Optimal joint management of a coastal aquifer and a substitute resource

Authors


Abstract

[1] This article characterizes the optimal joint management of a coastal aquifer and a costly water substitute. For this purpose we use a mathematical representation of the aquifer that incorporates the displacement of the interface between the seawater and the freshwater of the aquifer. We identify the spatial cost externalities created by users on each other and we show that the optimal water supply depends on the location of users. Users located in the coastal zone exclusively use the costly substitute. Those located in the more upstream area are supplied from the aquifer. At the optimum their withdrawal must take into account the cost externalities they generate on users located downstream. Last, users located in a median zone use the aquifer with a surface transportation cost. We show that the optimum can be implemented in a decentralized economy through a very simple Pigouvian tax. Finally, the optimal and decentralized extraction policies are simulated on a very simple example.

1. Introduction

[2] Except for arid and semiarid regions, surface water has been traditionally the primary water source for consumption. The main reason explaining this intensive use of surface water is its easy access, i.e., in fact its low cost. However both the increasing pressure on surface resources, generated by economic and population growth, and a badly controlled pollution development have required to diversify the water supply sources. Thus during the second half of the 20th century, groundwater withdrawals have regularly increased. They now amount to about one-third of the world freshwater consumption. This increase, often higher than natural recharge thresholds, has resulted in an important fall in aquifer levels and in a need for populations to switch toward other water sources with lower quality or higher cost. In coastal aquifers where freshwater is hydraulically connected to seawater, withdrawals can endanger the long-term use of the resource. Under natural equilibrium conditions the hydraulic gradient ensures a water run-off toward the sea protecting freshwater. However, the gradient is often relatively small and any excessive net withdrawal can alter the hydrostatic balance. Then seawater enters more deeply into the aquifer and replaces freshwater. This phenomenon, known as seawater intrusion, prohibits access to the resource for the users in the coastal areas.

[3] Although seawater intrusion is a critical problem for many coastal aquifers, the economic literature has rarely addressed this issue. Some models consider an aquifer as a “bathtub” where water consumption results in quality degradation. This is the kind of model used by Cummings and McFarland [1974] or, more recently, by Koundouri [1997] to study the Kiti aquifer on the island of Cyprus. Tsur and Zemel [1995] consider saline intrusion as an irreversible event occurring when the groundwater table declines below some threshold level. Because of a lack of knowledge and measure precision, this threshold level is unknown. In this kind of “Doomsday model,” Tzur and Zemel show that exploitation policies under uncertainty are more conservative than under certainty. Hart and Parker [1995] have proposed a spatial modeling of seawater intrusion in a coastal aquifer based on a standard moving interface model. They study the water allocation resulting from a competitive market and from taxes and quota policies. In a dynamic setting, Zeitouni and Dinar [1997] consider two aquifers hydrologically interrelated. They develop an optimal control model of this groundwater system in which water quality varies across the aquifers, due for example to seawater intrusion. Zeitouni and Dinar [1997] identify conditions under which a joint management of the whole water system is preferable to an independent one. Last, Green and Sunding [2000] consider a microparameter model for analyzing the efficiency of different taxes (i.e., a Pigouvian tax per unit of pollution, a per-unit charge on water pumping and a land tax) on seawater intrusion in California's Salinas Valley. They show in particular that the Pigouvian tax is nearly twice as efficient as the land tax. None of these papers integrates both the pumping location choice on the aquifer and the decision to use a costly water substitute. This is quite surprising as water supply from an expensive substitute constitutes a solution that makes possible to mitigate the detrimental effects of seawater intrusion on the aquifer water quality. This has been for example the solution put forward at the end of the Seventies in the Netherlands where water from the Rhine has been used for replenishing the main aquifer in Amsterdam. The present paper tries to fill this gap by analyzing the joint use of an aquifer under seawater intrusion condition with a substitute.

[4] The specific objective of this study is to characterize the optimal withdrawal rate from a coastal aquifer under saline intrusion conditions when there exists a costly substitute. The economic analysis is based on an analytical model of a sharp interface between seawater and freshwater. We limit here our analysis to the stationary case and we determine the optimal water discharges from the aquifer and from the substitute. We focus on the cost externality created by users spread over the aquifer. When total withdrawal increases, the hydraulic interface moves landward and users located in the coastal area who can no longer use the aquifer must be water supplied from the costly substitute. It follows that users located far away from the coast can create a negative externality on users located in coastal areas. How is it possible to make people internalize this externality? May a single price for all users allow to implement an optimal allocation of water? This is the kind of questions we address in this paper by assuming that the social planner objective is to maximize the net social surplus.

[5] This article is organized as follows. In the next section, we present the model and we describe the characteristics of the two water resources, i.e., a renewable aquifer under seawater intrusion and a substitute. We point out in particular the spatial constraint created by seawater intrusion on the aquifer rate of extraction. In section 3, we describe the optimal use of these resources when the location of pumping on the aquifer is given a priori. In section 4, we relax this assumption and we consider an endogenous pumping location. We show that a simple Pigouvian tax allows to implement the optimum in a decentralized economy. The last section presents some numerical simulations of the model.

