A mathematical programming model is built to simulate farmers’ behavior under the proposed water rights regime. The simulation exercise enables a comparison of the potential performance of the proposed allocation regime with that of the current proportional allocation rule in terms of economic efficiency.
4.1 Proportional and Priority Allocation Rules
The basin (subscript b) taken as a case study is represented by a sample of seven IDs (subscript id), each of which has a different number of farm types (subscript f) which are considered as the decision-making units. Each farm type represents a number of farms \({nf}_{id,f}\) with an average size of \({s}_{id,f}\) irrigated hectares, with \({s}_{id}\) being the total irrigated area in each ID (\({s}_{id}={\sum }_{f}{nf}_{id,f}\cdot {s}_{id,f}\)). Moreover, each ID in the sample represents \({m}_{id}\) irrigation districts with similar features within the GRB, and \({s}_{b}\) is the total irrigated area in the basin (\({s}_{b}={\sum }_{id,f}{m}_{id}\cdot {nf}_{id,f}\cdot {s}_{id,f}\)).
Under current management rules, the annual aggregate volume of water available for irrigation at the basin level (
\({WA}_{b}\) measured in cubic meters) is shared among IDs proportionally according to the water rights granted to each district. In hydrological years when
\({WA}_{b}\) is higher than or equal to the sum of water rights granted to every rights holder in the basin (full water allotment,
\({FWA}_{b}\)), the volume of water allocated to each ID is equal to the water rights granted (
\({FWA}_{id}\)). In years when
\({WA}_{b}\) is lower than
\({FWA}_{b}\), the volume allocated to each ID (
\({WA}_{id}\)) is proportionally reduced as follows:
$${WA}_{id}={FWA}_{id}\cdot \frac{{WA}_{b}}{{FWA}_{b}}$$
(1)
Similarly, IDs share the water annually allocated proportionally among all their farm types, i.e., water allocations in cubic meters per hectare (\({wa}_{id,f}\)) are the same for all farm types within the ID (\({wa}_{id,f}={wa}_{id},\forall f\)), such that \({wa}_{id}={WA}_{id}/{s}_{id}\). Thus, the annual water allocation for farm type f measured in cubic meters is \({WA}_{id,f}={wa}_{id}\cdot {s}_{id,f}\).
Under the proposed new distribution rules, each farm type would hold water rights defined as portfolios of two different types of water rights, priority and general rights. Thus, the annual water allocation to each farm type measured in cubic meters would be calculated as the sum of the annual water allocation for holding priority and general water rights, as follows:
$${WA}_{id,f}={wapr}_{id}\cdot \left[{PR}_{id,f}\cdot {s}_{id,f}\cdot {nf}_{id,f}\right]+{wagr}_{id}\cdot \left[{GR}_{id,f}\cdot {s}_{id,f}\cdot {nf}_{id,f}\right]$$
(2)
where
\({wapr}_{id}\) is the annual water allocation per hectare for every farm type within an ID for priority rights and
\({wagr}_{id}\) is the same for general rights, while
\({PR}_{id,f}\) and
\({GR}_{id,f}\) are the shares of priority and general rights held by farm type
\(id,f\), which add up to one (
\({PR}_{id,f}+{GR}_{id,f}=1\)) for every farm type. This implies that the same identity applies at the ID level (
\({PR}_{id}+{GR}_{id}=1\)) and the basin level (
\({PR}_{b}+{GR}_{b}=1\)).
According to the latter expression, the volume of water annually allocated to each farm type measured in cubic meters per hectare is:
$${wa}_{id,f}={wapr}_{id}\cdot {PR}_{id,f}+{wagr}_{id}\cdot {GR}_{id,f}$$
(3)
It is worth explaining that the maximum or full water allotments per hectare and year for priority (\({fwapr}_{id}\)) and general (\({fwagr}_{id}\)) water rights are the same for every farm type within an ID, according to the water rights granted at the ID level. This means that \({fwa}_{id}={fwapr}_{id}={fwagr}_{id}\).
