Open access peer-reviewed chapter - ONLINE FIRST

On the Design of Total Water Use-Based Incentive Schemes for Groundwater Management

By Wided Mattoussi, Mohamed Salah Matoussi and Foued Mattoussi

Reviewed: September 20th 2021Published: November 30th 2021

DOI: 10.5772/intechopen.100540

Downloaded: 17

Abstract

Groundwater over-pumping by manipulating water meters may constraint the efficient use of the resource, leading to the potential aquifers’ deterioration. Well designed institutional arrangements might be effective at reducing over-exploitation. The objective of this research was to shed light on the design of various incentive schemes to face groundwater over-pumping ranging from individual water use-based incentive schemes, where individual withdrawals are the users’ private information, to total water use-based incentive schemes, where the aggregate withdrawal is publicly observable. For the latter setting, two schemes were proposed. The first one is within the framework of moral hazard in teams, where the Water Authority administers monetary incentives that do not balance the budget, restoring thereby the full-information outcome. The second scheme promotes a cooperative management governed by a collective responsibility rule that induces peer monitoring by members. We show that groundwater overuse is more likely when monitoring costs are high, punishments are weak and cooperatives are large. We also show how the cooperative size and punishments are determined endogenously by constraints on monitoring. We extend the basic analysis to study collusion in monitoring between cooperative members and compare different monitoring structures. The results confirm that well-designed incentives and institutions can reduce groundwater over-exploitation, and that constraints on monitoring costs affect institutional design.

Keywords

  • groundwater over-pumping
  • moral hazard in teams
  • cooperatives
  • peer monitoring
  • cooperative size
  • collusion
  • monitoring structure

1. Introduction

Groundwater resources are important for at least two reasons: Firstly, they are well appropriate for drinking1 due to their (generally) high quality [2, 3]. Secondly, groundwater reservoirs constitute very important long-term storage [4, 5, 6], particularly useful during long periods of droughts characterizing arid and semi-arid regions.

The two major threats to groundwater are over-exploitation and pollution [7]. The rapid demographic growth, urbanization, industrialization, intensification of farming practices and climate change pressures have led to an increasing demand for groundwater as a reliable source of water supply2.

Intensive abstraction can deplete the groundwater in an aquifer. It might be possible to over-pump any aquifer temporary during periods of droughts, but durable over-exploitation would certainly lead to irreversible degradations [9], due for example to saline intrusion, sea water intrusion and quality deteriorations generated by declining water tables. The artificial recharge of aquifers is appealing, but it might not be implemented on a large scale, meaning that only the long-term natural replenishment guarantees the groundwater conservation. The only course left open is then a better management of the resource.

A large set of policy instruments have been developed for a better groundwater management, including economic instruments and institutional arrangements. Several economic instruments are used to limit over-exploitation such as water quotas, pumping taxes or marketable pumping permits. They however focus on individual withdrawals which are assumed to be publicly observable. This is rarely the case in the real world as legal and administrative settings are generally insufficient to perfectly monitor individual withdrawals, meaning that groundwater may present open-access resource features [10]. Groundwater is hence withdrawn in an imperfect informational context and the above instruments become ineffective.

Economists now agree on the fact that resource allocation in less developed economies is profoundly influenced by non firm institutions such as group lending, credit cooperatives, sharecropping, Water Users Associations and so forth. In developing countries various water institutions coexist. They range from centralized regulation, where management responsibility is entirely delegated to government agencies, to markets3 for tradable water rights where farmers can sell their water shares to higher value uses. In between lies the entire spectrum of water allocation methods characterized by levels of decentralization, including Water Users Associations (referred to as water cooperatives), where users are involved in the decision-making process, and are thereby entrusted with part of the management responsibility normally held by government agencies. Water institutions can be effective at improving the resource allocation whenever they are well designed. Institutions influence individual behavior through incentives they give rise to, but institutions themselves evolve endogenously in part because of their incentive properties [11].

This paper sheds light on the design of various incentive schemes to face groundwater over-exploitation by farmers who can over-pump water typically by manipulating their individual water meters in an asymmetric information context4. In the sequel of this work, we refer to groundwater over-pumping as groundwater theft. The study in particular shows how the effective design of water institutions in response to a perceived problem of theft can help to reduce theft of water, and thereby improve water use efficiency. The response of the Water Authority (hereafter, WA), to tackle theft will differ according to whether it uses an incentive scheme based on the individual farmer’s withdrawal which is her private information, namely the centralized management which dominated water management in developing countries for several decades [13] or it resorts to total water use-based incentive schemes, where the total amount of water withdrawn by farmers is publicly observable and payments from farmers can be conditioned on it.

In the centralized scheme, farmers who steal, do so directly from the WA, and thereby do not impose negative externalities on each other. The WA tries to reduce theft by directly monitoring the farmers’ behavior, punishing observed instances of theft. We show in the model that some monitoring is always required in equilibrium. The WA tolerates some theft in order to save in monitoring costs.

In the decentralized management, where unobserved farmers withdrawals can be regulated through instruments based on collective performance (observed aggregate withdrawals) two schemes will be proposed. The first one corresponds to the framework of moral hazard in teams’ problem where the WA administers incentive schemes that do not balance the budget, restoring thereby the full-information outcome [14]. Such scheme works independently of the team size, but it may be infeasible when farmers have endowment constraints. This is why one may resort to an alternative team-based incentive scheme that might not violate individual endowment constraints and in which the WA makes use of the informational advantages farmers have over the WA because of their long standing and high trade links (especially in close-knit societies). This corresponds to cooperative institutions which are very likely to be well suited to deal with a variety of collective action problems associated with water management, though their success in doing so depends on some particular features of their design. In this research we show that such institutions may also be well suited to dealing with groundwater theft; we discuss the features of their design that enable them to do so. We show that the incentives for theft vary considerably in response to these features and discuss implications for policy. We in particular consider the properties of cooperatives characterized by a collective responsibility rule, which makes all members jointly liable for aggregate withdrawals, and show that this feature is likely to induce peer monitoring by cooperative members5 which is likely to be more effective at reducing theft than any means available to more centralized structures. We in particular show that groundwater theft is more likely when monitoring costs are high, punishment levels are weak and cooperatives are large. Moreover, straightforward comparison of the two team-based schemes shows that with sufficiently stringent punishments, the two schemes achieve the full-information water use level. Otherwise, theft occurs in cooperatives and a positive monitoring effort is required in equilibrium, meaning that the first team-based incentive scheme outperforms the cooperative management.

The results in the cooperative setting are obtained for given levels of punishment and cooperative size, but cooperatives are typically able to influence both of these variables. The model shows that these institutional characteristics are endogenously determined by constraints on monitoring: Higher monitoring costs increase punishment levels and reduce the cooperative size. Simulations also show that cooperatives can be neither too small because of the “monitoring costs savings” effect nor too large because of the “stealing” effect.

