**3.2 The Gisser-Sanchez model and groundwater management**

 Problems of groundwater allocation have been studied basically in the context of the theory of mine [26–29]. The basic model by Gisser and Sanchez is a simplified representation of the economic, hydrologic, and agronomic facts that must be considered relative to the irrigator's choice of water pumping [1]. The validity of the GSE model rests on the key assumption that the aquifer has to be quite large and on the secondary assumption of a small slope in the water-demand function.

 A separate literature should also have to be taken into account, which deals with groundwater quality. Some papers in this line can be found such as [30–32].

 Groundwater allocation problems have been studied mainly in the context of mine and economists like [33–35]. Some principles of inventory management to derive decision rules for the optimal temporal allocation in a dynamic programming format can also be found in such papers. The effects of different policy instruments that could correct misallocation of commonly owned groundwater can be found in papers such as [31, 35–39], which studied the effects of different policy instruments that might correct the misallocation of commonly owned groundwater. One of the main results of this chapter is that net benefits from groundwater management could amount to over \$100 per acre, but noted that these benefits could decline with increases in interest rate. One of the solutions to this problem was obtained by authors such as Allen and Gisser [40], who derived a formula for a tax that should be imposed on groundwater which was pumped in order to yield the optimal control solution. Finally, in papers such as [41], it can be recognized the issue of congestion externality in aquifers with open access characteristics and suggested a charging tax to accommodate this externality.

When this point is achieved, farmers will either import supplemental water or be restricted to use a smaller amount of water by being assigned water rights. Nevertheless, some changes in the hypothesis related to regulation of water pumping in the aquifer could be made. This case allows to model consistently an optimal control problem and also allows one kind of clarification that should be related with the case of no control. This is the departure point for the works [9, 10] by Gisser and Sanchez.

The basic model analyzed by Gisser and Sanchez is a simplified representation of the economic, hydrologic, and agronomic facts that should be considered for the irrigator's choice of water pumping. An irrigator benefit function could be represented using this function suggested by [44]:

$$
\pi(t) = V(\langle wt \rangle) - C(H(t))w(t) \tag{1}
$$

 where π(t) denotes profits at time t. Net farm revenue from water use π(t) neglecting pumping costs is denoted by

$$V(w) = \int\_0^w p(\mathbf{x})d\mathbf{x} \tag{2}$$

 where p(x) is the inverse demand function for water. C(H) is the average and marginal pumping costs per acre-foot of water and H(t) is the height of water table above some arbitrary reference point at time t [1, 40]. The change in the height of

water is given by differential Eq. (2), which represents the hydrologic state of the aquifer (or equivalently, the environmental constraint of the problem)

$$
\dot{H} = \frac{1}{AS} (R + (a - 1)w),
\\
H(0) = H0 \tag{3}
$$

In this equation, R exemplifies a constant recharge determined in acre feet per year; a is the constant return flow coefficient (which could be considered to be just a simple number); H0 is the initial level of the water table measured in feet above sea level; A is the surface area of the aquifer (uniform at all depths), measured in acres per year; and S is the specific yield of the aquifer. These equations are based on the UNESCO-Encyclopedia Life Support Systems and also on the papers by [1, 43, 45] on the Gisser-Sanchez effect.

More precisely, the aquifer in Gisser and Sanchez's work is modeled as a bathtub, unconfined aquifer, with infinite hydraulic conductivity. It is necessary to point out that infinite hydraulic conductivity implies that the aquifer will never dry up, irrespective of groundwater extraction rates, which is equivalent to the assumption of a bottomless aquifer. The adoption of this hypothesis can be acknowledged by the hypothesis that it is implied by an standard hypothesis which is related to the literature and which implies that time goes to infinity [1]. Nevertheless, if this is not this way, a steady-state solution might not be reached. Besides, Provencher [43] showed that the optimal pumping rate can be substantially lower when the hydraulic conductivity is small enough to result in a significant cone of depression around the well. The assumption of constant return flow in the presence of fixed irrigation technology suggests a constant rate of water application.

 The hypothesis of deterministic and constant recharge in conjunction with the hypothesis of constant return flow suggests constant types of land use [44], independence of surface water and groundwater systems, and constant average rainfall. Besides, sunk costs, replacement costs, and capital costs in general are overlooked, and it is implicitly assumed that energy costs are constant. It is also indirectly accepted that the well pump capacity constraint is nonbinding. Finally, refinement in Gisser and Sanchez's model could be also achieved by assuming that only land superimposing the aquifer can be irrigated. That is, the demand curve does not shift to the right over time. This implies that, the unambiguous recognition of the fact that the main hypothesis behind the GSE indicates that the result should be carefully when working on real aquifer systems.