2. A Model of Coastal Aquifer Use

[6] We consider a territory located on the seaside. The population can be water supplied from two resources: a renewable coastal aquifer and another resource called substitute.

2.1. Aquifer Under Seawater Intrusion Conditions

[7] The coastal aquifer is recharged by freshwater entering at its landward boundary, at a distance X0 from the coast. R denotes the instantaneous recharge rate, i.e., the net water inflow excluding extraction. At the seawater boundary, an influx of seawater migrates toward the bottom of the aquifer and displaces freshwater. H0 denotes the elevation of the sea above the substratum of the aquifer. Figure 1 describes this standard two-dimension model of coastal aquifer under saline intrusion conditions.

Figure 1.

Model of the coastal aquifer under saline conditions.

[8] Under natural equilibrium conditions, freshwater denser than seawater forms a lens that floats on seawater. At the freshwater and seawater contact, mechanical dispersion causes mixing and there exists a transition zone. As a simplification, we neglect this brackish zone and we assume that the interface is sharp. Under some circumstances depending upon aquifer physical properties, this mixing zone is small compared to the aquifer's thickness and the transition zone can be approximated by a sharp interface; see Bear et al. [1999]. Pumpings, however, reduce the net freshwater flow which is discharged into the sea. This determines a new hydrostatic equilibrium between freshwater and seawater. As the water extraction rate increases, the interface moves landward and upward. Saltwater moves to the bottom of the aquifer and displaces the freshwater upward. In the same time, the water table goes down and the freshwater lens goes thinner. As seawater intrusion progresses, pumping wells close to the coast become saline and have to be abandoned. Indeed standards established by the Environmental Protection Agency require drinking water to contain no more than 0.5 g/L of total soluble salts (TSS), a common measure of salinity. Seawater contains approximately 30 g/L of TSS. According to the recent report of the Food and Agriculture Organization, Food and Agriculture Organization [1997], a two to three percent mixing with seawater renders freshwater inadequate for human consumption.

[9] Krulce et al. [1997] have shown that the displacement of the freshwater-seawater interface creates a spatial constraint on resource availability: the proportion of natural recharge that can be withdrawn without risk (i.e., without seawater intrusion into the pumping well) increases with the distance to the coast. We can then define for any point x, x ∈ [0, X0] the maximum feasible extraction flow when no pumping occurs elsewhere in the aquifer. This maximum flow, denoted by equation image(x), is strictly increasing and concave in x; see Bear et al. [1999] for a complete presentation of the hydrologic model and Moreaux and Reynaud [2001] for a more extensive derivation of the function equation image(x). This maximum flow is null below a critical location x, near the coast, and we denote by equation image0 with equation image0 < R the maximum feasible pumping rate at x = X0. We denote by equation image(Y) the inverse function equation image−1. This function defines the nearest point to the coast from which a flow Y is available. This spatial constraint creates a tradeoff between the total rate of pumping from the aquifer and the proportion of users that can not directly access to this resource. Only users located upstream equation image(Y) have a direct access to the aquifer; a high rate of pumping will exclude a high proportion of users from a direct access to the aquifer (i.e., those located in the coastal area).

[10] The maximum feasible flow from x when total pumping elsewhere is Y is given by:

equation image

The feasible flow from x is null when total withdrawals on the aquifer exceed equation image(x) otherwise a maximum flow equation image(x) − Y can be withdrawn. This creates a constraint similar to the constraint defined by Cummings [1971]. In that paper, freshwater available for consumption depends on the length of the corner of seawater under the aquifer and depends on the capital intensity of pumping.

[11] A user located at x ∈ [0, X0] and who pumps water from the aquifer at a rate y must bear two costs: a cost of extraction and, under some circumstances, a cost of surface transportation. The cost of extraction depends on the flow as well as the elevation required to bring water to the surface. Elevation cost is neglected as it significantly complicates the model without modifying the qualitative results of the paper. The interested reader may refer to Moreaux and Reynaud [2001] for a model with water elevation cost. We assume here that the cost is linear with respect to the flow and we denote by α the cost of extraction per unit of flow. Therefore the total cost of extraction for a flow y is αy. The surface transportation cost is a consequence of seawater intrusion: high withdrawals generate an upstream move of the interface that prohibits a direct access to the resource for users in the coastal area. Those users wishing to be water supplied from the aquifer must locate their pumping well upstream their location. Thus they have to bear a transportation cost. This cost depends both on the flow and the distance. We assume that the cost per unit of distance and flow, denoted by β, is constant. The transportation cost of a flow y on a distance d is βdy.