We extend the analysis thereafter to tackle the issue of collusion in monitoring efforts of cooperative members and show that collusion is welfare enhancing. We then compare among different monitoring structures, mutual and localized monitoring. Although in practice the mutual monitoring structure - whereby each farmer in the cooperative is being simultaneously monitored by all of her peers - is commonly observed [15], other monitoring structures deserve consideration. An interesting departure from the mutual structure is the “localized monitoring” in which every farmer monitors (and is monitored) by only one of her peers avoiding therefore the duplication of monitoring. We show that in equilibrium the localized monitoring effort is higher than twice the mutual monitoring level. This result is driven by the distributional character of peer monitoring, where monitoring allows cooperative members to shift the cooperative fine on others in addition to reducing theft.

The paper is organized as follows. In Section 2, we review the relevant literature. Section 3 sketches our model. In Section 4, we present an individual water use-based incentive scheme. In Section 5, we propose two total water use-based incentive schemes. We state a number of propositions describing the dependence of groundwater theft on a number of determinants, some of which are themselves determined by more fundamental factors including costs of monitoring. Section 6 provides two extensions of the basic model: the analysis first allows for collusion in monitoring efforts of cooperative members and then compares different monitoring structures. Section 7 proposes some policy recommendations. In Section 8, we present some potential extensions for further research. Section 9 concludes. Mathematical details are relegated to an Appendix.

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2. Literature review

Our research relates to various types of literature emphasizing the team nature of a problem. It relates to the peer monitoring on group lending programs and to decentralized groundwater management literatures, where peer monitoring is recognized as an effective instrument in mitigating the moral hazard behavior of individuals linked by a collective responsibility rule. It also relates to the non-point source pollution and to the non-point groundwater withdrawals, where unobserved individual emissions (withdrawals) can be regulated through instruments based on collective performance, which is the level of observed aggregate (ambient) pollution (withdrawals).

In the peer-monitoring literature, peer monitoring is an important means to mitigate free riding in groups of borrowers related by a joint-liability clause that creates an incentive mechanism in which each group member has an interest in screening and monitoring the other members. In the case of non-repayment by the group, all members will be denied future access to loans from the program, and defaulters who are caught may face fixed social sanctions. The seminal publications in this area are Stiglitz [19] and Varian [21], who show that the (costless) peer monitoring within groups can prevent members’ shirking in their productive efforts (Varian), and reduce poor project selection (Stiglitz), improving thereby repayment rates and reducing the costs of lending. More recently several papers including Ghatak and Guinnane [20], Armendariz [15], Che [17], and Conning [18], elaborate on the Stiglitz-Varian models, relax the assumption of the costless peer monitoring and deal with various extensions including the optimal group size, monitoring structures and the dynamic aspect of contractual relationship between group members.

In the context of decentralized groundwater management, Montginoul et al. [22] mentioned the use of a mechanism that consists of providing incentives for all groundwater users getting involved in the monitoring of groundwater abstraction to monitor each other, in order to increase the probability of control. The cost of decentralized monitoring (peer monitoring) is expected to be lower, since agents have more information on the actions of other agents (areas and crops irrigated, irrigation practices and frequencies, etc.) than any centralized structure. The incentive to participate in such a decentralized monitoring system can be provided by redistributing a share of the fine to the person who discovers the defaulter. This system has been used for centuries for regulating access to forests and common pastures in the Italian Alps [23]. This mechanism is likely to be rejected in many cultural contexts as it may be strongly assimilated to denouncement.

The literature on non-point source pollution follows the pioneering work of Segerson [24], whose analysis built on the earlier theoretical analysis of Holmström6 [14], who addressed the problem of free riding in teams in a more general environment.

Segerson [24] proposed a target based mechanism (TBM) where a regulator should monitor ambient pollution concentrations of a well-defined group imposes to each group member a tax (or a subsidy) proportional to the difference between observed group emission level and the group target. She shows that for a sufficiently high level of the ambient tax, the Nash equilibrium yields an aggregate pollution level equal to the group target.

Segerson’s work has inspired several intriguing extensions (e.g., Xepapadeas [25]; Miceli and Segerson [26]; Karp [27]; Millock and Salanié [28]). Xepapadeas [25] proposed a scheme of subsidies and two fining regimes: collective and random fining. Under collective fining, all firms are charged a fine whenever the observed ambient pollution level lies above some predetermined standard. Under the random fining scheme, only one firm is randomly chosen to be punished, irrespective of being responsible for the whole group’s deviation from the standard level.

Miceli and Segerson [26] proposed the introduction of collective responsibility rules among group members that create incentives similar to the ones created by ambient taxes. However, Litchenberg [29] noted that these liability rules are not likely to be first-best and are probably best-suited for controlling pollution related to the use of hazardous materials, or for non-frequent occurrences of environmental degradation like oil spills. Karp [27] suggested a model in which polluting firms behave strategically with respect to the regulator and found that their tax burden is lower under an ambient tax than taxes based on individual emissions, provided that the tax adjusts quickly, firms are patient, and the number of firms is small. Millock and Salanié [28] proposed a model of ambient taxes, where polluters might cooperate, and show that ambient taxes give strong incentives towards cooperation. However, when the degree of cooperation is unknown, the optimal regulation requires the regulator to offer a choice between a standard Pigouvian tax and a much lower ambient tax.

Although theoretically appealing, ambient-based schemes are rarely implemented in the field (an exception is presented in Ribaudo, Horan, and Smith [30]) for numerous technical, practical and political reasons [31].

The “ambient tax” instrument proposed by economic literature to solve diffuse pollution problems can be well suited to manage unobserved groundwater withdrawals since withdrawals of a well-defined group can be approximated by the groundwater table level monitored at some observation points [9]. The ambient tax (subsidy) would be charged (paid) to all users if the groundwater table falls below (above) the target level set by a regulator which was decided to not overpass.

In the decentralized management of groundwater several authors including Giordana [32], Lenouvel et al. [12] and Figureau et al. [33], show that contract-based instruments may play a significant role in reducing groundwater over-pumping.

Giordana [32] proposed three incentive instruments to fight groundwater over-pumping: a tax/subsidy over reported individual withdrawals with a random audit and penalties in case of non-compliance by groundwater users, an ambient tax, and a mixed instrument combing both instruments. He shows that the latter scheme outperforms the former two schemes.

Lenouvel et al. [12] proposed an optional target-based mechanism to reduce groundwater over-exploitation when farmers’ behavior is imperfectly monitored. The mechanism combines a classical ambient tax, paid by all farmers of the area when the water table level falls below a pre-defined target, with an optional individual contract signed with the regulator in which signatory farmers commit to provide true information to the regulator concerning the location of their wells, irrigated fields, and volume pumped, and to facilitate the control of this information. These farmers avoid the collective sanction if they comply with an individual quota. This mechanism is tested experimentally in the lab with a contextualized protocol and results show that it reduces withdrawals but that subjects are able to coordinate in a repeated setting to extract an informational rent.

