## Abstract

We set up a model of land-use and irrigation water choices to assess the impact of dry weather conditions and possible restriction policies on farmers' payoffs in the Beauce area in France. Given the informational context, we construct a dynamic two-period model in which farmers make conjectures on the water abstraction by other users and take into account variations in the height of the water table. We solve the problem using dynamic programming. We simulate different restriction policies, proposed in the literature and tested in the field. We show that these restrictions, although efficient with respect to hydrological criteria, result in serious economic losses for the farmers.

## Introduction

In the second half of the twentieth century, the increasing use of tube wells and mechanical pumps has led to significant groundwater depletion in many parts of the world (Shah *et al*., 2007). With the help of irrigation, high-performance agricultural production areas have been created above aquifers, contributing to the growing pressure on groundwater resources (Shah *et al*., 2007). As a consequence, water managers need to counteract dropping groundwater levels to secure water for other uses as well as for future generations. To do so, they may rely on institutions (Ostrom, 1990), regulatory tools (Pérez-Blanco and Gómez, 2014) or economic instruments, such as pricing and water markets (see for example Easter, Rosegrant and Dinar, 1999; Koundouri, 2004).

In this article, we focus on the Beauce area, one of the most important agricultural production regions in France and one of the biggest cereal producing regions in Europe. The Beauce aquifer is a typical example of an aquifer depleted by individual pumping for irrigation and an interesting example because of the restriction policies already in place. Restrictions in the Beauce area are proportional to reductions in farmers' individual quotas and are contingent on the aquifer level. Less precipitation and increasing water demand may render potentially drastic restriction policies necessary (Lejars *et al*., 2012a, 2012b). The aim of this article is to evaluate the impact of different restriction policies on farmers' land allocation and irrigation decisions under dry weather conditions.

Similar to the work of Madani and Dinar (2012), we make rather unusual but realistic assumptions about the informational context in which farmers take their decisions. Concerning the time horizon, farmers are assumed to be neither completely myopic, i.e. maximising their instantaneous welfare, nor completely farsighted, i.e. taking the long-term outcomes of payoffs and resources into account. Concerning the actions of other resource users, they are neither completely smart, i.e. learning about the behaviour of other users, nor completely ignorant of other resource users' water abstractions.^{1}

Completely myopic agents justify the use of static models, and such agents have been extensively used in programming models explaining optimal crop choices (see for example Howitt, 1995; Heckelei, 2002; Heckelei and Wolff, 2003; Graveline *et al*., 2012 or Graveline and Mérel, 2014). Good knowledge of future changes in the resource and farsighted agents justify the use of dynamic resource models, explaining the optimal choice of water use over time, and have been extensively used in the resource economics literature (see for example Burt, 1967; Gisser and Sánchez, 1980; Roseta-Palma, 2002; Moreaux and Reynaud, 2006 or De Frutos Cachorro, Erdlenbruch and Tidball, 2014).^{2}

In this article, we construct a model in which farmers choose crop allocation and irrigation water volumes. Individual farmers do not consider long-term changes in the resource and are therefore modelled as decision makers with a limited planning horizon. However, when taking their decisions in spring, farmers consider the potential impact of restrictions not only on the spring crop but also on the summer crop. They are hence farsighted over two periods. In addition, farmers in the Beauce area can observe the level of the resource and make conjectures about the total abstraction volume made by other resource users, based on information from the previous years. In contrast to other hydro-economic models in the literature (see for example Britz, Ferris and Kuhn, 2013; Erfani, Binions and Harou, 2014), we presume the farmers have imperfect information on other users' actions but do make some best guesses about their water abstractions. In addition, there is no formal water market that can adjust water demand and supply (but see Erfani, Binions and Harou, 2014 for a model including water markets). Because farmers are somewhat farsighted when restrictions are in place and because they can monitor changes in the resource over the year, we construct a dynamic (two-period) hydro-agro-economic model.

Situated south-west of Paris, the Beauce aquifer extends over 9,700 km^{2} (see Lejars *et al*., 2012c). With less than 600 mm of rainfall per year, it is one of the driest regions in France (see Lejars *et al*., 2012c). As a consequence, about 50 per cent of the agricultural land is irrigated (MAAF, 2012), mainly with water from the aquifer. The aquifer is also a crucial resource for drinking water in the region. The management of the Beauce aquifer is therefore an important issue that has been addressed through several governance schemes. In particular, since 1995, irrigation restrictions depend on the state of the aquifer and since 1999, individual irrigation quotas have been introduced, which are adjusted every spring through a reduction coefficient calculated as a function of the state of the aquifer.^{3}

Future weather conditions may render drastic restrictions necessary. This is why Lejars *et al*. (2012a) discussed the potential impact of restrictions representing 40 and 70 per cent of quota in force today. In this article, we first assess how the farmers adjust to dry weather conditions and what this implies for the aquifer. We then introduce restrictions of 20, 40 and 70 per cent of individual quotas under dry conditions in our model. While Lejars *et al*. (2012a) studied such restrictions in focus groups with farmers, we test the impact of such restrictions in an analytical model. Similar stringent restrictions have been found to be necessary in other regions of Europe. For example, Pérez-Blanco and Gómez (2014) reported that drought management plans in the Guadalquivir river basin in Spain lead to restrictions of 30 per cent when the drought alert index is reached and up to 70 per cent in emergency situations. Our study shows that, although restrictions are efficient in preserving water-table levels, they result in serious economic losses for farmers, representing almost one-third of gross annual value added in the most extreme scenario.

The article is organised as follows. In Section 2, we present the hydro-agro-economic model and the solution approach we use. In Section 3, we describe the existing data and the transformations we undertook to be able to apply the model to the Beauce study area. We model choices of a representative farm specialised in field crops and sugar beet production, which is one of the four main farm types in the study area. We estimate the water response of the underlying yield functions. We also consider how yield responses and water use by competitive sectors change, depending on weather conditions. In Section 4, we present the baseline case, a normal year corresponding to 2010, and results for different scenarios, namely, a dry year with no policy intervention, and four policy scenarios, in which quotas and restrictions are used to cope with dry conditions. In Section 5, we discuss the impact of some key parameters of our model. Finally, in Section 6, we present our conclusions and ideas for further research.