2.2. The Substitute

[12] At any point of the segment [0, X0] users have access to a substitute. This substitute can be a long-distance water transfer (such transfers are already an important source of water in numerous Mediterranean countries) or it can correspond to a nonconventional resource (i.e., seawater desalination or reuse of wastewater). We assume that the substitute is available at a unit cost γ which is the same for all users. Thus the total cost of supplying a flow z of substitute is γz. Assuming that the cost of the substitute is the same for all users means that the distribution cost is negligible compared to the long-distance transfer cost or compared to the desalination cost. Last, the individual maximum permanent flow of substitute is denoted by equation image.

2.3. Social Planner Program

[13] We consider a continuum of users located on the surface of the aquifer according to a probability distribution function f(x), xequation image [0, X0] with cumulative distribution function F(x). We denote by U(.) the utility function of a representative user. U(.) is assumed to be of class &#55349;&#56478;2 on [0, +∞], strictly increasing and concave. Moreover water is an essential good: equation image = +∞ where w denotes the per capita water consumption. The social planner must define for each user located at x with x ∈ [0, X0] the location of his pumping point on the aquifer equation image(x) ∈ [0,X0], the aquifer extraction rate equation image(x) ∈ [0, equation image0] and the supply from the substitute equation image(x) ∈ [0, equation image]. These location and extraction rate choices maximize the net aggregate social surplus defined as:

equation image

The first term of the aggregate social surplus corresponds to the gross utility of the representative user located at x and consuming a flow of water equation image(x) + equation image(x). The second term is the cost of its supply from the aquifer. It includes the extraction cost and the transportation cost on a distance ∣xequation image(x)∣. The next term corresponds to the cost of its water supply from the substitute. The sum of the three terms into brackets defines the net utility of a user located at x.

3. Optimal Use of Resources Without Choice of Pumping Location

[14] In this section, pumping location is taken to be exogenous and a user can be water supplied from the aquifer only if this resource is available at the user location: equation image(x) = x, ∀ x ∈ [0, X0]. This situation may correspond to riparian property rights on land. The problem then consists in defining for each user x ∈ [0, X0] the aquifer optimal extraction rate, equation image*(x) and the substitute flow equation image*(x) which maximize the aggregate surplus. We assume that the substitute is abundant in the following sense. We denote by z* the solution of U′(z) = γ. Then the substitute is said to be abundant if equation imagez*, a user only supplied from the substitute is not constrained by resource availability. We also assume that the substitute is more costly than the aquifer, even without any surface transportation γ ≥ α. Note that when the substitute is abundant and the unit cost of the substitute is lower then all users exclusively consume the substitute at a rate z*. In such a case, the aquifer is not used at the optimum and it does not have any economic value.

3.1. Social Planner Program

[15] By definition the users located downstream x cannot use the aquifer, even if there is no withdrawal elsewhere. Thus they must be supplied from the substitute. Some other users upstream x may also only use the substitute. We denote by Θ with x ≤ Θ ≤ X0 the location threshold such that all users located downstream exclusively consume the substitute. Users xequation image [0, Θ] are said to be “downstream users” and users x ∈ [Θ, X0] are called “upstream users”. Given the assumption of an abundant substitute, each downstream user uses a flow z* of this resource. The aggregate net surplus of downstream users is defined as:

equation image

and is an increasing function of Θ. Thus characterizing the optimal water supply consists in determining the threshold Θ, the withdrawals from the aquifer and from the substitute, equation image(x) and equation image(x) for x ∈ [Θ, X0]. The optimum is the solution of the program equation image1:

equation image

[16] The first constraint of program equation image1 means that, given the total rate of pumping on the aquifer, water must be available at Θ. The maximum feasible flow from Θ without any pumping elsewhere must be higher or equal than the global extraction rate that is the aggregate withdrawal rate for users located between Θ and X0. Let us notice that upstream Θ this spatial aquifer availability constraint is de facto satisfied since equation image(.) is an increasing function. The tradeoff defining the optimal use of the aquifer is the following. At the optimum, aquifer users must internalize the fact that their withdrawals oblige users located in the coastal area to consume the more expensive substitute. This effect gives upstream users some incentives to limit their withdrawals. However, the proportion of natural recharge that can be extracted from the aquifer decreases as we get closer to the coast. This gives some incentives to locate the optimal threshold Θ in a more upstream area of the aquifer. This can be achieved only through a high pumping rate of upstream users. It follows that the value in situ of the aquifer is reduced by the spatial availability constraint. This tradeoff is similar to the one defining the optimal use of water from a canal with losses, a problem studied by Chakravorty and Umetsu [2003]. The quantity of resource available at a point of the canal is equal to the flow available upward less the network losses between these two points. At the optimum, upstream users must take into account the fact that their withdrawals decrease the resource available for users located downstream. This gives upstream users incentives to limit their withdrawals. However, the network losses along the canal provide incentives for high withdrawal level for upstream users. The optimal use of water results from the tradeoff between these two effects.

