2.1. Background Literature
In the case of a surface irrigation network, the water provider usually applies flat rates, especially when there are no limitations to the availability of water resources and differences in water uses cannot be, or are too costly to be, assessed. This condition calls for the imposition of flat rates by the water provider, even if the level of use varies. With flat rates, users are taken to have similar access and are charged equally across farms and land [
21], i.e., the tariff is the same per hectare of land for all farms. Indeed, the regions supplied by surface irrigation network are usually very large and comprise huge extensive farms irrigating only a small quota of the cultivated agricultural land or specialized small fruit and vegetable farms irrigating most of the cultivated land. As a result, farmers benefit differently from the water supplied by the WA and pay tariffs as a part of total overall water supply costs which are, however, proportional to the total agricultural farmland and not to the irrigated farmland. Moreover, flat rates do not usually incorporate the environmental costs generated by irrigation activities, which threaten the status of water resources, especially due to nutrient leaching. Under such conditions, missing to link tariffs to water use and disregarding the total costs generated by the use of water resources, the water provider cannot expect that tariffs provide incentives to farmers to rationalize the use of water for irrigation. Tariffs play just the role of recovering supply costs.
A typical agency problem surrounding the use of water resources in agriculture arises when the WA decides to apply incentive tariff schemes for the water supplied through surface irrigation networks. In this case, farmers may own private information on water use which is unknown to the WA (e.g., water use profitability) and they may take opportunistic actions totally or partially undetected by the WA (e.g., a different amounts of water withdrawn compared to that agreed or assigned to the farm). These actions lead to increasing the WA’s water supply and the management costs [
32].
In particular, when a certain amount of irrigation water is assigned or self-reported by the farmer, the WA often faces difficulties in verifying whether farmers are complying with the amount reported. Under such condition, monitoring is costly and not fully effective. To avoid non-compliance, the WA might apply a sanction to farmers [
27]. Thereby, farmers actions remain solely the choice of the farmer, but depending on incentives to take the action. These incentives against cheating not only depend on the sanction, but also on the efficacy of monitoring in detecting their action. In this respect, different technology options may be available. Direct monitoring by WA operators may be very costly, while the use of information technologies could be much cheaper and hence help to discourage cheating and free riding due to information asymmetries.
If the WA monitoring capacity is perfect (the WA is in a position to perfectly detect who is complying or not with the agreed amount of water at zero cost) the incentive mechanism is fully efficient and non-compliance is avoided with no sanction. If this is not the case, the WA needs to design an incentive water pricing scheme, including a monitoring and sanctioning strategy, to boost compliance [
34]. The optimal monitoring strategy depends on the cost needed to enforce such mechanisms and on the effect on water use efficiency.
Given this context, the model below simulates the behavior of a WA the aim of which is to maximize the social benefit incentivizing rational water use. It is considered that the water authority is acting on behalf of a group of farmers: it seeks to maximize total farmer profits minus the costs of water provisions (including environmental costs); it also shares costs among users according to water use and may provide sanctions for non-compliant farmers. To incentivize rational water use the WA may apply incentive tariffs linked to some observable characteristics correlated to water use.
In order to analyze these contract design issues, we developed a methodology based on the Principal-Agent Theory [
35,
36], taking into account potential instrument design based on the asymmetric information literature. Specifically, the analytical approach developed in this study makes it possible to estimate the costs faced by the WA in setting up different pricing mechanisms in those circumstances where water is not metered.
2.2. Model Setting and Flat Rate Pricing Schemes
The sequence of decisions for the flat rate scenario works as follows: (1) During the irrigation season farmers take decisions regarding how much to irrigate; (2) At the end of the irrigating season the regulator recovers supply costs by imposing a flat rate. In this framework, farmers’ decisions with respect to water uses is independent of the cost faced by the water regulator to supply the service; on the contrary, the supply cost and hence the tariff depends on water uses. This occurs because farmers sign for water uses ex ante, while decisions on pricing are taken by the regulator ex post, at the end of the irrigation season, and depend on what farmers have subscribed to ex ante.
Consider that farm type when has a cultivated area with different crop water requirements. Without loss of generality we assume that each farm has a land area equal to 1. Supplying the water to the farm is costly for the WA and the farmer, as a result of irrigation receives a profit of . A quadratic production function is assumed for input factors concurring in generating the farms profit with regard to a cultivated crop. The water supply cost function represents total water costs for delivering the water to the farm for irrigating each crop of farm type The character indicates the share of irrigated area of the farm type . From now on, we will consider the share of irrigated area as a proxy for water use, while farm profits and regulator supply costs are assumed to be a function of the share of irrigated area and are assumed to be increasing and concave in with and .
