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Optimal water allocation: a slightly diﬀerent perspective
There is a relatively abundant literature on designing, planning, and operating water re-sources systems. These include the withdrawal and allocation of water across diﬀerent resources and users. The majority of works are focused on the optimization of surface irri-gation networks (Reca et al., 2001; Bharati et al., 2008) or on the optimization of cropping patterns from a planners’ or irrigation association perspective e.g. Montazar and Rahimikob (2008). Lehmann et al. (2013) provides a recent example of optimal farm management in the context of climate change. Their model incorporates detailed information on crop re-sponses to both irrigation and fertilizers application and accounts for risk, which is assumed to increase with climate change.
The main diﬀerence from the models discussed in this paper is that they follow a normative or engineering perspective i.e. what is best? whereas calibrated models seek to represent the behaviour of farmers or farming. The objective, here, is to examine how to improve the allocation of water in time and space to satisfy optimization criteria. Particular applications may be irrigation scheduling (McGuckin et al., 1987), dam operations, and conjunctive surface and groundwater use. For this type of application the time scale applied is often less than a year to account for variation in water supply and crop needs according to the season. These applications often involve dynamic optimization procedures and sometimes account for uncertainty in weather-related parameters (precipitation, water requirement per crop for a given year). Models are often connected to hydrological and engineering models such as reservoir operation models, to account for the processes of water supply and of the behaviour of other water use sectors, e.g. industry and drinking water. As such they are often referred to as hydro-economic models.
The objective function
The objective function is maximization or minimization of the utility, often assimilated to profit or expected profit, in the case of risk. More elaborate indicators are sometimes used when more than one attribute is considered in the objective function. Gòmez-Limón et al. (2003) suggest that the decision-making process is driven by other criteria than simple profit maximization, for instance, minimization of family labor. Bartolini et al. (2007) develop LP with multi-attribute utility theory (Keeney and Raiﬀa, 1993) by specifying an alternative objective function that is the sum of weighted attributes including profit, labor and diversification. Agriculture is a specific production process since some farms employ only on-farm labour and have limited resources and liquidity that could be formalized as constraints. However this is rarely represented in the objective function. In the large majority of cases yearly profit (or net revenue) is considered: the yearly revenue (prices and quantities of harvested products) less the variable costs. Variable costs are those that are attributable to a specific field production or livestock. Few studies provide details of what they actually consider to be variable costs (machinery costs for instance can be viewed to a certain extent as structural, i.e., attributable to the whole farm or as specific variable costs, i.e, attributable to a single crop).
The formulation of the general model has been discussed in Chapter 1.
Two diﬀerent activities i are always considered for the rain fed or irrigated technology of the same crop. Activities may also distinguish diﬀerent irrigation technologies. To our knowledge, there are no livestock activities that have been incorporated into WPM. WPM should be able to represent (i) the change in land allocation between crops and alternative activities (sometimes there are several technologies that may suit one crop, for instance diﬀerent irrigation technologies), (ii) the substitution between inputs, particularly between several water resources or between irrigated land and water. If there is a substitution possibility between water and land, the water application per unit of land can be adapted. The first ability always exists in WPM, but the second is not always present. In the latter case, production function are of Von Liebig type i.e. inputs are specified in fixed proportion. In other words for a given output, inputs are required in fixed proportions whatever the costs may be and qi = xiyi and a unique input is considered in the production term (qi) (often land). Frisvold and Konyar (2012) call this a « rationing model » ( see Paragraph 2.4).
Normative versus positive models and the calibration issue
The perspective and philosophy behind normative and positive models diﬀer completely, in theory at least. The normative perspective which is a deductive approach assumes an apri-ori objective function and derives the optimality conditions . The aim of the model is not to represent the real observed situation but to represent the optimal situation. The inter-pretation of the diﬀerence between model and observed result is the sub-optimal character of the real observed situation. The modeled output is the « optimal » allocation of inputs.
This perspective appears to be relevant in exploring optimal water management solutions such as those discussed in Paragraph 2.2.6. The various model outcomes can be compared with each other even if the baseline diﬀers from the real observed situation. An example of a normative linear programming model is provided by Doppler et al. (2002) who explore the improvement potential of farming in relation to water allocation on crops, together with the impact of water pricing strategies.
The limit of this approach is when significant costs or benefits are unobserved or unknown, and are not accounted for in the model. This can be the case with non-market costs such as leisure time for the farmer or unobserved rents such as those provided by contracts between farmers’ and industry for specialty crops. In this case the model does not account for all costs and benefits and the optimal picture it provides is erroneous. The implication for calibration is that normative models are not calibrated: in consequence they do not replicate the observed situation. In practice these models specify linear costs and yields (Leontief technologies) and are referred to as linear programming (LP) models.