[17] We can now write the first-order necessary conditions associated to program equation image1. Pointwise maximization of the Lagrangian with respect to equation image and equation image yields:

equation image
equation image

Maximization with respect to Θ yields:

equation image

The optimal Θ is such that the social loss from marginally moving landward the threshold Θ (a higher proportion of the population must be supplied from the more costly substitute, first term into parentheses in the first-order condition (6)) is exactly balanced by the social gain (a higher rate of pumping from the aquifer is possible as equation image′(.) > 0, second term in the first-order condition (6)). The complementary slackness conditions are:

equation image
equation image
equation image
equation image
equation image

3.2. Characterization of the Optimum

3.2.1. Optimal Water Consumption

[18] The water consumption of an upstream user does not depend on its location:

equation image

(hereinafter referred to as lemma 1). Demonstration of lemma 1 is given in Appendix A. The intuition of this result is that all upstream users are identical both in term of utility and costs of access to the water resources. In order to exhaust any Pareto improving reallocations, upstream users must consume the same quantity of water. Next, we determine for a given threshold Θ the optimal water consumption of an upstream user as a function of Θ, equation image*(Θ).

[19] If Θ is large enough, upstream users are supplied from the aquifer and their individual extraction rate is y* defined as the solution of U′(y*) = α. Such a rate is possible only if the mass of the upstream users given by [1 − F(Θ)] is sufficiently small compared to the available aquifer resource given by equation image(Θ). This requires that Θ > equation image where equation image is the unique solution of [1 − F(x)]y* = equation image(x). To summarize:

equation image

[20] If Θ ≤ equation image, a positive rent is attributed to the aquifer. Let us consider first the case where all upstream users extract water exclusively from the aquifer. The optimal per capita water consumption is equation image(Θ)/[1 − F(Θ)]. This situation is optimal as long as U′(equation image(Θ)/[1 − F(Θ)]) < γ, i.e. as long as equation image < Θ ≤ equation image where equation image is the solution of equation U′(equation image(x)/[1 − F(x)]) = γ. To summarize:

equation image

[21] Last, if x < Θ < equation image, it is optimal for an upstream user to use the substitute. It follows that equation image(x) > 0 and equation image(x) > 0 and thus ξ(x) = 0. The necessary condition (5) gives equation image(x) + equation image(x) = z*. Thus

equation image

3.2.2. Optimal Threshold Θ*

[22] The optimal threshold Θ* belongs to the interval [equation image, equation image] (hereinafter referred to as lemma 2). The proof of lemma 2 is given in Appendix A. The intuition is the following. A too high value for Θ is not optimal as too much users must be water supplied from the costly substitute. A too low value for Θ is also not optimal as a too small fraction of the aquifer recharge can be used in that case. The optimal threshold Θ* belongs to an intermediate range of values. The preceding lemmas allow to fully characterize the optimal exploitation of the two resources.

[23] If the substitute resource is costly and abundant, there exists a threshold Θ* equation image [equation image, equation image] such that the optimal exploitation of the two resources is defined by users x < Θ* exclusively consume the substitute, equation image* = 0 and equation image* = z*, and users x ≥ Θ* exclusively consume the aquifer, equation image* = equation image and equation image* = 0 (hereinafter referred to as proposition 1). Given the very general functional forms considered (utility function and population distribution over space), we cannot prove the uniqueness of the threshold Θ* without imposing some additional restrictions. However, in the case of a uniform population distribution, it can be shown that the surplus on interval [equation image, equation image] is first increasing and then decreasing. The threshold Θ* is then unique whatever the shape of the utility function.

3.3. Scarcity Rent

[24] Using the results of proposition 1, we can rewrite the necessary conditions of optimality (4) and (5) as

equation image

where the scarcity rent λ* and the optimal threshold Θ* are determined as the solution of the following system of equations:

equation image

The optimal exploitation of the aquifer is such that the marginal utility of water is equal to the marginal social cost which includes the unit cost of extraction and the scarcity rent. The scarcity rent λ gives the value of an additional unit of aquifer resource available from Θ. The right-hand side term of the second equation (12) comes from the aquifer spatial availability constraint. If the initial flow, entering the aquifer, that can be pumped at X0 (i.e., equation image0) were available at any point of the aquifer, then this term would be null since in that case dequation image/dx = 0. The scarcity rent associated to the aquifer would be equal to the cost differential between the substitute and the aquifer, i.e., γ − α. This is a well-known result of the natural resource economic literature. However, since dequation image/dx > 0, the right-hand side term is negative. Thus we have λ* < γ − α and the scarcity rent is lower than the cost differential. The aquifer resource must be attributed a scarcity rent because first, this resource is available in limited quantity and second, upstream withdrawals oblige downstream users to switch to the more costly substitute. However, reducing withdrawals at some location does not make this water available downstream because the maximum available flow decreases when one approaches the coast. This effect constitutes a social incentive to use the aquifer where it is available. It depresses the scarcity rent and hence we have λ* < γ − α. Last, notice that the water consumption of a representative individual admits a discontinuity at Θ*. Water consumption drops from U−1(α + λ*) down to U−1(γ).

