*2.1. Theoretical Perspective of the Analytical Model*

We refer to a region that is composed of *N* subareas (*n* = 1, 2, ... , *N*). Each subarea *n* could comprise water/irrigation districts that incorporate several agricultural producers regulated by individual water district mandates. Each subarea *n* includes *Kn* (*kn* = 1, 2, ... , *Kn*) agricultural producers that are considered nonpoint source polluters of a given pollutant, or of a set of several pollutants (for simplicity we will refer to salinity as the pollutant in question). Each agricultural producer applies water on agricultural crops to produce market products. A byproduct in the form of agricultural pollution is the irrigation return flow that may contain a regulated water quality pollutant, which we will specify to be salinity.

Each of the *k* producers in the *n*-th subarea may have different factors affecting agricultural production conditions (natural and technical) that can lead to different cropping patterns, crop yields, net revenue, and the salt concentration and salt load of the return flow. We define a production function of agricultural yield and return flow for producer *k* as (for simplicity we drop the indexes *k* and *n*):

$$\begin{cases} \begin{array}{c} \mathcal{Y} = f(\mathcal{W}, \mathcal{C}, T | \underline{\mathbf{X}})\\ \mathcal{Q} = \mathcal{g}(\mathcal{W}, \mathcal{C}, \mathcal{T} | \underline{\mathbf{X}})\\ \mathcal{S} = h(\mathcal{W}, \mathcal{C}, \mathcal{T} | \underline{\mathbf{X}}) \end{array} \tag{1} $$

where *Y* is yield per acre of a given crop, *W* is water applied per acre, *C* is salinity level of applied water, *T* is irrigation technology used (expressed in integer values to indicate various irrigation technologies available to each agricultural grower within a designated subarea), *Q* is volume of return flow produced on that farm, *S* is the salt concentration of the return flows, and *X* is a vector of all fixed effects related to the location of that producer. We will discuss later the first and second order conditions of the production function derivatives, namely the shape of these three components of the production function.

Given Equation (1), agricultural producers within a designated subarea maximize their net revenue under constraints imposed by both natural and regulatory conditions:

$$
\pi\_{k\_n} = \sum\_{\mathbb{C} \text{rops}} p \cdot Y \cdot L - w \cdot \mathcal{W} - t \cdot T \tag{2}
$$

*s*.*t*.:

$$\sum\_{Crops} L \le \overline{L} \tag{3}$$

$$\sum\_{Crops} L \cdot \mathcal{W} \le \overline{\mathcal{W}} \tag{4}$$

Additional constraints imposed on each agricultural producer within a designated subarea by subarea management, are summarized in (5), and discussed below

$$\begin{array}{c} \text{Land following constraints} \\ \text{Irrigation technology constraints} \\ \text{Specific crop constraints} \end{array} \tag{5}$$

where for each agricultural producer within a designated subarea, *p* is the price per unit of crop, *L* is the area grown with that crop, *w* is the price of water, *t* is the per-acre cost of the technology, *L* is the total cultivable land of the agricultural producer, and *W* is the water quota imposed by the subarea on the agricultural producer. Net revenue is defined as the revenue from crop sales minus the variable costs of production and payments of fees for exceedance of pollution load.

The solution to (2)–(5) provides for each agricultural grower within a designated subarea: the area under production with each crop selected; the total amount of water applied; the technology selected for each crop; the total profit; the total volume of return flow from the designated subarea; and the salt concentration of the return flow that can be used to compute drainage salt (mass) loads. While we may predict the volume *Q* and salt loading with associated *S* for each subarea, such information is not available to either the subarea management or to the federal regulator.

The subarea managers have access to monitoring data that provides the total volume of *Q* from all agricultural producers and its quality, *S*, that can be used to estimate salt loading. Salt loading is the factor each subarea manager is obligated not to exceed on a monthly and annual basis by the regulator, as defined within the Total Maximum Daily Load (TMDL) allocation for each subarea. TMDLs are the policy vehicles that are used by the US Environmental Protection Agency to limit nonpoint source pollution to levels that do not exceed the assimilative capacity of the receiving water body. TMDLs are keyed into water quality standards or objectives at a compliance monitoring station for the pollutant in the receiving water, and are designed to be protective. The agricultural non-point source pollutant management problem is a typical principal–agent problem under circumstances of asymmetry of information. Hence, we need to introduce several simplifying assumptions. We start by drawing (Figure 1) a schematic regional setting, using four agriculturally dominated subareas located on the valley floor, and a water body in the form of a river (describing the actual situation in the region that we will empirically analyze). The remaining three subareas are tributary river watersheds where water flow is controlled by upstream dams and reservoirs and whose operation is largely independent of agricultural drainage decision making.

**Figure 1.** A schematic representation of the region with subareas, agricultural producers, and a regulated receiving water body (river).
