is based on simulations using data from an agricultural region in Northern Greece. The main contribution of the paper is the introduction of heterogeneity in crops’ production and market characteristics. We show that the more diverse are the crops, in technology and market prices, and the stricter is the water constraint, the higher are the benefits from using a tradable water permit system.

Appendices

Appendix 1: Time Path Under Myopic Behavior

Using the time-adjusted Eq. (6), we derive water use for time period $t+1$, $q_{r,t+1}$, by substituting the corresponding height of the water table, $H_{t+1}$, from Eq. (5). Then we derive the change in water use between the two successive time periods,

$$\begin{aligned} q_{r,t+1}-q_{r,t}=\frac{c_{0}}{2ASb_{r}p_{r}}\left[ N-\left( 1-\alpha \right) M\sum _{r=1}^{2}q_{r,t}\right] . \end{aligned}$$

(17)

Similarly we obtain the change in aggregate water use over the two periods,

$$\begin{aligned} Q_{t+1}-Q_{t}=\frac{c_{0}\Omega }{AS}\left[ N-\left( 1-\alpha \right) M\sum _{r=1}^{2}q_{r,t}\right] . \end{aligned}$$

(18)

The above two equations express the time path for individual and aggregate groundwater use in irrigated agriculture as discrete-time functions.

Utilizing the initial condition $H(0)=H_{0}$, Eq. (6) and (7) yields the initial individual and total pumping water volumes, $q_{r,0}$ and $Q_{0}$. The initial conditions allow us to formulate a first-order difference equation for the total groundwater use $Q_{t+\Delta \tau }=f(Q_{t})$.

In order to solve a first order difference equation for the total groundwater use it is necessary to find a formula that satisfies $Q_{t+\Delta \tau }=f(Q_{t})$.

Equation (9) can be used as the base for formulating the first order difference equation, which can be written as,

$$\begin{aligned} Q_{t+1}+1=\Delta Q_{t}+K\ , \end{aligned}$$

(19)

where by (18), $\Delta =1-\left( 1-a\right) \frac{c_{0}\Omega }{AS}M$ and $K=\frac{c_{0}\Omega }{AS}N$.

The first step in solving Eq. (19) is to find a particular solution, denoted as $Q_{s}$, which is actually any solution to the above first order difference equation. A constant over time variable is applied in Eq. (19) (Pemberton and Rau 2001), yielding the following particular solution,

$$\begin{aligned} Q_{s}=\frac{N}{1-\alpha }. \end{aligned}$$

The associated homogenous equation of (19) is $Z_{t+1}=\Delta Z_{t}$ ; hence the complementary solution is $\Phi \Delta ^{t}$, where $\Phi$ is an arbitrary constant. Therefore, the general solution to the difference equation is,

$$\begin{aligned} Q_{t}=\frac{N}{1-\alpha }+\Phi \Delta ^{t}. \end{aligned}$$

(20)

The value of the constant $\Phi$ is derived using the boundary condition $Q_{0}$ and thus, the final solution concerning the time path of the aggregate groundwater use is,

$$\begin{aligned} Q_{t}=\frac{N}{1-\alpha }+\left( Q_{0}-\frac{N}{1-\alpha }\right) \left[ 1-\left( 1-\alpha \right) \frac{c_{0}\Omega }{AS}M\right] ^{t}. \end{aligned}$$

(21)

Appendix 2: Time Path Under Tradable Water Permits

The current value Hamiltonian for the optimal control problem presented in (11) is:

$$\begin{aligned} \mathcal {H}=M\sum \limits _{r=1}^{2}NB_{r,t}\left( \overline{Q}_{t}\right) + \frac{\mu }{AS}\left[ N-\left( 1-\alpha \right) \overline{Q}_{t}\right] \end{aligned}$$

(22)

where $\mu$ is the costate variable reflecting the shadow value of groundwater (i.e. the change in the marginal use cost of groundwater as the height of the water table changes over time). This parameter differentiates the social from the private optimal solution. Given the current value Hamiltonian and assuming an interior solution, the necessary conditions for optimization (optimality condition and adjoint equation respectively) are,

$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial \overline{Q}_{t}}&= 0, \end{aligned}$$

(23)

$$\begin{aligned} \dot{\mu }&= \delta \mu -\frac{\partial \mathcal {H}}{\partial H_{t}}=\delta \mu -c_{0}\overline{Q}_{t}. \end{aligned}$$

(24)

Condition (23) requires that the total marginal net benefits from water use are equal to the shadow value of the actual volume of water pumped from the aquifer. This condition is solved for $\mu$ as function of $\overline{Q}_{t}$ and $H_{t}$.²⁵ Solving the state equation for $\overline{ Q}_{t}$ as a function of $\dot{H}$ and substituting, yields the shadow value of groundwater,

$$\begin{aligned} \mu =\frac{AS}{\left( 1-\alpha \right) }\left[ \frac{\Psi }{\Omega }-\frac{ N-AS\dot{H}}{M\Omega \left( 1-\alpha \right) }-c_{0}\left( S_{L}-H_{t}\right) \right] . \end{aligned}$$

(25)

