Regional blending of fresh and saline irrigation water: Is it efficient?


  • Iddo Kan,

    Corresponding author
    1. Department of Agricultural Economics and Management and Center for Agricultural Economic Research, Hebrew University of Jerusalem,Rehovot,Israel
    • Corresponding author: I. Kan, Department of Agricultural Economics and Management and Center for Agricultural Economic Research, Hebrew University of Jerusalem, PO Box 12, Rehovot 76100, Israel. (

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  • Mickey Rapaport-Rom

    1. Natural Resource and Environmental Research Center, University of Haifa,Haifa,Israel
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[1] Blending fresh and saline irrigation waters is implemented in many countries facing water scarcity. However, when analyzed at the field level, previous economic studies have indicated that blending fresh and saline water is suboptimal. This paper examines the blending issue on a regional scale, where both water sources and land are concurrently allocated to crops. Regional water distribution networks that enable salinity adjustment at the field level are compared to networks that allow controlling water salinity on a regional scale only, such that salt concentrations cannot differ by crop. We characterize the conditions for blending to be an optimal strategy under regional salinity control networks, and show that these conditions can be met by an empirical water production model commonly used in the literature. Empirical analysis of 16 regions in Israel revealed optimal blending in six of them. However, regardless of whether blending is optimal or not, the optimal fresh-water application is higher under regional salinity control networks, implying that blending does not support freshwater conservation. The paper analyzes the relationship between water and land constraints' shadow values, and the properties of the two water distribution networks. We show that although farming revenues are higher under networks that allow assignment of specific water salinities to crops, regional salinity control networks can become more profitable to farmers who face prices set endogenously so as to be the binding factor on the use of constrained water and land. The implications of the network selection on intraregional water supply costs are discussed.

1. Introduction

[2] Water scarcity, which an estimated 60% of the world's population is expected to face by the year 2025 [Qadir et al., 2007], is causing increasing reliance on various higher-salinity water sources as substitutes for freshwater irrigation. One of these sources is aquifers containing brackish water, as utilized, for example, in Israel [Pasternak and De Malach, 1995], Texas [Mehta et al., 2000], and Argentina [Foster and Chilton, 2003]. Another source is drainage emitted from subsurface tile lines, which are installed to lower the water table in waterlogged areas, for example, in Pakistan [Ghassemi et al., 1995], California [Oster and Grattan, 2002], and Australia [National Water Commission, 2006]. Irrigation by wastewater, which carries salts added through domestic, industrial, and animal production uses of fresh water, constitutes both a reliable substitute for an unstable supply of (scarce) fresh water, and a way to avoid costly alternative disposal methods. Therefore, treated wastewater application in agriculture is on the rise in California [State Water Resources Control Board, 1999], Australia [Schaefer, 2001], Europe [Angelakis and Bontoux, 2001], the Middle East, and North Africa [FAO, 1997].

[3] Under water scarcity, a regional blending problem arises whenever a few water sources with differing salinity levels are available for irrigation within a given region (a region is considered an area wherein multiple crops are grown, be it a farm, a district, etc.; whereas a field is assigned to a single crop). Irrigating all of the crops in a region with a mixture of all of the water sources is one option. Alternatively, each source can be applied unmixed to a specific group of crops; e.g., brackish and fresh water to salinity-tolerant and salinity-sensitive crops, respectively. Yet these are only two of numerous water management alternatives, e.g., some sources may be blended, others applied unmixed, and the rest not consumed at all.

[4] In fact, water blending has been used in many regions throughout the world, including the North China Plain [Sheng and Xiuling, 1997], the Broadview water district in California [Wichelns et al., 2002], Australia [Hamilton et al., 2007], Egypt [Tanji and Kielen, 2002], and Pakistan [Sharma and Minhas, 2005]. Some review studies mention blending as a strategy used for, or having the potential for saving, fresh water [Aslam et al., 2002; Hamdy, 2003; Kulkarni, 2011]. Blending options have also been the subject of extensive agronomic studies examining their impacts on yield and soil properties (see Dudley et al. [2008] for review). However, those studies do not tell us whether the widespread blending practice is an economically efficient strategy.

[5] Perhaps the first to analyze the efficacy of mixing waters were Parkinson et al. [1970]; however, they considered only predetermined blending combinations. For Feinerman and Yaron [1983] and Knapp and Dinar [1984] blending was endogenous and, depending on prices and crops' salinity tolerance, could become optimal at the regional and field levels, respectively. However, in both of those studies, salinity was the only factor affecting production.

[6] Dinar et al. [1986]analyzed blending at the field level, based on the crop-water-salinity production model developed byLetey et al. [1985], wherein larger quantities of water can compensate for yield reductions caused by higher salinity levels. Using the quadratic production functions estimated by Letey and Dinar [1986], Dinar et al. [1986]showed that field-level blending can be optimal in the case of salt-tolerant crops such as cotton.Kan et al. [2002] utilized the same production model [Letey et al., 1985] to estimate sigmoid production functions. They concluded that blending becomes optimal only when field-level water application constraints are introduced. On the other hand, incorporating the positive impact of salinity on the quality of output in some crops may render field-level blending economically justifiable [Kan, 2008].

[7] This paper presents the blending problem as being related to regional water infrastructures. In a given region, farmers' flexibility with respect to controlling the salinity of the water applied to each of their fields depends on the number of discrete water types to which they have access. This accessibility, in turn, depends on both the farmers' direct access to water sources (e.g., to aquifers and on-farm drainage systems) and the properties of intraregional water supply networks; such networks may deliver wastewater from treatment plants, fresh water from snowmelt-runoff catchments, etc. As accessibility to a larger number of water sources entails higher water distribution costs, to achieve efficiency, a regional planner should weigh the benefits derived from the flexibility provided by the distribution network against the associated water distribution costs. This paper analyzes the properties of the benefit side of this cost-benefit equation and infers regarding the implications of network selection on water use, farming profits, and options for saving on intraregion water delivery costs.

[8] Specifically, two kinds of water distribution networks are considered: networks that provide every field in a region with access to all available water sources, thereby enabling field-level salinity control (FSC) of water; and networks that supply a single type of water with uniform salinity level to all fields, thereby allowing only regional salinity control (RSC). Note that while FSC networks are expected to generate the highest farming profits, they may also carry the highest distribution costs, particularly in regions where, in order to enable separate conveyance of each water source to every field, parallel pipe systems must be installed.

