Hubert Stahn
email: hubert.stahn@univ-amu.fr

Agnes Tomini
email: agnes.tomini@univ-amu.fr

On the Environmental E¢ ciency of Water Storage: The Case of a Conjunctive Use of Ground and Rainwater

Keywords: groundwater management, rainwater harvesting, optimal control, conjunctive use JEL classi…cation: Q15, Q25, C61, D61

Rainwater harvesting, consisting in collecting runo¤ from precipitation, has been widely developed to stop groundwater declines and even raise water tables. However this expected environmental e¤ect is not self-evident. We showin a simple setting that the success of this conjunctive use depends on whether the runo¤ rate is above a threshold value. Moreover, the bigger the storage capacity, the higher the runo¤ rate must be to obtain an environmentally e¢ cient system. We also extend the model to include other hydrological parameters and ecological damages, which respectively increase and decrease the environmental e¢ ciency of rainwater harvesting.

Introduction

The depletion of groundwater has generated new interest in the oldest human technology developed to provide a water supply: rainwater harvesting systems. The …rst evidence of the collection of rainwater dates back 6,000 years in the Gangsu region of China, which has a semi-arid to arid continental climate [START_REF] Gautier | Oil, Water, and Climate : An Introduction[END_REF]. Rain storage was also particularly important in Southern India, where dams were built by the villagers to capture rainwater from the monsoons [START_REF] Ranganathan | Modeling Economics of Conjunctive Surface an Groundwater Irrigation Systems[END_REF], in small islands with no signi…cant river systems [START_REF] Krishna | Cistern Water Systems in the US Virgin Islands[END_REF] and in remote and arid locations [START_REF] Perrens | Collection and Storage Strategies for Domestic Rainwater Systems in Australia, Hydrology Papers[END_REF]. Although this traditional technique was largely abandoned in favor of large-scale projects, it has enjoyed a revival in popularity since the early 20th century. For instance, Gibraltar has one of the largest rainwater collection systems in existence and rainwater is still the primary water source on many US ranches. Furthermore, the United Nations Environment Program 3 reported that in 2000, the Gansu province in China had built 2,183,000 rainwater tanks with a capacity of 73.1 million cubic meters, supplying drinking water and supplementary irrigation. More recently, the Rainwater Partnership, registered with the UN Commission on Sustainable Development, was established in 2004 to promote this practice and its integration into water resource management policies. In parallel, a lot of international cooperation encourages rainwater harvesting by …nancing projects all around the world. For example, Canada …nanced the "Hebei Dryland Project" in China between 1989 and 2000, while the UN Food and Agriculture Organization supported projects in Ghana, Kenya, Tanzania and Zambia, and more recently a cooperative project between Germany and Tunisia was carried out. 4This renewed popularity is at least partially due to the fact that this technology is often presented as the simplest and most a¤ordable way to stop groundwater declines and even raise water tables [START_REF] Roebuck | A Whole Life Costing Approach for Rainwater Harvesting Systems[END_REF]. However, to the best of our knowledge, no study, except Stahn and Tomini [START_REF] Stahn | Rainwater harvesting under endogenous capacity of storage: a solution to aquifer preservation?[END_REF], examines whether this solution is actually environmentally acceptable, in terms of both reducing extraction and raising water tables.

However, because, the ‡ow of collected rainwater is constrained by the capacity of the storage technology,. rainwater harvesting can only substitute for part of existing water sources. This capacity -constraint implies that groundwater may still be extracted as a supplemental source of water to meet water demands. We should therefore consider rainwater harvesting with some caution within the context of conjunctive use management.

Moreover, this method is dependent on local conditions like precipitation and surface runo¤ [START_REF] Boers | A review of rainwater harvesting[END_REF]. since rainwater harvesting involves either collecting rainfall directly where it falls or collecting the runo¤ originating from it. This allows to mitigate water losses and consequently increases water supplies. However, this additional water supply may lead to a decline in the value of groundwater, since the dependence on its stock decreases, and this may in turn encourage people to increase current extraction from the ground and thereby contribute to a long-term decline in the water table. At the opposite extreme, when all water runs overland, meaning that the aquifer is nonrenewable, some stock of reserve will remain in the ground when rainwater is collected at the surface, whereas if groundwater was the only water supply, the aquifer would be depleted. As local conditions may vary greatly from one area to another, the environmental success or failure of this technology is therefore far from self-evident.

The purpose of this paper is to extend the literature on conjunctive use to adequately consider the environmental potential of rainwater harvesting. Our results will di¤erentiate between the situations in which conjunctive use has a positive long-term impact on the water table, and those in which it does not. More precisely, the discussion is formulated to compare the long-term water table level resulting from optimal conjunctive use of groundwater and rainwater storage with the level when no rainwater is collected. For optimal use, the collection of rainwater must begin before the steady state is reached, and it must be e¢ cient to use both water sources simultaneously over the long run. Our model allows us to de…ne the conditions under which such a conjunctive use optimally emerges. We then show that below a threshold value of runo¤, rainwater harvesting is economically optimal but not environmentally e¢ cient, because the water table is lower than it would be without any rainwater storage. To re…ne the analysis, we …rst introduce our results in a basic setting, and then extend our framework to include the hydrological in ‡uences of other parameters a¤ecting storage and the recharge of the aquifer, as well as ecological damages.

The rest of the paper is organized as follows. In the next section, we provide some background on models of groundwater management, which provide various explanations of aquifer depletion. Sections 3 and 4 introduce the model of conjunctive use of groundwater and rainwater under a capacity constraint in a basic setting, and derive the optimal use of groundwater when there is no collection of rainwater. Section 5 presents an analysis of environmental e¢ ciency with an implicit determination of the threshold value of runo¤ rates above which rainwater harvesting allows to raise the water table over the long term. Sections 6 and 7 derive some extensions to incorporate improvements regarding the water system and ecological damages. Finally, section 8 provides some concluding remarks. An appendix contains all the proofs. They work out in a completely integrated case which covers the results of the di¤erent sections5 .

Background of groundwater management

Groundwater management has generated a substantial literature in resource economics. Since the seminal work of Gisser and Sánchez [START_REF] Gisser | Competition versus Optimal Control in Groundwater Pumping[END_REF], many studies have focused on the e¤ects of the absence of central control (e.g., Allen and Gisser [START_REF] Allen | Competition versus Optimal Control in Groundwater Pumping when Demand is Nonlinear[END_REF], Feirnermann and Knapp [START_REF] Feinerman | Bene…ts from Groundwater Management: Magnitude, Sensitivity, and Distribution[END_REF] or more recently Koundouri [START_REF] Koundouri | Potential for Groundwater Management: Gisser-Sánchez E¤ect Reconsidered[END_REF]). Such absence intuitively leads to the depletion of groundwater, because agents fail to take various externalities into account (Provencher and Burt [START_REF] Provencher | The externalities associated with the common property exploitation of groundwater[END_REF]). Most studies compare myopic farmers, under perfect competition, with a social planner's solution focusing on the stock externality (Feirnerman and Knapp [START_REF] Feinerman | Bene…ts from Groundwater Management: Magnitude, Sensitivity, and Distribution[END_REF], Nieswiadomy [START_REF] Nieswiadomy | The Demand for Irrigation Water in the High Plains of Texas, 1957-1980[END_REF], Brill and Burness [START_REF] Brill | Planning versus Competitive Rates of Groundwater Pumping[END_REF]). Others address strategic behavior (Negri [23], Provencher and Burt [START_REF] Provencher | The externalities associated with the common property exploitation of groundwater[END_REF] or Rubio and Casino [START_REF] Rubio | Competitive versus E¢ cient Extraction of a Common Property Resource: The Groundwater Case[END_REF]). However, the magnitude of the adverse consequences on resource stocks and the potential bene…ts from groundwater management are still controversial.