## A model of irrigation and land-use choice

### The model

We consider a two-period and *k*-crop model for a representative farm with a surface area *S*. We call *t*= 0 the first time period (spring) and *t*= 1 the second time period (summer) for which decisions are taken. At the beginning of spring, the farmer chooses the share of land, *α _{k}*(

*t*), with 0 ≤

*α*(

_{k}*t*) ≤ 1 and the (per hectare) irrigation water volume,

*w*(

_{k}*t*), for each crop

*k*and each period

*t*. These are the decision variables.

*M*represents the number of representative farmers in the study area, which covers a total surface area of

*S*hectares. These farmers share the same aquifer, described by the height of the water-table,

_{b}*H*(

*t*), which is the state variable. The water-table changes as a function of all the farmers' irrigation decisions (see equation (5)). In the following, we describe all the parameters and variables of the model.

^{4}

First, the (per hectare) yield response to water, *y _{k}*, for each crop, is given by:

*a*,

_{k}*b*and

_{k}*x*are positive parameters.

_{k}Each farmer aims to maximise the present value of gross values added, $\u2211t\beta t\pi (t),$ given the price for each crop *p _{k}*, the discount rate

*β*and variable costs. For per hectare variable costs, we distinguish operating expenses $cko,$ which depend on the share of surface area allocated to each crop, from pumping costs $ckp,$ which depend on the water-table height and on the per hectare water volume used for each crop. Hence

*d*and

_{k}*e*are positive parameters of operating expenses and

_{k}*z*and

*c*are positive parameters of the cost of pumping. In particular, the quadratic form of operating expenses is due to implicit management costs associated with a given land allocation. As shown by Carpentier and Letort (2012), quadratic costs occur because of the constraints associated with quasi-fixed inputs (machinery and labour peak loads) and crop rotations (see also Heckelei and Wolff, 2003). Concerning the pumping cost function,

*z*measures the marginal costs of maximum possible lift and

*c*is the unit energy cost (see for example Gisser and Sánchez, 1980). Thus, gross value added in period

*t*is given by

*W*, corrected by the withdrawal coefficient

*γ*and increases according to the return flow coefficient

*σ*and the net recharge over the period concerned,

*r*(

*t*). The storage capacity of the aquifer is represented by the surface area of the study area,

*S*and the aquifer stock coefficient,

_{b}*η*. The height of the water-table in the second period thus depends on the height of the water-table in the first period in the following way:

Total extractions are given by:

*M*the number of representative farmers,

*w*(

^{j}*t*) is the irrigation water volumes of non-representative farms and

*w*(

^{o}*t*) is the water extraction for other uses, namely, drinking water and industrial uses.

We consider that the representative farm does not know the value added of the other players who share the aquifer. However, the farmer guesses the volume used by other water users, for example, based on the total amount of water used in a previous agricultural campaign. Finally, we assume that the value of the resource at the end of the planning horizon, *V*(*H*(2)), is constant. This means that the implicit price of the water resource at that time is zero. The farmer's planning horizon is indeed only one agricultural campaign with two irrigation periods, and the value of water at the end of these seasons is nil for the production process considered here.

The general problem for the representative farmer is hence the following:

*π*(

*t*) described in equation (4) and constraints (equation (9)) above.

### A simpler case

In the following, we consider a simpler case representing a typical situation in the Beauce area. We use a model with three crops, of which two are grown in spring. Because there is only one main summer crop, which is grown on a contractually fixed proportion of land, we assume the case where the share of the summer crop is fixed. The contract also implies that the summer crop cannot be grown without a minimum amount of irrigation. Hence, we have

#### Land use and water volumes in summer

We can now solve the dynamic programming problem using backward induction. As *V*(*H*(2)) = *V _{T}* (constant), we have

The necessary condition of optimality is

*p*

_{3}

*a*

_{3}− 2

*p*

_{3}

*b*

_{3}

*w*

_{3}(1) is the marginal benefit derived from the summer crop and

*z*-

*cH*(1) is the marginal cost of water use in summer. Hence, equation (12) describes the optimal irrigation water choice as the one that equalises marginal benefit and marginal costs for the summer crop. Moreover, given the relation between the water table and irrigation water use (see equation (13)), marginal costs for water use in summer depend on the optimal irrigation water choice in spring. Substituting equations (12) and (13) in equation (10), we can compute the maximum value of the resource in summer as a function of the choices made in spring

#### Land use and water volumes in spring

Next, we maximise the value of the resource in spring in *t*= 0. We have to solve

*V*(

*H*(1)) described in equation (14). One necessary condition of optimality is

*P*(1) and

*P*(2) the value added from crops 1 and 2

Equation (16) describes the optimal share of land-use used for crop 1 in spring. Notice that this solution depends on the difference between the gains from crop 1 (equation (17)) and crop 2 (equation (18)) and the impact of the choice of land-use in spring, *α*_{1}(0), on the discounted value of the resource in summer $\beta (\u2202\pi \u2217(1)/\u2202\alpha 1(0))$ (see equation (14)). Clearly, the greater the difference between the gains obtained from crops 1 and 2, and/or the smaller the irrigation volume used in summer, the greater the share chosen for crop 1.

The other conditions for a maximum are

Following equations (19) and (20), optimal irrigation water volumes for crop 1 (crop 2, respectively) depend on the share of land used for crop 1 (crop 2), the difference between marginal benefits and costs of water use for crop 1 (crop 2) and the value of the resource in summer given the irrigation water choice for crop 1 (crop 2) in spring.

We have a system of three equations: equations (16), (19) and (20), with three unknowns which we can therefore determine and find $\alpha 1\u2217(0),$$w1\u2217(0)$ and $w2\u2217(0).$ Finally, we have to substitute $\alpha 1\u2217(0),$$w1\u2217(0)$ and $w2\u2217(0)$ in equation (12) to find $w3\u2217(1)$ the optimal irrigation water choice for crop 3.

At this point, we have only described the optimal interior solution of the problem. To take into account corner solutions, we need to consider different cases, depending on whether or not water use quotas and restrictions are implemented (see Table 1). Quotas reduce the total water amount available. They can be reduced by a coefficient *ω* (0 < *ω* ≤ 1), depending on the water-table level at the beginning of the irrigation season. Without quotas or restrictions, we have to consider the 15 cases in Table 1. If quotas and restrictions are implemented, we have to consider the additional constraint

*V*(

*H*(0)).

