Water clubs: a new model for managing freshwater in small catchments

Over the past 30 years market-based mechanisms have been upheld as the gold-standard of water governance approaches. However, while some water markets have been successful, many struggle to meet the strict institutional preconditions for efficiency and effectiveness, which has implications for social welfare and public good provision. As pressures on water resources increase, there is a growing need to consider alternative market designs that foster more cooperation between water users. Drawing on club theory and game theory, this paper investigates the economic and environmental benefits of (re)designing water markets as clubs. It finds that in small catchments, the introduction of group-level trading can increase provision of the public good, improve social welfare, and reduce free-riding when compared with a regulatory status quo. It also finds that the club model performs best when the number of active traders is low—a result that challenges the common assumption regarding group size and effective market performance. In local contexts, the group-level trading may also help characterise the optimal group size by weighing the benefits from more trading opportunities against the losses from free-riding.


Introduction
Managing demand over limited water resources has become a defining characteristic of catchment management (Garrick et al 2020).For the past 30 years water managers have focused on addressing inefficiencies by increasing competition between water users through trading (Leonard et al 2019).Voluntary trading between willing buyers and sellers can direct water from low to higher value uses and provide information on relative values that is useful for informing policy decisions (Hanemann 2006).Under certain conditions, theory predicts that the trade and exchange of rights can facilitate more efficient outcomes (Olmstead 2010), and, indeed, some econometric evidence from water markets in large catchments, such as the Murray-Darling Basin and the Rio Grande, support this proposition (Debaere and Li 2020, 2022, Rafey 2023).However, markets have rarely been recommended for managing water allocation in small catchments (a notable exception being (Rosegrant and Binswanger 1994)).The institutional preconditions under which a formal water market can operate efficiently are strict: property rights need to be well defined, defended, and divestible, transaction costs need to be low, and scarcity needs to be present (Coase 1960, Pujol et al 2006).Further, markets are costly to design, implement, and regulate, and can crowd-out the tendency of local actors to cooperate and provide for the public good aspects of water, such as instream flows (Agneman and Chevrot-Bianco 2022, Grafton et al 2022).For these reasons, traditional market models have been viewed less suitable for small catchments where transaction costs associated with implementation and monitoring and enforcement can outweigh efficiency gains from trade (Colby 1990, Wheeler et al 2017).In Aotearoa New Zealand, a geographical region dominated by small catchments with few water users (typified by the catchment shown in figure 1), regulation has been the preferred policy tool for managing water (Booker et al 2022).Markets are costly and culturally difficult to implement due to unresolved issues around rights and interests in water (Talbot-Jones and Grafton 2021).However, regulation is not delivering efficient outcomes: issues associated with incomplete information and regulatory capture have led to many small catchments on the east coast of Aotearoa New Zealand experiencing scarcity as a result of declining water quality and overallocation (Joy 2022).
To overcome the challenges of market failure and government failure in Aotearoa New Zealand and elsewhere, there is a need for innovation in water market design.This could be particularly useful in small catchments, which are rarely identified as suitable contexts for formal markets because of transaction cost barriers to implementation and operation (Grafton et al 2011, McCann and Garrick 2014, Womble and Hanemann 2020).A little-known, but important area of the social sciences, that could assist in the innovation of market design, is the economic theory of clubs (Buchanan 1965, Cornes andSandler 1996).A club is a voluntary group deriving mutual benefits from sharing the costs of producing an activity that has public good characteristics (Prakash and Potoski 2007).The gains from a successful club are sufficiently large that members will pay dues, whether monetary or non-monetary, and adhere to club rules in order to gain the benefits of membership (Prakash and Potoski 2007).
As outlined in Nordhaus ( 2015), the major conditions for a successful club include the following: (i) that there is a public-good-type resource that can be shared; (ii) that the cooperative arrangement, including the dues, is beneficial for each of the members; (iii) that non-members can be excluded or penalised at relatively low cost to members; and (iv) that the membership is stable in the sense that no one wants to leave.
Versions of the club model (CM) have been used to understand how voluntary environmental programmes can be used as policy instruments for environmental governance (Potoski and Prakash 2013).They have also been used to aid in 'just transitions' in cases where policy settings are forced to change as a result of crises or rapidly increasing costs (Decaro et al 2017).There has been little attention paid to understanding how they could improve formal water market design, however, or how they could assist in cases where water scarcity is present.This paper seeks to address this gap, by examining how designing water markets as trading clubs could increase cooperation and the provision of instream flow in small catchments.
In this paper we show conceptually and mathematically how incorporating club theory into water market design can nullify some of the challenges faced by more traditional market models.In doing so we make three novel contributions to the literature.First, we incorporate both the private and public aspects of water into a trading model allowing water users to account for the social externality arising from leaving water instream.Second, by introducing a trading option that also features a public good, we are able to quantify the extent to which trade can offset the negative impacts of free-riding.Third, we show the impacts of group size on voluntary trading arrangements and produce a result that contravenes the common argument that efficient water markets require many active traders.Instead, we show that the benefits of a club are greatest when the group size is small and that the individual benefits that arise from investing in the public good dissipate as free-riding becomes the norm.In essence, this points to a possible bliss point or optimal group size for efficient operation of the club that is dependent on catchment parameters and an area for future research.
The paper proceeds as follows.In the following section we outline the underlying argument and assumptions.We then provide details of the model and report our results.The discussion considers the benefits of using a CM across contexts.We then close with conclusions and final remarks.