Another part of the literature deals with issues of groundwater quality; either from a pollution control perspective (Vickner et al. [START_REF] Vickner | A dynamic Economic Analysis of Nitrate Leaching in Corn Production under Nonuniform Irrigation Conditions[END_REF], Yadav [START_REF] Yadav | Dynamic optimization of nitrogen use when groundwater contamination is internalized at the standard in the long run[END_REF] or Xepapadeas [START_REF] Xepapadeas | Environmental policy design and dynamic nonpoint-source pollution[END_REF]) or explaining the relationship between contamination and water-use decisions (Roseta-Palma [START_REF] Roseta-Palma | Groundwater management when Water Quality is Endogenous[END_REF], Hellegers et al. [START_REF] Hellegers | Dynamics of Agricultural Groundwater Extraction[END_REF]). This last set of studies highlights additional externalities due to the e¤ect of pumping on aquifer contamination. Roseta-Palma [START_REF] Roseta-Palma | Groundwater management when Water Quality is Endogenous[END_REF] shows that the steady state value of the water table level is always higher in quantity-quality models than in quantity-only models.

Finally, another strand of the literature investigates the optimal management of groundwater under the conjunctive use of di¤erent water resources: surface water and groundwater (e.g., Burt [6], Provencher [START_REF] Provencher | Issues in the Conjunctive Use of Surface Water and Groundwater[END_REF], Tsur [START_REF] Tsur | The Economics of Conjunctive Ground and Surface Water Irrigation System : Basic Principles and Empirical Evidence from Southern California, in Decentralization and Coordination of Water Resource Management[END_REF]), multiple aquifers (Zeitouni and Dinar [START_REF] Zeitouni | Mitigating negative water quality and quality externalities by joint management of adjacent aquifers[END_REF], Roumasset and Wada [START_REF] Roumasset | Ordering the extraction of renewable resources: The case of multiple aquifers[END_REF]), or rainwater and groundwater (Stahn and Tomini [35]). However, most of the models that analyze the conjunctive use of groundwater and surface water di¤er very little from typical groundwater-only models. They assume that groundwater is the single control variable used to supplement a …xed amount of surface water, as a back-up supply. There are a handful of papers that explore an additional externality when surface water is stochastic (Knapp and Olson [START_REF] Knapp | The Economics of Cojunctive Groundwater Management with Stochastic Surface Supplies[END_REF], Gemma and Tsur [START_REF] Gemma | The Stabilization Value of Groundwater and Conjunctive Water Management under Uncertainty[END_REF], Tsur and Graham-Tomasi [START_REF] Tsur | The Bu¤er Value of Groundwater with Stochastic Surface Supplies[END_REF]). Ignoring the role of the resource as a bu¤er creates ine¢ ciencies and therefore aquifer depletion. Moreover, few papers consider water sources of di¤erent qualities. Dinar and Xepapadeas [START_REF] Dinar | Regulating water quantity and quality in irrigated agriculture[END_REF] empirically show that an individual monitoring regime is superior to central monitoring in preventing degradation of the quality and depletion of the quantity of water in the aquifer when farmers use water from a central surface supply and simultaneously extract from the ground.

However, neither the early literature nor more recent studies consider the hydrologic link between groundwater and surface water. Since conjunctive use is often considered in discussing water conservation and the harmonious use of water resources to avoid undesirable e¤ects (FAO [START_REF] Fao | Land and water integration a driver basin management[END_REF]), this caveat deserves more attention. To the best of our knowledge, few papers study the interactions between several di¤erent water supplies. Zeitouni and Dinar [START_REF] Zeitouni | Mitigating negative water quality and quality externalities by joint management of adjacent aquifers[END_REF] explore the contamination ‡ows between several connected aquifers. Burness and Martin [START_REF] Burness | Management of a tributary aquifer[END_REF] consider the links between groundwater and surface water, in their modeling of the instantaneous rate of aquifer recharge caused by groundwater pumping, by river e¤ects. They model these e¤ects asas externalities that increase intensive pumping. Azaiez [START_REF] Azaiez | A model for conjunctive use of ground and surface water with opportunity costs[END_REF] develops a multi-stage model to determine how much surface water must be imported for irrigation and to arti…cially recharge the aquifer, which is also and simultaneously used for irrigation. Pongkijvorasin and Roumasset [START_REF] Pongkijvorasin | Optimal Conjunctive Use of Surface and Groundwater with Recharge and Return Flows: Dynamic and Spatial Patterns[END_REF] develop an optimal control model where groundwater users enjoy positive externalities from canal conveyance loss and from return ‡ows to the aquifer from irrigation. But, these studies still consider surface and groundwater as two separate entities, only related by the fact that some surface water is lost to deep percolation or is used for aquifer recharge, representing positive externalities on the aquifer. None of them considers the idea of both water resources coming from the same source, so that the use of one may limit the availability of the other. Such situations obviously occur when rainwater is harvested directly, because it prevents replenishment of the aquifers; but generally speaking they can also arise when water is diverted from rivers that contribute to groundwater recharge 6 or with dam constructions that stop ‡ows towards the downstream parts of water basins, where aquifers may exist. We help to …ll this gap in the literature by closely linking groundwater and rainwater harvesting. As a consequence, this model contributes to a deeper understanding of conjunctive use and complements the existing literature by analyzing two additional features: hydrological relationships and ‡ow constraints of surface water.

Modeling the conjunctive use of rainwater and groundwater

At each time period t, water can be abstracted from the system by collecting rainwater, w r (t), and extracting groundwater, w g (t). Rainwater is directly captured from precipitation, R, falling within the water basin, but the storage of this water is limited by the capacity of the available technology, w r W . We also assume that it is not possible to collect the entire precipitation: W < R. Consequently, part of the remaining rain ‡ows over the land as surface runo¤, with 2 (0; 1) denoting the runo¤ coe¢ cient, while the other part replenishes the aquifer. The dynamics of the groundwater elevation, h, can then be written:

_ h = (1 ) (R w r (t)) w g (t) (1)
The total cost of collecting rainwater is K w r , where K > 0 is the unit cost. Following the literature, the cost of extracting w g units of groundwater depends on the stock and is given by C(h)w g . As usual, the unit pumping cost, C(h), is decreasing with water elevation at a decreasing rate. This means that the dependence of the extraction cost on the stock, C 0 (h)w g , increases with higher water head levels.

C 0 (h) < 0; C 00 (h) 0 (2)
The capacity of the aquifer is known with certainty and the maximum elevation is denoted by h > 0. At this level, the unit and marginal pumping costs are zero, while for an empty aquifer, the unit cost becomes very large.