[9] Regarding these two networks, our objectives are (1) to explore the basic links between efficient water management at the field and regional levels, particularly with respect to the blending problem, (2) to examine the networks' implications on the use of fresh water, and (3) to evaluate the difference in farmers' profits between these networks under various water- and land-pricing schemes. These are essentially empirical issues, and as such, are analyzed based on specifications: The modeling approach developed byKan et al. [2002] is adopted and applied to the case study of Israel.

[10] We develop a regional-scale positive mathematical programming (PMP) model that allocates constrained land and water sources among crops. The spatial units of the analysis areecological regions; this partitioning of Israel into 21 zones is commonly used by the authorities for data collection and spatial analyses. Each ecological region is characterized by specific geological, topographical, demographic, and climatic attributes, which affect both the conditions for agricultural production and the availability of water sources. Our data are for the year 2002. The analysis focuses on the 16 regions that had access to at least two of the four potential sources of irrigation water: fresh water, brackish water, and secondary- and tertiary-treated wastewater.Table 1 reports the total land and irrigation water used for agricultural production in 2002 in the 16 ecological regions, ordered from north to south.

Table 1. Regional Land and Water Consumptiona
RegionLand (103 ha)Fresh Water (1 dS m−1) (106 m3)Secondary Wastewater (2 dS m−1) (106 m3)Tertiary Wastewater (2 dS m−1) (106 m3)Brackish Water (4 dS m−1) (106 m3)Total Water (106 m3)Average Salinity (dS m−1)
  • a

    These amounts are considered constraints in the empirical analyzes.

Hula Basin17.859.83.663.41.06
Western Galilee24.739.
Beit She'an12.353.30.110.664.01.50
Harod Valley4.116.00.716.71.13
Jordan Valley2.121.91.923.71.24
Lower Galilee12.917.
Jezreel Valley25.016.339.40.456.01.73
All regions322.2473.7182.4139.581.0876.61.64

[11] While southern regions face lowered access to fresh water for irrigation, they also have at their disposal more alternative sources with higher salinity levels; hence, on average, the salinity of the applied irrigation water is higher in these regions (Table 1, right-hand column). This situation reflects Israel's spatial distribution of water scarcity: Israel is located on the boundary of a desert, its north rainy, and its south dry. Rain falls in the winter only, yet water consumption is highest in the summer. Rainy years alternate with dry years in no discernible pattern. The role of the national water system is therefore to collect the winter rainfall from rainy years and store it for use in dry years, and to deliver fresh water from the north southward to Israel's populated center and to the Negev. In 2002, nearly 50% of Israel's wastewater was treated [Feinerman et al., 2003] and reused for agricultural irrigation, particularly in the Jezreel Valley and the Negev.

[12] All water sources in Israel are state property, and as such, the state controls their consumption by setting nontransferable user quotas, as well as administrative prices that are uniform nationwide. As in many other countries, prices in Israel do not reflect water scarcity [Kislev, 2006; Molle, 2009]. The state also owns the vast majority of the agricultural land, and charges leasing fees based on cropping acreage and regions, regardless of land-market prices [Israel Land Authority, 2010]. In our empirical analysis we simulate regions with FSC and RSC networks and compare farmers' profits under the observed administrative water and land prices, as well as under prices determined endogenously, i.e., those that incorporate the regional water and land constraints' shadow values, such that these prices are both binding and impose consumption of the entirety of land and water available to the regions.

[13] Section 2describes the regional water management model and its versions vis-à-vis the distribution networks' structures. Insection 3 we analyze the link between efficient water management on the field and regional scales. Section 4 describes the specifications and data of the empirical analysis, the results of which are discussed in section 5. Section 6 concludes.

2. Model

[14] Consider a small region j, inline image in a small, open economy. Let i, inline image, denote a crop where each of the region's farmers can potentially grow the I crops. Farmers in each region are assumed acting in a competitive environment subject to regional conditions and constraints. Our model analyzes the regionally aggregated effects of various water distribution networks in terms of regional land and water allocations among crops, and the associated regional crop production profits.

[15] The region has access to N types of water sources, which differ only in their salinities, prices, and regional availability constraints. We denote by inline image (m3 yr−1) the regional quantity constraint of water source n, inline image. We let inline image (dS m−1) and inline image ($ m−3) be the salinity and price of source n, respectively; both are uniform across regions, and exogenous from the regional farmers' point of view.

[16] A regional water supply network enables provision of only K, inline image, types of water, which are separately distributed to all fields in the region. Suppose that, of the N available water sources, the first inline image sources are restricted to being distributed only after they have been blended, whereas each of the other inline image sources can be distributed separately. Let us index by k, inline image, a specific water type available for irrigation, such that inline image stands for the mixture of the first M sources; inline image is water source number inline image; and inline image is water source inline image. The salinity and price of the mixed water ( inline image) are

display math

respectively, where inline image (m3 yr−1) denotes the regional use of water source n.

[17] Let inline image (mm yr−1) be the precipitation during crop i's growing season in region j, the salinity of which is inline image (dS m−1); and let inline image (mm yr−1) denote the application of water type k to crop i, inline image, in region j. Then, the quantity of water applied to crop i in region j is inline image, and the salinity and price thereof are

display math


[18] Let inline image ($ ton−1) be crop i's output price, which is similar across all regions in our small, open economy. Denote by inline image ($ ha−1 yr−1) the annual per-hectare revenue minus water purchasing costs associated with cropi:

display math

where inline image (ton ha−1 yr−1) is crop i's production function specific to region j. The regional farming profits inline image ($ yr−1) are given by

display math

where inline image (ha) is the land allocated to crop i; inline image is the vector of land allotments; and the function inline image ($ yr−1) represents the annual nonwater production costs associated with crop i. The function inline image indirectly reflects the impact of various unobserved factors considered by farmers when contemplating land allocation among crops, including the spatial variability of the soil quality, marketing and agronomic risks, managerial limitations, etc. Under the PMP modeling approach, inline image is assumed continuous and strictly convex with respect to all inline image, and therefore, when adequately calibrated to reproduce an observed land allocation in a base year, land allocation changes smoothly in response to exogenous shocks that affect the crops' relative per-hectare profitability levels [Howitt, 1995a], i.e., of the relative values of inline image; these per-hectare profits are affected by the optimal quantities and qualities of the water applied to the crops, which in turn depend on the regional water distribution strategies.