Economic justification for a water club
The idea of a water club should be viewed as an innovation in water market design and a solution to the policy challenge facing the application of water markets to small catchments.By designing and treating a water market as a club, many of the institutional limitations of existing water market regimes are overcome, most notably the requirement for many active traders, high transaction cost barriers to trade, monitoring, and enforcement, and insufficient provision of the public good elements of water, such as instream flow, that can arise from the political complexities of setting an efficient limit on extraction using the cap and trade model.
In doing so, the CM offers a new perspective on the value of self-governing trading regimes at the group level.That this proposition is yet to be formalised in water market design is an oversight for two related empirical reasons: one, there is sufficient evidence to show that informal water markets operate across a range of contexts when group enforcement strategies are present (Easter et al 2018); and two, small groups can develop long-standing relationships through repeated interactions that lower costs and encourage group members to allocate more to the public good than occurs with large groups (Ostrom et al 1992, Ostrom 2010).
From a theoretical perspective it is also clear why self-governing trading regimes may work in small catchments.First, the marginal return from investing in the public good has an inverse relationship with group size.Second, public goods provided in small group settings can be characterised by 'large' marginal returns.This means that in large catchments with many users there is little incentive for users to provide for the public good elements of water; however, in small catchments, users have a greater incentive to voluntarily contribute to the public good thereby creating positive social externalities.
In small catchments, which are characterised by fewer users, we propose that club theory can guide the design of self-governing trading regimes that encourage water users to internalise the social externality values of instream flow.If the marginal returns on investment are sufficiently high and water users can achieve efficiency gains through trade, water users could be motivated to independently invest in the public good.Club theory provides a strong theoretical and conceptual foundation to motivate this regime.
A brief description of the proposed water club is as follows: the club is an agreement by participating water users to maximise their utility subject to others' earnings.Entry to the club is voluntary, but users are required to contribute some portion of their endowment to the environment as payment for membership.Users are incentivised to join because the benefits of trading brings sufficient net benefits to club members.In line with the theory of voluntary clubs (Prakash and Potoski 2007), the central purpose of the club is to produce positive social externalities beyond the status quo (SQ).This delivers important welfare gains to society and the environment.
The private benefits of voluntary club membership accrue only to individual club members, not to other members, and certainly not to non-members.In the case of the water club, the private benefit arises from being able to participate in water tradesan arrangement not available to non-members.The additional social welfare gains come from permitting club members to contribute to the public good beyond the minimum level that is required and identify their own socially optimal level of take based on their own levels of demand.
While a central government may have a comparative advantage in obtaining water supply data, users are better able to determine their own demand.This club arrangement circumvents the incentive of users to capitalise on information asymmetry by enabling local communities to act as conduits for information over repeated periods.In doing so, it also reduces potentially high transaction costs by devolving decision-making responsibility and monitoring and enforcement responsibility to the group level.The remainder of this paper examines how the structure of incentives will encourage water users acting in their own self-interest to choose to enter the club and contribute to the public good beyond the equilibrium level observed under the SQ.