3.4. Decentralized Use of Water Resource

[25] We are now interested in the exploitation of these two resources within the framework of a decentralized economy where each user maximizes his private surplus. A consequence of the spatial structure of this model is that upstream users can oblige users located in the coastal area to use the expensive substitute, the opposite not being true. Users are thus in very asymmetric situations according to their location. Let us assume that all users pump the aquifer without taking into account its scarcity. The non cooperative extraction rate for a user having a direct access to the aquifer is ync defined by U′(ync) = α. We thus have ync > y*. We are facing the traditional problem of common resource exploitation with possibility of exclusion of some users. Let Θnc denotes the most downstream individual using the aquifer in a decentralized situation. This threshold is defined by the following equality: equation imagenc) = [1 − Fnc)]ync. As Θnc > Θ*, decentralized exploitation of the aquifer leads to overexploitation. A Pigouvian tax τ = λ* per unit of flow withdrawn allows to restore the optimum. The unitary price is however differentiated over space. The price paid by an individual who uses the aquifer α + λ* is lower than the one faced by users of the substitute γ. The tax revenue collected by the regulator can be used in order to subsidize the substitute via lump-sum transfers. However, the average price for the substitute excluding the lump-sum transfer will not be equal to the aquifer water price, in general.

4. Optimal Use of Resources With Endogenous Pumping Location

4.1. Optimal Location of the Pumping Point

[26] Given a total pumping rate Y from the aquifer, the optimal location of the pumping point is

equation image

(hereinafter referred to as lemma 3). The intuition is straightforward. The more downstream users wishing to be supplied from the aquifer minimize their surface transportation cost by locating their pumping point at equation image(Y), the nearest point to their location from where the aquifer resource is available. The more upstream users who have a direct access to the aquifer pump where they are located.

[27] Let Δ denote the nearest point to the coast at which the aquifer is available. We define now the downstream and upstream users as those respectively located at x ∈ [0, Δ] and x ∈ [Δ, X0]. By definition, the constraint equation image(Δ) − imageequation image(x) f (x)dx ≥ 0 must hold in order to make water from the aquifer available from Δ.

[28] The program of the social planner writes:

equation image

The first term of aggregate welfare is the net surplus of users who bear a surface transportation cost. The second term corresponds to the net surplus of users who do not bear such a cost.

4.2. Optimal Use of the Two Resources

4.2.1. Substitute is Too Costly to be Used

[29] We first characterize the optimal exploitation of the aquifer when the substitute is too expensive to be used. In this case we have equation image(x) = 0, ∀ x ∈ [0, X0]. The necessary conditions of optimality give:

equation image
equation image
equation image

Let us notice that since water is assumed to be an essential good, we have at the optimum equation image(x) > 0 ∀ x ∈ [0, X0] and thus η(x) ≡ 0. At the optimum, users who do not bear any transportation costs are allocated the same rate of withdrawal that takes into account the cost externality they create on downstream users. The withdrawal rate allocated to the users who must transport water on the surface decreases as we go closer to the coast since the marginal cost of the resource increases. Remember that λ is the multiplier associated to the spatial availability constraint. It measures the value of an additional unit of aquifer available from Δ. Condition (14) means that at the optimum the value in situ of an additional unit of aquifer at Δ multiplied by the effect on the quantity available induced by a marginal modification of Δ, must be equal to the marginal cost borne by the downstream users, i.e., β ∫0Δequation image(x) f(x)dx. Let (equation image*w(x), λ*w, Δ*w) be the optimum solution when there is no substitute. This solution is also the optimum in the presence of a substitute if the unit cost of the substitute is higher than the social marginal cost of the aquifer for all the users. The social marginal cost of the aquifer resource is maximum for x = 0. One should not used the substitute if γ > α + βΔ*w + λ*w = equation image.

4.2.2. Substitute is Moderately Costly, α < γ ≤ equation image

[30] If α < γ ≤ equation image, it is now optimal for users in the coastal zone to use the substitute. We denote by Θ ∈ [0, X0] the threshold location such that the downstream users (where downstream is now relative to Θ) exclusively use the substitute. Their individual consumption is z* and their aggregate surplus is: SDW(Θ) = F(Θ)[U(z*) − γz*].