 Given the above hydroeconomic model, Gisser and Sanchez used a linear water demand function (estimated by [31, 32]) using parametric linear programming, hydrologic parameters that were considered realistic in the 1960s, and a discount rate of 10%, and simulated the intertemporal water pumpage for Pecos Basin in New Mexico, once under the assumption of no control and once under the assumption of optimal control. The most interesting result is that the trajectories under the two regimes are almost identical. This result leads to the main conclusion that there is no substantive quantitative difference between socially optimal rules for pumping water and competitive rates. Therefore, the welfare loss from intertemporal misallocation of pumping effort is negligible. This conclusion amounts to the GSE.

An important effect to consider is that, solving analytically the model, Gisser and Sanchez main result is that, if Eq. (3) is true, then the difference between the two strategies is so small that it can be ignored for practical consideration, where Eq. (3) is

$$
\left[\frac{k \, C\_t \, (a - 1)}{AS}\right] \simeq \mathbf{0} \tag{4}
$$

*Groundwater Management Competitive Solutions: The Relevance of the Gisser-Sanchez Model DOI: http://dx.doi.org/10.5772/intechopen.85507* 

In Eq. (4), k can be considered to be the reduction in demand for water per \$1 intensification in price (that is, the slope of the uncompensated demand curve for groundwater), Ct is the intensification in pumping cost per acre-foot per 1-foot decline in the water table, and AS are given in Eq. (2). If Eq. (3) holds, then the rate of discount will be practically identical with the exponent of the competition result. Therefore, as long as the slope of the groundwater demand is small relative to the aquifer's area times its storativity [1], GSE will persist. From this, the main conclusion is that, if differences between optimal and competitive rates of water pumping are small, then policy considerations can be limited to those which ensure that the market operates in a competitive fashion, and concerns relative to rectifying common property effects could be removed.

### **3.3 Robustness of the GSE effect**

 The GSE effect presents important policy implications. Some empirical papers discussing the robustness of this effect are, Noel et al. [35] found that control increases the value of groundwater in the Yolo basin in California, by 10%. This result is fairly different from [37], who found that control raised the net benefit of groundwater in the Ogallala basin by only 0.3% empirical estimates of benefits from groundwater management in Kern county (California, USA) do not exceed 10%. Nevertheless, in works such as [39], it can be found that groundwater management in the Texas High Plains would be unwarranted, and he proceeded with a sensitivity analysis of present value profits using different slopes and intercept values for the groundwater-demand curve. It is interesting to point out that this analysis indicated that benefits from groundwater management do not increase monotonically as the absolute value of the slope increases.

A basic hypothesis of the Gisser and Sánchez model is that the demand curve for water is linear. This is a fairly conventional hypothesis in most economic demand models. In order to study the relative importance of this hypothesis for the GSE, optimal control and no-control strategies are compared, using a nonlinear demand curve [40]. This comparison confirmed that, for the case of the nonlinear demand function, what had been demonstrated by the GSE for the case of a linear demand function.

However, in works such as [20], it can be found that the differences between the two regimes may not be trivial if the relationship the average extraction cost and the water table level and/or if there exist significant differences in land productivity, applying dynamic programming to a model of a confined aquifer underlying the Crow Creek Valley in South-Western Montana.

It is essential to take into account that when land is assumed to be homogeneous, the gross returns function with respect to water use tends to be nearly linear. Nevertheless, with greater heterogeneity in productivity, the returns function is more concave, and differences in the optimal use policy under a common property setting are more pronounced [1]. Hence, the need for more theoretical work is to determine an asymmetric groundwater pumping differential game, where differences in land productivity are taken into account.

#### **3.4 Variable relations and endogenous rates of change**

Implicit in GSE model is the hypothesis of nonvariable economic relations (that is, time-independent demand) and/or exogenous and constant rates of change (that is, constant and fixed exogenous crop mix, constant crop requirements, fixed irrigation technology), and some significant exceptions can be found such as [43, 44], with constant exogenous kinds of land use and nonvariable hydrologic conditions.

 Nevertheless, in studies with a long run perspective, predictable results could turn out to be weaker as the steady state is approached. Estimated benefit and cost functions used in the simulations of GSE may bear little relation to the actual benefit and cost functions when economic, hydrologic, and agronomic conditions are much different. More complex representations of increasing resource scarcity incorporate opportunities for adaptation to the rising resource prices which are a main indicator for scarcity. In the long run, adoption of new techniques, substitution of alternative inputs, and production of a different mix of products offer rational responses to increasing scarcity [1], [38].