[19] The regional planner's decision variables include inline image, inline image, and inline image, where inline image is the vector of regional water utilizations of the M blended sources and inline imageis the set of vectors of per-hectare applications of theK water types to the I crops. The optimization problem is

display math

where inline image is the intraregional water distribution costs; and Xj (ha) is the regional land constraint.

[20] The FSC and RSC distribution networks constitute extreme cases of this model. Under FSC, inline image, i.e., the regional distribution network enables access by all fields to all N water sources available to the region. The RSC scenario is obtained by setting inline image, such that inline image, i.e., the network provides all fields with access to one water type only, be it one of the N sources or a mixture of all or part thereof.

[21] While we formulate and study the properties of the agricultural profits function inline image, explicit analysis of the features of the intraregional distribution cost function inline imageis beyond the scope of this work. We do, however, refer to options to reduce costs by changing the spatial distribution of water types, as well as to the costs' impacts on the economic feasibility of switching between RSC and FSC networks. Also note that the analyses are based on a one-year static model, and therefore overlook dynamic processes such as salt accumulation in soils and regional water sources, as well as intragrowing-season options for water applications.

3. Interrelations of Water Management on Field and Regional Levels

[22] In this section we conduct a microlevel analysis of optimal water management at the field level, and use the findings for inferences with respect to optimal strategies at the macro level, i.e., management on a regional scale. The analysis aims at answering the following questions: Is it optimal to blend water when the regional distribution network enables FSC? And if not, then how does this outcome affect options for decreasing supply costs? Moreover, if indeed field-level blending is suboptimal, can blending under RSC networks ever become optimal? In other words, does the presence of an RSC network inevitably imply that only one of the sources available to a region should be utilized?

[23] For simplicity's sake, in this section we assume the availability of only two water sources (N = 2), one of which is fresh, the other saline, with availability constraints of inline image and inline image; salinity levels of inline image and inline image; and prices of inline image and inline image, respectively. To obtain a nontrivial case, the relationships inline image and inline image are assumed.

3.1. Field-Level Salinity Control

[24] We begin with the FSC scenario. Denote by inline image and inline image, respectively, the per-hectare quantities of fresh and saline water specifically applied to cropi (regional indices are omitted). Let us define inline image as the share of fresh water applied to crop i: inline image, i.e., inline image, where cases of inline image represent blending; and inline image and inline image are the two nonblending options. If the economic problem in equation (3) is formulated in terms of inline image and inline image, then the variable inline image constitutes a convenient instrument for analyzing the optimality of blending the fresh and saline waters specifically applied to crop i. These two decision variables determine both the salinity of the water applied to crop i,

display math

and its price inline image, such that equation (1) becomes

display math

where inline image is crop i's per-hectare annual revenue.

[25] We set aside the supply cost inline imageand assume for simplicity's sake that water constraints are not binding, i.e., under optimization, each water source's value of marginal production (VMP) is equal to the water's price. Under FSC, the optimal per-hectare annual water application inline image and the blending variable inline image are set to maximize inline image; we define this maximum as inline image.

[26] The features of the function inline imageare a key factor in the blending issue. Production functions are fitted to each of the crops incorporated in our empirical analysis based on the meta-modeling method used byKan et al. [2002], as set forth in section 3.2. In this section we make use of a function resulting from this procedure. To illustrate, we take the case of watermelons in the Beit She'an region, where the salinities of the fresh and saline water are 1 and 4 dS m−1, respectively. The discussion is based on the graphical exposition presented in Figure 1.

Figure 1.

Graphic illustration of the relationships between the conditions for optimal blending under the FSC and RSC scenarios, based on the case of watermelons in the Beit She'an region. Note: In (a), (b), and (c), lighter-shaded contours are associated with higher values.

[27] In Figure 1a isoquants of inline image are plotted in the inline image plane (lighter shaded contours are associated with higher values) under the observed water prices in the Beit She'an region: inline image ¢ (US cents) and inline image ¢ per cubic meter (all monetary values are in US dollars as of 2002), i.e., a water price ratio of inline image. Under these specific conditions, inline image mm yr−1 and inline image (point a), i.e., optimality entails application of pure fresh water. In Figure 1b we keep inline image, while inline image increases to inline image quadrupled, which yields the pair inline image mm yr−1 and inline image (point b), i.e., another freshwater corner solution. Note, however, the emergence of a local maximum at inline image mm yr−1 and inline image (point c). In Figure 1c, wherein the freshwater price is further amplified to inline image, this pure saline water local maximum becomes the global maximum (point d), i.e., inline image mm yr−1 and inline image. Thus, as already shown by Kan et al. [2002], as long as the field-level water applications of both water sources are not constrained, field-level blending does not constitute an optimal strategy under any water-price ratio; i.e., cases of inline image do not appear. This finding, which is shown here for watermelons, holds for all 45 crops across the 16 ecological regions in our study.

[28] What are the implications of this field-level analysis for regional water management in general, and for saving on distribution costs in particular? Since field-level blending is suboptimal, the regional farming profits would be maximized by applying each water source separately to a different set of crops. This gives rise to the question: If water sources should be applied separately in any case, then why should every single field in a region have access to all water sources, when nonuniform spatial distribution of water types might reduce supply costs?

[29] Suppose that a regional planner splits a region into N subregions, providing each with access to only one of the N water sources. If the area devoted to each subregion is set equal to the aggregated land allocated to the crops corresponding to its water type under the FSC optimal solution, then this regional splitting strategy could yield the same maximal farming profits as in the FSC network; concomitantly, there may be a considerable savings in water distribution costs since parallel pipe systems need not be installed. On the other hand, splitting may raise problems overlooked by our model. For example, advantages associated with cyclic use of water are precluded [see Qadir et al., 2007]. Moreover, by applying the splitting strategy, the planner is essentially choosing the types of crops to be grown in each subregion; hence, soils may be eroded due to repeatedly planting salinity-tolerant (sensitive) crops, and because there may be an insufficient number of crops in each group for crop rotation to replenish nutrients and mitigate the buildup of pathogens. In addition, farmers may resist the splitting option because of the income inequality that might be created among the water source-related subregions. Such drawbacks should be considered and weighed against the savings in water distribution costs.