C( h) = C 0 ( h) = 0; lim h!0 C(h) = +1 (3)
Total water abstraction,w = w g + w r , is used for irrigation. As usual in the literature, irrigation demand is a positive, continuous and strictly decreasing function of price, P . We can then deduce that social bene…ts from water consumption are given by the area under the inverse demand curve, P (w) net of water costs. We assume that the marginal social bene…t is decreasing with water consumption at an increasing rate:

P 0 (w) < 0; P 00 (w) 0 (4)
The social bene…t from water consumption, net of costs, is thus given as follows:

Z w 0 P (x)dx C(h(t))w g (t) Kw r (t) (5)
We can now analyze the optimal management of these interconnected water resources.

The optimal use of groundwater with rainwater storage

The social planner chooses the optimal consumption of both water resources in order to maximize the sum of discounted net bene…ts (Eq 5), given the law of groundwater dynamics (Eq 1) and subject to the constraints that both extraction (denoted GW hereafter) and rainwater collection (denoted RW hereafter) are non-negative and that RW is upper-bounded by the capacity limitation. Given a discount rate > 0, the problem can be written as follows:

max wg(t);wr(t) R +1 0 exp t R wg(t)+wr(t) 0 P (x)dx c(h(t))w g (t) Kw r (t) dt w.r.t 8 < : _ h = (1 ) (R w r (t)) w g (t) w g (t) 0; W w r (t) 0 h 0 given and h(1) free (6)
The current value of the Hamiltonian for this optimal management problem is:

H = Z (wg(t)+wr(t)) 0 P (x)dx c(h(t))w g (t) Kw r (t)+ (t) ((1 ) (R w r (t)) w g (t)) (7)
where (t) 0 is the shadow value of a unit of water held in the aquifer. The social planner seeking optimal water consumption has three options to choose from: only GW, only RW, or a combination of both. Let us add these additional constraints on controls, w g (t) and w r (t), to de…ne the Lagrangian of the Hamiltonian (Eq 7):

L = H + g (t) w g (t) + r (t) w r (t) + W (t) (W w r (t)) (8)
where g (t) and r (t) are the Lagrangian multipliers associated with the non-negativity constraints on, w g (t) and w r (t) respectively, and W (t) that associated with the capacity constraint W . This formulation leads us to identify three possible regimes. 7 Optimal water uses must therefore be chosen to satisfy the following …rst order conditions:

P (w g (t) + w r (t)) C (h(t)) (t) + g (t) = 0 (9) P (w g (t) + w r (t)) K (1 ) (t) + r (t) W (t) = 0 (10)
g (t)w g (t) = 0; r (t)w r (t) = 0; W (t) (W w r (t)) = 0 (11) g (t); r (t); W (t) 0; w g (t) 0; and w r (t) 2 [0; W ] (12)
Condition (Eq 9) represents the optimality condition, which states that the marginal bene…t of GW is equal to the marginal social cost of GW extraction and its corresponding Lagrange multiplier. The social cost of GW extraction, or the full marginal cost of GW, is determined by the sum of the extraction cost and the shadow value of groundwater resource. Likewise, condition (Eq 10) represents the optimality condition for RW, taking into account the implicit price associated with the capacity constraint on RW harvesting. Conditions (Eq 11) and (Eq 12) correspond to the standard slackness conditions. We must also take into account the dynamics of the resource and its shadow price, characterized by the two following equations:

_ h(t) = (1 ) (R w r (t)) w g (t) (13) _ (t) = (t) + C 0 (h(t)) w g (t) (14)
Let us …rst explain the structure of the stationary solution. RW can be thought of as a backstop resource for GW, since it is a perfect substitute that is developed as GW gets depleted. Remember that for full aquifers, the unit pumping cost is negligible (see Eq 3). Moreover, the concepts of backstop resource and the ordering of resource extraction are already established in the literature (Koundouri and Christou [START_REF] Koundouri | Dynamic adaptation to resource scarcity and backstop availability : Theory and Application to Groundwater[END_REF], Krulce et al. [START_REF] Krulce | Optimal Management of a Renewable and Replaceable Resource: The Case of Coastal Groundwater[END_REF], Roumasset and Wada [START_REF] Roumasset | Ordering the extraction of renewable resources: The case of multiple aquifers[END_REF]). As a consequence, we know that optimality requires resources to be used following a least-cost-marginal-opportunity-cost rule …rstextraction rule, where the marginal opportunity cost includes extraction and marginal user costs. As such, the optimal steady state may or may not entail groundwater use with RW harvesting. Typically, RW harvesting will not be used as long as the marginal bene…t of water consumption is lower than the RW marginal opportunity cost at the steady state. However, we know that the water table declines before the steady state is reached. If the water table falls to a level such that the marginal pumping cost is higher than the harvesting cost, i.e., c(h) > K, but the marginal bene…t of water consumption is higher than the GW marginal opportunity cost, we know for sure that we will end up with the conjunctive steady state in which RW harvesting is used at capacity. 8 This discussion is summarized in the following su¢ cient condition for the emergence of a conjunctive use system:

P (R) > K 1 C 0 C 1 (K) (R W ) (15)
Given this emergence condition (Eq 15) 9 , we know that RW harvesting is economically feasible. We can now analyze equations ( 9) to [START_REF] Gale | The Jacobian Matrix and Global Univalence of Mapping[END_REF] in order to characterize the steady state equilibrium of the optimization problem [START_REF] Burt | The Economics of Conjunctive Use of Ground and Surface Water[END_REF], and examine its qualitative properties. All the results are introduced below.

Proposition 1. Given assumptions on water demand (4) and on cost functions (2) and (3), and the emergence condition [START_REF] Hellegers | Dynamics of Agricultural Groundwater Extraction[END_REF], the optimal management problem ( 6): (i) has a unique steady state which is a local saddle point; (ii) leads to the conjunctive use RW and GW, with w r = W and w g = ( 1)(R W ), so that the total water consumption is w = ( 1)R + W ; (iii) yields the long term water table h r ( ; R; W ) solving the following condition:

P ((1 )R + W ) = C (h) (1 ) C 0 (h) (R W ) (16)
provided the shadow value of the resource:

r = (1 ) C 0 (h) (R W )
Proposition 1 states that sustainable water consumption is given by the sum of the GW that can be sustainably extracted from the aquifer and the part of runo¤ that can be captured. From this conjunctive use system, Eq. ( 16) hows that the groundwater elevation stabilizes in the long run at a level for which the water price is equal to the social cost of GW pumping. The determination of the water table level will consequently depend on the hydrologic parameters and R, and the size of the RW reservoir W .

According to the literature, this solution is not far from the groundwater-only solution. It is as if the storage capacity becomes zero, W = 0, so we do not collect RW at the surface. More formally, the social planner maximizes the sum of discounted net bene…t (5) from water consumption with respect to the single control, i.e. groundwater pumping, and subject to the state equation ( 1) without any possibility of collecting RW. In particular, conditions ( 9) and ( 14) look similar (given w r (t) = 0), while condition [START_REF] Feinerman | Bene…ts from Groundwater Management: Magnitude, Sensitivity, and Distribution[END_REF] vanishes. And, one of the main modi…cations comes from water consumption which will be equivalent to water pumping. This then implies changes in condition [START_REF] Huang | Changes in groundwater induced by water diversion in the Lower Tarim River, Xinjiang Uygur, NW China: Evidence from environmental isotopes and water chemistry[END_REF]. All these modi…cations are summarized in the following corollary.