[31] The program of the social planner is

equation image

Pointwise Lagrangian maximization with respect to equation image and equation image yields:

equation image
equation image

The first-order conditions with respect to Θ and Δ are:

equation image
equation image

Condition (19) means that the value in situ of one unit of aquifer resource multiplied by the variation of the available resource induced by a marginal modification of Δ must be equal to the marginal cost paid by users who transport this resource, i.e., those located between Θ and Δ. An increase of Δ allows to increase the total rate of withdrawal. This has a positive effect on welfare. However, the increase in Δ results in higher transportation costs and so in a negative effect on social welfare. Condition (18) means that at the optimum, as long as the constraints Θ ≥ 0 and X0 − Θ ≥ 0 are not binding, there is no jump of utility at Θ. Last, the complementary slackness conditions are:

equation image
equation image
equation image
equation image
equation image
equation image
equation image

[32] Let us analyze in a more precise way the optimal use of water sources according to the location of consumers. If the substitute is abundant and moderately costly, α < γ ≤ equation image, each user uses only one type of resource (hereinafter referred to as lemma 4). If the substitute is abundant and moderately costly, α < γ ≤ equation image, users x ∈ [Δ, X0] and x ∈ [Θ, Δ] use exclusively the aquifer, without and with surface transportation respectively. Those located on segment [0, Θ] use the costly substitute (hereinafter referred to as lemma 5).

[33] Demonstrations of lemmas 4 and 5 are similar to that of lemma 1 and are available from the authors upon request. The intuition is the following one. The aquifer is less costly than the substitute. Thus it is socially costly not to consume a unit of the aquifer resource where it is available. When a consumer reduces its withdrawal from one unit, the water available downstream is smaller than this unit because of the spatial availability constraint (equation image(x) decreases as we move toward the coast). This creates an incentive to use the aquifer where it is available. The two preceding lemmas imply that the optimal structure of withdrawals is as represented in Figure 2.

Figure 2.

User location and type of water resource.

[34] We can then characterize the optimal exploitation of the two water resources by rewriting the necessary optimality conditions. The optimal exploitation of the two resources is summarized in proposition 2 as follows.

[35] If the substitute is abundant and moderately costly, α < γ ≤ equation image, the optimum {equation image*(x), equation image*(x), equation image*(x), λ*, Θ*, Δ*} is:

equation image

with

equation image

[36] At the optimum, aquifer's users must internalize the externalities they create on the other users. Their marginal utility must be equal to the social marginal cost including the scarcity rent. The rent is such that the value in situ of one aquifer water unit multiplied by the variation of resource resulting from the marginal modification of Δ* is equal to the marginal cost borne by the users who transport this resource, i.e., those located in [Θ*, Δ*]. To conclude, notice that once again given the general assumptions on the utility function, we cannot demonstrate the uniqueness of the thresholds Θ* and Δ*.

4.3. Decentralized Management

[37] In a decentralized economy the users do not take into account the resource scarcity. They equate their marginal utility to their private marginal cost. The noncooperative exploitation of water resources {equation imagenc(x), equation imagenc(x), equation imagenc(x)} is characterized by:

equation image
equation image
equation image

and the critical thresholds Δnc and Θnc satisfy the equations:

equation image
equation image

Comparing equations (21)(25) to proposition 2, it can easily be shown that Δnc ≥ Δ*. It means that the decentralized management of the water system leads to a global over-exploitation of the aquifer. A corollary is that the proportion of users supplied from the aquifer without transportation (i.e., users located on the interval [Δnc, X0]) is suboptimal in a decentralized economy. In addition, Δnc − Θnc ≥ Δ* − Θ*: aquifer water is transported over too long distances in a decentralized economy. Last, depending on the value of model's parameters we can have either Θnc ≥ Θ* or Θnc < Θ* which means that the proportion of population using the substitute is not optimal in most of the cases. Hence decentralized exploitation leads to two sources of inefficiency. First the aquifer is over-exploited and second, the allocation of users on the two resources is not optimal.

[38] Surprisingly, a constant tax per unit of withdrawal τ = λ*, still allows to restore the optimum. The reason is that the repartition of users on the different water resources is the same in a decentralized economy and at the optimum: users in the upstream area use the aquifer directly and those located downstream use the substitute and intermediate populations use the aquifer with a transportation cost. Hence implementing the optimum is possible by making the thresholds (Δ*,Θ*) and (Δnc, Θnc) coincide and the individual water consumptions identical. These two conditions can be achieved through a single tax since the scarcity rent associated to the aquifer resource is the same whatever the location considered. Last, although the tax per unit of withdrawal is constant, the price of the water varies according to the location on the aquifer. The unit price of water is equal to α + λ* for users who use the aquifer without surface transportation, it is equal to α + β(Δ* − x) + λ* for those who must bear a transportation cost and it is equal to γ for users who must use the costly substitute.

5. Simulation of the Model

[39] In order to illustrate the above results, we simulate the optimal exploitation of the two resources when users can choose the location of their pumping point on the aquifer.