3.2. Regional Salinity Control

[30] We turn now to the RSC scenario: The water distribution network enables access of every field in the region to only one type of water, the salinity of which is set prior to distribution by a mixture of all available water sources. In our example of a dual-water source region, this constraint implies inline image for all inline image. Moreover, cases of inline image and inline image mean that the regional planner chooses to avoid utilization of one of the sources—fresh water or saline water—respectively; if such an outcome is optimal with respect to farming profits, it may also significantly reduce water supply costs since the expenses associated with extracting one of the sources and blending the two water types are avoided. The question is: Can solutions of inline imageemerge? In other words, does the suboptimality of field-level blending preclude the optimality of blending on a regional scale? Can we infer that installation of an RSC network will automatically render utilization of only one water source optimal? This is our next subject.

[31] Setting aside the intraregional supply costs inline image, the regional-scale optimization problem inequation (3) becomes

display math

[32] Suppose that this problem is solved in two stages: first, for any given level of η, the optimal land allocation inline image and water applications inline image are found. This yields the regional profit function inline image. Then, in the second stage, the optimal blending level inline image is computed so as to maximize inline image. This process may mimic an optimization procedure carried out by a regional planner who controls η: While searching for inline image, the planner takes into account that η is considered a given by farmers and affects their decisions regarding allocation of land and application of water to crops.

[33] Thus, given η, and (to simplify the theoretical analysis) assuming an internal solution with respect to both variables inline image and inline image for all inline image, the maximization of inline image with respect to w and xunder the regional land and water constraints yields the first-order conditions (FOCs):

display math
display math

where inline image, inline image (both in $ m−3), and inline image ($ ha−1) are the shadow values of the regional fresh water, saline water, and land constraints respectively; accordingly, inline imageis the optimal annual per-hectare revenue minus water purchasing costs.

[34] To simplify the theoretical analysis we suppose that the administrative prices inline image and inline image and the water availability constraints inline image and inline image are high enough to render both water constraints nonbinding, such that inline image and inline image. By substituting inline image and inline image into the objective function in equation (6), we get inline image, which is the output of the first optimization stage. In the second stage we search for inline image, which maximizes inline image. Using the envelope theorem we obtain

display math
display math

[35] A necessary condition for the optimality of blending is the existence of inline image, such that inline image, for which both the FOC inline imageand the second-order condition (SOC) inline image are met, and inline image, inline image. In Appendix Awe prove that a necessary condition for the fulfillment of these internal optimal-solution requirements is strict concavity of the inline image function for at least one of the I crops. We now turn our attention back to Figure 1, wherein the achievement of this particular condition is analyzed empirically.

[36] The dashed curves in Figures 1a, 1b, and 1c represent the function inline image, plotted in the inline image plane under the aforementioned inline imagewater-price ratios of 1.4, 4, and 10, respectively. Each of these curves is composed of the points at which the isoquants of the function inline imageare tangent to the iso-salinity lines inline image, as illustrated in Figure 1a for the cases of inline image and inline image. Our focus is on Figure 1d, wherein these three inline image functions are plotted versus η. The aforementioned absence of an internal field-level optimum with respect to watermelon ( inline image) is evidenced by the optimal corner solutions indicated in Figure 1d by points a, b, and d commensurate with the optima in Figures 1a, 1b, and 1c, respectively. However, does the absence of optimal internal field-level solutions rule out the potential for an optimal internal solution on the regional scale (i.e., inline image)? The answer is no: In Figure 1d the curve related to the inline imagewater-price ratio of 1:4 indicates that the function inline image for watermelon is strictly concave under this price ratio, as required for the optimality of regional blending.

[37] Together, the three curves in Figure 1d show that concavity of the function inline image emerges only when it is constantly positively affected by the increase in the freshwater fraction η inline image, as seen for the inline image curve associated with the inline imagewater-price ratio of 1:4, and vanishes otherwise. That is, the concavity disappears as the water price ratio inline imageincreases; this empirical phenomenon is valid for all crops analyzed herein that do exhibit concavity (e.g., watermelons, potatoes, tomatoes, and celery). This is the fundamental feature eliminating the optimality of field-level blending, yet enabling blending to become optimal under the RSC scenario. Given this characteristic, the FOC for optimal regional blending inline imagecan be satisfied provided that, under the same water-price ratio, some crops exhibit positive responses of inline image to increases in the freshwater fraction η inline image, and others negative inline image, where, as required by the SOC, the concavity associated with the former group overcomes the convexity of the latter.

[38] Thus, we conclude that the optimality of blending under RSC networks cannot be rejected a priori; empirical water management analyses should allow for the appearance of blending options.

[39] The optimality of blending under RSC depends on a range of factors, among them the water and land constraints' shadow values. If a water source's usage is limited by a binding constraint, then under optimal conditions, the water's VMP is equal to the sum of the water purchasing price and the constraint's shadow value (see equation (7)). Another factor is the inline image functions: As shown in Appendix A, their convexity levels play a role in fulfilling the SOC for an internal optimal solution. This convexity depends on the land's shadow value inline image (see equation (A3) in Appendix A). Thus, binding water and land constraints affect the appearance of blending as an optimal strategy through their respective shadow values.

4. Specifications, Calibration, and Data

[40] Returning to the general optimization problem (equation (3)), an empirical, regional-scale PMP analysis based on any ecological regionj, inline image, requires parameterization, estimation, and calibration of the production function inline image and the nonwater cost function inline image with respect to each crop i, inline image.

[41] Beginning with the production side, the production function represents yield responses to variations in water application and salinity, which should be based on solid agronomic theory and validated by well-designed field experiments. At the same time, it should have enough degrees of freedom to enable calibration based on microeconomic principles, so that the PMP model will reproduce water allocations observed in a base year. To achieve these two goals we adopt the composite production function developed byKan et al. [2002]:

display math

where inline image (ton ha−1 yr−1) and inline image (ton m−3) are parameters, and inline image (m3 ha−1 yr−1) is a function relating evapotranspiration (ET) to water application and salinity:

display math

Here inline image (m3 ha−1 yr−1) is crop i's potential ET in region j, and inline image through inline imageare parameters. The production function's parameters are estimated and calibrated by a four-stage meta-analysis procedure, as described inAppendix B. At the heart of this analysis lies the plant-level agronomic model developed byShani et al. [2007] as a refinement of Letey et al.’s [1985] model. While in the latter, additional water quantities can perfectly offset yield reductions caused by increases in salinity, under Shani et al. [2007], such perfect offsets are limited to low ranges of water application.

