Julia De

Frutos Cachorro

Katrin Erdlenbruch

Mabel Tidball

Sharing a groundwater resource in a context of regime shifts

Keywords: Groundwater resource, dierential game, strategic interactions, exogenous shock, regime shift. JEL: C72, Q25

We study the occurrence of shocks in a common groundwater resource problem using a dierential game. In particular, we use Rubio and Casino's adaptation of the Gisser and Sánchez model where we introduce a sudden change in the dynamics of the resource, namely a decrease in the recharge rate of the aquifer. We compare the pareto optimal solution with open-loop and feedback equilibria. First, we show analytically how dierent solutions, at the steady state, depend on the intensity of the shock. Moreover, we show that the cost and the strategic eects are decreasing functions of the intensity of the shock, i.e. that all the solutions get closer at the steady state for more intense shocks. We nally apply the game to the particular case of the Western La Mancha aquifer. The aim of this application is to estimate how shocks inuence the ineciency of open loop and feedback strategies in terms of welfare. We show that this ineciency decreases the earlier the shock occurs or the higher the intensity of the shock.

Introduction

In this paper, we study the exploitation of a common groundwater resource as a dierential game in order to take into account the strategic and dynamic interactions between the users of the resource. Specically, we consider a groundwater resource used for irrigation by several farmers. Common groundwater resources are often exploited under a common property regime, that is the access is restricted to land owners situated over the aquifer. Numerous papers have studied this issue (for example [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF] [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF], [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF] [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF], [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF] [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF], [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF] [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF]) and have concluded that private exploitation is inecient, in terms of stock and welfare, in comparison to optimal exploitation. This ineciency is due to the various externalities which appear because of the sharing of this type of resource, namely the pumping cost externality which characterizes the fact that withdrawals made by one farmer lower the water-table level, resulting in an increase in pumping costs for the other users. On the other hand, the stock externality, also called strategic externality, represents competition between farmers because of the limited availability of the water stock (see [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF] [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF]). [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF] [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF] showed the ineciency of private exploitation for the Pecos River Basin, New Mexico. They also characterized the analytical dierence between the optimal and private solutions, and they concluded that the dierence is negligible if the capacity of the aquifer is large. [START_REF] Nieswiadomy | Input Substitution in Irrigated Agriculture in the High Plains of Texas 1970-80[END_REF] [START_REF] Nieswiadomy | Input Substitution in Irrigated Agriculture in the High Plains of Texas 1970-80[END_REF] called this consideration the Gisser and Sánchez eect (GSE), see [START_REF] Koundouri | Current Issues in the Economics of Groundwater Resource Management[END_REF] [START_REF] Koundouri | Current Issues in the Economics of Groundwater Resource Management[END_REF] for an overview. The most important policy implication derived from this study is that regulation of a common groundwater resource is not justied if the dierence of welfare from private and optimal exploitations is insuciently important. However, authors assume that farmers behave myopically in the calculation of the private solution, that is, farmers take decisions over a short period of time, without considering the impact of the other users on the available stock.

Several studies have used game theory to take into account the strategic and dynamic interactions between the resource users when computing the private solution (for example [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF] [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF], [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF] [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF], [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF] [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF]). In [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF],

Negri characterizes analytical solutions of the water-table level at the steady state for two types of Nash equilibria, open-loop and feedback solutions, and for the socially optimal case, also referred as pareto optimal case in the literature. He shows that the dierence between the socially optimal solution and the open-loop solution is positive and captures the pumping cost externality. Moreover, he shows that the dierence between the open-loop solution and the feedback solution is also positive and captures the strategic externality. The dierence between the socially optimal and the feedback solutions is then positive and represents the ineciency of private exploitation. Provencher and Burt [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF] take up Negri's ideas to prove, in a general way, that if the objective function of the problem is concave, the feedback solution is inecient, in comparison with the socially optimal solution. In [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF], Rubio and Casino adapt the Gisser and Sánchez model as a dierential game and derive analytical solutions of the socially optimal, open loop and feedback cases over an innite planning horizon. They also conrm Negri's result: strategic behaviour exacerbates the ineciency of private solutions. Moreover, they conrm the Gisser-Sánchez rule when the strategic externality is considered: for large aquifers, the dierent solutions get closer at the steady state.