Under such a condition, a rational farmer will choose to irrigate a share of area that will allow him to maximize his profit:
The irrigated share is the decisional variable and is the percentage of total cultivated area of the farm. Thus, the profit function is a per hectare profit function. Then, the optimal level of the farm’s irrigated share is the level for which marginal profits equal zero: . Let us call this level .
Ex post, the regulator must recover supply costs by imposing water tariffs to farmers. It is also assumed that the WA does not face any enforcement and monitoring costs, nor other transaction costs. Moreover, the WA, by assumption, is not in a position to monitor the farms’ water use and consequently to allocate supply costs among farmers based on actual uses.
Under such condition, the per hectare tariff paid by farmers will be:
where
is the tariff paid by each farmer
and the superscript FR indicate flat rate
The farmer pays the water tariffs based on the overall water supply costs
and there is no link between farms’ water consumption and the tariff paid to the regulator.
2.4. Incentive Pricing with Full Compliance and Perfect Detection
In this section we analyze the contract offered to the farmer that combines the irrigated share
and the water tariff
, assuming the WA fully observes the farm’s action. In such a situation, the WA’s problem is to recover water supply costs.
s.t.
The maximization of social benefit makes it possible to maximize the aggregate profit (i.e., the WA’s and farm’s profit) and involves the farm’s profit , the WA’s water supply costs and transaction costs linear on tariff . The objective function is subject to a cost recovery constraint (CR), indicating that the water tariffs must cover at least the water supply costs and the transaction costs generated by implementing incentive water pricing.
Given the transaction costs generated by the water tariff, it can be supposed that the CR constraint is always binding in optimum. Rearranging Equation (4) we are able to determine the level of the tariff , which is in function of the irrigated share and transaction costs.
Substituting in the objective function Equations (5):
And taking the derivative with respect to
, the First Order Condition (FOC) yields the following optimal solution:
By solving Equation (6), where the farm’s marginal profits equal marginal costs Weighted by the level of transaction costs , we determine the optimal share of irrigated land which we can therefore replace in Equation (5) to determine water tariff .
The result of Equation (6) implies that when transaction costs are high the optimal irrigated share decreases and the tariff increases. The optimal level of reaches its maximum when , in the absence of transaction costs, and the marginal benefit equals the marginal social cost of water.
2.5. Incentive Pricing with Effective Detection
In the absence of water metering, under the incentive pricing scenario the farmers’ decisions may either to participate and comply with the agreed rules or to participating and cheat, e.g., irrigating higher irrigated shares than those allowed by the contract. Compliance implies a disutility for the farmer. This disutility is equal to the difference between the maximum profit that the farmer would obtain in the absence of incentive pricing and the profit the farmer would obtain by irrigating the share of irrigated area declared at the beginning of the irrigation season, . If the farmer chooses not to comply with his statement, his disutility would be equal to zero.
With the purpose of discouraging false reporting, the regulator monitors farm actions. If the regulator deems that there are no problems, the farmer will pay to the regulator the agreed tariff Otherwise, the farmer is obliged to pay a sanction, , in addition to the tariff. Assuming that the farmer is risk neutral, sanctions can be considered the utility that the farmer obtains when complying with the rules.
In this assumption it is considered that monitoring costs are involved in transaction costs and no explicit costs from monitoring. The monitoring strategy introduced by the WA to detect farmers’ actions in the absence of water metering is not perfect. That is, the WA might fail to detect farmers’ behavior . Without loss of generality, we introduce a discrete probability setting, where: is the probability that the farmer is found to comply with his statement when he is actually complying and is the probability that farmer is found to be non-compliant with his statement when he is actually not complying. Likewise, and are the probabilities of failing to capture the right signal. The incentive strategy is a viable strategy when dominates , otherwise the prerequisite to implement an incentive pricing strategy fails. That is, the range of possible values for is and for , .