Cases | Values |
---|---|

Case 1 | $\alpha 1=0\u21d2w1=0$ |

Case 2 | $\alpha 2=0\u21d2w2=0$ |

Case 3 | w_{1} = 0 |

Case 4 | w_{2} = 0 |

Case 5 | $w3=w-$ |

Case 6 | $w1=w2=0$ |

Case 7 | w_{1} = 0 and $w3=w-$ |

Case 8 | w_{2} = 0 and $w3=w-$ |

Case 9 | w_{1} = w_{2} = 0 and $w3=w-$ |

Case 10 | $w1=w2=\alpha 1=0$ |

Case 11 | $w1=w2=\alpha 2=0$ |

Case 12 | $w1=\alpha 1=0$ and $w3=w-$ |

Case 13 | $w2=\alpha 2=0$ and $w3=w-$ |

Case 14 | $w1=w2=\alpha 1=0$ and $w3=w-$ |

Case 15 | $w1=w2=\alpha 2=0$ and $w3=w-$ |

Cases | Values |
---|---|

Case 1 | $\alpha 1=0\u21d2w1=0$ |

Case 2 | $\alpha 2=0\u21d2w2=0$ |

Case 3 | w_{1} = 0 |

Case 4 | w_{2} = 0 |

Case 5 | $w3=w-$ |

Case 6 | $w1=w2=0$ |

Case 7 | w_{1} = 0 and $w3=w-$ |

Case 8 | w_{2} = 0 and $w3=w-$ |

Case 9 | w_{1} = w_{2} = 0 and $w3=w-$ |

Case 10 | $w1=w2=\alpha 1=0$ |

Case 11 | $w1=w2=\alpha 2=0$ |

Case 12 | $w1=\alpha 1=0$ and $w3=w-$ |

Case 13 | $w2=\alpha 2=0$ and $w3=w-$ |

Case 14 | $w1=w2=\alpha 1=0$ and $w3=w-$ |

Case 15 | $w1=w2=\alpha 2=0$ and $w3=w-$ |

## Data on the Beauce area

Our study area, the ‘Central Beauce’ area, which was defined by Lejars *et al*. (2012c), occupies an area of 300,600 ha of agricultural land and can be considered to be representative of the whole Beauce region in terms of farm types. The Beauce region is one of the driest regions in France, with <600 mm rainfall per year. More than half the farms depend on individual water extractions from the Beauce aquifer. Since 1999, the aquifer has a well-established volumetric management system consisting of individual irrigation quotas, which are adjusted each year by a reduction coefficient as a function of water-table levels and are communicated to farmers at the beginning of the irrigation season (see Petit, 2002). Farmers can observe the water-table level in their wells, or they can learn about the water-table level from the water-basin manager and official statistics. In addition, each spring, they are informed whether additional restrictions will be introduced in the region. Whether restrictions apply or not depend on the level of the aquifer. Severe restrictions apply when the crisis threshold is reached, and some restrictions may even apply earlier, when the alert threshold is reached. In 2010, the crisis threshold was at 110.75 m NGF,^{5} the alert threshold at 112.19 m NGF. Relatively high variability of water-table levels led to variations in restrictions ranging from 4.5 to 55 per cent from 1999 to the present (see Bouarfa *et al*., 2011 or Lejars *et al*., 2012a). In the future, severe restrictions could be necessary under certain climate change assumptions (Lejars *et al*., 2012a). Following Lejars *et al.*, we test restrictions corresponding to 40 and 70 per cent reductions in individual quotas. In the following sub-sections, we describe the agronomic, hydro-geological and economic data we use to inform our model of irrigation and land-use choice. Our baseline case is the year 2010, which corresponds to a year with normal precipitation in the study area. We also consider a scenario of a dry year, with and without restrictions on irrigation water use, for which some of the parameters change.

### Agronomic data

#### Types of farms

On the basis of RGA^{6} land-use data in 2010, Lejars *et al*. (2012a, 2012b, 2012c,) identified four types of field crop farms in the study area. All of them cultivate over 45 per cent of winter crops (mainly wheat) but differ from each other in the spring or summer crops in which they specialise: sugar beet in the first group, rapeseed in the second, special crops in the third and maize in the fourth. Here, we focus on the most common type of farm in our study area, which accounts for 679 farmers specialised in field crops and sugar beet. Land use of the representative field crops sugar beet farm consists of winter cereals, winter barley and sugar beet, with 48, 17 and 16 per cent of the land-use share, respectively. The general agronomic data are available in the

#### Yield response to water

We compute the yield response to water based on simulation data from the agronomic PILOTE model (see Mailhol *et al*., 2011). The data accounts for the water balance in the irrigation season [rain, real evapotranspiration (ETR) and irrigation at different dates] and for the yields of different types of crops and soil for the period 1997–2001. We aggregate data using different regressions according to the type of crop, the type of soil and weather conditions. Regression results are given in Tables 2–4. We focus on results for average/deep soil, which is the most common soil found on specialised sugar beet farms, and on normal and dry weather conditions. Weather conditions are defined as a function of efficient rainfall (rain minus real ETR) and computed for the most representative crop in each irrigation season, i.e. wheat in spring and sugar beet in summer. In spring, dry conditions correspond to an ETR ≤−60 mm and normal conditions to an ETR between −60 and 35 mm. In summer, the dry condition corresponds to an ETR ≤−220 mm and normal conditions to an ETR between −220 and −120 mm.

Variable | Coefficient | Std error | t-value | $P>|t|$ | 95% Confidence interval |
---|---|---|---|---|---|

Dry weather conditions^{a} | |||||

w | 0.0051176 | 0.0012194 | 4.20 | 0.000 | [0.0027036, 0.0075315] |

w^{2} | −2.14e−06 | 9.86e−07 | −2.17 | 0.032 | [−4.09e−06, −1.87e−06] |

Constant | 7.144896 | 0.3032466 | 23.56 | 0.000 | [6.544589, 7.545203] |

Normal weather conditions^{b} | |||||

w | 0.0031337 | 0.0005348 | 5.86 | 0.000 | [0.0020779, 0.0041895] |

w^{2} | −1.71e−06 | 4.76e−07 | −3.60 | 0.000 | [−2.65e−06, −7.71e−07] |

Constant | 9.415315 | 0.1214989 | 77.49 | 0.000 | [9.175474, 9.655156] |

Variable | Coefficient | Std error | t-value | $P>|t|$ | 95% Confidence interval |
---|---|---|---|---|---|