Model
Economic principles justify the basic structure of the water club.In this section, we build on the existing water market literature (Madani 2010, Dinar and Hogarth 2015) and set up a game theoretical model to investigate the extent to which water users operating as a small catchment can be encouraged to voluntarily invest in the public good, alongside making their own production decisions.We envision a scenario in which water users within a catchment are allocated an endowment and are presented with an opportunity to join a water club.The cost of joining the club is that all members must voluntarily contribute some of their endowment to the environment.It is assumed that the social benefits of contributing at least some of a users' endowment to the environmental pool will increase social welfare outcomes; however, the incentive for individual club members is to invest in private production or trade their endowment up to the point where the marginal net benefits of investing in the public good equal the marginal net benefits of trade.
In our model, we assume there are N ⩾ 2 water users in a catchment who require water for production on a given amount of land (L i for user i).These water users are entitled to an initial endowment (W i ) from which they must choose how to best allocate it across three main uses.First, as members of the club, water users must invest some portion of their water (e i ) in the environment.As the environment is a public good, we assume that this investment provides social benefits, but that users have an incentive to free-ride on their contributions beyond the baseline investment requirements.Second, water users will have the option to use some portion of their water (y i ) for agricultural production.Third, club members can choose whether or not to trade with other members.
In our model, it is assumed that all water users will take the agricultural output prices as given and that there is diminishing marginal returns from using too much water on their given amount of land.Finally, water users may trade their remaining endowment with others.We denote (s i ) as the supply traded away and (d i ) as the extra water demanded from others.Water users choosing to augment their supply may do so in order to contribute more to the environment or to expand their agricultural production.
Given the other users' investment decisions {e 1 , . . ., e i−1 , e i+1 , . . ., e N }, user i maximizes her utility as follows: where G(e 1 , e 2 , . . ., e N ) is the positive externality produced by water users investing in the environment.
It is noteworthy that if G(•) = 1 for any investment level, then the CM reduces to a standard water market, with only private benefits taken into account by members.The term F i (y i ) indicates production function for user i, which can be different for all users.Therefore, the first term in (1a) is the user's utility from her own production, amplified by the benefit from the public good.The second term is the additional benefit from net trade (s i − d i ), where P is the endogenously determined trade price resulting from the water users' actions.The last term illustrates the cost of water transfers and can be a proxy for the transaction costs associated with trade.Although these costs can be shared by the buyer and seller, we assume the supplier bears the burden in this model and reflects this in the trade price at equilibrium.Finally, (1b) is the resource constraint for water user i, where the left-hand side is the total supply of water, and the right-hand side represents the total demand for water.
Given that our model is a game among water users in a catchment, the equilibrium behaviour of the agents is determined via their best-response functions.Since the users choose how much of their endowment to allocate to the environment, to their own production, and to trade, we can use the first order conditions to estimate optimal allocations, assuming that the production, investment, and cost functions are twice differentiable and the firm profit function is concave.Given the setup, it is not possible to provide an analytical derivation for the bestresponse functions in closed form.Regardless, the equilibrium behaviour can be characterised by the following set of equations: ∂G(e 1 , . . ., e N ) ∂e i F i (y i ) = ∂F i (y i ) ∂y i G(e 1 , . . ., e N ); for i = 1, 2, . . ., N (2b) The first equation above (2a) is the resource constraint introduced in (1b), while (2b) tells us the optimal allocation of water between the public investment and the private use.Intuitively, the left-hand side is the marginal benefit from allocating the last unit of water in the environment, while the righthand side is the marginal benefit from allocating the last unit of water for production.This is illustrated conceptually in figure 2, which shows the trade-off faced by the water user in terms of public investment and private use.As shown by the diagram, water users will aim to invest at point A for private production and the rest for public investment, where the marginal benefits of investing in the public good are equal to the marginal benefits of private production.
The decision of whether to trade depends on (2c): the water user will sell water up to the point where the marginal benefit of the last unit traded away equals the trade price minus the marginal cost of transfer.This relationship no longer holds if the user is a buyer, however.In this case, the trade price is simply the marginal benefit of the last unit bought via trade.Due to the cost of transfer, it is inefficient for a water user to buy and sell water simultaneously, which is captured by (2d).Finally, (2e) sets the equilibrium for the trade market: the units supplied via trade must equal the units demanded by all users.