[42] There are three noteworthy properties of the production function. First, while the function is calibrated to reproduce base-year water applications, the responses to variations in water and salinity are dominated by the experimentally based estimates of their impacts on ET, formulated byequation (11). Second, for a crop lacking such experimental information, the ET function can be borrowed from another crop of the same botanical family (e.g., from barley to wheat), while the conversion to yield is accomplished by calibration based on data of the specific crop under consideration. Finally, the functional form of ET transforms the yield function inline image into an increasing sigmoid function with respect to water. In contrast to the quadratic function (used, for instance, by Letey and Dinar [1986]), for each crop i there is some strictly positive water application, denoted inline image, under which the water's VMP equals its value of average production (VAP); if the corresponding VAP exceeds the water price, then inline imageconstitutes a lower bound on the set of optimal per-hectare water applications. This property has a potential impact on regional-scale optimization under regional water-quantity constraints: A change in the salinity of the water applied to crops—either through blending or through assignment of crops to water types that differ from those to which they were assigned in the calibration stage—shifts the crops' water production function, and may cause the selection of inline image for all i, inline image, eventually resulting in land fallowing. Thus, land fallowing under our PMP model may be a result of both the sigmoid structure of the production function inline image and the convex nonwater production cost function inline image.

[43] For the nonwater cost function inline image we adopt the commonly used quadratic specification [e.g., Howitt, 1995b; Röhm and Dabbert, 2003]

display math

where inline image and inline imageare parameters calibrated by the two-stage PMP calibration procedure developed byHowitt [1995a]. Note that this simplified cost function, like other more general cost functions that require implementation of more sophisticated calibration procedures [e.g., Paris and Howitt, 1998; Heckelei and Wolff, 2003], has the convexity property that affects the emergence of optimality of blending waters on the regional scale (see Appendix A).

[44] Data were collected from publicly available information sources for the year 2002 (the model and the entire data set are available from the authors upon request). Regional-scale data included the cropping areas of 45 crops in each ecological region, the average price, and total consumption of water from each of the four sources. Average per-hectare yield, output prices, and average per-hectare nonwater costs were available only for the nationwide level, and were obtained from the Israel Central Bureau of Statistics [ICBS, 2004] and various production-instructional bulletins published by the Israel Ministry of Agriculture and Rural Development [IMOARD, 2002]. To calibrate a nonwater cost function for the lowest profitable crop [see Howitt, 1995a], the associated maximum yield reduction below average yield was calculated, using nationwide yield levels for the period 1992–2002; precipitation and potential ET levels during the growing season of each crop; and the Christiansen uniformity coefficient associated with each crop, all from the ICBS [2004] and IMOARD [2002] bulletins. The nationwide average water prices were 17.5, 16.1, 14.0, and 9.7 ¢ m−3for fresh water, tertiary- and secondary-treated wastewaters, and brackish water, respectively. The salinities of the fresh and brackish waters were 1 and 4 dS m−1, respectively, and the salinity of both the secondary- and tertiary-treated wastewaters was 2 dS m−1. The typical soil characteristics incorporated into the agronomic model [Shani et al., 2007] for each region were obtained from Ravikovitch [1992].

[45] However, our regionally aggregated data provide information neither on the types and quantities of water applied to each crop, nor on the regional availability constraint on each water source. Therefore, additional assumptions are required for calibrating our model.

[46] For the allocations of water to crops we employed a hierarchical procedure. Given the current Israeli regulations prohibiting the blending of fresh water and treated wastewater (blending is expected to be permitted when proposed new regulations on wastewater treatment come into effect [Israel Ministry of Environmental Protection, 2003]), and the relative rarity of brackish water applications, we assume that, in the base year, only one type of water source was used to irrigate each crop. Initially, the regional quantity of treated wastewater was allocated to crops meeting the regulations regarding agricultural use of wastewater [Israel Ministry of Health, 1999]. Then, brackish water was allocated to the most salinity-tolerant crops using the salinity tolerance ranking suggested byMaas and Hofmann [1977]. The remaining crops were assumed irrigated with fresh water. Then, the base-year per-hectare water applications were calculated for each crop based on factorization of the quantities indicated byIMOARD's [2002] production bulletins so as to match the computed regional total water consumptions to the observed ones.

[47] Regarding water constraints, according to Feinerman et al. [2003], water prices constituted the binding consumption factor for 85% of the agricultural water consumers in 2002. We therefore calibrate the model while assuming that under the observed water consumption, a water source's VMP is equal to its price (see Appendix B). For runs of the model under the FSC and RSC scenarios, the observed total regional water consumptions (Table 1) are considered the water constraints. Thus, while the water constraints' shadow values are assumed zero in the calibration, they may become positive under the FSC and RSC simulations.

[48] The programming model was built on an Excel worksheet and run by the Premium Solver Platform V6.5 instrument. The nonconvexity of the objective function (as demonstrated in Figure 1), driven by the sigmoid nature of the water's production function, required seeking the global optimum by employing a multistart search procedure. However, our calibrated model can reproduce the observed (and assumed) regional water and land allocations in the base year only if it is run so as to search for a local optimum. Therefore, in our empirical analyses of the FSC and RSC scenarios, we consider the calibrated base-year model a common starting point, and compare the optimization procedure's outputs to one another only, and not to the calibrated model.

5. Empirical Analysis

[49] The objectives of the empirical analysis were to compare the FSC and RSC scenarios with respect to utilization of land and water resources and farming profitability under exogenous and endogenous pricing schemes of water sources and land.

[50] As already noted, no cases of optimal blending were found under the FSC scenario, i.e., each crop is irrigated by only one type of water. Nevertheless, blending emerges in the case of RSC, as can be culled from the right-hand section ofTable 2, which presents the region-level use of land and water resources under the FSC and RSC scenarios in terms of percentage of the regional amounts reported inTable 1. Since under our RSC scenario, water of one quality only was delivered to all fields, any exploitation of more than one water source implies blending, which occurs in six of the 16 regions.