Figureau et al. [33] have proposed three policy instruments, which can be used to enhance farmers’ compliance with individual groundwater allocations for irrigation in a decentralized management context. The first policy couples economic incentives by combing the use of a penalty with a reward. The penalty consists of a tax charged to farmers who exceed their allocation and is proportional to the over-pumping. The revenues from this penalty system are then shared between farmers who withdraw less than their entitlement, each one receiving a share proportional to their water saving.

The second policy is a “pooling agreement” through which farmers would agree to mutualize their quotas, in the sense that some farmers agree to relinquish part of their individual water allocation to help other farmers confronted by unusual situations. The volume given back to the Groundwater Users Association (GWUA) is then redistributed to farmers who have an exceptional need for extra water. The internal redistribution follows general principles and rules, which have been validated by the farm community. The contract is favorable to the agents as a team relative to the standard penalty system provided that the team does not exceed the targeted abstraction level, but unfavorable to the team if the target is exceeded. The third policy combines payments and fines. Farmers exceeding their quota pay an increasing block fine for the extra water pumped. Revenues from fines are then redistributed among farmers who use less than their quota; the amount received by farmers is proportional to their water saving efforts.

The three policies are tested through experiments with farmers and results reveal a preference for the third scheme that combines economic and social incentives, as it is expected to meet water and budget balance simultaneously.

Our cooperative model differs from most of the existing theoretical literature on peer monitoring in two respects. First, in their models the punishment is fixed: in the case of non-repayment by the group, all members will be denied future access to loans from the program, and defaulters who are caught may face fixed social sanctions. However, in our model the punishment depends continuously on the extra water pumped by farmers. Second, peer monitoring in this paper is quite specific in that farmers are competing in monitoring, which gives rise to a distributional effect in addition to an incentive effect. Indeed, peer monitoring may allow each cooperative member to shift the cooperative fine on others in addition to mitigating the moral hazard behavior of group members.

As for the ambient tax literature, it differs from the cooperative model in several respects. First, in our model the joint liability clause creates incentives for peer monitoring by group members, while ambient taxes do not. Second, in the ambient tax mechanism each individual is taxed according to the socially marginal damage when ambient emissions deviate from some predetermined level of emission. In our study, however, the distribution of the punishment burden is endogenously determined by peer monitoring. Third, in our study whether efficiency is obtained or not depends on the stringency of the punishment burden. However, most mechanisms suggested in the ambient tax literature are theoretically suitable for implementing the efficient allocation of abatement efforts in a Nash equilibrium.

In the decentralized groundwater management using mechanisms creating incentives for peer monitoring, such incentives are created through joint liability clauses based on rewards rather than punishment sharing rules as in our model.

Our cooperative model differs from decentralized groundwater management using contract-based instruments in three respects. Firstly, their mechanisms do not create incentives for peer monitoring as in our model. Secondly, in our model there are no rewards for farmers using less than the allocated quota. Third, in their models, they mainly use economic instruments as rewards and punishments to mitigate aquifer over-exploitation, but they do not come close to institutional design of the group of groundwater users such as the optimal size of the group and the optimal punishment or reward.

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3. The problem

Consider two identical farmers who pump water from a shared renewableaquifer to produce a homogeneous farm good. Even in this restricted setting certain features emerge that we believe to be interesting for policy implications and might be relevant for empirical investigations. Suppose that the yield yresponse to water qcan be described by the relation y=gq,where g.is increasing and concave. Each farmer bears a cost c,measured in units of output for every unit of water used. In addition, the farmer pays a linear price pper unit of water, which is set by the Ministry of Agriculture. The quantity of water maximizing the farmer’s profit equates the marginal value product of water to the marginal cost of generating such a quantity

qfi:g'q=c+p.E1

The superscript firefers to the full-information setting. In the complete information setting and when we abstract from any shadow cost of public funds that might imply Ramsey-pricing considerations, the WA can implement the first-best efficient outcome by setting pequal to δ. δrepresents the full public cost of mobilizing water to irrigated areas, which includes investment costs, operation and maintenance costs, extraction externalities associated with pumping from a shared aquifer, and any shadow cost associated with the scarcity of water.

The farmer is allocated a quotaequal to her full-information water use7, qfi. When the farmer’s water use is her private information (unlike the aggregate amount of water used by all farmers which is publicly observable), the farmer who has an individual water meter can well exceed her quota by manipulating her meter. The amount of water used in excess (referred to as goundwater theft) can be written as α=qqfi.

The response of the WA will differ according to whether there is centralized management or management based on total water use which is publicly and costlessly known. We consider these two cases in turn.

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4. Centralized scheme

Here we present the centralized management in an incomplete informational context. In what follows, a few assumptions necessary to the analysis are listed.

  • Assumption 1: The WA invests in monitoring devices aiming at making individual withdrawals observable. Monitoring incurs a social cost denoted by Ψm, which is increasing, convex and satisfies Ψ0=0. The cost of monitoring should be understood as including not only measurement devices, but other costs such as the wages of monitors.

  • Assumption 2: If the farmer is not monitored, then she pays the mandated water fee associated with her allotment, pqfi.Otherwise, she is discovered exceeding her quota (stealing) with a probability πmwhich increases with the intensity of monitoring. To simplify the exposition the probability π.is assumed to be commonly known and takes the form

πm=minκm1.E2

where κ>0(we assume henceforth that it is sufficiently small to generate an interior solution, which is realistic).

  • Assumption 3: When the farmer is detected stealing, her individual withdrawal is established without error and she pays pqfiplus a penalty, Fcsproportional to the level of theft. The punishment is a monetary transfer from the farmer to the WA and takes the form

Fcs=fmaxα0.E3

where the punishment rate fis positive, greater8 than pand given outside the model. There are no rewards for using less than the allocated quota. The solutions to the centralized and cooperative managements will be indexed with the superscript csand superscript c, respectively.

  • Assumption 4: Let Qbe the aquifer storage capacity or the stock of the resource in situ. For feasibility requirements we assume throughout that the price of water is lower than the marginal yield of using half of the storage capacity from which is deducted the private cost of one unit of water

pg'Q2c.E4

Rewriting this condition yields 2g'1c+pQ(where g'1c+p=qfi), which states that the total quota allocated to farmers must be lower than the storage capacity. This is to guarantee a rate of utilization that does not exceed the rate of replenishment to avoid the depletion of aquifers.

The order of events is that the WA chooses monitoring9, mthen each farmer chooses the quantity of water to use qcs. In what follows we focus on the subgame perfect equilibrium and solve the model by backward induction. In stage 2 of the game, the farmer chooses qcsso as to maximize her expected payoff:

maxqUcsqm=gqcqpqfiκmfqqfi.

Whose first-order condition is

g'qcs=c+κmf.E5

The performance of the centralized management relative to the full-information setting depends on the intensity of monitoring as summarized by corollary 1:

COROLLARY 1:

  1. Ifm<pκf,thenqcs>qfi;

  2. Ifmpκf,thenqcs=qfi.

Wherepκfis the minimum required level of monitoring that deters theft completely.