Key results
In our model, two main factors affect the performance of the CM: (1) the incentive for water users to free-ride on their environmental contributions, and (2) the impact of trade on user behaviour.To assess the benefit of forming a water club we therefore consider three settings and solve for each of the three specifications.
First, we consider a special case of the social planner (SP), who has full information, overlooks the complete set of actions, and allocates the resources to maximise total surplus.The optimal allocation can be solved through the following set of equations: ∂G(e 1 , . . ., e N ) ∂e i N ∑ j =1 F j (y j ) = ∂F i (y i ) ∂y i G(e 1 , . . ., e N ); where λ represents the marginal value of water, which is the opportunity cost of allocating water across various users and uses.This is analogous to the trade price (P) we set up for the CM.It is important to point out that (3) is very similar to (2), but the SP observes the full marginal benefit of allocating another unit to investing into the environment, as opposed to the private benefit observed under the club setting.More specifically, the term on the left-hand side of (3b) has the effect of a marginal increase in investment on all users' production, captured by the sum.This term differs from the left-hand side of (2b).There, a user only accounts for the marginal effect of the public benefits on her own production, which triggers free-riding.As the SP would internalise the positive externality, there is no free-riding problem in this case, and we interpret this specification as the first best option.Second, we solve the model without any trade component and refer to this case as the SQ.In this specification, we assume that under a standard regulatory framework water users are not allowed to trade, so they simply allocate their initial endowment across the respective public and private uses.In this case, the free-riding problem exists and has its full effect on the total surplus.We can solve this model using (2) and imposing s i = d i = 0 for all water users.
Our last specification is the CM, where the water users can only trade with each other if they are a member of the club.Since trading cannot make anyone worse off, being a member would be the equilibrium strategy (assuming negligible membership costs).Since the club allows for trade among members, the trade partially offsets the adverse effects of free-riding.
Finally, in the absence of positive externality (e.g.G(•) = 1), the left-hand side of equations ( 2b) and (3b) equals zero.This implies that, SP, CM, and a standard water market all coincide as there is no source of inefficiency left in the model.In some situations, such as when a water extraction cap is set at the efficient level, G(•) could be larger than one.

Numerical illustration
Because the notion of the CM is new and yet to be applied, we illustrate the potential impacts of the CM on behaviour numerically by estimating realistic parameters and functions.Taking the simple case of three users (N = 3), we envision a scenario where all water users have identical production technology, but b There is no unique marginal value in this case as each user allocates their endowments without any trading.
different initial endowments 3 .It is worth noting that this case can be extended to incorporate more users and different production technologies.We assume a constant elasticity of substitution function for investment in the public good, where the elasticity of substitution is 0.5 (indicating the users' contributions are complements), and a Cobb-Douglas production function with share parameter equal to 0.75. 4We consider three sets of initial endowments: S1 allocates the total available water supply equally across the users, S2 gives more to the third user from the first user, and S3 further limits the first user by giving even more to third user 5 .Table 1 shows that under these conditions, the CM partially offsets the negative impacts of free-riding observed under the regulatory setting.This improves social welfare (measured by the total producer surplus) beyond SQ, although it does not reach the levels attained by SP.We also see that under these scenarios, water users operating under CM invest in the public good beyond what is observed under the SQ (shown as public investment in table 1).This can be understood as a response to the positive marginal returns received 3 A case with only two users has the ancillary simplification that whenever there is trade, both users have to be involved. 4The parameter values are chosen to illustrate a standard case, but they do not impact the results qualitatively.The parameter values can easily be substituted for alternative scenarios, informed by local contexts. 5The numerical illustration is done in Matlab.All the Matlab codes and other supplementary files are available from authors upon request.
from investing in the public good.As outlined in figure 2, water users can be expected to invest in the public good up to the point that the marginal benefit of allocating water for private production and for public investment are equal, at which point, the water users will use water for some other purpose.
Although not evident in table 1, our analysis and robustness tests show that dispersion in the initial endowment further facilitates trading, suggesting that the CM will have greater application in catchments where there are large variations in user allocations.We also find a group size effect, such that as N increases, free-riding increases and there is less incentive to contribute to the public good.Although this might suggest that the CM is therefore better suited to small catchments, our results also point to the presence of a counter effect where a greater number of water users effectively lowers the barriers to trade6 .
Barriers to trade are important to consider as they can negate the benefits of joining the club.In our model, these barriers are treated as transaction costs and are assumed to include transfer costs, administration costs, enforcement costs, and so-on.The size of these transaction costs, as well as other catchment specific parameters, will determine how much trading will offset free-riding when the CM is applied in a local context.