Table 2. Utilization of Water and Land Under FSC and RSC Networks, Expressed in Terms of Percent of the Amounts Reported in Table 1
RegionField-Level Salinity ControlRegional Salinity Control
Fresh WaterSecondary WastewaterTertiary WastewaterBrackish WaterTotal WaterLandFresh WaterSecondary WastewaterTertiary WastewaterBrackish WaterTotal WaterLand
Hula Basin9710097100100094100
Western Galilee9010092100100087100
Beit She'an10010047911001000083100
Harod Valley100096100100096100
Jordan Valley911009299941009599
Lower Galilee1003988100100081100
Jezreel Valley100100099100100002983
All regions91761005386999422100387599

[51] With respect to the water constraints, overall optimal water use under FSC amounts to 86%, compared to only 75% under RSC (Table 2, last row). On the other hand, total utilization of fresh water under FSC is 3% less, a difference attributed to the FSC network's ability to separately irrigate salinity-tolerant crops with higher-salinity water sources. In contrast, in the RSC case, any utilization of saline water increases the salinity of the single-water type delivered to all crops. Therefore, to reduce the salinity of the supplied irrigation water, the application of fresh water is higher in regions where it is blended with saline water (e.g., in the Jordan Valley and Ra'anana). Fresh water use is also higher in regions where irrigation by saline water is avoided altogether (e.g., in the Hula Basin and Western Galilee), due to the need to compensate therein for the associated reduction in the total quantity of water utilized for irrigation. We therefore conclude that RSC networks increase the optimal quantity of fresh water consumption, regardless of whether blending is optimal or not. Thus, RSC networks do not support fresh water savings.

[52] Land fallowing appears in four cases (see Table 2): in the Rehovot and Jordan Valley regions under FSC; and in the Jordan Valley and Jezreel Valley regions in the RSC scenario. Examination of the waters' VMP versus VAP reveals that for all irrigated crops, VAP > VMP, implying that the appearance of the uncultivated area is related to the quadratic nonwater cost functions rather than to the sigmoid structure of the water's production function.

[53] Under what conditions is switching from the RSC to FSC network warranted from the point of view of farmers who are charged for the use of water and land? Our analysis reveals that the pricing methods for agricultural water and land play a crucial role in this matter. Specifically, we consider prices set endogenously in the sense that they incorporate the land and water constraints' shadow values, and thereby become the binding factor of the water and land uses. We first analyze the water and land constraints' shadow values under the two networks, and then examine the associated network preferred by farmers under endogenous pricing methods.

[54] Table 3presents the differences between the FSC and RSC networks vis-à-vis regional water and land constraints' shadow values.

Table 3. Differences (FSC Minus RSC) of Water and Land Shadow Values
RegionFresh Water (¢ m−3)Secondary Wastewater (¢ m−3)Tertiary Wastewater (¢ m−3)Brackish Water (¢ m−3)All Waters Weighted Average (¢ m−3)Land ($ ha−1)
Hula Basin−0.10.5−0.10.2
Western Galilee−
Beit She'an−−0.50.8
Harod Valley0.
Jordan Valley0.
Lower Galilee9.30.08.0−7.3
Jezreel Valley−−18.22.6

[55] As aforementioned, the contribution of fresh water to production is greater under RSC; therefore, the FSC minus RSC differences in fresh water's shadow values are negative in most regions. Positive differences in the shadow values of the other more saline sources reflect their smaller contribution to production under RSC relative to FSC; exceptions are two cases related to tertiary wastewater usage: the Besór region, wherein application of tertiary wastewater only is optimal under RSC; and the Negev, wherein the constraint of brackish water is binding under FSC, yet not under RSC. The overall difference in water scarcity can be evaluated by the average water shadow values weighted by their respective consumptions (Table 3, column 5). In most cases, the sign of the weighted average shadow values corresponds to that of fresh water; water is scarcer under RSC in nine of the 16 regions. Scarcity of land, on the other hand, is higher under FSC in 11 regions. As pointed out by Schwabe et al. [2006], water and land scarcities are both nonseparable, and are expected to exhibit opposite trends: As water availability decreases, crop production becomes less profitable; therefore, the VMP of the agricultural land diminishes. However, our analysis shows that when a few water types with differing salinities are utilized, the scarcities of both water and land can be higher under FSC relative to RCS; examples are the Western Galilee and Arava regions.

[56] From a social point of view, switching from an RSC to an FSC network is worthwhile only in regions where the difference in farming revenues, net of the (unknown) water supply and land's farming opportunity costs, exceeds the (unknown) associated additional intraregional distribution costs ( inline image). However, from the individual point of view of the regional farmer, the justifiability of such a switch depends on the farming profits, which in turn depend on the land and water prices administratively set by the state. Here we analyze this issue from the farmer's individual point of view.

[57] Consider the case where the state limits regional water consumption to some predetermined constraints; however, instead of setting quotas, it induces consumption at these constraining levels by adjusting the regional water prices. To achieve this goal, prices should be equal to the observed base-year prices plus the shadow values of the water constraints; i.e., prices are set endogenously so as to incorporate the waters' scarcity levels. Compared with the base-year prices, this pricing method would reduce farming profits and cause income transfers from the farming sector to the state. Similarly, the state can increase the base-year land-lease fees by the land constraint's shadow value, and thereby increase its income without causing land fallowing. Under such scenarios, the relative regional profitability of crop production under the FSC and RSC networks is affected by the differences in land's and waters' shadow values.

[58] Table 4presents the differences in regional farming profits (FSC minus RSC) under four schemes of water and land pricing: (1) both the water prices and land lease fees are the base-year observed ones; (2) water prices only are set endogenously so as to incorporate the regional water constraints' shadow values [e.g., in the two-water model (seeequation (7)), the prices of fresh and saline water become inline image and inline image, respectively]; (3) land prices only are set endogenously [i.e., the lease fees are increased by inline image (see equation (8))]; (4) both water and land prices are set endogenously. Switching from an RSC to an FSC network pays only in regions where the difference in profits exceeds the (unknown) associated additional distribution costs ( inline image).