The full-information outcome obtains if the farmer is intensively monitored; Otherwise, theft occurs, as it becomes a privately profitable activity, i.e., the expected net benefit from stealing is pκmfqcsqfi>0.

Now let us turn to the initial contracting stage, where the WA anticipates the farmer’s behavior and picks monitoring, mthat maximizes the social welfare function. Specifically this function is the sum of the farmers’ surpluses 2gqcscqcspqfiκmfqcsqfiand the water supplier surplus equal to the revenue from the expected payments for water use, 2pqfi+κmfqcsqfifrom which is deducted the cost of mobilizing the resource to the irrigated area, 2δqcsand the cost incurred by monitoring, 2Ψm

Wcsmf=2gqcsc+δqcsΨm.E6

The WA must also consider two major constraints. The first one is the water availability constraint

2qcsQ.E7

which reflects the scarcity of the resource: farmers can at most use what is available. And the second one is the replenishment constraint

2qfiQ.E8

which states that the rate of groundwater utilization should be lower than the replenishment rate. In what follows proposition 1 characterizes the solution to the WA’s problem:

PROPOSITION 1:Suppose that assumptions(1), (2), (3)and(4)hold and constraints(C1)and(C2)bind10, then the optimal monitoringmcssolves

mcs:hκfg''qcs=Ψ'mcs.E9

whereμis the Lagrangian multiplier on constraint(C1)andh=κmcsfδμ.

Proof: See the appendix.

The proposition reveals that some monitoring is always required in equilibrium. However, because monitoring is costly, the optimal response of the WA would be to tolerate some theft in order to save in monitoring costs. Moreover, the equilibrium monitoring effort responds directly to the degree of water scarcity, captured by parameter μ(the scarcity rent or shadow value of water).

mcsμ=κfg''qcs1κf2g''qcshg'''qcsg''qcsΨ''mcs>0.E10

The more severe the shortage of water is, the higher is the required monitoring effective at reducing theft.

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5. Team-based incentive schemes

5.1 First scheme

We assume that the aggregate water use of the two farmers, Q=q1+q2,is publicly and costlessly known, and can be contracted for directly. In particular the WA designs a team-based incentive scheme where it asks the farmer to pay the fixed water fee associated with her allocated quota, pqifiand a share of the full extra amount if actual water use exceeds the total quota allocated to the group, sip.Q2qififor i=1,2, where si.is differentiable. We use a very restricted strategy set for the agents, and we shall look only at symmetric equilibria which implies that

qinbb=qnbbfori=1,2.

and that every team member is allocated the same quota qifi=qfifor i=1,2. The solution to this problem is indexed with the superscript nbb, referring to non balanced budget.

The order of events is that taking the price of water11pfor given, each farmer chooses the quantity of water to use qnbbthat maximizes her expected payoff

maxqUq=gqcqpqfisip.Q2qfi.

Whose first-order condition is

g'q=c+si'p.Q2qfi.p,E11

A mere comparison of (1) and (11) gives

si'p.Q2qfi.p=p,

and thereby

si'p.Q2qfi=1,

Implying that each farmer has to pay the total liability, p.Q2qfito the WA.

sip.Q2qfi=p.Q2qfi.E12

The WA can restore the full-information outcome by administering incentive schemes that do not balance the budget12 since both farmers will be paying the full extra amount p.Q2qfi.

This incentive scheme works independently of the team size, but its implementation may be constrained by the farmers’ limited liabilities. That is why the WA may promote cooperative behavior.

5.2 Cooperative management

Similarly to the previous scheme we assume that the total water use by cooperative members, Q=q1+q2, is publicly observable, and aggregate payments from the cooperative to the WA can be conditioned on it. In particular, this feature allows for a collective responsibility rule: when theft occurs, the cooperative as a whole receives a punishment proportional to the total amount of water stolen:

Fc=fi=1,2qiqifi.E13

Now suppose that, relative to the WA, farmers have informational advantages in monitoring each other, as a result of social ties and/or spatial proximity and neighborhood and/or long term trade relations.

We assume that peer monitoring brings about only evidence of the occurrence theft but not of its amount13. The WA may then contemplate the possibility of inducing peer monitoring between cooperative members by setting a collective responsibility rule that makes all members jointly liable: If theft occurs, the fine inflicted on the cooperative as a whole is shared equally between farmers who are caught stealing; otherwise it is shared by all members.

Performing peer monitoring is costly, we denote by ψmthe associated cost, which is assumed to be increasing, convex and satisfies ψ'0=0. Each farmer commits to a level of monitoring14 (observable by the other members) before the choice of the level of water to use is made. The probability that a farmer iis caught stealing water is then given by:

πimj=minκmj1,E14

where κ>0. Farmers do not collude in either their production or monitoring decisions15.

The order of events is that taking for given the price of water p, cooperative members choose mi, then having observed each others’ choice of mithey choose the level of water to use qi. Suppose that both farmers steal water, i.e., αk>0for k=i,j.The expected share16 of farmer ifrom the cooperative fine is decreased by her peer’s monitoring, and is in turn increased by her own monitoring.

siexp=121κmi+κmj.E15

The subgame perfect equilibrium is the profile m1cm2cq1cq2cof monitoring efforts mic0and water use levels qicmapping from the set of monitoring decisions into the set of water use decisions, qic:0+20,Q). In what follows we shall focus on symmetricequilibria which imply that

qic=qcandmic=mcfori=1,2.

and that every cooperative member is allocated the same quota qifi=qfifor i=1,2.

Similarly to the centralized structure, in cooperatives we shall restrict attention to the punishments that are higher than the price of water, i.e., f>p,because otherwise farmers will always have an interest in stealing everything. The outcome will depend on the stringency of the punishment rate. If it is sufficiently high, i.e., when flies above 2p,farmers will neither steal, nor monitor in equilibrium (as high punishments ensure that the collective penalty will be severe enough to deter theft). Otherwise17, i.e., when fp2p,farmers will steal and monitor in equilibrium. Summarizing:

PROPOSITION 2: Iff2p, there exists a unique symmetric subgame perfect equilibriumqcmcsuch that

qc=qfiandmc=0.E16

Ifp<f<2p.Then, the unique symmetric subgame perfectqcmcsatisfies

qc:g'q=c+12f,E17

and

mc=ϕk2.E18

Where, k2=fqcqfi14fg"qcandϕ=ψ'1.

Proof: See the appendix.

Peer monitoring reduces groundwater theft18 (incentive effect) and it may allow every cooperative member to shift19 the cooperative fine on the others (distributional effect).