Corollary 1. A GW-only management problem provides a unique saddle point equilibrium de…ned as follows: (i) In the long run we extract exactly the natural recharge, w g = ( 1)R, (ii) the stationary value for aquifer elevation, h g ( ; R) solves the condition:

P ((1 )R) = C (h) (1 ) C 0 (h) R (17)
provided the shadow value of the resource: g = (1 )R C 0 (h).

Corollary 1 gives some interesting insights into the relevant di¤erences between solutions of a GW management problem with and without RW storage. First, water consumption is higher in the system with RW storage than in the GW-only system, because a part of surface runo¤ -which is lost if there is no storage -is consumed in addition to the pumped water. Consequently, the marginal bene…t is lower with RW storage, which should imply a positive e¤ect on the water table level, as in the case of increased precipitation.

Second, RW harvesting enables the social planner to reduce pumping by an equivalent amount to the storage that would have replenished the aquifer, which may alleviate the pressure on groundwater. Lower extraction rates mitigate the dependence on groundwater stocks, c 0 (h)w g , and consequently reduce the long-term value of the aquifer. Thus, RW harvesting reduces the opportunity cost of extracting an additional unit of groundwater, which creates an incentive to extract more water, with a negative impact on the water table level.

So RW harvesting has two opposite e¤ects on the water table elevation, which suggests that care must be taken in analyzing the trade-o¤ between an increase in water consumption by W and a decrease in groundwater pumping by ( 1)W . Whether RW harvesting is a hydrologically-e¢ cient strategy (with respect to the water table level) actually depends on the magnitude of these two e¤ects.

The long-term e¤ects of RWH on groundwater resource

Following the literature, the capture of runo¤ is central to the de…nition of this technique. 10 And the potential for application is appraised essentially according to the quantity of water that can be collected. We have underlined the fact that RW harvesting can increase the water supply by collecting water that would otherwise have been lost. However, we have also shown that this process modi…es incentives with regard to extraction, so that the e¤ects of RWH are not self-evident. The purpose of this section is therefore to analyze the extent to which runo¤ collection a¤ects bene…ts from water consumption and decreases the burden on groundwater. This situation is considered to be environmentally e¢ cient, rather than economically e¢ cient, in the light of our optimal management approach.

To gain intuition of this, consider the extreme case in which there is no runo¤: = 0. IIt is straightforward that water consumption w will be the same with and without RW storage, since there is perfect equivalence between the amount of water available on the surface and in the ground. Marginal bene…ts from water consumption are thus identical, while the long-term marginal social costs are di¤erent. Given our assumptions on the cost function, the water table level in a GW-only management problem will be higher to satisfy simultaneously Eqs ( 16) and [START_REF] Koundouri | Potential for Groundwater Management: Gisser-Sánchez E¤ect Reconsidered[END_REF]. Here, there is only a cost e¤ect, which reduces the incentives to conserve water for later uses.

Conversely, with an extreme rate of runo¤: = 1, not one drop of rain will soak into the soil to recharge the aquifer. The problem is sharply modi…ed since the groundwater resource is now non-renewable. In a scenario without any RW storage, the aquifer will be exhausted and water consumption will end in the long run; the long-term marginal cost becomes in…nite, as does the water price. However, with RW storage, the water price is bounded by the RW supply, meaning that extraction will stop before the stock is exhausted. In this speci…c case, the result depends only on a demand e¤ect whereby RW harvesting conserves a higher water table level. This brief analysis shows that for extremely low values of runo¤, the cost e¤ect may o¤set the demand e¤ect, while for extremely high values of runo¤, the demand e¤ect may be stronger. This raises the question of the existence of a threshold value of runo¤, which identi…es the environmentally e¢ cient situation. This result is summarized as follows.

Proposition 2. There exists a unique threshold value of runo¤, 2 (0; 1), such that the level of the water table in a system without RW harvesting is the same as the level with RW storage, h g ( ) = h r ( ). Moreover, 8 > , RW storage is environmentally-e¢ cient.

Proposition 2 highlights the fact that there exists a speci…c runo¤ rate such that the decrease in the marginal bene…t is perfectly balanced by a decrease in the marginal social cost of GW. In this case, the collection of RW does not preserve more water in the ground. However, for the geographic areas with higher rates of runo¤, the collection of RW will enable the social planner to preserve larger groundwater stocks. For this range of values, RW harvesting is economically and environmentally-e¢ cient. Otherwise, this technique will increase the long-term depletion of the resource.

However, this threshold value is de…ned for speci…c values of storage capacity and precipitation. In light of Proposition 1, an increase in the size of the reservoir W will increase water consumption and reduce the pumping rate. This implies a modi…cation in the trade-o¤ between a decrease in the marginal bene…t and a decrease in the stock e¤ect, while there is no e¤ect for the system relying on GW alone. However, we can expect smaller and smaller demand e¤ects, because the water price decreases at a lower rate with higher levels of consumption. This implies a larger threshold value, as summarized in the following proposition. Proposition 3. Since the threshold is increasing with the storage capacity W , we can say that in order to be environmentally e¢ cient:: (i) the implementation of large storage capacity requires a high rate of runo¤; (ii) for a given rate of runo¤, the policy-maker should bear in mind the existence of an upper-bound to the storage capacity.

The determination of the impact of changes in precipitation may appear easier at …rst sight, since water consumption and water pumping move in the same direction. In light of Proposition 1 and Corollary 1, we know that the aquifer head level will be raised separately in each water system. 11 However, it is more complicated to determine the resulting e¤ect on the threshold. Whether the threshold value rises or falls depends on the relative strengths a¤ecting the system with and without RW storage. Hence, it can increase, decrease or remain unchanged, as summarized in the proposition below: 11 We can easily compute the partial derivatives: @hr @R =

(1 )

h P 0 ((1 )R+ W )+ C 0 (h) i C 0 (h)+ (1 ) C 00 (h)(R W )
> 0 and @hg @R =

(1 )

h P 0 ((1 )R)+ C 0 (h) i C 0 (h)+ (1 ) C 00 (h)R > 0.
Proposition 4. The e¤ect of higher precipitation, R, on the threshold value, is ambiguous within our general setting. Nevertheless, with the following restrictions, we can state that the e¤ect is: (i) positive with a linear unit pumping cost, C(h);

(ii) negative with a linear demand function, P (w);

(iii) zero using the framework of Gisser and Sánchez [START_REF] Gisser | Competition versus Optimal Control in Groundwater Pumping[END_REF], which is a combination of the two previous items.

In contrast to the e¤ect of the storage capacity on the threshold, the impact of changes in climatic conditions is less clear-cut. They raise the question not only of which areas experience an increase in precipitation, but also of economic speci…cities. In regions subject to climate change, more targeted studies are required to evaluate the environmental e¢ ciency of an RW harvesting policy.