Table 4. Differences (FSC Minus RSC) of Agricultural Profits (106 $ yr−1) Under Observed and Endogenous Water and Land Prices
RegionObserved PricesEndogenous Water PricesEndogenous Land PricesEndogenous Water and Land Prices
Hula Basin0.
Western Galilee0.
Beit She'an0.050.27−0.010.21
Harod Valley0.
Jordan Valley0.
Lower Galilee0.78-0.821.37−0.23
Jezreel Valley2.033.831.613.42

[59] In the first pricing scheme, the calibrated PMP model is run under the base-year observed administrative prices, and essentially improves the water allocation obtained through the previously described hierarchical procedure employed to assign water types to crops in the base year (seeAppendix B); the land allocation is concomitantly ameliorated as per the changes in relative profitability of crops caused by the water allocation adjustments. As expected, the FSC version yields better improvements than the RSC one, i.e., the greater flexibility provided by FSC networks enables higher regional farming profits in all regions except the Haród Valley, wherein the solutions coincide. Lachìsh region exhibits the largest profit difference: $9.69 million per year, which on average amounts to 26 ¢ per cubic meter of the water consumed under the FSC scenario; this average gain is 50% higher than the fresh water's price, and therefore may exceed the added costs associated with installing and operating a dual water delivery system in order to enable the FSC solution in this region. For comparison, in all other regions the average per-water-unit profit increase does not exceed 8 ¢ m−3.

[60] In the three endogenous pricing scenarios, farming revenues are similar to those obtained under the base-year prices. However, the costs associated with water purchase and land lease are increased by the shadow values, and differently affect farmers' profits under the FSC and RSC networks. The shadow values under FSC may be higher than those under RSC to the extent that the latter would even become more profitable. In this case, since FSC regional water distribution costs ( inline image) are expected to exceed those of RSC, RSC becomes the preferred network from farmers' point of view, regardless of the difference in the distribution costs between the two networks (note that this situation cannot emerge in similar analyses based on a social point of view, in which land and water prices do not play a role in the question of which of the two networks should be favored).

[61] Setting only water prices endogenously increases the farming profit difference in the nine regions, wherein water's weighted average shadow values are lower under FSC (Table 3). In six of the other seven regions, the advantage of the FSC network is reduced, and in one region—the Lower Galilee—the RSC network is even more profitable to farmers than is the FSC; thus, RSC should be favored by farmers in this region if the state would set endogenous water prices. Following this rationale, if land prices only are set endogenously, RSC becomes the farmers' clear-cut preferred network in three of the aforementioned 11 regions wherein land shadow values are lower under RSC; obviously, FSC becomes more appealing to farmers in the four regions exhibiting the opposite phenomenon. The overall effect of resource scarcity is obtained when both land and water prices are set endogenously; in this case, RSC is unquestionably the farmers' preferred scenario in three regions.

[62] These analyses indicate that pricing methods can affect farmers' preferences with respect to regional distribution networks. In view of equations (7) and (8), shadow values' impact on the relative profitability of the two networks is an empirical result that depends on the model's calibrated parameters and is therefore sensitive to the combined effects of base-year conditions.

6. Concluding Remarks

[63] By examining interrelations between field and regional levels of irrigation management, and performing an empirical analysis for the case of Israel, this paper derives the following five main conclusions:

[64] 1. Despite the suboptimality of blending water sources with differing salinity levels when water salinity can be controlled at the field level, the blending option should not be ruled out where a regional water distribution network allows water salinity to be controlled solely on a regional scale; blending with such networks may be optimal.

[65] 2. Due to the suboptimality of field-level blending, splitting regions into subregions, each assigned to irrigation by a different water source, may reduce water distribution expenses without diminishing farming profits.

[66] 3. Contrary to that of saline water, freshwater consumption under RSC networks is frequently higher than under FSC networks, which is attributed to the salinity increase associated with the use of saline water when blending is the optimal strategy; and to the lower overall regional water utilization where optimality entails application of fresh water only; thus, blending networks around the globe, if aimed at saving fresh water by blending it with saline water, might end up missing this target.

[67] 4. When water sources of a few differing salinity levels are utilized, variations in land scarcity due to switching from RSC to FSC may run in the same direction as that of the overall water scarcity, as measured by the average shadow values of the waters weighted by their respective consumptions.

[68] 5. Only when water and/or land prices are set endogenously so as to be the binding factors of the use of these constrained resources might RSC networks become more profitable than FSC networks from farmers' point of view, rendering RSC the farmers' preferred network without needing to account for the differences in intraregional distribution costs.

[69] These findings can serve as general guidelines for planners of water policies and intraregional distribution networks. However, the design of region-specific networks requires consideration of a range of factors overlooked by our analyses; for example, the supply costs associated with various water distribution options, region-specific feasibility constraints, intraregional heterogeneity (e.g., in microclimate, soil quality, and topography), scenarios involving intermediate salinity control options (i.e., cases of inline image), and efficiency versus parity considerations. Furthermore, while our model may constitute a useful instrument for optimizing regional farming profits, it may also be extended along various avenues. Here we mention two potential subjects for future research.

[70] First, the presence of contaminants in treated wastewater, such as pharmaceutical compounds and heavy metals, may give rise to regulations limiting the field-level application of wastewater; under such regulations, blending may become optimal under FSC networks [Kan et al., 2002]. For example, in our two-water analysis, if the application of saline water to watermelons had been restricted at the field level to a maximum of 160 mm yr−1, blending of fresh and saline water for watermelon irrigation would have become optimal in the Beit She'an region (see point e in Figure 1c). Such restrictions may also alter the relative profitability of both FSC and RSC.

[71] Second, as aforementioned, the time unit of our model is one year, and therefore only the annual average salinity of the applied water is considered. For shorter time frames, Shani et al. [2009]developed a dynamic model for optimizing intragrowing-period distribution of field-level water applications; however, that model incorporates only one type of water. Extending their framework to the case of various water qualities may reveal the optimality of applying more than one water type throughout the growing season. Such a dynamic model would enable analyzing various strategies of intragrowing-season water applications; e.g., comparing blending with cyclic use on the bases of economic efficiency and fresh water conservation under FSC and RSC networks.

[72] In a longer time frame, irrigation by saline water may alter soil characteristics, thereby reducing productivity [e.g., Shani and Ben-Gal, 2005]. Moreover, deep-percolation flows may gradually change the salinity of intraregional water sources [Knapp and Baerenklau, 2006], and in turn the optimal solutions with respect to both blending and assignment of differing water types to crops.