The motivation of our work is based on the idea that some exogenous threats are not taken into account in previous studies, as for example the occurrence of regime shifts. We focus on the study of the ineciency of the private solution with strategic and dynamic interactions and in presence of an anticipated shock. More specically, we take Rubio and Casino's game in [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF] and we introduce a sudden change in the dynamics of the resource, which leads to an abrupt decrease of the water availability for the users of the resource. We model this shock as a decrease in the recharge of the aquifer. More generally, such a shock corresponds to a regime shift. We then compare the pareto optimal solution with open-loop and feedback Nash equilibria.

In [START_REF] De Frutos Cachorro | Optimal adaptation strategies to face shocks on groundwater resources[END_REF], de Frutos Cachorro et al. (2014) study how information about this type of shock aects the optimal management of the water resource by a centralized regulator (a water agency). For the deterministic case, when the date of the shock is known, the water agency would prepare for the event by applying an incautious extraction strategy before the occurrence of the shock. Such a result can already be found in the literature dealing with the impact of irreversible events (see [START_REF] Tsur | Dynamic and stochastic analysis of environmental and natural resources[END_REF] [START_REF] Tsur | Dynamic and stochastic analysis of environmental and natural resources[END_REF]), where the phenomenon is known as the "impatience eect".

In this paper, we combine Rubio and Casino's game theory approach and de Frutos Cachorro et al. study on the eect of regime shifts, in order to assess the dierence between the pareto optimal and private solutions with strategic and dynamic interactions between users.

We show that the combined eect of strategic interactions and this type of shock leads to an overexploitation of the resource in the short, medium and long run. Moreover, we study the ineciency of private exploitation with respect to the intensity and date of occurrence of the shock. From an economic point of view, we could expect that the higher the intensity of the shock, i.e. the lower the quantity of water available, the higher the overall pumping costs and the competition between users. However, we show that cost and strategic eects are particularly important for low-intense or later occurring shocks. Finally, we estimate the ineciency of private exploitation in terms of welfare for a particular case, the Western la Mancha aquifer. This aquifer is situated in the South of Spain, under a semi-arid climate where dry periods are frequent. Moreover, in the last decades, the aquifer has suered from various inecient regimes of exploitation. Our results suggest that some regulation of the aquifer is justied. Indeed, although eciency gains from following the pareto optimal solution are lower than in absence of shocks, they still reach several millions of Euros, for example 37 millions of Euros in the situation of a mid-intensity shock. The pareto optimal solution could be implemented by imposing licenses, such that extractions over time correspond to the pareto optimal extractions. This paper is organized in the following way. In section 2, we present Rubio and Casino's game and introduce the case of an exogenous and deterministic shock therein. In section 3, we describe analytical resolutions of the problem for dierent information structures. In section 4, we compute the pareto optimal solution corresponding to the problem. In section 5, we rst compare the dierent analytical solutions, in terms of long-term stocks. We then make a numerical application of the model to the Western La Mancha aquifer with the aim to estimate, in terms of stocks and welfare, the ineciency of private solutions. Finally, in section 6, we conclude and give some perspectives for future research.

2

The model First, we present the adaptation of the Gisser and Sánchez model (1980) [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF] as a dierential game developed by Rubio andCasino (2001, 2003) ( [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF], [START_REF] Rubio | Strategic Behavior and Eciency in the common Property Extraction of Groundwater[END_REF]). In [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF], the demand for irrigation water is a linear function,

g = a -bp, a, b > 0, (1)
where g represents water pumping, p, the price of water and a and b are parameters.