Introducing additional hydrologic parameters

We now extend the previous model to incorporate some improvements regarding the water system, and we will then discuss the hydrological in ‡uence of these new parameters on the threshold value. Up to this point, we have been working with two strong assumptions: (i) all irrigation water is consumed, so that there is no return ‡ow to the aquifer, and (ii) there is no loss of collected water. Now, we consider that there are return ‡ows to the aquifer from both rainwater harvesting and groundwater extraction (consequent to irrigation) at a rate 2 (0; 1), and that a part of collected rainwater is lost at the rate " 2 (0; 1). This loss encompasses evaporation, leakage and quality degradation. Intuitively, we state that the runo¤ rate is higher than the loss rate, otherwise RW storage could not capture any runo¤:

". We also assume that the proportion of rainwater naturally percolating to the aquifer is higher than the proportion of stored rainwater which is not lost and which percolates to the aquifer after use: 1

(1 "). This slightly modi…es our optimization problem (Eq 6). First, groundwater and rainwater are no longer perfect substitutes, and the water supply becomes w = w g + (1 ")w r . Second, the dynamics of the water head becomes:

_ h = (1 )(R w r (t)) w g (t) + (w g (t) + (1 ")w r (t)) (18)
Third, condition (Eq 15), which ensures the existence of a steady state with conjunctive use must be modi…ed to take into account return ‡ows and losses. It becomes:

P R 1 > K 1 " 1 C 0 C 1 K 1 " (R (1 )W ) (19)
The steady state solutions are obtained through a similar optimization problem taking into account all the modi…cations, and are characterized by the following proposition.

Proposition 5. The optimal management problem always has a unique steady state which is a local saddle point. In this conjunctive use system, RW collection is still w r = W , but: (i) the steady state pumping rate is given by w

g = (1 )R W [1 (1 ")] 1
, so that the total consumption is w = (1 )R+( ")W 1 ;

(ii) the long term water table level, h r ( ; R; ; W; ") , solves the following condition:

P (1 )R+( ")W 1 = C (h) (1 )R W [1 (1 ")] C 0 (h) (20)
As for the GW-only system, we merely have to consider additional return ‡ows, so the qualitative results of Corollary 1 are unchanged, apart from the …rst item. We state that the long-term pumping rate is w g = (1 )R 1 and the stationary value for the aquifer, h g ( ; R; ), is obtained from the following condition:

P (1 )R 1 = C (h) (1 )R C 0 (h) (21)
Likewise for the basic model, which is extended by the qualitative results on the existence of the threshold value in Proposition 2 extend: there is still a unique threshold value, 2 ["; 1

(1 ")], above which RW storage is environmentally e¢ cient. In fact, there are additional restrictions on the runo¤ rate , due to the reasonable assumptions induced by the introduction of the loss rate and the return ‡ow rate. At the lower bound, i.e., = ", the water consumption is the same with or without RW harvesting (see Eqs [START_REF] Krishna | Cistern Water Systems in the US Virgin Islands[END_REF] and ( 21)), and there is only, as previously, a cost e¤ect which reduces the incentive to conserve water. At the upper bound, the result is slightly di¤erent. Since 1

(1 "), the proportion of rainwater which naturally percolates to the aquifer is equal to the proportion of stored rainwater which is not lost and which percolates to the aquifer after use, so that RW harvesting does not a¤ect the dynamics of the aquifer. The long-term pumping cost is the same with or without RW harvesting, but more water is available in the …rst case. There is only a demand e¤ect which increases the incentives to conserve water.

Let us now investigate the impact of the two new parameters, f"; g. Water loss in the RW harvesting technology works as runo¤, reducing the available amount of water in the system. So if we assume the technology just becomes environmentally e¢ cient and the loss rate increases, then this additional water loss must be compensated for by a higher amount of runo¤s. Since at the steady state, RW harvesting is at capacity, this means that we need a higher rate of runo¤: the threshold must increase. Return ‡ows increase the recharge of the aquifer. Consequently, as for the natural recharge R, an increase in the return ‡ow rate modi…es the system with and without RW harvesting: the same indeterminacy occurs. To summarize: Proposition 6. (i) There always exists a unique threshold, which now belongs to ["; 1

(1 ")].

(ii) The threshold increases with a higher rate of RW losses, ".

(iii) The e¤ects of return ‡ows on the threshold depend on economic speci…cities. For instance, with linear marginal pumping cost and iso-elastic demand, the result depends on the degree of iso-elasticity.

A conjunctive use model with ecological damages

In this …nal step, we now add the values of GW-dependent ecosystems to our model, along the lines of Esteban and Albiac [START_REF] Esteban | Groundwater and ecosystems damages: Questioning the Gisser-Sánchez e¤ect[END_REF]. Depletion, _ h < 0, creates environmental damages to associated ecosystems measured by additional costs. Otherwise, for _ h > 0, higher water table levels improve ecosystems, generating additional bene…ts for society. This e¤ect is captured by an increasing function G( _ h) with G(0) = 0. Unlike these authors, we do not introduce linearity. This is why we also require this function, as a standard cost function, to be convex: G 00 ( _ h) > 0. The social planner now maximizes the following net social bene…t:

Z wg(t)+(1 ")wr(t) 0 D(x)dx c(h(t))w g (t) Kw r (t) G( _ h) (22)
with respect to GW dynamics (Eq 18).

The emergence condition (Eq 19) must now be modi…ed by adding the marginal environmental cost at the steady state mitigated by return ‡ows. It becomes:

P R 1 > K 1 " 1 C 0 C 1 K 1 " (R (1 )W ) + (1 )G 0 (0) (23)
The study of this maximization problem is summarized in the following proposition:

Proposition 7. The optimal management problem always has a unique steady state which is a local saddle point. Moreover, (i) Conjunctive use is not a¤ected by ecological damages. RW collection is still w r = W , pumping rate is given by w

g = (1 )R W [1 (1 ")] 1
, so that the total consumption is again w = (1 )R+( ")W 1 , (ii) GW elevation solves the new condition:

P (1 )R+( ")W 1 = C (h) (1 )R W [1 (1 ")] C 0 (h) + (1 )G 0 (0) (24)
The GW-only system will be modi…ed in the same way, so that only Eq. ( 21) changes by adding the new term G 0 (0). The water table level is therefore obtained from the new condition:

P (1 )R 1 = C (h) (1 )R C 0 (h) + (1 )G 0 (0) (25)
Since the term G 0 (0) is a constant and has the same e¤ect on Eqs. ( 24) and ( 25), the analysis of the threshold value, , in Proposition (6) holds, but both the threshold value and the water table will now also be dependent on ecological damages. By adding a …xed cost, the water head level is higher, whatever the system, when we take ecological damages into account. 12 But the e¤ect on the water level with RW harvesting will be lower, because this technique reduces the dependence on the aquifer. If the runo¤ rate corresponds to the threshold, and environmental damages increase, we typically move to an environmentally ine¢ cient situation. This means that the threshold is increasing with damages.

Proposition 8. If we compare equations ( 24) and ( 25), we observe that: (i) there again exists a unique threshold; (ii) the water table increases in both cases when the damages increase, but it increases less under RW harvesting; (iii) the threshold increases with damages.

The last point may seem surprising. In ecosystems severely a¤ected by intensive pumping, we would expect RW harvesting to alleviate the pressure on aquifers. But, here we …nd that the environmental e¢ ciency of RW harvesting is reduced. This could result from the way we consider damages: they are linked to changes in the water table. It would be interesting to include the level of the water height in the formulation of damages proposed by Esteban and Albiac [START_REF] Esteban | Groundwater and ecosystems damages: Questioning the Gisser-Sánchez e¤ect[END_REF].