Appendix A:  

[73] Based on equation (5) and employing the envelope theorem gives

display math

The first right-hand element inequation (A1) is positive, since inline image (outputs decline with higher salinities) and inline image, which follows from equation (4). Given inline image, the sign of inline image is indeterminate. Thus, there may be η, inline image, under which the linear combination inline image equation (9a) is zeroed, i.e., the FOC can be met.

[74] Turning to the SOC, using equation (8), we obtain

display math
display math

where inline image and inline image is the set of inline image crops, excluding crop i. Substituting into equation (9b) and rearranging yields

display math

where inline image and inline image is the set of inline image crops, excluding crops i and u.

[75] Under the PMP assumption of inline image for all inline image, the right-hand element in the right side of(A4)is non-negative. Regarding the left-hand element,equation (A1) can be redifferentiated to obtain

display math

[76] The sign of which is undetermined due to the indeterminateness of inline image and inline image [Kan et al., 2002], as well as that of inline image, which follows from the involvement of the two formerly mentioned elements, as well as others:

display math

This implies that equation (A4) may end up negative, i.e., the SOC, inline image, may also be satisfied. A necessary condition for this occurrence is the negativity of the element inline image, which necessitates strict concavity of the function inline image for at least one crop i, inline image.

[77] Note that the convexity levels of the functions inline imageaffect the value of the right-hand element ofequation (A4), and thereby influence the value of the sum inline image required for inducing the sufficient condition for the optimal internal solution.

Appendix B:  

[78] In the first calibration stage, Shani et al.’s [2007]model is run to generate a data set wherein ET values are calculated for various combinations of annual water applications and salinities. These plant-level data are translated into field-level quantities by assuming a lognormal spatial distribution of water infiltration, as suggested byKnapp [1992]. This distribution's mean value equals 1 for mass balance [Feinerman et al., 1983], and the standard deviation is calculated to fit Christiansen's uniformity coefficient [see Knapp, 1992] of the irrigation system typically used for each crop. Then, in the second stage, the produced data set is used to estimate the parameters inline imageinline image by nonlinear regression, subject to the feasibility constraint inline image. In the third stage, the parameter inline image is calibrated based on the FOC with respect to water application: Let inline image, inline image, and inline imagebe the observed base-year quantity, salinity, and price of the water applied to cropi in region j, respectively. Then, optimality requires equality between the irrigation water's VMP and its price:

display math

Finally, in the fourth stage, the calibrated inline imageparameter, the base-year yield, and inline image, as well as inline image and inline image, are substituted into equation (10) for calibration of the parameter inline image.


General Model

j ( inline image)

region index.

i ( inline image)

crop index.

n ( inline image)

water source index.


number of blended water sources.

k ( inline image)

irrigation water type index.

inline image

salinity of water source n, dS m−1.

inline image

price of water source n, $ m−3.

inline image

quantity constraint of water type n in region j, m3 yr−1.

inline image

regional usage of water source n in region j, m3 yr−1.

inline image

vector of regional water utilizations of the M blended sources in region j, m3 yr−1.

inline image

salinity of blended waters (k = 1) in region j, dS m−1.

inline image

price of blended waters (k = 1) in region j, $ m−3.

inline image

precipitation during growing season of crop i in region j, mm yr−1.

inline image

salinity of rainfall, dS m−1.

inline image

quantity of water type k applied to crop i in region j, mm yr−1.

inline image

set of vectors of per-hectare applications ofK water types to I crops in region j, mm yr−1.

inline image

quantity of water applied to crop i in region j, mm yr−1.

inline image

salinity of water applied to crop i in region j, dS m−1.

inline image

price of water applied to crop i in region j, $ m−3.

inline image

production of crop i in region j, ton ha−1 yr−1.

inline image

price of output of crop i, $ m−3.

inline image

revenue minus water purchasing costs of crop i in region j, $ ha−1 yr−1.

inline image

land of crop i in region j, ha.

inline image

agricultural land constraint in region j, ha.

inline image

vector of land allocation in region j, ha.

inline image

nonwater production costs of crop i in region j, $ yr−1.

inline image

regional profit in region j, $ yr−1.

inline image

distribution costs in region j, $ yr−1.

inline image

salinity of fresh water, dS m−1.

inline image

salinity of saline water, dS m−1.

inline image

salinity of precipitation, dS m−1.

inline image

price of fresh water, $ m−3.

inline image

price of saline water, $ m−3.

inline image

quantity of fresh water applied to crop i, mm yr−1.

inline image

quantity of saline water applied to crop i, mm yr−1.

inline image

total water applied to crop i, mm yr−1.

inline image

share of fresh water applied to crop i.

inline image

salinity of water applied to crop i, dS m−1.

inline image

price of water applied to crop i, $ m−3.

inline image

revenue of crop i, $ ha−1 yr−1.

inline image

revenue minus water purchasing costs of crop i, $ ha−1 yr−1.

inline image

share of fresh water applied to all crops under RSC.


inline image

production function parameter, ton ha−1 yr−1.

inline image

production function parameter, ton m−3.

inline image

evapotranspiration of crop i in region j, m3 ha−1 yr−1.

inline image

potential evapotranspiration of crop i in region j, m3 ha−1 yr−1.

inline imageinline image

evapotranspiration function parameters.

inline image

base-year quantity of water applied to cropi in region j, mm yr−1.

inline image

base-year salinity of water applied to cropi in region j, dS m−1.

inline image

base-year production of cropi in region j, ton ha−1 yr−1.

inline image

base-year price of water applied to cropi, $ m−3.

inline image

base-year land allocated to cropi in region j, ha.


[79] We are grateful for the funds given for this research by the German Federal Ministry of Education and Research (BMBF) in collaboration with the Israeli Ministry of Science and Technology under the GLOWA Jordan River Project, the Deutsche Forschungsgemeinschaft (DFG), the Israeli Ministry of Agriculture and Rural Development, the Israeli Governmental Authority for Water and Sewage, and the Center for Agricultural Economic Research at the Hebrew University. We thank Mordechai Shechter for valuable comments and David Haim for assistance. We also thank the editor and two anonymous reviewers for insightful comments.