In [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF], Rubio and Casino assume that the number of farmers, M, is xed and nite over time. The individual demand for irrigation water can be described as a linear function,

g i = θ i (a -bp), i = 1..M, (2)
where 0 < θ i < 1 and M i=1 θ i = 1. Thus,

M i=1 g i = M i=1 θ i (a -bp) = a -bp = g. (3)
Moreover, the revenue of the farmer i is equal to

g i p(x)dx = g i a -g i θ i b dx = a b g i - 1 2bθ i g 2 i .
We assume that the marginal cost of extraction is a linear function that depends on G, the stock of the aquifer. Total costs of extraction are then

C = (z -cG)g, z, c > 0, ( 4
)
where z is the sum of xed costs and the maximum marginal cost of extraction and c the slope of the marginal pumping cost function. As z and c do not depend on the rate of extraction, the individual pumping cost of the i th farmer is Ci = (z -cG)g i , z, c > 0.

(5)

The dynamics of the aquifer can be described as

Ġ = -(1 -α)g + r = -(1 -α) M i=1 g i + r, ( 6
)
where r is the recharge rate and α the return coecient, α ∈ [0, [START_REF] De Frutos Cachorro | Optimal adaptation strategies to face shocks on groundwater resources[END_REF].

Assuming that interactions between farmers are rational and non-cooperative, the problem of the i th farmer is to maximize individual welfare, dened as the present value of his future prots, where ρ is the discount rate, taking into account the dynamics of the aquifer (equation ( 6)) and given initial conditions and positivity constraints:

max g i (.) ∞ 0 F i (G, g i ) e -ρt dt, (7)
where,

F i (G, g i ) = a b g i - 1 2bθ i g 2 i -(z -cG)g i , (8)
Ġ = -(1 -α) M i=1 g i + r, (9)
G(0) = G 0 given, ( 10
)
g i ≥ 0 G ≥ 0. (11)
Now, we disturb the system of the resource by introducing an exogenous and deterministic shock in the dynamics of the aquifer, as proposed by de Frutos Cachorro et al. (2014) [START_REF] De Frutos Cachorro | Optimal adaptation strategies to face shocks on groundwater resources[END_REF]. This disturbance represents a sudden reduction on the recharge rate, r, at time t a , known to the users. Thus, from t a on, the recharge rate switches from r = r 1 to r = r 2 , with r 1 > r 2 . This can happen because of an exceptional extraction of groundwater for other uses from t a on, as for example the construction of a reservoir or a transfer to another river basin.

The problem of the i th farmer becomes then [START_REF] Sanz | El acuífero 23 de la Mancha Occidental y el acuífero 24 del Campo de Montiel : Funcionamiento, Gestión, problemática y alternativas[END_REF], constrained by the dynamics:

Ġ =    -(1 -α) M i=1 g i + r 1 if t ≤ t a -(1 -α) M i=1 g i + r 2 if t > t a , (12)
with r 1 > r 2 , F i (G, g i ) from equation [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF], and conditions [START_REF] Cantos | Tipología de sequías en España[END_REF] and [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF].

In what follows, we are going to solve the game including this shock when players have dierent information structures: open-loop (OL) and feedback (FB), as in Rubio andCasino (2001, 2003) ( [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF], [START_REF] Rubio | Strategic Behavior and Eciency in the common Property Extraction of Groundwater[END_REF]). In every case, we solve problems in two steps: between t a and ∞ and between 0 and t a . We anticipate that equilibria of the various problems will be dierent according to the structure of information used by players. We call adaptation behavior the extraction decisions that are implemented by farmers from t=0 until the end of the planning horizon, knowing from t=0 that a shock will occur at some xed t a1 .

3

Non-cooperative cases

Resolution of the open-loop case

We assume that farmers made a commitment at the initial instant (t = 0) about their path of extractions over time. This is an open-loop information structure. The Hamiltonian corresponding to the problem of the ith farmer is:

H i =    F i (G, g i ) + π i (t)(r 1 -(1 -α) M i=1 g i ) if t ≤ t a F i (G, g i ) + π i (t)(r 2 -(1 -α) M i=1 g i ) if t > t a , (13)
with F i (G, g i ) from equation [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF], and π i (t), the adjoint variable. G(t) and π i (t) are continuous functions in the interval [0, ∞). We have detailed the analytical resolution of the open-loop game in appendix A.