The aggregate volume of water needed to satisfy all priority rights holders is denoted as
\({FWAPR}_{b}\), and is calculated as the sum of all priority rights in the basin:
$${FWAPR}_{b}={\sum }_{id,f}{fwa}_{id}\cdot {m}_{id}\cdot {PR}_{id,f}\cdot {nf}_{id,f}\cdot {s}_{id,f}$$
(4)
while the quantity annually available to allocate among priority rights holders (
\({WAPR}_{b}\)) is:
$${WAPR}_{b}={\sum }_{id,f}{wapr}_{id}\cdot {m}_{id}\cdot {PR}_{id,f}\cdot {nf}_{id,f}\cdot {s}_{id,f}$$
(5)
Likewise,
\({FWAGR}_{b}\) and
\({WAGR}_{b}\) denote the volume of water needed to fully meet water demands from all general rights holders and the water annually available to be allocated among these rights holders, respectively:
$${FWAGR}_{b}={\sum }_{id,f}{fwa}_{id}\cdot {m}_{id}\cdot {GR}_{id,f}\cdot {nf}_{id,f}\cdot {s}_{id,f}$$
(6)
$${WAGR}_{b}={\sum }_{id,f}{wagr}_{id}\cdot {m}_{id}\cdot {GR}_{id,f}\cdot {nf}_{id,f}\cdot {s}_{id,f}$$
(7)
The share of priority rights at the basin level (
\({PR}_{b}\)) relates the aggregate volume of water needed to satisfy all priority rights holders (
\({FWAPR}_{b}\)) to the aggregate volume of water needed to satisfy all water rights (
\({FWA}_{b}={FWAPR}_{b}+{FWAGR}_{b}\)) as follows:
$${PR}_{b}={FWAPR}_{b}/{FWA}_{b}$$
(8)
Priority rights are served first with the full water allotment granted to the corresponding ID (
\({fwa}_{id}\)) as long as the water availability
\({WA}_{b}\) is enough to cover all water demands from priority rights holders (
\({WA}_{b}\ge {FWAPR}_{b}\) or, following Eq. (
8),
\({WA}_{b}\ge {FWA}_{b}\cdot {PR}_{b}\)). After that, the remaining water available for sharing is allocated proportionally among general rights holders. In years when there is not enough water to meet full allotments for priority rights, the available water is rationed among priority rights holders following the proportional rule according to their full water allotments. In these years, general rights holders would not receive any water.
The abovementioned priority allocation rules can be expressed mathematically considering the following formulae:
$${WA}_{b}\ge {FWAPR}_{b}\left\{\begin{array}{c}{wapr}_{id}={fwa}_{id}\\ {wagr}_{id}=\frac{{WA}_{b}-{FWAPR}_{b}}{{FWAGR}_{b}}\cdot {fwa}_{id}\end{array}\right.$$
(9)
$${WA}_{b}<{FWAPR}_{b}\left\{\begin{array}{c}{wapr}_{id}=\frac{{WA}_{b}}{{FWAPR}_{b}}\cdot {fwa}_{id}\\ {wagr}_{id}=0\end{array}\right.$$
(10)
In Eq. (
9), when the aggregate volume of water available for irrigation at the basin level (
\({WA}_{b}\)) is greater than or equal to the water requirements to satisfy all priority rights holders (
\({FWAPR}_{b}\) or
\({FWA}_{b}\cdot {PR}_{b}\)), priority rights holders receive allocations corresponding to the full water allotment for the ID (
\({fwa}_{id}\)). General rights holders receive a proportion of their full water allotments. This proportion depends on how much water remains available for general rights related to the total volume of water needed to cover them (
\(\frac{{WA}_{b}-{FWAPR}_{b}}{{FWAGR}_{b}}\)). Similarly, when there is not enough water available at the basin level to satisfy the demand from priority rights holders (Eq. (
10)), they receive a proportion of their full water allotment (
\({fwa}_{id}\)), while general rights holders receive no water at all.
The optimal share of priority rights at the river basin level is initially unknown, so it will be parametrized to determine the \({PR}_{b}\) that yields the most efficient outcome.