Conclusion

This paper analyzes the extent to which rainwater harvesting is environmentally e¢cient. More precisely, we investigate whether this technique, used in conjunction with groundwater extraction, is more e¤ective in limiting aquifer depletion than a purely groundwater-based system. We emphasize the importance of considering the hydrological cycle in its entirety, so that ground and surface water are interdependent and must be managed as such. In a basic setting, we highlight the fact that rainwater harvesting should be seen not only as a means to collect surface runo¤ in order to alleviate pressure on conventional water sources, but also as an economically optimal solution, which will only be environmentally e¢ cient when the surface runo¤ rate is high enough.

Moreover, the runo¤ rate needs to be even higher when the storage capacity is bigger. This draws attention to the fact that water policies may have an adverse impact if they encourage larger storage capacities with no consideration for speci…c parameters. We then examine how this result is modi…ed by the inclusion of additional parameters like evaporation, return ‡ows and ecological damages due to aquifer depletion. Variations in these parameters a¤ect the likelihood that rainwater harvesting will be environmentally e¢ cient. This technique could therefore be a long-term solution for groundwater depletion under speci…c conditions, which must be taken into account in water policies.

Our approach remains focused on certain speci…c aspects, and could be extended in a number of directions. Firstly, we could enrich the ecological part of the model. For instance, we could link the environmental damages to the level of the water table. This could even be introduced in a nonlinear way, since the impact on biodiversity is stronger when the aquifer is depleted. Other characteristics of the aquifer could be included, such as stochastic recharge, leakage and GW quality.

Secondly, in this paper we assume a …xed capacity. However, the RW harvesting technology can be improved over time. This requires investment strategy and therefore the inclusion of additional dynamics in the model. This problem was explored by Stahn and Tomini [START_REF] Stahn | Rainwater harvesting under endogenous capacity of storage: a solution to aquifer preservation?[END_REF], but they do not explore the question of the threshold in runo¤ rates.

Finally, it is well-known that strategic behavior modi…es the aquifer level, and its e¤ect on the threshold should therefore be taken into account in the model. This analysis quickly becomes very complex, especially when feedback strategies are introduced. A …rst exploration is provided by Soubeyran et al. [START_REF] Soubeyran | Anti-)coordination problems with scarce water resources[END_REF].

The Hamiltonian conditions are given by systems (CTR1) and (DYN1):

(CT R1) 8 > > > > < > > > > : P (w(t)) C (h(t)) a (t) + G 0 _ h(t) + g (t) = 0 (GW 1 ) P (w(t)) K " b 1 " (t) + G 0 _ h(t) + r (t) W (t) = 0 (RW 1 ) g (t)w g (t) = 0, r (t)w r (t) = 0, W (t) (W w r (t)) = 0 ( S1 1 )
g (t); r (t); W (t) 0, w g (t) 0, and w r (t)

2 [0; W ] (S2 1 ) (DY N 1) _ h(t) = R aw g (t) bw r (t) _ (t) = (t) + C 0 (h(t)) w g (t)

A.2. The construction of the steady state

Let us consider a steady state satisfying (CTR1) and (DYN1) with _ h(t) = _ (t) = 0. There are eight possible cases,since each of the three Lagrange multipliers may or may not be strictly positive. We nevertheless immediately observe that: Remark 1. At a steady state: (i) both constrains on w g cannot be simultaneously binding since W > 0, (ii) there must be some GW consumption (i.e , g = 0) otherwise _

h(t) = 0 implies that w r = R b = (1 ) (1 )
(1 ") R > R, which is impossible since w r W < R.

Now let us observe:

Lemma 1. If the capacity constraint on RW harvesting is not binding (i.e., W = 0) at the steady state, then K " > C(h).

Proof of Lemma 1

Since g = W = _ h = 0; the di¤erence between Eqs. (GW 1 ) and (RW 1 ) is:

K " C (h) + b 1 " a [ + G 0 (0)] r = 0 (28)
Using (i) r 0 and (ii) = 1 C 0 (h) w g since _ = 0, it follows that:

K " C (h) b 1 " a 1 C 0 (h) w g G 0 (0) (29)
(iii) Notice that b

1 " a = " 1 " < 0 since > ", and (iv) by assumption C 0 < 0; G 0 > 0: The lhs of Eq [START_REF] Ranganathan | Modeling Economics of Conjunctive Surface an Groundwater Irrigation Systems[END_REF] is therefore positive and we can conclude that K " > C(h).

From Lemma 1, we can assert : Lemma 2. If we assume that

P R a > K " 1 C 0 C 1 (K " ) (R aW ) + aG 0 (0) (30)
the RW storage capacity is always binding at a steady state.

Proof of Lemma 2

Assume the contrary, w r < W , so that W = 0: Since = 1 C 0 (h) w g , Eq. (RW 1 ) at the steady state can be written as:

r = P (w) + K " b 1 " 1 C 0 (h) w g G 0 (0) (31)
Using Lemma 1, i.e., K " > C(h) , C 1 (K " ) < h, and since C" 0, we deduce that:

r P (w) + K " b (1 ") C 0 C 1 (K " ) w g + b 1 " G 0 (0) (32)
Moreover from system (DYN1) at steady state we know that:

(i) R aw g bw r = 0 , aw g = R bw r < R bW since we havee assumed that w r < W . Moreover b = 1

(1 ") and 1 > > ", we can therefore say:

aw g < (1 ) (R W ) + (1 ")W < R (1 )W = R aW (33)
(ii) The total consumption of water is w = R bwr a

+ (1 ")w r = 1 a (R (b a(1 ")) w r ). Since b a(1 ") = " , we have w = 1 a [R + ( ") w r ].
If we now remember that w r < W < R and " < < 1, we can observe that:

w < 1 a ((1 ) + ( ")) R < R 1 ( 34
)
Under assumptions P 0 < 0 and C 0 < 0 and items (i) and (ii), we can rewrite condition (32) as:

r < P R 1 + K " b a (1 ") C 0 C 1 (K " ) (R aW ) + b 1 " G 0 (0) (35)
Finally, since 0 < b (1 ")a < 1 (see (iii) of the proof of lemma 1), we can state that:

r < P R a + K " 1 C 0 C 1 (K " ) (R aW ) + aG 0 (0) (36)
Given the assumption introduced in Eq (30), this implies that r < 0 which is a contradiction.

If we now combine Remark 1 and Lemmas 1 and 2, we can say that a steady state, if it exists, veri…es:

:

P (w) C (h) a ( + G 0 (0)) = 0; w g = 1 a (R bW ) W = P (w) K " b 1 " ( + G 0 (0)) ; = 1 a C 0 (h) (R bW ) W 0 and w r = W; w = 1 a (R + ( ")W ) (37)
In other words, a unique state exists, if there exists a unique solution h 2 0; h to

1 (h ) = P 1 a (R + ( ")W ) C (h ) + 1 C 0 (h ) (R bW ) aG 0 (0) = 0 (38)
which veri…es:

W = P 1 a (R + ( ")W ) K " + b a (1 ") C 0 (h ) (R bW ) b 1 " G 0 (0) 0 (39)
So let us …rst observe:

Lemma 3. If condition [START_REF] Roebuck | A Whole Life Costing Approach for Rainwater Harvesting Systems[END_REF] is satis…ed, any solution (it is exists) to 1 (h ) = 0 veri…es W > 0.