The open-loop solution supposes that the farmer does not change his extraction decisions over the whole time horizon. This assumption is not very realistic whenever the farmer can observe the resource stock, and then indirectly the actions of the other users of the resource. If he acts strategically he might want to adapt to the other users' actions.

As discussed in Negri [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF], the open-loop solution does not allow to account for strategic interactions among resource users, but the feedback solution does.

3.2

Resolution of the feedback case

In the feedback information structure, farmers observe the level of the resource during the planning period, i.e. they have information about the state (or the water-table level) of the resource over time. Indeed, in many cases of groundwater management, the water-table level can be observed from the individual wells. Thus, it is more credible for the farmers to maximize their prot assuming that actions or strategies made by the other farmers depend not only on time but on the state of the groundwater resource. We are going to solve this case on the basis of the principle of dynamic programming. The full resolution of the problem is detailed in Appendix B.

4

The pareto optimum

One of the objectives of this paper being to estimate the ineciency of various equilibria dened previously, we need to dene the ecient solution of the problem: the pareto optimum. The dierence between the pareto optimal solution and any non-cooperative solution will dene ineciency, either in terms of stocks or in terms of welfare. Ineciency in terms of stocks is measured comparing steady state levels of stocks, while ineciency in terms of welfare is measured comparing gains obtained over the whole time horizon. For the pareto optimal solution, we suppose that a social planner decides how to manage the resource. The problem for the regulator is to maximize the social welfare, dened as the present value of the sum of future revenues of the M users of the resource.

The problem for the regulator is:

max {g i } M i=1 ∞ 0 M i=1 F i (G, g i )e -ρt dt, (14)
with F i (G, g i ) described in equation [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF], constrained by equation of motion [START_REF] Nieswiadomy | Input Substitution in Irrigated Agriculture in the High Plains of Texas 1970-80[END_REF], initial condition [START_REF] Cantos | Tipología de sequías en España[END_REF] and positivity conditions [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF]. Now, if a shock occurs at the known date t a , the problem for the social planner becomes [START_REF] Tsur | Endangered aquifers: groundwater management under threats of catastrophic events[END_REF], constrained by the equation of motion [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF], where r 1 and r 2 are values of the recharge rate before and after t a , respectively, with initial and positivity conditions described in equations [START_REF] Cantos | Tipología de sequías en España[END_REF] and [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF]. The full resolution of this problem is detailed in Appendix C. As discussed in de Frutos Cachorro et al. (2014) [START_REF] De Frutos Cachorro | Optimal adaptation strategies to face shocks on groundwater resources[END_REF], optimal adaptation behavior in presence of such shocks is characterized by a more intense extraction behavior in the short run (before the occurrence of the shock), namely "the impatience eect", and a less intense extraction behavior in the long run, compared to the situation without shocks. This is because the presence of a shock reduces the recharge rate, i.e. the water available in the future, thus reducing extraction possibilities in the long run. In addition, the "impatience eect" induces farmers to accelerate extraction in the short run to oset future losses in the long run. One important issue is now to see whether this compensating short-term behavior might be changed in case of non-cooperative solutions.

In what follows, we analyze and compare the pareto optimal solution with the dierent equilibria (open-loop and feedback) obtained when such a shock takes place. [START_REF] Koundouri | Current Issues in the Economics of Groundwater Resource Management[END_REF] Results

Theoretical Results

In this section, we compare analytically the eciency of the dierent stock solutions at the steady state. We remind that ineciency in terms of stocks is dened as the dierence between steady-state stock levels obtained from the pareto optimal solution and dierent private equilibria. From equations (67), (30), (52) in the Appendices, we obtain solutions of the stock for the pareto optimum (PO), the open-loop (OL) and the feedback case (FB), with M, the number of symmetric farmers (M > 1):

G P O ∞ = r 2 cb(1 -α) + r 2 ρ - a cb + z c , (15)
G OL ∞ = r 2 cb(1 -α) + r 2 M ρ - a cb + z c , (16)
and