4.2 Farmers’ Decision-making
We assume that farmers try to maximize farming profits as a function of their water allocation (
\({\pi }_{id,f}=f\left({wa}_{id,f}\right)\)). As explained above, farmers’ annual water allocations vary depending on the availability of irrigation water. Thus, since
\({\tilde{WA }}_{b}\) is a stochastic variable, water allocations
\({\tilde{wa }}_{id}\),
\({\tilde{wapr }}_{id}\),
\({\tilde{wagr }}_{id}\), and
\({\tilde{wa }}_{id,f}\) are also stochastic variables ranging from
\({fwa}_{id}\) to 0, which in turn means that farming profits are stochastic variables (
\({\tilde{\pi }}_{id,f}\)). Within this stochastic framework, it is assumed that farmers make decisions about whether to upgrade their water rights to maximize their expected (or average) profit. To simulate the risk from water supply variability,
N = 1000 probabilistic values for
\({\tilde{WA }}_{b,n}\) are considered (the subscript
n denotes each irrigation water availability scenario). For the current climate scenario, these values have been taken from the hydrological simulation model built by Gómez-Limón (
2020). For the climate change scenario, the values for
\({\tilde{WA }}_{b,n}\) have been obtained by modifying the values taken for the current climate scenario to reflect more frequent and intense drought episodes, as explained in Sect. 3.2. In both cases, these scenarios (
n = 1, …,1000) have been considered equally probable.
We use the expected total gross margin (\({GM}_{id,f,n}\)) as a proxy of profit in the short run. Gross margin is a mathematical function of the area covered by the different crops (i.e., farmers’ decision variables), denoted by \({X}_{c,id,f,n}\), where \(c\) denotes the crop. In addition, farmers can decide what percentage of their water rights will become priority rights (\({PR}_{id,f}\)), with the remaining rights being kept as general water rights (\({GR}_{id,f}=1-{PR}_{id,f}\)). However, it is worth noting that these last two decision variables do not depend on the water availability scenario \(n\), since both denote long-run decision-making (i.e., this farmer’s choice is considered to remain the same for the N = 1000 scenarios). Thus, the simulation of farmers’ decision-making maximizes the expected total gross margin considering both kinds of decision variables, \({GM}_{id,f,n}=f\left({X}_{c,id,f,n},{PR}_{id,f}\right)\).
The modeling approach is based on the standard Positive Mathematical Programming (PMP) formally introduced by Howitt (
1995) and the average cost approach proposed by Heckelei and Britz (
2005).
In case of drought, it is assumed that farmers react by changing their cropping pattern, replacing water-intensive crops with others that have lower water needs or even rainfed crops (i.e., no irrigation water is required). Thus, three rainfed alternatives (wheat and sunflower plus olive when this permanent crop is present in a farm type) have also been considered as decision variables for simulations under drought scenarios.
Considering the priority allocation rule proposed, decision-making for the different farm types can be integrated into a single model at the basin level, where optimum values for the variables
\({X}_{c,id,f,n}\) and
\({PR}_{id,f}\) (and
\({GR}_{id,f}\)) are to be found for every value considered for the parameter
\({PR}_{b}\):
$$\underset{{X}_{c,\mathit{id},f,n},{\mathit{PR}}_{\mathit{id},f}}{\mathrm{Max}}Z=\frac{1}{N}{\sum }_{id,f,n}{GM}_{id,f,n}\cdot {m}_{id}\cdot {nf}_{id,f}$$
(11)
$${{}^{s.t}\;\;\;\;\;GM}_{id,f,n}=\sum\nolimits_c\left[\left(p_{c,id}\cdot y_{c,id,f}+s_{c,id}-\alpha_{c,id,f}-1/2{\cdot\beta}_{c,id,f}\cdot X_{c,id,f,n}\right)\cdot X_{c,id,f,n}\right]\;\;\;\;\;\forall id,f,n$$
(12)
$$\sum\nolimits_cX_{c,id,f,n}=s_{id,f}\;\;\;\;\;\forall id,f,n$$
(13)
$$\sum\nolimits_c{wr}_{c,id,f}\cdot X_{c,id,f,n}\leq{wa}_{id,f,n}\cdot s_{id,f}\;\;\;\;\;\forall id,f,n$$
(14)
$${wa}_{id,f,n}={wapr}_{id,n}\cdot{PR}_{id,f}+{wagr}_{id,n}\cdot{GR}_{id,f}\;\;\;\;\;\forall id,f,n$$