Differentiating Eq. (25) with respect to time and equating to the right hand side of (24) yields, $\frac{AS}{\left( 1-\alpha \right) }\left[ \frac{AS\ddot{H}_{t}}{M\Omega \left( 1-\alpha \right) }+c_{0} \dot{H}_{t}\right] =\delta \mu -c_{0}\overline{Q}_{t}$. Substituting $\mu$ from (25) and $\overline{Q}_{t}\$from the state equation, and rearranging terms gives the following second order differential equation,

$$\begin{aligned} \ddot{H}_{t}-\delta \dot{H}_{t}-\frac{M\Omega \left( 1-\alpha \right) \delta c_{0}}{AS}H_{t}+\frac{M\Omega \left( 1-\alpha \right) }{AS}\delta c_{0}\left( \Theta +\frac{N}{\delta AS}\right) =0. \end{aligned}$$

(26)

The general solution of the above differential equation can be estimated by reducing it to a first order equation after factorization,

$$\begin{aligned} H_{t}^{p}=H_{0}-\frac{\widetilde{Q}}{Mc_{0}\Omega }+\frac{N}{\delta AS} +X_{1}e^{\rho _{1}t}+X_{2}e^{\rho _{2}t}, \end{aligned}$$

(27)

where, $X_{1}$ and $X_{2}$ are arbitrary constants, while $\rho _{1}$, $\rho _{2}$ are the roots of the polynomial function, after the factorization of differential operators, defined as,

$$\begin{aligned} \rho _{1,2}^{p}=\frac{\delta }{2}\pm \sqrt{\frac{\delta ^{2}}{4}+\frac{ M\Omega \left( 1-\alpha \right) \delta c_{0}}{AS}} \end{aligned}$$

(28)

where the superscript $p$ denotes the equilibrium under the tradeable water permits system. Applying the boundary conditions $H(0)=H_{0}$ and $H(T)=H_{min}$ to (13), yields $X_{i}$, $i,j=1,2$,

$$\begin{aligned} X_{i}^{p}=\frac{H_{0}e^{\rho _{j}^{p}T}-H_{min}-\left( e^{\rho _{j}^{p}T}-1\right) \left( \Theta +\frac{N}{\delta AS}\right) }{e^{\rho _{j}^{p}T}-e^{\rho _{i}^{p}T}} \end{aligned}$$

(29)

Then, the aggregate annual allowable use of groundwater resources is,

$$\begin{aligned} \overline{Q}_{t}^{p}=\widehat{Q}-\frac{AS}{\left( 1-\alpha \right) }\left( \rho _{1}^{p}X_{1}^{p}e^{\rho _{1}^{p}t}+\rho _{2}^{p}X_{2}^{p}e^{\rho _{2}^{p}t}\right) . \end{aligned}$$

(30)

Appendix 3: Time Path Under Non-Tradable Water Quotas

The policy maker solves again the optimal control problem defined in (11). The necessary conditions for the Hamiltonian’s maximization (22) are given by Eqs. (23) and (24). Noting that, $v_{t}=\frac{\overline{Q}_{t}}{MQ_{0}}$, and using (14), we have $\frac{\partial q_{r,t}}{\partial \overline{Q}_{t}}=\frac{ q_{i0}}{MQ_{0}}$. Solving the optimality condition $\frac{\partial \mathcal {H }}{\partial \overline{Q}_{t}}=0$, yields the shadow value of groundwater $\mu$ under the quota management system. Following the same steps as in the previous Section, we derive the first order equation,

$$\begin{aligned} H_{t}^{q}=H_{0}-\Theta +\frac{N}{\delta AS}+X_{1}e^{\rho _{1}t}+X_{2}e^{\rho _{2}t}, \end{aligned}$$

(31)

where, $\Theta =\frac{N\Omega ^{q}}{M\left( 1-\alpha \right) c_{0}}+S_{L}+- \frac{\Psi ^{q}}{c_{0}}$, $\Omega ^{q}=2\frac{ p_{1}b_{1}q_{1,0}^{2}+p_{2}b_{2}q_{2,0}^{2}}{\left( q_{1,0}+q_{2,0}\right) ^{2}}$ and $\Psi ^{q}=\frac{p_{1}a_{1}q_{1,0}+p_{2}a_{2}q_{2,0}}{ q_{1,0}+q_{2,0}}$. The roots of this function are,

$$\begin{aligned} \rho _{1,2}^{q}=\frac{\delta }{2}\pm \sqrt{\frac{\delta ^{2}}{4}+\frac{ M\left( 1-\alpha \right) \delta c_{0}}{AS\Omega ^{q}}}, \end{aligned}$$

(32)

where the superscript $q$ denotes the equilibrium under the non-tradeable water quota system. Applying the same as before boundary conditions $H(0)=H_{0}$ and $H(T)=H_{min}$ to (31), yields $X_{i}^{q}$, $i,j=1,2$,

$$\begin{aligned} X_{i}^{q}=\frac{H_{0}e^{\rho _{j}^{q}T}-H_{min}-\left( e^{\rho _{j}^{q}T}-1\right) \left( \Theta ^{q}+\frac{N}{\delta AS}\right) }{e^{\rho _{j}^{q}T}-e^{\rho _{i}^{q}T}} \end{aligned}$$

(33)

Then, the aggregate annual allowable use of groundwater resources is,

$$\begin{aligned} \overline{Q}_{t}^{q}=\widehat{Q}-\frac{AS}{\left( 1-\alpha \right) } \left( \rho _{1}^{q}X_{1}^{q}e^{\rho _{1}^{q}t}+\rho _{2}^{q}X_{2}^{q}e^{\rho _{2}^{q}t}\right) . \end{aligned}$$

(34)