**2. Methods**

The Israeli water supply system was designed to cope with challenges associated with temporal and spatial water distribution. Natural freshwater sources are enriched during the winter, whereas most of the consumption occurs in the summer; this pattern requires water storage. The water delivery system was originally designed to transfer water from the rainier northern parts of the country to the populated center and for irrigating the large agricultural lands in the south. Since 2005, with the installation of desalination plants on the Mediterranean coast, the supply has been gradually shifted to a west–east direction. As a public property, water is centrally managed by the government, which designs the supply and controls consumption through a set of prices, quotas, and pumping licenses [36]. These physical and legislational structures imply that the government is facing a water management optimization problem that integrates dynamic and spatial dimensions. The MYWAS-VALUE model was developed to solve such problems.

MYWAS is a dynamic model of the Israeli water system, and VALUE represents the activities in the vegetative agricultural regions as incorporated into MYWAS. MYWAS encompasses 21 urban regions that consume freshwater for domestic and industrial uses and 18 agricultural regions that can consume freshwater, TWW, and brackish water. The water sources are represented in the model by 19 naturally enriched freshwater stocks, 5 seawater desalination plants, 4 non-enriched brackish water aquifers, 19 wastewater treatment plants, 163 freshwater pipelines, and 74 pipelines for sewage, TWW, and brackish water. MYWAS determines the socially optimal allocation of water types to the demand regions during each period throughout a predetermined planning horizon while also accounting for the welfare of the water users in those regions, the variable supply costs, the constraints associated with water availability in the sources, and the infrastructural capacity constraints. In addition, the model determines the extent to which each infrastructural water element is extended during each period while weighing the investment costs versus the net benefits associated with the extended capacities in future periods.

VALUE is a positive mathematical programming (PMP) model of MYWAS's 18 agricultural regions. Each region incorporates 55 crops whose output prices constitute equilibrium in the statewide markets for industrial, export, and local fresh vegetative products that are assumed to be competitive. The crop production functions account for the salinity of the water supplied to each crop. The land allocations to the crops in the regions maximize social welfare subject to regional input constraints, where social welfare incorporates the surpluses of the consumers of agricultural products minus the production costs. The constrained inputs in each region include land, foreign workers (who are allocated to farmers based on cropping patterns), and the amounts of water delivered to the region from accessible sources; the latter is determined by MYWAS.

Population growth shifts the demand for water in urban zones and the country-wide demand for vegetative agricultural products to the right, thereby driving the dynamic expansion of water supply infrastructures. The model tracks the salinity concentrations along the water supply system and can control the salinity of the irrigation water in each agricultural region by increasing desalination capacities and/or changing the shares of allocated water from the different sources accessible to the region. The model reports the efficiency water prices at any node of the water distribution network, which are equal to the shadow price of the water at this node. In addition, it reports the irrigation water's VMP, which depends on its salinity. Of course, for an optimal allocation, the efficiency water price and the VMP are equal. Based on the efficiency prices, the model reports the allocation of welfare among the urban and agricultural water users, the water suppliers, and the consumers of agricultural outputs (this presumes that prices are the exclusive water allocation instrument in the water economy; in practice, this is the case in Israel, although prices are higher than optimal because of cost recovery regulations [29]). Our scenarios span a 30-year period, which we divide into 10 equal sub-periods to reduce the computational burden; each period is referred to by its last year.

The version of MYWAS-VALUE that is employed in this study was calibrated based on 2019 data (the model is available as a Supplementary Material to this paper). A detailed description of the model is provided by Slater et al. [35]; the rest of this section describes the recalibration of the VALUE model under the FB assumption, which replaces the RB specification based on which the version of the model in Slater et al. [35] is calibrated.

Our challenge is to calibrate VALUE in the absence of field-level information regarding the actual allocation of water types to the crops in each region—an allocation that is assumed to be optimal in the prevailing situation. To that end, we employ a multistage calibration procedure that involves the optimal assignment of the water sources accessed by a region to the crops grown therein while accounting for the crops' relative salinity tolerance and profitability. Specifically, we introduce a preliminary stage to the commonly used two-stage procedure applied for the calibration of classical PMP models [37]. In this preliminary stage, water types are optimally allocated to each crop subject to their respective regional water availability constraints, where the land allocated to each crop is kept constant at its baseline level. Note that our production function for each crop is a nonlinear function that relates the per hectare yield to the per hectare annual water application and the salinity level of the applied water (as in Slater et al. [35], the per hectare annual amount of water applied to each crop is constant, and therefore only changes in the salinity of the water assigned to each crop affect its per hectare yield); thus, changes in the type of water applied to a crop vary its per hectare outputs. Therefore, the preliminary calibration stage optimally assigns the water sources to the crops and sets the production function parameters so as to reproduce the per hectare yield reported in crop budget reports (see Appendix A for a formal description of the preliminary calibration stage). Then, we apply the first stage of the PMP calibration procedure, which elicits the dual values of the perturbed crop-specific land constraints. Note that to obtain the correct dual values, one should also incorporate the water allocations to the crops as decision variables, which renders the optimization problem of that stage nonlinear (in contrast to the first-stage linear problem of the original PMP procedure). The rest of the calibration process follows the second stage of the PMP procedure as well as the calibration of the demand functions for agricultural products and urban water usage (see [35]).

The outcome of the preliminary calibration stage with respect to the optimal allocation of irrigation water types to crops involves minimal blending; that is, each crop is irrigated by only one water type, where mixtures are assigned to a few crops to meet the availability constraints associated with the regional water sources. While this qualitative result was already shown by Kan and Rapaport-Rom [22], here, the water allocation to crops is optimal rather than imposed by other criteria (e.g., Kan and Rapaport-Rom [22] employed a hierarchical procedure to assign water types to crops). Figure 2 presents the allocation of the irrigation water types—desalinated freshwater (EC = 0.25 dS m<sup>−</sup>1), fresh groundwater (EC = 1 dS m−1), TWW (EC = 1–1.77 dS m−1), and brackish water (EC = 2.35–4.0 dS m<sup>−</sup>1)—to four groups of crops classified according to their sensitivity to salinity: sensitive, moderately sensitive, moderately tolerant, and tolerant [38]; as expected, the higher the salinity tolerance of the crops, the higher the salinity of the irrigation water allocated to them.

**Figure 2.** Optimal allocation of the four irrigation water types that differ in terms of their salinity levels to four groups of crops with different salinity tolerances.