The immediate corollary of proposition 2 is that the comparative evaluation of the performance of the two team-based incentive schemes depends on the stringency of punishment rates. For sufficiently stringent punishments (i.e., f2p), the incentive problem is eliminated altogether and the full-information outcome obtains. Otherwise, theft occurs in cooperatives and a positive monitoring effort is required in equilibrium; consequently, the first team-based incentive scheme outperforms the cooperative management. The following corollary states this point.

COROLLARY 2:

  1. Iff2p,then

    qc=qnbbandWc=Wnbb,E19

  • Ifp<f<2p,then

  • qc>qnbbandWc<Wnbb.E20

    Where,qnbb=qfiandWc=2gqcc+δqcψmcandWnbb=Wfi=2gqfic+δqfiare the cooperative and full-information social welfare levels, respectively.

    5.3 Comparative statics

    To obtain explicit solutions where possible we assume that monitoring costs take the quadratic form ψm=12bm2where b>0. We first investigate the impact of water price and punishment on the equilibrium level of theft. As one intuitively expects, theft increases with water price and decreases with punishments:

    qcqfif=qcf=12g''qc<0.E21
    qcqfit=1g''qc>0.E22

    We now show how the intensity of monitoring will be related to monitoring costs, punishments and the price of water. Monitoring is decreasing with the costs of monitoring and increasing with water price and punishment20.

    mcb=k2b2<0.E23
    mcp=ϕ'k2fg''qc>0.E24
    mcf=1bqcqfi+f28g'''qcg''qc3>0.E25

    A higher water price increases the incentives for theft and the punishment burden that would be incurred by a member who was the only one to be caught, inducing farmers to perform more monitoring to shift the cooperative fine on the others.

    In the range of non stringent punishments, there is a higher scope for theft, and an increase in the punishment rate renders the punishment burden for a given level of theft high, and also the total punishment that would be incurred by a member who was the only one to be caught. This would increase the farmers’ incentives to compete more in monitoring to shift the cooperative fine on the others. The results above suggest that the distributional effect of peer monitoring is very likely to always dominate the incentive effect.

    It is interesting to compare the equilibrium monitoring effort to the socially optimal one. We compare equilibrium outcomes to those that would occur in a second-best problem faced by the WA as a social planner who can decide about monitoring levels of farmers but not their water use choices once monitoring decisions have been made. In addition, we assume that the WA cannot affect the farmers’ incentives to steal water for given monitoring levels. In particular, the WA cannot ensure that farmers do not steal the resource. The WA chooses a monitoring level, mwhich maximizes the social welfare function

    maxm02gqcc+δqcψm.E26

    It is socially optimal not to monitor in cooperatives governed by these rules (m=0) whenever monitoring is costly. Farmers over-monitor in equilibrium, mc>m,because of their rent seekingbehavior which results from the dominance of the distributional effect of peer monitoring over the incentive effect.

    5.4 Endogenous punishment

    Here the basic analysis is extended to allow for endogenous punishment, where the punishment rate fis collectively chosen by cooperative members at an initial contracting stage. Inflicting punishment fis costly, we denote by φfthe associated cost that can be either pecuniary or manifest in nature when there is deterioration of social relations from inflicting punishment on members of a close-knit society. φfis increasing and strongly convex (i.e., φ'''f>0in addition to φ''f>0), and satisfies φ0=0. The strong convexity of φis driven by the increased complexity and difficulty of enforcing more and more stringent punishments on relatives, neighbors and friends.

    Cooperative members choose the punishment rate fcthat maximizes an objective function defined as the sum of the cooperative members surpluses, 2gqccqcpqfifαc12bmc2φfand the WA’s surplus, which is equal to its revenue from water proceeds, 2pqfifrom which is deducted the cost of proving water to the cooperative area, 2δqc.

    maxfWcf=2gqcc+δqcfαc12bmc2φf.E27

    This has a first-order condition21:

    fc:1g''qcγ+12f2qcqfi2k2bqcqfi+f28g'''qcg''qc3=φ'fE28

    which is also sufficient22 to identify a global maximum.

    From this condition one can show that the punishment level is decreasing with monitoring costs. Totally differentiating the first order condition with respect to fand band rearranging yields:

    fcb=2k2Gb2qcqfi+f28g'''qcg''qc3<0.E29

    where23G=d2Wcdf2<0. This result confirms that the two instruments, monitoring and punishment complementeach other, as an increase in the cost of one reduces the level of the other.

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    6. Cooperative size

    The analysis thus far has remained restricted to the two-farmer cooperative. However, in practice, most cooperatives irrigating from aquifers involve up to as many as 40 farmers, and most cooperatives using surface water involve more than 100 farmers [20]. We investigate here the optimal cooperative size, where the basic set-up is extended from the two-farmer cooperative to the n-farmer one24.

    We characterize the symmetric subgame perfect equilibrium qncmncfor the relevant range of non stringent punishments, p<f<np(assuming that the second-order condition for a maximum holds25):

    qnc:g'q=c+1nf,E30
    mnc:knΦnκm=n1ψ'n1m.E31

    Where

    kn=fqncqfi1n2fg"qnc>0andΦnκm=1κmn2n1n1k=11κmk1=1κm1κmn2n111κmn1.

    From the necessary conditions one can see that the farmer withdraws more water as the cooperative becomes larger:

    qncn=1n2fg"qnc>0.E32

    meaning that larger groups increase the incentives for theft. However, it is not clear whether the equilibrium monitoring level tends to increase or decrease with the cooperative size. The intuition suggests that the group size affects the incentive problem in two ways. A larger group discourages monitoring, as the evidence about the farmer’s theft could be established when she is detected stealing by at least one of her peers. Monitoring all together might hence become useless, as the same outcome could be achieved with a smaller number of farmers, avoiding thereby the useless duplication of monitoring. This free-riding problem reduces the farmers’ incentives for monitoring. On the other hand, a larger group may increase the total amount of water stolen in the cooperative, increasing therefore the maximum punishment that would be incurred by a member who was the only one to be caught. This would rather increase the farmer’s incentives to monitor more intensively to catch the other members stealing, which may reduce her expected share from the total fine. This rent-increasing effectwill thus counteract the above free-riding effectby encouraging more intense monitoring as the cooperative becomes larger.

    It is very difficult to derive an analytical expression of monitoring level in equilibrium, as monitoring is implicitly given by (31). In order to get some insights, we will proceed in the remainder of this section to the following simplification: we restrict attention to sufficiently small values of κ,which implies that all terms in κnfor n2,become of the second-order and can thereby be dropped from our calculations. Consequently (31) reduces to26

    mnc:n1ψ'n1mknn11n1n2κm.E33

    To obtain explicit solutions where possible we assume that monitoring costs take the quadratic form ψm=12bm2where b>0. Rewriting (33) yields the approximated equilibrium monitoring effort

    mncknn1b+knκn2,E34

    Monitoring decreases with monitoring costs which is straightforward

    mncb=knn1b+knκn22<0,E35

    As for the impact of the cooperative size on monitoring it is given by

    mncnθknn,E36

    Where θ=bn1b+knκn22>0, which means that the signs of the two partial derivativesmncnand knnare the same. knnis equal to

    knn=f1+g'''qncn2g''qnc2qncn+2fng''qnc,E37

    Plugging the expression of qncngiven by (32) into (37) yields

    knn=f2ng''qnc21ng'''qcn3g''qnc2.E38

    which sign is ambiguous because the term in the bracket parenthesis has an ambiguous sign (the terms 21nand g3qcn3g"qnc2are both strictly positive and it is not clear which term overcomes the other). Because of the analytical complexity of the problem, we will address this issue via a numerical example - the example is:

    • The production function is gq=q;

    • The per-unit private cost and price of water are c=p=0.2;

    • The transaction costs related to monitoring take two different values b=3and b=10.