In addition, the sanction applied by the regulator to dis-incentivize non-compliance is assumed to contribute to increasing transaction costs. With such a hypothesis, the following problem includes a sanction item in the objective function and an incentive compatibility constraint (IC) besides the CR constraint discussed above.
s.t:
where
and
represent probabilities of detection,
indicates the farm profits in function of irrigated share,
indicates the tariff and
represents sanction. The IC guarantees a utility for compliance which is higher than the utility of being non-compliant. The left hand side of the newly introduced IC is the reduced form of
. Likewise, the right hand side is the reduced form of
. The reason behind this constraint is that in order to make sure the farmer complies with the rules, the benefit obtained by the farmer when he observes the rules must be greater than the benefit obtained by the farmer when he does not observe the rules. The IC can be further rearranged highlighting that to incentivize compliance, the utility the farmer obtains by complying with rules,
, must be higher than the relevant disutility,
. It is worth noting here that the utility that the farmer obtains by complying with rules is influenced by the fact that the farmer has some probability of being detected as non-compliant.
Overall, differences in utilities are conditioned by the WA’s monitoring capacity (probability to correctly detect farmers’ behavior) and by the magnitude of the losses experienced when complying with rules.
When the IC constraint holds with strict equality it is possible to estimate the level of the sanction:
The level of the sanction is obtained by the difference between the profit obtained with no restriction on irrigated land use and the profit obtained with restriction on irrigated land use divided by the difference between the probability that compliance is detected when the farmer is compliant and the probability that the farmer is detected compliant when he is non-compliant .
We obtain the following solution by substituting in the objective function
determined from Equation (9) and
determined from Equation (4) when both constraints are satisfied with strict equality and taking the FOC with respect to
:
The equilibrium reached in Equation (10) (see
Appendix A) is contingent of probabilities of detection and transaction costs. The variation of its components influence the optimal level of the irrigated share, the level of tariff and the level of the sanction, contributing in conditioning the magnitude of the social benefit.
By increasing the accuracy of monitoring (probability to correctly detect farmers’ actions), farms’ irrigated share decreases, the tariff decreases and the sanction needed to discourage non-compliance decreases.
Given the transaction cost levels, the maximum impact on a farm’s irrigated share is obtained when and , that is, when monitoring is perfect. The farmer is complying with the rules of the contract and the WA’s capacity to determine that farmers are complying with the rules is maximized. Under such a hypothesis, the equilibrium solution is subject to the level of transaction costs. The higher the level of the lower the irrigated share.
On the contrary, when then . That is, the equilibrium solution regresses to the flat rate case as the incentive mechanism has no effect on irrigated land use.
Finally, for and there are infinite intermediate solutions between the above-discussed probability scenario limits.
With reference to transaction costs, with increasing transaction cost levels the tariff level is increased from Equation (4) and, as a result, increases the marginal profit level in Equation (10) and decreases the share of irrigated area. The farmer might be wishing to decrease the irrigated share to pay less. Under such conditions, the cheating option may become more attractive and the moral hazard problem is more likely to prevail. As a reaction, the WA increases the sanction to discourage non-compliance. In addition, the value of the sanction is also influenced by the accuracy of the instruments adopted by the WA to monitor uses and increases with the reduction of the accuracy level.
2.6. Evaluating Strategies under the Two Pricing Schemes
As discussed above, the WA might face additional transaction costs and suffer some inefficiency due to imperfect monitoring to implement an incentive pricing strategy in the absence of water metering.
Because of this, the WA might decide to keep the flat rate tariff if the social benefits generated by the implementation of such pricing regimes are higher than the social benefits brought about by the implementation of the incentive pricing schemes:
where
and
stand respectively for the social benefit under the flat rate pricing scenario and the social benefit under the incentive pricing scenario. For the flat rate scenario, transaction costs are assumed to equal zero.
As stated previously, the prerequisite to implementing an incentive tariff is that the probability of detecting farmers as compliant when they are actually complying, must dominate the probability of detecting farmers as compliant when they are actually not complying. Such a prerequisite of dominance is a necessary condition to implement an incentive tariff, but is not a sufficient condition to justify the transition from the flat rate regime to the incentive pricing regime. The transition is favored for high levels of supply costs recovered by pricing water, for high degrees of accuracy of the instruments adopted by the WA to monitor water usage and for low levels of transaction costs faced by the WA with the implementation of the incentive pricing scheme.
Another aspect motivating the transition from the flat rate regime to the incentive pricing regime is the presence of a heterogeneous population of farmers. Unlike the flat rate, incentive pricing enables the WA to allocate supply costs among users on the basis of actual uses. This effect might positively impact overall social benefits and make it possible to tie supply costs to the benefits generated by the provision of water to irrigation.