Dry weather conditions^{a} | |||||

w | 0.0051176 | 0.0012194 | 4.20 | 0.000 | [0.0027036, 0.0075315] |

w^{2} | −2.14e−06 | 9.86e−07 | −2.17 | 0.032 | [−4.09e−06, −1.87e−06] |

Constant | 7.144896 | 0.3032466 | 23.56 | 0.000 | [6.544589, 7.545203] |

Normal weather conditions^{b} | |||||

w | 0.0031337 | 0.0005348 | 5.86 | 0.000 | [0.0020779, 0.0041895] |

w^{2} | −1.71e−06 | 4.76e−07 | −3.60 | 0.000 | [−2.65e−06, −7.71e−07] |

Constant | 9.415315 | 0.1214989 | 77.49 | 0.000 | [9.175474, 9.655156] |

^{a}Number of observations: 125. Adjusted *R*^{2} 0.3071.

^{b}Number of observations: 173. Adjusted *R*^{2} 0.2750.

Variable | Coefficient | Std error | t-value | $P>|t|$ | 95% Confidence interval |
---|---|---|---|---|---|

Dry weather conditions^{a} | |||||

w | 0.004653 | 0.0007293 | 6.38 | 0.000 | [0.0032086, 0.0060974] |

w^{2} | −1.99e−06 | 6.00e−07 | −3.32 | 0.001 | [−3.18e−06, −8.01e−07] |

Const. | 5.876013 | 0.1789212 | 32.84 | 0.000 | [5.521637, 6.231389] |

Normal weather conditions^{b} | |||||

w | 0.002735 | 0.0003649 | 7.50 | 0.000 | [0.002014, 0.0034559] |

w^{2} | −1.25e−06 | 3.57e−07 | −3.50 | 0.001 | [−1.95e−06, −5.44e−07] |

Const. | 7.238088 | 0.0763662 | 94.78 | 0.000 | [7.087203, 7.388972] |

Variable | Coefficient | Std error | t-value | $P>|t|$ | 95% Confidence interval |
---|---|---|---|---|---|

Dry weather conditions^{a} | |||||

w | 0.004653 | 0.0007293 | 6.38 | 0.000 | [0.0032086, 0.0060974] |

w^{2} | −1.99e−06 | 6.00e−07 | −3.32 | 0.001 | [−3.18e−06, −8.01e−07] |

Const. | 5.876013 | 0.1789212 | 32.84 | 0.000 | [5.521637, 6.231389] |

Normal weather conditions^{b} | |||||

w | 0.002735 | 0.0003649 | 7.50 | 0.000 | [0.002014, 0.0034559] |

w^{2} | −1.25e−06 | 3.57e−07 | −3.50 | 0.001 | [−1.95e−06, −5.44e−07] |

Const. | 7.238088 | 0.0763662 | 94.78 | 0.000 | [7.087203, 7.388972] |

^{a}Number of observations: 119. Adjusted *R*^{2} 0.5088.

^{b}Number of observations: 154. Adjusted *R*^{2} 0.4916.

Variable | Coefficient | Std error | t-value | $P>|t|$ | 95% Confidence interval |
---|---|---|---|---|---|

Dry weather conditions^{a} | |||||

w | 0.0554281 | 0.0048382 | 11.46 | 0.000 | [0.00458902, 0.064966] |

w^{2} | −0.0000141 | 2.83e−06 | −4.97 | 0.000 | [−0.0000196, −8.48e−06] |

Const. | 42.94781 | 1.710531 | 25.11 | 0.000 | [39.57571, 46.31992] |

Normal weather conditions^{b} | |||||

w | 0.0325382 | 0.004551 | 7.15 | 0.000 | [0.023583, 0.0414934] |

w^{2} | −7.43e−06 | 2.88e−06 | −2.58 | 0.010 | [−0.0000131, −1.75e−06] |

Const. | 65.02174 | 1.462459 | 44.46 | 0.000 | [62.14399, 67.89948] |

Variable | Coefficient | Std error | t-value | $P>|t|$ | 95% Confidence interval |
---|---|---|---|---|---|

Dry weather conditions^{a} | |||||

w | 0.0554281 | 0.0048382 | 11.46 | 0.000 | [0.00458902, 0.064966] |

w^{2} | −0.0000141 | 2.83e−06 | −4.97 | 0.000 | [−0.0000196, −8.48e−06] |

Const. | 42.94781 | 1.710531 | 25.11 | 0.000 | [39.57571, 46.31992] |

Normal weather conditions^{b} | |||||

w | 0.0325382 | 0.004551 | 7.15 | 0.000 | [0.023583, 0.0414934] |

w^{2} | −7.43e−06 | 2.88e−06 | −2.58 | 0.010 | [−0.0000131, −1.75e−06] |

Const. | 65.02174 | 1.462459 | 44.46 | 0.000 | [62.14399, 67.89948] |

^{a}Number of observations: 212. Adjusted *R*^{2} 0.7105.

^{b}Number of observations: 309. Adjusted *R*^{2} 0.4112.

We find that the quadratic relationship between water and yields gives the overall best results, which is in line with results in the literature (see for example Bozkurt *et al*., 2006 or Ali, 2011 for a survey). We also tested linear and cubic relationships, but the fit was less good. Note that we use simulated data as the basis for our regressions. All the scenarios we use are assumed to be equiprobable. We can therefore compare the goodness of fit of different model specifications. The values of the regression coefficients are listed in

*b*.