    Simulations suggest that the shape and the value of monitoring, mcnas a function of the cooperative size considerably changes when monitoring costs vary. When b=3,the monitoring function gradually decreases as the cooperative becomes larger, i. e., for n3. This means that the free ridingeffect tends to always dominate for small monitoring costs. Whereas, for b=10,monitoring levels become smaller and, the function increases for small values of nand starting from n=4it gradually declines. This implies that for large monitoring costs the rent seekingeffect might come into play. Simulation results suggest the existence of a monitoring cost, b¯such that:

    • For any b<b¯,the equilibrium monitoring mcnis decreasing with the cooperative size and;

    • For any b>b¯,the equilibrium monitoring mcnincreases up to some level n˜and then gradually declines.

    We now explore the optimal cooperative size. Farmers may seek a group size nmaxthat maximizes the average27 cooperative welfare function WAcn

    nmaxargmaxn2WAcn=gqncc+δqncψmE39

    The first-order condition for an interior solution (assuming that the second-order condition holds) is given by:

    g'qncc+δqcnnψ'mncmncn=0,E40

    The (first-order) change in social welfare attributable to a marginal entrant is composed of two terms. The first term implies that the new entrant causes every member to better free ride on her peers and thus to contract her monitoring effort. This would increase the opportunities for theft for everyone. This stealingeffect causes a reduction in social welfare of

    g'qncc+δqncn<0,E41

    On the positive side, free riding results in monitoring cost savings. This cost savingseffect brings about an increase in social welfare of

    ψ'mncmncn>0.E42

    The optimal cooperative size, nmaxequates the social marginal benefit stemming from monitoring cost savings to the social marginal losses caused by a higher occurrence of theft. The net benefits of peer monitoring are maximized when the cooperative size is neither too small (due to the “monitoring cost savings” effect) nor too large (due to the “stealing” effect).

    The effect of the group size on the cooperative welfare is found to be analytically complicated, that is why we use a numerical example to shed light on the intensity of stealingand cost savingseffects when one varies monitoring costs. Simulations are performed for the same production function and the same parameter-values considered above, to which we add the value of the external cost of water δ=3. Simulation results suggest that for b=3,the welfare function is maximized for nmax=4, while in the other case it is gradually decreasing. This means that the stealingeffect dominates almost everywhere and the best policy of the WA is to implement small and medium cooperatives.

    Finally, simulations suggest that monitoring costs reduce the cooperative size. The intuition is that higher monitoring costs make it more difficult to monitor, which gives more opportunities for theft, requiring smaller cooperatives to compensate.

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    7. Extensions

    7.1 Collusion

    The cooperative model described up until now corresponds to a non-cooperative game. Each cooperative member is out to maximize her expected payoff, and makes her decisions about monitoring and water use independently of the other members. What happens if we relax this assumption and consider possibilities of coordinated actions about monitoring? A natural model is to consider what happens if the two cooperative members choose their monitoring efforts in order to maximize joint payoffs, Uiqimi+Ujqjmj. The collusive outcome is given by corollary 3:

    COROLLARY 3: The collusive monitoring efforts are

    mi=mj=0.E43

    Proof:See the appendix.

    The collusive monitoring effort is efficient. In the absence of collusion, cooperative members compete on monitoring because of their rent-seekingbehavior, though monitoring is useless, as they will always split equally the cooperative fine between themselves28. Collusion is thus welfare enhancing, as it yields the same outcome and saves in monitoring costs.

    7.2 Monitoring structures

    Although in practice the mutual monitoring structure - whereby each farmer in the group is being simultaneously monitored by all of her peers - is commonly observed [15], other monitoring structures deserve consideration. An interesting departure from the mutual structure is the “localized monitoring” in which every farmer monitors (and is monitored) by only one of her closest neighbors, say each farmer monitors her left neighbor and is monitored by her right neighbor. There is a natural argument in favor of monitoring structures of the latter kind: it can potentially avoid duplication29 of monitoring. As a first and very tentative attempt to explore the issue of the optimal design of peer monitoring structures, we shall compare the mutual monitoring (MM) to the localized monitoring (LM) one with regard to the equilibrium water use and monitoring levels and thereby to the cooperative welfare level. The comparison will be held for a three-farmer cooperative.

    7.2.1 Mutual monitoring

    Consider a cooperative formed by three farmers i,jand kapplying mutual peer monitoring.We assume that a farmer monitors each of her peers with the same monitoring effort, which means that the cost of monitoring all others members is equal to ψ2mlfor l=i,j,k..

    The subgame perfect equilibrium is the profile mlcMMqlcMMl=i,j,kof monitoring efforts mlcMM0and water use levels, qlcMMmapping from the set of monitoring decisions into the set of water use decisions, qlcMM:0+30,Q)(where the superscript cMMrefers to cooperatives characterized by mutual monitoring). In what follows we shall focus on symmetricequilibria which imply that

    qlcMM=q3cMMandmlcMM=m3cMMforl=i,j,k.

    (assuming that the second-order condition for a maximum holds). The solution to this problem is summarized in proposition 3:

    PROPOSITION 3: Ifp<f<3p,there exists a unique symmetric subgame perfect equilibriumm3cMMq3cMMsatisfying

    q3cMM:g'q=c+13f,E44

    and

    m3cMM:f29g''q3cMMκ4m4+12κ4m32κ3m352κ3m2+3κ3mfq3cMMqfiκ4m33κ3m2+4κ2m2κ=2ψ'2m.E45

    Proof: See the appendix.

    7.2.2 Localized monitoring

    By the same token, consider a three-farmer cooperative in which farmers i,jand kapply localized monitoringwhereby each farmer monitors her left neighbor and is monitored by her right neighbor. We assume that farmer imonitors farmer jwith the monitoring level mij=mi, and farmer jmonitors farmer kwith the monitoring effort mjk=mj,and farmer kmonitors farmer iwith the monitoring effort mki=mk. We characterize the symmetric subgame perfect equilibrium (the superscript cLMrefers to cooperatives characterized by localized monitoring)

    qlcLM=q3cLMandmicLM=m3cLMforl=i,j,k.