Proof of Lemma 3 Using Eq.( 37) and W < R, we get w < 1 a (R + W ) < R a and by Eq. ( 33) we know that R bW < R aW . If we now bear in mind that P 0 < 0 and C 0 < 0, we can deduce from condition (30) that:

0 < P R a K " + 1 C 0 C 1 (K " ) (R aW ) aG 0 (0) < P (w) K " + 1 C 0 C 1 (K " ) (R bW ) aG 0 (0) 1 (C 1 (K " )) (40)
But 1 (h) is increasing in h since C 0 < 0 and C" 0. We deduce that the steady state h < C 1 (K " ) or C(h ) > K " . We can therefore say from equation ( 38) that:

P 1 a (R + ( ")W ) K " + 1 C 0 (h ) (R bW ) aG 0 (0) > 0 (41)
Since b a(1 ") 2 (0; 1), C 0 < 0 and G 0 (0) > 0, we can even state:

P 1 a (R + ( ")W ) K " + b a (1 ") C 0 (h ) (R bW ) b 1 " G 0 (0) > 0 (42)
or, in other words, that W > 0.

We can …nally a¢ rm that: Proposition 9. Under condition [START_REF] Roebuck | A Whole Life Costing Approach for Rainwater Harvesting Systems[END_REF], the optimization problem (26) admits a unique steady state given by equations [START_REF] Tsur | The Bu¤er Value of Groundwater with Stochastic Surface Supplies[END_REF].

Proof of Proposition 9:

It simply remains to show that 1 (h) = 0 admits a unique solution h 2 0; h . This is immediate since (i) 0 1 > 0 because C 0 < 0 and C" 0, (ii) lim h!0 1 (h) = 1 because lim h!0 C(h) = 1 and (iii) lim h! h 1 (h) > 0. However this last point is less obvious. So let us …rst observe that lim h! h 1 (h) = P 1 a (R + ( ")W ) aG 0 (0) since we have assumed that

C( h) = C 0 ( h) = 0. Now remark that 1 a (R + ( ")W ) < R 1
since W < R and use condition [START_REF] Roebuck | A Whole Life Costing Approach for Rainwater Harvesting Systems[END_REF] in order to obtain:

lim h!0 1 ( h) > P R a aG 0 (0) > K " 1 C 0 C 1 (K " ) (R aW ) > 0 (43)
(remember C 0 < 0 and R aW > 0).

A.3. The local saddle point property

Let us return to the systems (CTR1) and (DYN1) and set g = 0, and r = 0. Since W > 0 at the steady state we can choose a neighborhood around the steady state such that W (t) > 0 and therefore set w r (t) = W . In this case equation (GW 1 ) becomes:

2 (w g (t); h(t); (t)) = P (w g (t) + (1 ") W ) C (h(t)) a ( (t) + G 0 ( R + aw g (t) + bW )) = 0 (44)
Moreover @ wg 2 = P 0 a 2 G" < 0 by our assumptions. We can therefore apply the implicit function theorem around the steady state and build w g (t) = 3 (h(t); (t)) > 0. We even know that

@ h 3 = C 0 P 0 a 2 G" > 0 and @ 3 = a P 0 a 2 G" < 0 (45)
It follows that the dynamics of the system is locally given by:

_ h(t) = R a 3 (h(t); (t)) bW _ (t) = (t) + C 0 (h(t)) 3 (h(t); (t)) (46)
It remain to verify that the Jacobian J of this two dimensional system has a positive trace T r(J) and a negative determinant det(J). By computation, we obtain:

J = " aC 0 P 0 a 2 G" a 2 P 0 a 2 G" C" 3 + (C 0 ) 2 P 0 a 2 G" + aC 0 P 0 a 2 G" # (47)
We can therefore say, under our assumptions, that :

T r(J) = > 0 and det(J) = aC 0 + a 2 C" 3 P 0 a 2 G" < 0 (48)

B. The study of the threshold

Let us introduce the system: 4 ( ; h; R; ; ) = 0 5 ( ; h; W; R; "; ;

) = 0 = P 1 a R C (h) + 1 C 0 (h) R a = 0 P 1 a (R + ( ")W ) C (h) + 1 C 0 (h) (R bW ) a = 0 (
49) with = G 0 (0). The …rst equation gives us the stationary water table without RWH while the second includes this option. So if we solves this system in h and we obtain the threshold value, for which the stationary water table h is the same without or with rainwater harvesting. Moreover this threshold can then be linked to the di¤erent parameters (W; R; "; ; ).

B.1. Existence of a threshold

Let us observe that for any admissible (W; R; "; ; ), 8fi = 4; 5g, (i) @ h i > 0 since C 0 < 0 and C" 0, (ii) lim h!0 i = 1 since lim h!0 C(h) = 1 and (iii) lim h! h i > 0 (apply the same argument as in the proof of Proposition 9). We can therefore, by the implicit function theorem, de…ne two continuous functions h g ( ) and h r ( ) which respectively solve in h; 4 ( ; h; R; ; ) = 0 and 5 ( ; h; W; R; "; ; ) = 0. It remains now to study (h g ( ) h r ( )) since the threshold solves h g ( ) = h r ( ).

Since 2 ["; 1 (1 ")], let us …rst observe that (h g ( ) h r ( )) =" > 0. To verify this point, let us observe that, for = ", h g and h r respectively solve: 8 < :

P 1 " 1 R C (h g ) + 1 " C 0 (h g ) R (1 ) = 0 P 1 " 1 R C (h r ) + 1 " C 0 (h r ) (R (1 ) W ) (1 ) = 0 (50)
This implies:

C (h g ) + (1 ")R C 0 (h g ) = C (h r ) + 1 " C 0 (h r ) (R (1 ) W ) , h C (h g ) + (1 ")R C 0 (h g ) i h C (h r ) + (1 ")R C 0 (h r ) i = (1 ")(1 )W C 0 (h r ) (51)
It is straightforward that the right-hand side of Eq (51) is positive, so that:

h C (h g ) + (1 ")R C 0 (h g ) i > h C (h r ) + (1 ")R C 0 (h r ) i (52)
Finally since C 0 < 0 and C" 0, we deduce that h g (") > h r (").

Second, observe that (h g ( ) h r ( )) =1 (1 ") < 0: In this case, the quantities h g and h r solve: 8 < :

P (1 ") 1 R C (h g ) + (1 ") C 0 (h g ) R (1 ) = 0 P (1 ") 1 R + (1 ") W C (h r ) + (1 ") C 0 (h r ) R (1 ) = 0 (53)
Since P

(1 ") 1

R > P

(1 ") 1

R + (1 ") W (remember that P 0 < 0) we can write:

C (h g ) (1 ") C 0 (h g ) R > C (h r ) (1 ") C 0 (h r ) R (54) Since C (h) (1 ") C 0 (h) R is decreasing in h, we conclude that h g (1 (1 ")) < h r (1 (1 ")).
Finally by continuity, we can a¢ rm that there exists at least one 2 ["; 1 (1 ")] such that h g ( ) h r ( ) = 0.

B.2. Uniqueness of the threshold

Since this threshold and the associated water table are also obtained as a solution to system (49) for a given set (W; R; "; ; ) of parameters, we can use the Gale-Nikaido theorem (see Gale-Nikaido [START_REF] Gale | The Jacobian Matrix and Global Univalence of Mapping[END_REF] or Mas-Colell [START_REF] Mas-Colell | Homeomorphims of Compact, Convex Sets and the Jacobian Matrix[END_REF]) which states that if every principal minor of @ ;h ( i=4;5 ) is positive and the domain K = 0; h [0; 1] of the function is a rectangle, the solution, if it exists, is unique. Let us verify this point.