5.1. Specification of the Model

[40] Users are distributed on the surface of the aquifer according to a uniform distribution, F(x) = x/X0 and f(x) = 1/X0, ∀ x ∈ [0, X0]. The utility function of a representative user is assumed to be quadratic: U(w) = equation imagemw2 + nw with m = −1 and n = 6000. The unit cost of extraction of the aquifer α is 2000 monetary units. The unit cost of the costly substitute γ varies from 2000 to 5000 monetary units. The physical parameters describing the coastal aquifer and allowing to parameterize the function equation image(x) are the following. The hydraulic conductivity k in meter per day is equal to 40, the recharge R in cubic meter per day and per kilometer is equal to 2500, X0 is equal to 25 km, H0 is equal to 50 m and a, the density differential between seawater and freshwater measured in kilogram per liter is 0.035.

5.2. Optimum

[41] At the optimum, the downstream users x ∈ [0, Θ*] consume the substitute:

equation image

Users x ∈ [Θ*, Δ*] use the aquifer with a transportation cost from Δ*:

equation image

Finally, users x ∈ [Δ*, X0] use the aquifer resource without transportation:

equation image

The scarcity rent and the critical thresholds are defined by:

equation image

5.3. Decentralized Exploitation

[42] In a decentralized economy the downstream users x ∈ [0, Θnc] use the substitute:

equation image

Users x ∈ [Θnc, Δnc] use the aquifer with transportation from Δnc:

equation image

Users x ∈ [Δnc, X0] use the aquifer without transportation:

equation image

The critical thresholds are defined by:

equation image

5.4. A Comparison of the Optimal and the Decentralized Exploitation

[43] Figures 3 and 4 present the solutions of equation image4 and equation image5 in the (Θ, Δ)-plane. These numerical solutions are parameterized by the unit cost of the substitute, γ. The comparison between the optimal and the decentralized exploitation leads to the following remarks. First, whatever the level of the cost of the substitute, we have Δnc ≥ Δ*: decentralized exploitation leads to an overexploitation of the aquifer. Moreover, in the decentralized case the threshold Δ is an increasing function of the cost of the substitute whereas it is decreasing at the optimum. In a decentralized economy, an increase in the cost of the substitute makes the aquifer more profitable. Thus some users who previously used the substitute may prefer to use the aquifer with a surface transportation cost. As the cost of the substitute increases, some users switch toward aquifer extraction. As a result, the rate of aquifer extraction increases with γ. It necessarily follows that Δnc increases with γ. In the case of an optimal aquifer exploitation, an opposite effect is internalized by aquifer users. At the optimum, the cost of the substitute increase results in an increase of the value in situ of the aquifer water (which is now more profitable). This corresponds to an increase of the scarcity rent associated to the aquifer. This effect, internalized by all aquifer users, tends to reduce the rate of withdrawal from this resource. It follows that in the case of an optimal resource exploitation, the threshold Δ* may or may not increase with γ. In Figure 3, Δ* is a decreasing function of the unit substitute cost.

Figure 3.

Thresholds Θ* and Δ* as a function of the unit cost of the substitute, optimal exploitation.

Figure 4.

Thresholds Θnc and Δnc as a function of the unit cost of the substitute, decentralized exploitation.

[44] Let us also note that Θnc ≥ Θ*, which means that in a decentralized economy, too many users use the substitute. Finally since Δnc − Θnc ≥ Δ* − Θ*, too many users bear a transportation cost on a too long distance in the decentralized economy.

[45] Table 1 shows how the scarcity rent evolves according to the unit cost of the substitute. The rent is increasing with the cost of the substitute: the more costly the substitute is, the higher is the value in situ of the aquifer water. For an upstream user who only bears directly the cost of extraction, the rent may represents 45% of its social marginal cost (when γ = 5000 monetary units). For a user located in Θ* who bears the largest cost of transportation, it can represent 32, 2%. When aquifer water is scarce and when the substitute is very costly, the rent represents a significant share of the social marginal cost. Thus the decentralized exploitation of the aquifer results in a loss of surplus. As mentioned previously, this is the consequence of two sources of inefficiency. The first is the overexploitation of the aquifer. The second is the nonoptimal distribution of the users on the two water resources.

Table 1. Share of the Scarcity Rent to the Cost
Substitute Cost γRent λ*Upstream User,a %User x ∈ [Θ*, Δ*],b %Downstream User,c %
  • a

    Ratio of the rent to the social marginal cost for an upstream user, λ*/(α + λ*).

  • b

    Ratio of the rent to the social marginal cost for x ∈ [Θ*, Δ*], λ*/(α + β(Δ* − x) + λ*).

  • c

    Ratio of the rent to the social marginal cost for a downstream user, λ*/γ.