(15)
$${PR}_{id,f}+{GR}_{id,f}=1\;\;\;\;\;\forall id,f$$
(16)
$$\left\{\begin{array}{cc}{wapr}_{id,n}={fwa}_{id},&\mathrm{if}{WA}_{b,n}\geq{FWAPR}_b\\{wapr}_{id,n}=\frac{{WA}_{b,n}}{{FWAPR}_b}\cdot{fwa}_{id},&\mathrm{if}{WA}_{b,n}<{FWAPR}_b\end{array}\;\;\;\;\;\forall id,n\right.$$
(17)
$$\left\{\begin{array}{cc}{wagr}_{id,n}=\frac{{WA}_{b,n}-{FWAPR}_b}{{FWAGR}_b}\cdot{fwa}_{id},&\mathrm{if}{WA}_{b,n}\geq{FWAPR}_b\\{wagr}_{id}=0,&\mathrm{if}{WA}_{b,n}<{FWAPR}_b\end{array}\right.\;\;\;\;\;\forall id,n$$
(18)
$${PR}_{b}=\frac{{FWAPR}_{b}}{{FWA}_{b}}$$
(19)
$${{\varvec{A}}}_{id,f}{{\varvec{X}}}_{id,f,n}\le {{\varvec{B}}}_{id,f}\;\;\;\;\;\forall id,f,n$$
(20)
$$X_{c,id,f,n}\geq0;{\;PR}_{id,f}\geq0;{\;GR}_{id,f}\geq0\;\;\;\;\;\forall c,id,f,n$$
(21)
In the above expressions, \({GM}_{id,f,n}\) represents the farm’s expected gross margin for farm type \(id,f\) in scenario \(n\) calculated as the sum of total income, including both product sales (expected crop price, \({p}_{c,id}\), multiplied by expected crop yield, \({y}_{c,id,f}\)) and coupled subsidies (\({s}_{c,id}\)), minus the variable cost function (\({\alpha }_{c,id,f}+1/2{\cdot \beta }_{c,id,f}\cdot {X}_{c,id,f,n}\)) for every crop \(c\), where \({\alpha }_{c,id,f}\) and \({\cdot \beta }_{c,id,f}\) are the PMP calibrating parameters.
The objective function (
11 and
12) allows the joint maximization of the average gross margin at the basin level (i.e., maximum economic efficiency solution), as a result of the optimum decision-making regarding the crop mixes in each scenario
\(n\) and the long-run choices about the upgrade into priority rights. Constraint (
13) is related to land availability and limits the total area covered by the different crop alternatives to the farm size (
\({s}_{id,f}\)). Equations (
14–
18) are related to water availability. Equation (
14) establishes that irrigation water use cannot exceed water availability, with the former being the sum of water requirements per crop (
\({wr}_{c,id,f}\)) and the latter the water allocation per farm type (
\({wa}_{id,f,n}\cdot {s}_{id,f}\)), while Eqs. (
15–
18), derived from Eqs. (
9) and (
10) explained above, describe how water availability is shared among farm types according to the priority rules proposed. Constraint (
19) just limits the maximum share of rights that can be upgraded to priority rights at the basin level, as fixed by the parameter
\({PR}_{b}\). Equation (
20) denotes the rest of the constraints defining the feasible solution set, which constitute agronomic (rotational and frequency requirements) and policy (cotton and sugar beet quotas) factors, with
\({{\varvec{X}}}_{id,f,n}\) being the matrix containing all variables
\({X}_{c,id,f,n}\),
\({{\varvec{A}}}_{id,f}\) the technical coefficient matrix for every variable and constraint of the irrigation district
\(id\) and farm type
\(f\), and
\({{\varvec{B}}}_{id,f}\) the vector of limit values for each constraint for the irrigation district
\(id\) and the farm type
\(f\). Finally, non-negativity constraints are imposed for
\({X}_{c,id,f,n}\),
\({PR}_{id,f}\) and
\({GR}_{id,f}\) (Eq. (
21)).
Considering the current proportional allocation rule, farm type decision-making can also be simulated for
N = 1000 water availability scenarios using a simplified version of model (
11), replacing Eq. (
15–
19) with a single expression (
22) representing how the proportional rule works for every irrigation district
\(id\) and farm type
\(f\).
$$\left\{\begin{array}{cc}{wa}_{id,f,n}={fwa}_{id},& \mathrm{if}\;\; { WA}_{b,n}\ge {FWA}_{b}\\ {wa}_{id,f,n}=\frac{{WA}_{b,n}}{{FWA}_{b}}\cdot {fwa}_{id},& \mathrm{if }\;\; {WA}_{b,n}<{FWA}_{b}\end{array}\right. \forall id,f,n$$
(22)
Simply by comparing the simulated results obtained for the two allocation rules, we are able to calculate variations in gross margins and water use at the farm, ID, and basin level, as well as other indicators, allowing an assessment of the proposed analysis.