_{k}### Hydro-geological data

We use hydro-geological data from Graveline (2013) concerning the Central Beauce part of the aquifer to set the withdrawal coefficient, the return flow coefficient and the aquifer storage coefficient.Water withdrawals for other uses than irrigation come from Lejars *et al*. (2012c). The water needs of other types of farms in spring and summer come from Lejars *et al*. (2012b), and the total surface area corresponds to the Central Beauce part of the aquifer. The initial water-table height for the baseline scenario is the one recorded in spring 2010.^{7} The initial water-table height in our example is thus 0.62 m above the alert threshold. Next, we set the net recharge in summer and spring to zero, as most recharge takes place in winter. Finally, water-table heights and withdrawals by other types of farms vary with the scenario. A summary of all these values is presented in

### Economic data

We use economic data from several sources, which are summarised in

. Prices for wheat and barley come from the national agency FranceAgriMer (2012) and price of sugar beet from sugar beet producer organisations (CGB, 2009). Operating expenses come from the farm data-base network, ROSACE (2010). Because we do not have enough data to regress operating expenses on farm area, we attribute all operating expenses to the quadratic term. Pumping costs correspond to the cost of energy required to pump water to the topsoil. For typical pump capacities, 0.136 kW is required to lift 1 m^{3}1 m. Considering pump efficiencies of 85 per cent

^{8}and energy costs of 0.07 euros/kWh, we obtain marginal pumping costs of 0.000224 euros/m

^{3}× m. For the largest potential pumping distance (considering the mean surface elevation to be 150 m above sea level and the deepest point of the aquifer to be 20 m above sea level), we obtain maximal (marginal) pumping costs of

*z*= 0.02912 euros per m

^{3}. Note that we do not consider neither water taxes nor investments nor payoffs for irrigation equipment. Therefore, our pumping costs correspond to a minimum bound. The final value of the resource is set to zero. Indeed, individual farmers do not internalise the consequences of long-term changes in the water-table. Finally, the discount rate is set at 5 per cent for each period considered.

^{9}

## Model results for the Beauce area

### Results for the baseline case: a normal year

Table 5 (second last column) shows the results of the simulation of the baseline case, a normal year corresponding to 2010. The representative sugar beet farmer chooses to allocate 60 per cent of his/her land to wheat and 24 per cent to barley, 16 per cent being used for sugar beet by assumption. Wheat is irrigated with 894 m^{3} per hectare, barley with 1,059 m^{3} per hectare and sugar beets with 2,167 m^{3} per hectare, leading to a total water volume of 138,782 m^{3} for one farm and 94.23 million m^{3} for all the field-crop sugar beet farms. This lowers the height of the water-table from the initial 92.81 to 92.09 m by the end of spring and to 91.67 m by the end of summer. Note that this water-table level (which corresponds to 111.67 m NGF) is above the crisis threshold (110.75 m NGF) that would lead to severe restrictions. Overall, a representative farm generates a gross annual value added of 89,717 euros.

Variables | Description | Unit | Baseline | Dry year |
---|---|---|---|---|

α_{1} | Share for wheat | Unitless | 0.60 | 0.56 |

α_{2} | Share for barley | Unitless | 0.24 | 0.28 |

w_{1} | Volume of water for wheat | m^{3}/ha | 894 | 1,178 |

w_{2} | Volume of water for barley | m^{3}/ha | 1,059 | 1,147 |

w_{3} | Volume of water for sugar beet | m^{3}/ha | 2,167 | 1,954 |

$w~$ | Total water volume | m^{3} | 138,782 | 157,800 |

V(H(0)) | Gross annual value-added | Euros | 89,717 | 84,043 |

$Mw~$ | Total water of sugar beet farms | 10^{6} m^{3} | 94.23 | 107.15 |

H_{1} | Aquifer level by end of spring | m | 92.09 | 91.93 |

H_{2} | Aquifer level by end of summer | m | 91.67 | 91.49 |

H_{0}− H_{2} | Decrease in the aquifer level | m | 1.14 | 1.32 |

Variables | Description | Unit | Baseline | Dry year |
---|---|---|---|---|

α_{1} | Share for wheat | Unitless | 0.60 | 0.56 |

α_{2} | Share for barley | Unitless | 0.24 | 0.28 |

w_{1} | Volume of water for wheat | m^{3}/ha | 894 | 1,178 |

w_{2} | Volume of water for barley | m^{3}/ha | 1,059 | 1,147 |

w_{3} | Volume of water for sugar beet | m^{3}/ha | 2,167 | 1,954 |

$w~$ | Total water volume | m^{3} | 138,782 | 157,800 |

V(H(0)) | Gross annual value-added | Euros | 89,717 | 84,043 |

$Mw~$ | Total water of sugar beet farms | 10^{6} m^{3} | 94.23 | 107.15 |

H_{1} | Aquifer level by end of spring | m | 92.09 | 91.93 |

H_{2} | Aquifer level by end of summer | m | 91.67 | 91.49 |

H_{0}− H_{2} | Decrease in the aquifer level | m | 1.14 | 1.32 |

### Results for a dry year

Table 5 compares simulation results for a dry year with the baseline case. Because the share of the summer crop is fixed, 16 per cent of land is still allocated to sugar beet, that is $\alpha -=0.16.$ However, the allocation of spring crops changes: compared with the baseline case, the representative farmer chooses to allocate less land to wheat (56 per cent compared with 60 per cent) and more to barley (28 per cent compared with 24 per cent). The intuition behind this change is that wheat is more sensitive to drought than barley, because yields are more responsive to water scarcity. This can be checked by computing the marginal productivity of water (MPW) at optimal values in normal and dry years. First, the MPW value for wheat decreases by around 70 euros/m^{3} per ha, whereas the MPW for barley decreases by only around 13 euros/m^{3}. Moreover, the difference in MPW between wheat and barley is around 346 euros/m^{3} per ha in a normal year and 288 euros/m^{3} per ha in a dry year. This explains the change in the farmers' choice of land-use.

Next, total irrigation water volume increases by 19,000 m^{3}. This is due to an increase in both wheat and barley irrigation (1,178 m^{3}/ha compared with 894 m^{3}/ha for wheat, 1,147 m^{3}/ha compared with 1,059 m^{3}/ha for barley), while irrigation for sugar beets is reduced. The resulting total water volume of a representative farm increases under dry conditions and amounts to 157,800 m^{3} (compared with 138,782 m^{3} in the normal year). This leads to a bigger drop in the water-table, to 91.49 m by the end of summer (compared with 91.67 m in the normal year), which corresponds to a drop of 1.32 m. Most of this additional decrease is due to withdrawals in spring. While in a normal spring, the water-table height was reduced by 0.72 m, in a dry spring, it is reduced by 0.88 m, i.e. by additional 0.16 m. Finally, despite these adaptations, gross annual value-added for the representative farmer decreases only slightly (by 5,674 euros) from 89,717 euros in the normal year to 84,043 euros in a dry year.