20000000
250038816.215.5–16.215.5
300073426.824.5–26.824.5
350010323429.5–3429.5
400012793932–3932
4500147242.432.7–42.432.7
5000161244.632.2–44.632.2

6. Conclusion

[46] The main objective of this work was to identify the specific problems posed by the optimal management of an aquifer under seawater intrusion when users can also be supplied from a substitute. In such aquifers, an increase in the withdrawal rate results in moving away from the coast the last feasible pumping point. This relation between extraction rate and location of pumping points generates a specific externality. We have analyzed the effects of this externality on the optimal use of the aquifer and the substitute. When the users cannot choose the location of their pumping point on the aquifer, the aquifer must be attributed a specific scarcity rent. We have shown that this scarcity rent is lower than the cost differential between the aquifer and the substitute. An efficient pricing of water must then take into account this rent. Implementing the optimum in a decentralized economy in which agents maximize their private surplus requires a uniform royalty on the aquifer. When users can choose their pumping location, the decentralized exploitation leads to two sources of inefficiencies. The first is the overexploitation of the aquifer and the second is the non optimal allocation of users on the two resources. A uniform royalty allows to correct these inefficiencies. However, the price of the water paid by users is differentiated according to their location. Last, the simulations suggest that the scarcity rent may be significant even in the case of a low cost differential between water from the aquifer and from the substitute.

[47] It is clear that a limit of the analysis suggested in this paper is the stationary framework. We have characterized the long-run stationary equilibrium (both in terms of water quantities and pumping location) but we do not analyze any dynamic path toward this equilibrium. Introducing the dynamics of seawater intrusion would allow to extend the analysis in many interesting directions. First, it would allow to consider intertemporal tradeoffs such between current withdrawals and future pumping. Second, it could be optimal for some users to have multiple pumping wells at different locations. In case of a low natural water recharge, having multiple wells could mitigate the pumping and transportation cost externalities. This kind of switching well analysis however requires to carefully model how the users invest in pumping wells.

Appendix A

A1. Demonstration of Lemma 1

[48] Let us first show that the aquifer pumping rate of an upstream user is necessarily positive. Let us assume that there exists at the optimum an interval Iequation image [Θ, X0] such that equation image(x) = 0, equation imagexequation imageI. Users on this interval necessarily use the substitute at a level z*. Since we are at the optimum, the aquifer total pumping is equation image(Θ) otherwise social surplus could be increased by making some users located downstream Θ use the less expensive aquifer. In addition ∀I, we can define J = [Θ, equation image] such that: F(J) = F(I). In other words, equation image is such that the mass of users on I is identical to the mass on J. Social surplus remains the same when allocating a flow y* of aquifer resource to users on I and a flow z* of substitute to those on J, the total flow withdrawn from the aquifer still being equation image(Θ) whereas the most downstream user of the aquifer is now equation image > Θ. It is then possible to strictly improve social surplus by allocating to some users downstream equation image the aquifer resource which is available since equation image(equation image) > equation image(Θ). The aquifer pumping rate for an upstream user is necessarily positive, the slack condition (7b) then gives η(x) = 0 for all then x ∈ [Θ, X0]. The first-order condition (4) is on this interval is U′(equation image(x) + equation image(x)) = α + λ, i.e.: equation image(x) + equation image(x) ≡ equation image, ∀ x ∈ [Θ, X0].

A2. Demonstration of Lemma 2

[49] On interval [x, equation image], aggregate surplus is strictly increasing with Θ. On this interval all upstream users consume water at a rate z*. By marginally increasing Θ, we increase the flow of aquifer water available for consumption without changing the consumption per capita of users: we marginally replace the costly substitute by the aquifer. The total cost of water supply decreases with Θ: aggregate surplus is thus increasing on this interval. On interval [equation image, equation image], the surplus change is unspecified: S′(Θ) = f(Θ)[(U(z*) − γz*) − (U(equation image*(Θ)) − αequation image*(Θ))] + (1 − F(Θ))[equation image′*(Θ)(U′(equation image*(Θ)) − α)]. A marginal increase of Θ prohibits direct access to the aquifer for users Θ. The effect on aggregate surplus (first term between brackets) is negative. However, the quantity of resource shared by upstream users increases. This results in a positive effect on the total surplus (second positive term between brackets). The effect on total surplus is thus ambiguous. On the interval [equation image, x0], the surplus decreases with Θ. On this interval, upstream users exclusively withdraw the aquifer resource at an unconstrained rate y*. A marginal increase of Θ prohibits access to the aquifer for users located at Θ and does not allow to increase the individual aquifer consumption of upstream users. Since equation image*(.) is a continuous function of Θ, social aggregate surplus is also a continuous function of Θ. From the preceding analysis, we can conclude that social surplus reaches its maximum on [equation image, equation image].

A3. Demonstration of Lemma 3

[50] Let the total withdrawal rate from the aquifer be Y, Y ∈ [0, equation image0]. By definition of the spatial availability constraint, the resource cannot be pumped downstream a critical position equation image(Y) which increases with Y. The surface transportation cost minimization imposes that users x ∈ [equation image(Y), X0] pump this resource where they are located. By doing this, they do not bear any transportation cost. Users x ∈ [0, equation image(Y)] cannot have a direct access to the aquifer. They have to be supplied from the nearest point to their location where the aquifer resource is available, that is, from equation image(Y).

Ancillary