    (assuming that the second-order condition for a maximum holds). The solution to this problem is characterized by proposition 4:

    PROPOSITION 4: Ifp<f<3p,there exists a unique symmetric subgame perfect equilibriumm3cLMq3cLMwhich satisfies

    q3cLM:g'q=c+13f,E46

    and

    m3cLM:12κ2m+κf29g''q3cCI+fq3cCIqfi=ψ'm.E47

    Proof:See the appendix.

    Comparing the monitoring levels in both settings is not straightforward, however under certain relevant circumstances one can say something. For sufficiently small κ, the terms κnfor n2, become of the second-order and equations (45) and (47) above reduce respectively to

    κfq3cMMqfiψ'2m3cMM.E48

    and

    κf29g''q3cLM+κfq3cLMqfiψ'm3cLM.E49

    A mere comparison of (48) and (49) suggests that the equilibrium localized monitoring is higher than twice the equilibrium mutual monitoring

    m3cLM>2m3cMM.E50

    This result may sound unlikely, one indeed expects farmers to monitor less in cooperatives characterized by localized monitoring, as this structure avoids the duplication of monitoring. But, the rationale for this result comes from the distributional character of monitoring, as cooperative members compete on monitoring to shift the cooperative fine on the others. In the mutual structure, a farmer may well reduce her expected share from the total fine as the cooperative fine might be shared with more than one member. However, in the localized structure, each farmer monitors only one of her neighbors increasing the risk to bear the whole punishment burden on her own. This acts as an incentive to monitor more to increase the probability of shifting the fine on that neighbor.

    An immediate implication of (44), (46) and (50) is that cooperatives using mutual monitoring welfare dominate those using localized monitoring

    WcMU3>WcCI3.E51

    Where WcMU3and WcCI3are welfare levels in the three-farmer cooperatives characterized by mutual and localized monitoring structures, respectively.

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    8. Policy implications

    Surface water is not only scarce but also highly uncertain and often of bad quality in particular in arid and semi-arid regions. Groundwater is seen as a unique guarantee (whenever it is well managed) for the long term viability of an agriculture sector crucial for giving those fragile countries a minimum food security.

    The principal aim of this research was to design the appropriate institutions and incentives to reduce groundwater over-exploitation. We have proposed various incentive schemes, some based on individual water use and others on aggregate water use. The model has in particular shown that total water use-based incentive schemes may well overcome the individual water use-based scheme (centralized management). Moreover, the comparative evaluation of the two team-based incentive schemes has shown that for a relevant range of non stringent punishments, the first scheme dominates the cooperative management since it restores the full-information outcome. Nevertheless such a scheme may suffer from a strong implementation problem: endowment constraints may render it infeasible. How to overcome this drawback? The solution could be imported from the theory of “positive” assortative matching, where farmers could be sorted into various homogenous groups giving the scope to policy makers to choose the team-based scheme to implement for each group. To illustrate this idea, suppose that the WA could sort farmers into two groups, a group of wealthy farmers and a group of all other farmers. How to screen wealthy farmers from others? Wealthy farmers may well be those who invest more in sophisticated agricultural machinery and/or grow high value crops and/or cultivate large plots of land,…etc. For the group of wealthy farmers, the optimal policy of the WA would be to implement the first team-based scheme; And for the other group, participatory management could be implemented through the creation of a collective responsibility rule that induces peer monitoring by members as modeled above.

    Simulations in our paper suggested that the cooperative size is weakly decreasing with monitoring, meaning that the best policy of the WA would be to implement non large cooperatives which would reduce the scope for groundwater over-exploitation.

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    9. Potential extensions for further research

    This research could be extended along the following directions. Firstly, we know from simulation results that it is socially beneficial to form small and medium cooperatives. But, what about irrigated areas with a large number of farmers? In practice, centralized structures are in charge of running such areas. In addition, a direct management transfer of these areas to cooperative institutions is unlikely to enhance water use efficiency, as the “stealing effect” a major source of inefficiency always dominate in large cooperatives. The cooperative management could however be implemented in a slightly different way through the creation of two hierarchies of cooperatives, primaryand secondarycooperatives. Farmers can form several groups of medium and/or small sizes depending upon whether peer monitoring involves low or high costs. These groups would constitute secondary cooperativeswhere each is run by a number of its members, called “cooperative manager”. The secondary cooperatives’ managers themselves would form what we call a “primary cooperative” playing the role of intermediary between secondary cooperatives and the WA. Among each secondary cooperative’s managers, some farmers would be chosen to form the primary cooperative’s managers and would be in charge of running it. Every cooperative is governed by similar rules as modeled above: members of the same group are related by some collective responsibility rule which would create incentives for two types of peer monitoring, (a.) peer monitoring-withinapplied within each secondary cooperative and (b.) peer monitoring-betweenapplied between secondary cooperatives.

    Secondly, in our theoretical study we have formalized the cooperative management where members interact only once, however in practice members interact repeatedly. The knowledge that pursuit of their short-term interests can harm their long-term aims by affecting the reaction of others in the future interactions may be a powerful inducement to behavior that displays apparent solidarity with the interests of the group. When cooperative members are homogenous, in particular they are all landlords and moreover the agricultural activity is their main source of income, their interactions can be modeled as an infinitely repeated game30. What one can learn from group lending literature is that such cooperatives might not need to have access to an exogenous penalty device since peer sanctions can be accomplished in the dynamic framework and they are self enforcing 31 [17]. It suffices to make the group members jointly liable for the payment of their total amount of water use, e.g., denying them future access to the public source of water supply or rising the price of water for next periods when theft occurs in the current period. In such a management design, a group member can be penalized by other members’ shirking (=stealing), in that a member’s shirking increases the payment burden of her peers, and thereby negatively affecting their payoffs. In turn, when cooperative members are heterogenous in that some members are landlords and others are rental contract holders and/or tenants, an exogenous penalty technology might be required, e.g., one may think for example of the exclusion of defaulting users from the cooperative or from the community or from certain kinds of input supply facilities32. Whereas, being excluded from the cooperative could be perceived as extremely harmful for a land owner, this might not be the case for rental contract holders who interact only for a finite number of periods and the pursuit of their short-term interests induce them to adopt the strategy of “take the money and run away33”. However, this may well harm their long-term interests - the loss of reputation may be very costly for these farmers: their exclusion from the cooperative might make it difficult for them to integrate other cooperatives in the same area and/or even in other areas. In short they might be excluded from the community. What might be critical here is the enforcement of such social sanctions in practice, especially how and why a farmer should ever impose a sanction on a “friend” or “relative” who has defaulted. One possible explanation on the face of it is the impossibility to keep the information about strategic defaults secret. An other explanation may be the harmful consequences of foregoing the punishment of defaulters, e.g., the absence of alternative sources of water supply if the cooperative is denied future access to her principal source of water supply.