### Results for a dry year with restriction policies

We now introduce restriction policies. In the study area, individual quotas are in place. We first analyse the case in which quotas restrict the water volume to amounts in a normal year. Quotas can be changed into restrictions in dry years when the level of the aquifer is low. Table 6 illustrates how the introduction of these policies changes the results. We consider four scenarios: the use of quotas alone and restrictions corresponding to 20, 40 and 70 per cent of the quotas. Lejars *et al*. (2012a) considered the 40 per cent and the 70 per cent restrictions as possible for future water policies. Graveline and Mérel (2014) considered 10 and 30 per cent as policy scenarios in a model on the Beauce aquifer. Hence, we consider the 20 per cent restriction a less extreme scenario. In all our restriction scenarios, the initial water-table levels are set below the crisis threshold, justifying policy intervention.

Var. | Description | Unit | Restriction policies | |||
---|---|---|---|---|---|---|

Quota | 20% | 40% | 70% | |||

α_{1} | Share for wheat | Unitless | 0.56 | 0.56 | 0.55 | 0.53 |

α_{2} | Share for barley | Unitless | 0.28 | 0.28 | 0.29 | 0.31 |

w_{1} | Volume of water for wheat | m^{3}/ha | 1,121 | 870 | 621 | 251 |

w_{2} | Volume of water for barley | m^{3}/ha | 1,077 | 768 | 460 | 3 |

w_{3} | Volume of water for sugar beet | m^{3}/ha | 1,300 | 1,300 | 1,300 | 1,300 |

$w~$ | Total water volume | m^{3} | 138,782 | 111,025 | 83,269 | 41,634 |

V(H(0)) | Gross annual value added | Euros | 83,811 | 81,090 | 75,360 | 61,175 |

$Mw~$ | Total water of sugar beet farms | 10^{6} m^{3} | 94.23 | 75.39 | 56.54 | 28.27 |

H_{0} − H_{2} | Decrease in the aquifer level | m | 1.14 | 0.91 | 0.77 | 0.56 |

Var. | Description | Unit | Restriction policies | |||
---|---|---|---|---|---|---|

Quota | 20% | 40% | 70% | |||

α_{1} | Share for wheat | Unitless | 0.56 | 0.56 | 0.55 | 0.53 |

α_{2} | Share for barley | Unitless | 0.28 | 0.28 | 0.29 | 0.31 |

w_{1} | Volume of water for wheat | m^{3}/ha | 1,121 | 870 | 621 | 251 |

w_{2} | Volume of water for barley | m^{3}/ha | 1,077 | 768 | 460 | 3 |

w_{3} | Volume of water for sugar beet | m^{3}/ha | 1,300 | 1,300 | 1,300 | 1,300 |

$w~$ | Total water volume | m^{3} | 138,782 | 111,025 | 83,269 | 41,634 |

V(H(0)) | Gross annual value added | Euros | 83,811 | 81,090 | 75,360 | 61,175 |

$Mw~$ | Total water of sugar beet farms | 10^{6} m^{3} | 94.23 | 75.39 | 56.54 | 28.27 |

H_{0} − H_{2} | Decrease in the aquifer level | m | 1.14 | 0.91 | 0.77 | 0.56 |

Let us first compare results for a dry year without restrictions to results for a dry year with restriction policies (see Tables 5 and 6). Concerning land-use allocation, the use of policies lead to lower land-use shares allocated to wheat and higher shares to barley. Land-use shares of sugar beet are fixed and hence not adjusted. Concerning the irrigation strategy, when restriction policies are implemented, the farmer has access to a smaller total water volume. Priority is then given to the contractual summer crop: sugar beet, for which a minimum amount of irrigation is required by contract, see Bouarfa *et al*. (2011). Optimal results show that volumes for wheat and barley are greatly reduced. With a restriction of 20 per cent (respectively, 40 per cent), the volume of water for wheat is reduced to 870 (respectively, 621) m^{3} per hectare (compared with 1,178 m^{3} per hectare without restrictions) and for barley to 768 (respectively, 460) m^{3} per hectare (compared with 1,147 m^{3} per hectare without restriction). The volume of water for barley is reduced more than for wheat, as wheat requires more water than barley. This is in line with results reported by Graveline and Mérel (2014). With a restriction of 70 per cent, barley is cultivated under dryland farming conditions. Indeed, an amount of 3 m^{3} per hectare is negligible as the volume applied in one water turn corresponds roughly to 55 m^{3} per hectare. Overall, water volume reductions are quite important, ranging for instance between 26 and 33 per cent of dry year amounts in the 20 per cent restriction scenario. Graveline and Mérel (2014) find water volume reductions that are smaller than 9 per cent for a 30 per cent restriction scenario (intensive margin) but report the move to less water intensive crops (extensive margin) already for 10 and 30 per cent restriction scenarios. Overall, total water volumes decrease to 28.27 (75.39 and 56.54) million m^{3} in the 70 per cent (20 and 40 per cent) restriction scenarios. Not surprisingly, restricting total water use has a beneficial effect on the height of the water-table, which drops by about 0.91 m (0.77 m) with a restriction of 20 per cent (respectively, 40 per cent) and by only 0.56 m in the most extreme scenario. Restrictions lead to changes in water-table levels of 1–2 per cent. However, such apparently slight variations correspond to large volumes of water, between 0.2 and 0.4 million m^{3}.^{10} Moreover, repeated withdrawals of 1–2 per cent can lead to substantial drops in the water-table level over longer time horizons, except when winter recharge is high. On the other hand, restrictions reduce gross annual value added: compared with the case in which only quotas apply, gross annual value added is reduced by about EUR 2,721 in the least restrictive scenario, EUR 8,451 with a 40 per cent restriction and EUR 22,637 with a 70 per cent restriction. Such losses correspond to 3, 10 and 27 per cent of the gross annual value added compared with when only quotas apply. For comparison, Lejars *et al*. (2012b) found reductions of 10 and 21 per cent of gross production under the 40 and 70 per cent restriction scenarios, which is very close to our results. In contrast, Graveline and Mérel (2014) report very moderate reductions in profits for the 30 per cent restriction scenario of a regional Beauce model. In line with our results, Reynaud (2009) or Bouarfa *et al*. (2011) find again important revenue reductions in their respective case studies. This underlines the fact that although restrictions adequately preserve groundwater levels, they have a significant impact on the farmer's economic situation in the short term, even assuming that he/she adapts optimally to the dry situation. A policy maker could thus only count on abundant winter recharge (which can exceed 1.5 m in wet years) to avoid too high economic losses for farmers (see Bruand *et al*., 1997 for data on recharge).