Discussion
The purpose of the water club is to address some of the limitations presented by the traditional models of market design, including high transaction costs and low levels of cooperation (Garrick et al 2020).An important question is: how would a top-down water club get started?It is plausible to envision a scenario where information asymmetries are causing inefficiencies in the governance of water and the costs of monitoring and enforcement are sufficiently high to act as a barrier to effectiveness.Institutional theory would argue that such a scenario would trigger a search for an alternative arrangement that could lower costs and overcome information problems (North 1990).
A scenario, which is familiar in Aotearoa New Zealand and across international contexts, sees group level actors in a small catchment endowed with a water allocation.Because users may not be permitted to trade, it is unlikely that the allocation arrangement is efficient or that it suitably encourages water users to optimise their consumption.Granting allocation rights on a first come first served basis, as occurs in Aotearoa New Zealand, or based on strict 'beneficial use' criteria, such as occurs in the United States, means that there is no guarantee that the users who value the water the most are the ones that have access to it, or that the most efficient users have access (Dinar et al 1997).
Under the CM proposed here, water users within a catchment are presented with an opportunity to join a water club.Because they do not have to join, this model can work in partnership with other governance arrangements that do not involve trading.The benefit of joining the club, however, is that members are permitted to trade some of their endowment with other members-an opportunity not available to non-members.This trading condition allows members to smooth their allocation across multiple uses and make choices that optimise their marginal returns.Unless transaction costs are prohibitively high, there is no reason water users would not be better off joining the club than abstaining.
A second critical question thus becomes how can club members be incentivised not to free-ride and to invest at least part of their allocation in the public good?Our model demonstrates that trading allows club members to independently approach the socially optimal outcome even when investing some of their endowment in the public good.Because club members receive some net benefit from investing in instream flow they have an incentive to contribute up to the point where the marginal net benefits of environmental investment equals the marginal net benefits of trade.Although the amount of investment will depend on the catchment specific parameters, our model shows that when water users are given the opportunity to cooperate in a club they donate more to the public good than would be invested under the SQ, even in the absence of monitoring and enforcement.
Research shows that informal water markets that rely on cooperation can be an effective and longlasting allocation mechanism when groups are small (Easter et al 2018).The CM presented here formalises this notion and in doing so offers decision-makers a governance regime that lowers transaction costs, improves social welfare outcomes, and improves outcomes for the environment.The success of this model depends on the social benefits of investing in the public good being evident to the water user, however.If these benefits are not able to be accounted for by the user, members may violate appropriate behaviour norms through shirking (Bowles and Gintis 2004).In small catchments with tight-knit communities and good communication, these risks could be mitigated by sociological pressures; however, in larger settings, monitoring and enforcement mechanisms may be required to compel members to adhere to club standards (Isaac and Walker 1988).
Given this, there is unlikely to be one universal CM that fits all settings.Specific catchment parameters are likely to affect institutional design across contexts, for instance, the number of active traders, the initial allocation distribution, general scarcity pressures, and so on.The model presented here offers a useful frame for thinking about how traditional market designs can be adapted to support the delivery of improved social and environmental outcomes in settings where formal markets have been rarely used todate.

Conclusion
Although yet to be implemented, the idea of a water club offers a promising alternative regime for addressing allocation challenges in small catchments.Drawing on the theory of institutions, as well as club theory and game theory, our results reveal how water markets can be designed to incentivise a small number of water users to consider both the public and private benefits of water in a way that delivers positive outcomes for the environment and local communities.Ultimately, the model shows how water markets can be (re)designed to incentivise voluntary investment in the public good and deliver improved welfare outcomes, increase the provision of instream flow, and improve cooperation at the group level.
An obvious limitation of this work is the absence of data to demonstrate the application of this concept in an empirical setting.However, this paper is the first step in a broader research agenda.It provides policymakers with a conceptual framework, model, and complementary open-access code (available on request), to aid in assessing the costs and benefits of applying a CM in a particular catchment.This will aid in future application of the model as policymakers will be able to assess the situations under which a CM could usefully complement or replace existing regulatory arrangements in an efficient and effective way.Future work that focuses on testing this model across empirical contexts will provide a useful companion to this piece of research.

Figure 1 .
Figure 1.A typical freshwater catchment in the South Island of Aotearoa New Zealand showing the catchment's geographical boundaries, surface water sources, and active water users.The Edendale groundwater management zone (GMZ) covers an area of approximately 12 400 ha on the Edendale Terrace in the Lower Mataura Valley.Source: NIWA.

Figure 2 .
Figure 2. Relationship between private production and public investment.