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    10. Conclusion

    This paper has investigated the design of the appropriate institutions and rules to enhance groundwater use efficiency by reducing over-pumping of aquifers. We have proposed various incentive schemes, some based on individual withdrawals which are the farmers’ private information and some based on the total water withdrawn by all farmers which is publicly observable. In the latter setting, two schemes are proposed. In the first scheme, the WA administers an incentive scheme that does not balance the budget, restoring thereby the full-information water use level. Such scheme works independently of the group size, but it may be infeasible when farmers have endowment constraints. This is why the WA resorts to a second total water use-based incentive scheme by promoting the cooperative behavior. We device cooperative institutions characterized by a joint liability clause that induces peer monitoring by members. We show that higher monitoring costs and larger cooperatives entail more theft and higher punishment levels reduce it. We also show how the cooperative membership and punishments are determined endogenously by constraints on monitoring. Higher monitoring costs increase punishment levels and reduce the size of the cooperative. The basic analysis is then extended to allow first for collusion in monitoring between cooperative members, and show that the collusive monitoring effort is efficient. Secondly, we explore a different monitoring structure “localized monitoring” and compared it with the mutual monitoring structure which is commonly observed in practice. Finally, we use the theoretical results to derive some useful policy recommendations that could help decision makers to implement the right policies to alleviate groundwater over-exploitation.

    Overall, these results provide strong confirmation of the ability of well designed incentives and institutions to reduce groundwater over-exploitation, and that constraints on monitoring costs affect institutional design.

    The details of mathematical demonstrations are available from the authors upon request.

    Classification

    JEL classification: Q13; Q15; Q25; R48

    Notes

    • Falkenmark [1] estimates that one-third of the world’s population rely on groundwater supply for drinking.
    • Groundwater constitutes about 89% of the freshwater on our planet (discounting that in the polar ice caps) [8] (p.1).
    • The most celebrated case of tradable water rights comes from Chile, where agrarian reforms and the Water Code of 1981 formalized water rights, and allowed for water sales separately from sales of land.
    • Despite the relevance of asymmetric information problems for water management, only a few studies examine the application of such concepts to irrigation water in general and to irrigation groundwater in particular [12].
    • There is now a substantial literature on peer monitoring [15, 16, 17, 18, 19, 20].
    • One main finding of Holmström is that in the absence of uncertainty, no budget balancing mechanism exists to solve the problem for avoiding individual free riding in teams.
    • It is hard to understand why the WA as a social planner should ever choose any quota but the full-information water use level.
    • Because otherwise the farmer will always have an interest in stealing everything. The net return from theft of water is equal to p−κmfα, which occurs with the probability κm<1. If f<p, one gets κmf<f<p, and therefore theft is strictly beneficial; this essentially implies that the net return is maximized when the farmer steals everything.
    • The WA is able to control the punishment rate, f in addition to controlling the monitoring decision. When punishment is endogenous and costly, some level of punishment is always required in equilibrium. However, because punishment is costly, the optimal response of the WA is to tolerate some theft of water in order to save in punishment costs.
    • Since qcs≥qfi, then (C2) is only binding if (C1) is also binding, and so we can ignore (C2).
    • Which is set by the Ministry of Agriculture.
    • If the WA has instead administered the incentive scheme where the total liability p.Q−2qfi was fully shared among the agents, this would result in an inefficient outcome [14]. The point is therefore not that group punishments is the only effective scheme, but rather budget-breaking is the essential instrument in neutralizing the free-riding problem.
    • All what a farmer may observe is whether the other cooperative members manipulate their meters and if they do, the evidence about these actions will be established with certainty. Indeed, only the farmer herself and the WA can have access to the farmer’s water meter.
    • One may think of observable sunk investments (such as tools and equipment) being made by members of the cooperative, and which would commit them to a higher monitoring effort. For instance, it is widely observed in developing countries like Tunisia that landlords build small houses in their farms where they can keep some farm equipment for daily use and where both landowners and agricultural laborers may spend some time.
    • For the moment we sidestep the issue of collusion in monitoring efforts, but we return to it later in Section 7.1.
    • The expected share of farmer i from the cooperative fine when everyone steals water is given bysiexp=12κmiκmj+κmj1−κmi+121−κmi1−κmj
    • It is worth noting that focusing on this range of punishment levels is less restrictive than it seems. Indeed, such a restriction holds only for the two-farmer cooperative; for the more general case of n-farmer cooperative, punishment rates will instead strictly lie between p and np.
    • The partial derivative of the level of theft undertaken by farmer i with respect to monitoring performed by her peer, mj is given by
    • This finding follows from equation (15). The expected share of farmer i from the total fine increases with the monitoring effort performed by her peer, mj:
    • From equation (B4) in the proof of proposition 2 in the appendix, we have
    • Differentiating the cooperative welfare function with respect to f yields
    • This follows from the strong convexity of the cost of inflicting punishment on cooperative members, ensuring therefore the concavity of the objective function Wcf.
    • The negative sign of the second-order derivative follows from the concavity of function Wc.
    • Unlike the two-farmer case where peer monitoring is necessarily mutual, the n-farmer case opens the scope for various kinds of monitoring structures. We will however, momentarily focus on the “mutual monitoring” structure, whereby each farmer monitors all her peers.
    • It is quite difficult to derive the second-order condition for this problem because the first-order conditions account for the following highly complicated term
    • This simplification involves no major loss of insights, as equation (33) captures the main qualitative aspects of the solution to the cooperative model.
    • The rational for the choice of the average social welfare function rather than the absolute one is that for the latter the group size effect might always dominate and the function is very likely to always increase with the cooperative size.
    • This finding comes from the symmetric equilibrium which implies that cooperative members perform the same level of monitoring, i.e., mic=mjc=mc. It follows that the expected share of farmer i from the total fine is equal to 12:
    • Duplication in the mutual structure obviously takes place when the number of farmers in the cooperative is larger than two.
    • Actually, landlords interact for a finite number of periods but, there is enough uncertainty about the end of their interactions that this can be modeled as an infinite repeated game.
    • Yeon-Koo Che [17] builds a group lending model where the incentive problem stems from the entrepreneurs’ unobservable effort decisions and their liquidity constraints. He shows that when the group members operate their projects repeatedly, the joint liability feature itself makes it credible for members to penalize others through their effort decisions. Under group lending, a member’s shirking in her productive effort increases the payment burden of her peers, i.e., a group member can be penalized by other members’ shirking.
    • Armendariz De Aghion [15] reports that social sanctions are observed in practice. For example, in agricultural cooperatives, social sanctions often involve the exclusion of defaulters from privileged access to input supplies and marketing facilities.
    • At the end of her contract, a rental contract holder may well benefit from water theft without incurring the cost of stealing because she can refuse to pay the punishment since she would leave the cooperative anyway.

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    Wided Mattoussi, Mohamed Salah Matoussi and Foued Mattoussi (November 30th 2021). On the Design of Total Water Use-Based Incentive Schemes for Groundwater Management [Online First], IntechOpen, DOI: 10.5772/intechopen.100540. Available from:

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