To summarise, we can confirm three general features of adaptation in the face of drought and restriction policies: first, land-use is affected by a reduction in the share of the most sensitive crop and an increase in the share of the less sensitive crop. Second, the total volume of irrigation water for all crops is reduced. Third, in each scenario, lowest water volumes are allocated to the less productive barley crop, higher volumes to the more water sensitive wheat crop and highest volumes to the contractual summer crop. We can also summarise the economic impacts of our simulations. The combined effect of a dry year and restrictions leads to very serious economic losses for the farmers: for example 10 per cent (16 per cent) of gross annual value added with a 20 per cent (40 per cent) restriction, corresponding to EUR 8,627 (EUR 14,357). The quota only policy leads to a loss of 7 per cent of gross annual value added (or EUR 5,906). The 70 per cent restriction in quota volumes would lead to a 32 per cent loss of gross annual value added. Concerning the level of the aquifer, restriction policies show lower aquifers than the baseline case, because initial aquifer levels were intentionally set very low when stringent restrictions are in place. By assumption, there is no recharge in spring and summer, and hence no restriction can enable recovery of the resource within a year. However, we can measure the performance of different restriction policies with respect to the drop in water-table levels they trigger. We can see that the more stringent the restriction, the smaller the drop in the aquifer level during the irrigation campaign. This confirms the importance of the implementation of restriction policies to preserve the resource.

## Discussion of key parameters

Finally, we need to analyse the importance of different parameters in the simulation results.^{11} One major limit of the model is the lack of information to estimate the quadratic function that represents operating costs. As final results could be driven by the choice of this cost function, we designed some scenarios with different operating cost parameters. These different scenarios are simulated in such a way that the marginal unitary cost per crop is the same, as can be seen in Table 7.

Variables | Baseline | Scenario 1 | Scenario 2 | |||
---|---|---|---|---|---|---|

d_{1} = 0 | e_{1} = 908 | d_{1} = 908/3 | e_{1} = 2 × 908/3 | d_{1} = 908/2 | e_{1} = 908/2 | |

d_{2} = 0 | e_{2} = 780 | d_{2} = 780/3 | e_{2} = 2 × 780/3 | d_{2} = 780/2 | e_{2} = 780/2 | |

d_{3} = 0 | e_{3} = 1,786 | d_{3} = 1,786/3 | e_{3} = 2 × 1,786/3 | d_{3} = 1,786/2 | e_{3} = 1,786/2 | |

α_{1} | 0.60 | 0.68 | 0.73 | |||

α_{2} | 0.24 | 0.18 | 0.11 | |||

w_{1} | 894 | 894 | 894 | |||

w_{2} | 1,059 | 1,059 | 1,059 | |||

w_{3} | 2,167 | 2,167 | 2,167 | |||

$w~$ | 138,782 | 137,431 | 136,082 | |||

V(H(0)) | 89,717 | 67,266 | 56,504 | |||

$Mw~$ | 94.23 | 93.32 | 92.4 | |||

H_{0} − H_{2} | 1.14 | 1.14 | 1.14 |

Variables | Baseline | Scenario 1 | Scenario 2 | |||
---|---|---|---|---|---|---|

d_{1} = 0 | e_{1} = 908 | d_{1} = 908/3 | e_{1} = 2 × 908/3 | d_{1} = 908/2 | e_{1} = 908/2 | |

d_{2} = 0 | e_{2} = 780 | d_{2} = 780/3 | e_{2} = 2 × 780/3 | d_{2} = 780/2 | e_{2} = 780/2 | |

d_{3} = 0 | e_{3} = 1,786 | d_{3} = 1,786/3 | e_{3} = 2 × 1,786/3 | d_{3} = 1,786/2 | e_{3} = 1,786/2 | |

α_{1} | 0.60 | 0.68 | 0.73 | |||

α_{2} | 0.24 | 0.18 | 0.11 | |||

w_{1} | 894 | 894 | 894 | |||

w_{2} | 1,059 | 1,059 | 1,059 | |||

w_{3} | 2,167 | 2,167 | 2,167 | |||

$w~$ | 138,782 | 137,431 | 136,082 | |||

V(H(0)) | 89,717 | 67,266 | 56,504 | |||

$Mw~$ | 94.23 | 93.32 | 92.4 | |||

H_{0} − H_{2} | 1.14 | 1.14 | 1.14 |

We observe significant changes in the share of land allocated to each crop and in the gross value added obtained by each farmer. For example, in scenario 2 in which the marginal unitary cost for each crop is shared equally between the linear and quadratic parameters, the share of land allocated to wheat (respectively to barley) increases (respectively decreases) by 13 points compared with the baseline scenario. Moreover, the gross value-added decreases by EUR 33,213 from the baseline scenario to the second scenario, which corresponds to an economic loss of 37 per cent. However, the simulation results provide some hints for the validation of our model. First, total volumes of water used by the farm do not vary significantly between scenarios (<2 per cent). This implies that changes in water-table levels are very low between scenarios. Concerning economic outputs, the values in the baseline scenario are more realistic, as reported in the different studies conducted in the study area (cf. Lejars *et al*., 2012a, 2012c).

Next, in our analysis, we use estimated parameter values, *a _{k}*,

*b*and

_{k}*x*, which contain uncertainty. We therefore conduct a sensitivity analysis with respect to these parameter values. More precisely, we draw 10,000 parameter values in a normal law with standard errors as estimated in the regressions shown in Tables 2–4. Results for normal and dry year scenarios

_{k}^{12}are given in Table 8.