@ ;h ( i=4;5 ) = " R 1 a P 0 g + 1 C 0 C 0 + R C" (R W ) 1 a P 0 r + 1 C 0 C 0 + R bW C" # with P 0 g P 0 r = P 0 1 a R P 0 1 a (R + ( ")W )
Concerning the principal minors of order 1, we notice that: (i) R 1 a P 0 g + 1 C 0 > 0 since C 0 < 0 and P 0 < 0 (ii) C 0 + R bW C" > 0 since C 0 < 0, C" 0 and R bW = (1 )(R W ) + (1 ")W > 0

It therefore remains to compute the determinant of @ ;h ( i=4;5 ). This quantity is given by: det (@ ;h ( i=4;5 )) = 1 a P 0 g + 1 C 0 RC 0 (1 )(R W )R C" (1 ")RW C"

+ 1 a P 0 r + 1 C 0 RC 0 + (1 )(R W )R C" + W C 0 = R a C 0 (1 )(R W ) C" P 0 g P 0 r | {z } >0 (1 ")RW C" 1 a P 0 g + 1 C 0 | {z } <0 + W C 0 1 a P 0 r + 1 C 0 | {z } >0
hence det (@ ;h ( i=4;5 )) > 0 (55) since C 0 ; P 0 g ; P 0 r < 0,C" 0 and because P " > 0, P 0 1 a R < P 0 1 a (R + ( ")W ) , i.e. P 0 g P 0 r < 0.

B.3. Comparative statics

Let us denote by one of the following parameters fW; R; ; "; g and let us compute @ @ . It we di¤erentiate system (49) with respect to (h; ; ) and solve the system, we know by the Cramer rule that: @ @ = det @ 4 @ h 4 @ 5 @ h 5 det @ 4 @ h 4 @ 5 @ h 5 = N D (56)

From Eq. ( 55), we know that the denominator D is positive. It follows that the sign of @ @ is simply given by its numerator N . Moreover a simple exercice of computation shows that : @ h 4 = C 0 (h) + 1 C 00 (h)R > 0 and @ h 5 = C 0 (h) + 1 C 00 (h) (R bW ) > 0 (57) and @ 4 @ 5 W 0 We can therefore identify, after computation, the following e¤ects:

(i) The threshold and the storage capacity, @ @W sign @ @W = sign

0 B B @ " a P 0 r b C 0 C 0 + R C" | {z } >0 since C 0 <0, C">0 1 C C A = sign " a P 0 r b C 0 (59)
Let us now observe that the computation of the di¤erence of the two equations (see Eq (49)) which de…ne the threshold gives:

P 1 a R P 1 a (R + ( ")W ) + bW C 0 = 0 (60)
and the convexity of the inverse demand gives:

P 1 a R P 1 a (R + ( ")W ) > ( ")W a P 0 r (61)
These two last equations immediately show that ( ") a P 0 r b C 0 > 0 hence that @ @W > 0:

(ii) The threshold and the evaporation rate, @ @" sign @ @" = sign W 1 a P 0 r + C 0 C 0 + R C" > 0 since P 0 r < 0, C 0 < 0 and C" > 0 (iii) The threshold and the marginal ecosystem damage, @ @ sign @ @ = sign a bW C" 0 (iv) The threshold and the recharge, @ @R Since @ h 5 = @ h 4 + bW C 00 (h) and 1 a > 0, a routine computation gives:

sign @ @R = sign 0 B @@h 4 | {z } >0 P 0 g + P 0 r | {z } 0 (since P " 0)

+ bW C 00 (h) | {z } 0 P 0 g + a C 0 (h) | {z } <0 1 C A
This quantity is clearly unsigned. For instance if the demand is linear, we have P 0 r P 0 g = 0 and @ @R < 0. But if the unit pumping cost is linear in h so that C" = 0, we observe @ @R > 0. Finally in the Gisser-Sanchez case (i.e. C" = 0 and P linear) it is immediate that @ @R = 0: (v) The threshold and the return ‡ow, @ @ sign @ @ = sign @ h 4 h R a 2 P 0 g + P 0 r + ( ")W a 2 so if > max n d ; 1 then @ @ > 0.

P 0 r + (1 ")W C 0 (h) i + bW C 00 (h)

R a 2 ( 1 "

 21 P 0 g + This quantity is again unsigned. To illustrate this point let us assume that C" = 0 so that C 0 = c 0 a constant with c 0 > 0. If we add that P (w) is linear (Gisser-Sanchez case) then sign @ )W c 0 < 0. If we now introduce an isoelastic demand P (w) = w with a

See http://www.unep.or.jp/ietc/publications/urban/urbanenv-2/9.asp)

For more details, see Annex 1 of the study Agricultural Technologies for Developing Countries of the European parliament.

The detailed proofs are available upon request.

For instance, water diversion in the Lower Tarim River in China induces a decline in groundwater recharge[START_REF] Huang | Changes in groundwater induced by water diversion in the Lower Tarim River, Xinjiang Uygur, NW China: Evidence from environmental isotopes and water chemistry[END_REF].

With 3 multipliers, we usually expect 2 3 cases. But (i) the capacity and non-negativity constraints on RW cannot both be satis…ed, (ii) by our boundary conditions at least one source of water is used, (iii) if only RW is used the capacity may or may not be reached, and (iv) there is a boundary regime for which the use of GW or RW is indi¤erent.

This is clearly a su¢ cient condition. Conjunctive use may occur for higher water tables, but this allows us to re…ne the analysis.

Observe that this su¢ cient condition is independent of the runo¤. This parameter can used later without additional restrictions.

For more details, see Boers and Ben-Asher[START_REF] Boers | A review of rainwater harvesting[END_REF]

Note …rst that the LHS of equality[START_REF] Krishna | Cistern Water Systems in the US Virgin Islands[END_REF] (respectively[START_REF] Krulce | Optimal Management of a Renewable and Replaceable Resource: The Case of Coastal Groundwater[END_REF]) and[START_REF] Nieswiadomy | The Demand for Irrigation Water in the High Plains of Texas, 1957-1980[END_REF] (respectively (25)) are unchanged, while the same positive constant is added to the RHS of (24) (respectively[START_REF] Perrens | Collection and Storage Strategies for Domestic Rainwater Systems in Australia, Hydrology Papers[END_REF]). If we remember that C 0 < 0 and C 00 > 0, the result follows.

APPENDIX

A. The solution to the general problem

Let us introduce some notations used below:

The problem

Let us consider the following optimal control problem:

The current value of the Hamiltonian for this problem is:

where (t) 0 denotes the shadow value of the groundwater resource. Moreover, if we introduce the Lagrange multipliers g (t) and 0 r (t) associated with the non-negativity constraints on w g (t) and w r (t), respectively, and the implicit price 0 W (t) of W , the capacity constraint, the Lagrangian function associated with this program is: L = H + g (t) w g (t) + 0 r (t) w r (t) + 0 W (t) (W w r (t)) [START_REF] Provencher | Issues in the Conjunctive Use of Surface Water and Groundwater[END_REF]