Calibrated linear programming models
Methods have been developed to calibrate LP models to observed allocations, although not perfectly. Accordingly, LP should not be systematically defined as a pure normative ap-proach. A first method consists of a micro-approach in the sense that a deep understanding of the technical and economic processes at farm level is required. In this case all con-straints related to the technical realities of farming (rotation, animal turnover, regulation etc.) are specified and should enable a closer approach to the observed situation. Another approach consists of increasing the degrees of freedom in the model specifications, to allow for partial calibration. In practice this can be done by integrating risk into the objective function which then becomes an expected profit function. This can be carried out via the MOTAD approach (Minimization of Total Absolute Deviation, Hazell and Norton (1986)) which calibrates the risk aversion parameter to minimize the discrepancy between modeled baseline and the observed situation. The risk aversion coeﬃcient can be interpreted as a pass-through coeﬃcient aﬀecting utility according to the perception of variability from the farmers’ point of view. Another solution is to build on multi-attribute utility theory and specify an objective function as the sum of weighted attributes: the weights can serve as calibration parameters, while minimizing the departure of the model from the observed allocation (Bartolini et al., 2007). However, the calibration of these models cannot be ex-act. Dono et al. (2010) recommend using the Finger and Kreinin similarity index (Finger and Kreinin, 1979) to evaluate the eﬀectiveness of calibration. Recent applications involv-ing water management in agriculture include Bazzani et al. (2004); Bartolini et al. (2007); Graveline et al. (2009); Acs et al. (2010); Graveline et al. (2012).
A major criticism is that LP often provides over-specialized responses (Gohin and Chantreuil, 1999), because crops that have the highest gross margins will be chosen at the maximum level until a constraint becomes binding and prevents the best crop from increasing further. Its inability to represent smooth reactions to policy shocks (« jumpy behaviour »), thus be-coming too far removed from reality at least in the short term, is sometimes mentioned. This has been overcome by using regression to smooth the step inverse demand function (Booker and Colby, 1995). In medium-term simulations, « jumpy behaviour » is not a prob-lem, because dynamics are not considered. Another disadvantage is the need of a large quantity of data at the farm level. The problem of imperfect calibration is however a minor issue when comparing policy alternatives, as long as as the model is a good representation of the farmer’s behaviour. It seems less suited for modeling detailed adaptations of farming in terms of input allocation (of water or nitrogen).
The LP cross-mix approach was developed by McCarl (1982) to avoid the overspecialization problem (See Appendix 1 for details). An example that focuses on agricultural water use is Graveline et al. (2013). The principle is to add a constraint to the LP problem to ensure a convex combination of historical crop mixes. By doing so, this constraint prevents unrealistic crop mixes, thereby implicitly accounting for crop rotation and other practices that aﬀect crop choice. However, the main problem is that this specification can be viewed as overly constraining and the drawback of this type of model is that it can not simulate allocations that have not been observed in the past. This is a major problem when dealing with water use and resources, since tendencies such as climate change and requirements for reducing agriucltural water use in response to other (environmental and urban) increases in demand have created a totally new regulatory and natural environment. In addition, the context of high price volatility observed in recent years (Persillet, 2009) makes the economic environment completely new as well. However, Chen and Önal (2012) circumvent this problem by adding to the bundle of likely crop mixes certain, new synthetic crop mixes that are simulated with the help of supply-price elasticities and varied commodity prices.
Table of contents :
1 Introduction – Synthesis chapter
1.2 Modeling agricultural sector input use behavior and its impact on water resources
1.3 Background and case studies
1.4 Economic model
1.5 Assessing the impact of farming on long-term nitrate contamination in the Upper Rhine aquifer
1.6 Impact of farming on water resources: assessing uncertainty with Monte Carlo simulations in a global change context
1.7 Intensive and extensive margin adjustments to water scarcity in France’s cereal belt
1.8 Trade-offs between irrigated agriculture and groundwater management for alternative water policies in Beauce (France)
1.9 Discussion: modeling agricultures’ use of inputs
2 Economic calibrated agricultural supply models to inform water management and policy: a review
2.2 Topics addressed by WPM
2.3 The model
2.4 Incorporating water into the production functions
2.5 Uncertainty and risk
2.6 Spatial modeling scales and the aggregation bias
2.7 Conclusion and research agenda
3 Combining economic and groundwater models for developing long term nitrate concentration scenarios in a large aquifer
3.2 Overview of the hydro-economic modeling chain
3.3 Farm models to assess the evolution of agricultural land use and nitrogen practices
3.4 Calibration of the economic model
3.5 Development of 2015 foresight scenarios
3.6 Simulating the impact of global change scenarios
3.7 Simulating the impact of price-based instruments
3.8 Conclusion and perspectives
4 Impact of farming on water resources: assessing uncertainty with Monte Carlo simulations in a global change context.
4.2 Materials and methods
4.3 Results at different scales on uncertainty and impact assessment
4.4 Discussion on deterministic versus stochastic simulations
5 Intensive and extensive margin adjustments to water scarcity in France’s cereal belt
5.4 Reference allocation
6 Trade-offs between irrigated agriculture and groundwater level for alternative water policies in Beauce (France)
6.2 Groundwater governance reform in the Beauce region
6.3 Current approaches in hydro-economic modeling
6.5 Data and calibration
6.6 Design of alternative management scenarios
6.7 Simulation results