Variables | Description | Unit | Baseline | Dry year | t-test |
---|---|---|---|---|---|

α_{1} | Share for wheat | Unitless | 0.60 (0.07) | 0.58 (0.17) | ** |

α_{2} | Share for barley | Unitless | 0.24 (0.07) | 0.26 (0.17) | ** |

w_{1} | Water volume for wheat | m^{3}/ha | 986 (409) | 1,440 (764) | ** |

w_{2} | Water volume for barley | m^{3}/ha | 1,174 (454) | 1,401 (580) | ** |

w_{3} | Water volume for sugar beet | m^{3}/ha | 2,225 (608) | 2,026 (436) | ** |

$w~$ | Total water volume | m^{3} | 154,392 (40,809) | 196,985 (73,219) | ** |

V(H(0)) | Gross annual value-added | Euros | 92,381 (8,033) | 98,937 (36,044) | ** |

H_{1} | Aquifer level at the end of spring | m | 92.04 (0.12) | 91.81 (0.23) | ** |

H_{2} | Aquifer level at the end of summer | m | 91.62 (0.13) | 91.37 (0.23) | ** |

Variables | Description | Unit | Baseline | Dry year | t-test |
---|---|---|---|---|---|

α_{1} | Share for wheat | Unitless | 0.60 (0.07) | 0.58 (0.17) | ** |

α_{2} | Share for barley | Unitless | 0.24 (0.07) | 0.26 (0.17) | ** |

w_{1} | Water volume for wheat | m^{3}/ha | 986 (409) | 1,440 (764) | ** |

w_{2} | Water volume for barley | m^{3}/ha | 1,174 (454) | 1,401 (580) | ** |

w_{3} | Water volume for sugar beet | m^{3}/ha | 2,225 (608) | 2,026 (436) | ** |

$w~$ | Total water volume | m^{3} | 154,392 (40,809) | 196,985 (73,219) | ** |

V(H(0)) | Gross annual value-added | Euros | 92,381 (8,033) | 98,937 (36,044) | ** |

H_{1} | Aquifer level at the end of spring | m | 92.04 (0.12) | 91.81 (0.23) | ** |

H_{2} | Aquifer level at the end of summer | m | 91.62 (0.13) | 91.37 (0.23) | ** |

**Cases where *H*_{0} of equal means is rejected with 95% confidence intervals.

We can see that land-use changes, as described in our example based on the year 2006, are robust to changes in parameters. Likewise, volumes of water increase for wheat and barley and decrease for the summer crop in the dry scenario, like in our example. Moreover, at the end of spring and summer, the levels of the aquifer are significantly lower under dry conditions than in a normal year. However, gross annual value added according to the uncertainty analysis is greater in a dry year than in a normal year in contrast to our example. This is probably due to a greater decrease in the yield of the contractual summer crop in our example. Results for the total water volumes are also robust as there is an increase in the dry scenario compared with the normal scenario. The implementation of restriction policies in dry years are then justified.

Finally, we ran other simulations with different values for prices, pumping costs and parameters of the dynamics of the resource.^{13} For example, cereal crop prices are key parameters in the economic model. Increasing the price of barley above that of wheat leads to a significant decrease in the share of land allocated to wheat (−20 points). Doubling cereal prices leads to higher revenues (+128 points) but does not influence the state of the aquifer at the end of summer. Doubling the price of sugar beet leads to higher irrigation water volumes used for this crop. Results are less sensitive to an increase in pumping costs. Pumping costs have to be multiplied by at least 10 to result in significant changes in the water volumes used and revenues obtained. Finally, a variation in hydrological parameters, for example in water volumes applied by other users or the total surface area of the study area, does not impact individual irrigation and land-use choices but plays an important role in aquifer levels.

## Conclusion

In this article, we assess the impact of dry weather conditions and restriction policies in the Beauce aquifer in France. To this end, we built a dynamic hydro-agro-economic model to simulate the choice of land-use and irrigation volumes made by farmers. We needed a dynamic model because we wanted to assess restriction policies that apply in spring and in summer, but which the farmers learn about and take into account at the beginning of spring. The dynamic effect is not very large in our model, because pumping costs are very low and the aquifer is very large. If the model were to be used in other study areas, the dynamic effect would be increasingly relevant. However, with large surface areas, such as Central Beauce, small drops in the level of the water-table lead to major reductions in water volumes and may harm the whole agricultural sector. Specifically, the estimated 1 to 2 per cent drops in the level of the water table during the irrigation period correspond to water volumes of between 0.2 and 0.4 million m^{3}.

The main contribution of the article is assessing the impact of dry weather conditions and water restrictions on farmers' decisions concerning optimal land-use and irrigation. We first consider a dry year scenario, in which there is an increase in water demand. We then consider a dry year scenario with different restriction policies. We show that, first, land-use strategies in the face of droughts involve decreasing the share of the most sensitive crop and increasing the share of the less sensitive crop. Second, total irrigation water volumes may increase in the absence of restrictions but are reduced when restrictions are implemented. Third, in the case of restrictions, water volumes are reduced in quite important proportions (with reduction greater than 26 per cent). Fourth, with restrictions, lowest water volumes are allocated to the less productive barley crop, higher volumes to the more water sensitive wheat crop and highest volumes to the contractual summer crop. Lastly, we show that the combined effect of a dry period and restriction policies results in significant losses for farmers, which can reach 16 per cent of gross value-added for a high but not implausible 40 per cent limitation on water use and up to 32 per cent for drastic restrictions of 70 per cent. The order of magnitude of these losses is in line with other studies on the Beauce aquifer (see for example Bouarfa *et al*., 2011 or Lejars *et al*. 2012b). Hence, the implementation of restriction policies comes at a cost, which our model can assess.

To conclude, although restriction policies are a satisfactory way of preserving water-table levels, they can lead to serious economic losses for farmers in the short term. To avoid such losses to farmers, a policy maker could count on abundant winter recharge. In wet winters, recharge can exceed 1.5 m (see Bruand *et al*., 1997), which allows, in the following spring to have water-table levels that are higher than initial levels, whatever the restriction scenario considered in the current year. However, as the Beauce aquifer is characterised by low winter recharges, this scenario is rather unlikely. Our results thus imply important future challenges for policy makers in our study area.

Several extensions of this work are possible: First, we could improve the dynamic model by considering more than two periods and a more complex crop rotation system. In addition, we could introduce uncertainty and show how a farmer can cope with it. Moreover, we could assume farmers are risk averse, for example by including farmers who minimise the variance of outcomes. Finally, we could introduce different types of farmers and the interactions between them and focus especially on how they learn about their respective behaviours.

## Supplementary data

## Acknowledgements

We are grateful to Sami Bouarfa, Jean-Louis Fusiller, Nina Graveline, Caroline Lejars, Sylvie Morardet and Farida Ouchiha, who helped on various parts of the work. We acknowledge financial support from the ANR project RISECO (ANR-08-JCJC-0074-01).

## References

^{3}but has been subject to quite high inter-annual variations over the last 30 years (see Coz, 2000).