G F B ∞ = r 2 M (1 -α)a * 1 - b * 1 a * 1 , (17)
with expressions b *

1 = b * 1 (r 2 ) < 0, a * 1 > 0 and lim r 2 →0 G F B ∞ = lim r 2 →0 G OL ∞ = lim r 2 →0 G P O ∞ = -a + zb cb > 0.
See Appendix D for more details about the positivity of the steady states and the proofs of propositions of this section. Moreover, solutions of the pumping rate at the steady state are:

g F B ∞ = g OL ∞ = g P O ∞ = r 2 (1 -α)M . (18)
Proposition 1 When the value of the recharge rate upon occurrence of the shock, r 2 , decreases (respectively increases), the level of the stock at the steady state decreases (respectively increases) for the dierent cases (PO, OL and FB) and reaches the same limit value when r 2 goes to zero. Moreover, solutions of pumping rates at the steady state are the same for the dierent information structures and decrease (respectively increase), the lesser (respectively the greater) the value of r 2 .

Hence, the shock decreases steady state stock levels for both the pareto optimal solution and the non-cooperative solutions. Negri shows in [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF] that open-loop solutions, and in a more important way feedback solutions, are inecient with respect to the pareto optimal solution. We have now to study the dierence of the various solutions to further quantify the eect of the shock on the dierent pumping strategies, or more specically, to analyze if the ranking of eciency of solutions is maintained in presence of such a shock. Dierences are calculated and described below. The dierence,

G P O ∞ -G OL ∞ = r 2 ρ 1 - 1 M , (19)
captures the pumping cost eect, while the dierence

G OL ∞ -G F B ∞ = r 2 1 M ρ + 1 (1 -α)cb - 1 M (1 -α)a * 1 - a cb + z c + b * 1 a * 1 , (20)
captures the strategic (or stock) eect.

Proposition 2 When a deterministic shock on the recharge rate takes place, the cost and strategic eects are positive.

Proposition 2 shows that the cost and strategic eects are positive when there is a shift on the recharge rate at a given date. This means that both non-cooperative solutions are inecient, in terms of stocks, in presence of the shock. An important issue is now to study how these ineciencies are inuenced by a change in the value of the shock. a non-transferable quota depending on the resource stock and take the example of the feedback case.

Proposition 4 There exists an optimal quota ḡ(G) with ḡ = g P O , such that g = g P O is a solution of the feedback problem, constraint by g ≤ ḡ.

Proposition 4 shows that an optimal quota ḡ(G) could be imposed in order to reach the pareto optimal solution, and this quota corresponds to the pareto optimal extraction path.

However, Gisser and Sanchez argued in [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF] that regulation of a common groundwater resource is not justied if the dierence of welfare from private and optimal exploitations is insuciently important. In what follows, we apply our game to the Western la Mancha aquifer in order to estimate ineciency in terms of welfare 3 and not only in terms of stocks. This application also allows us to estimate the magnitude of ineciencies, for shocks of dierent intensities and occurring at dierent dates.

Numerical application

In this section, we use parameter values of the Western La Mancha aquifer (WLM) from de Frutos Cachorro et al. (2014) [START_REF] De Frutos Cachorro | Optimal adaptation strategies to face shocks on groundwater resources[END_REF], Esteban and Albiac (2011) [START_REF] Esteban | Groundwater and ecosystems damages: Questioning the Gisser-Sánchez eect[END_REF] and Esteban and Dinar (2012) [START_REF] Esteban | Groundwater-dependent ecosystems: How does the type if ecosystem aect the optimal management strategy[END_REF]. The parameter values used are listed in Table 1.

Covering a surface area of 5 500 km 2 in the South of Spain, the WLM aquifer is located in a semi-arid region characterized by low and irregular rainfall and by high evapotranspiration due to the signicant number of sunny days (cf. López [START_REF] Sanz | El acuífero 23 de la Mancha Occidental y el acuífero 24 del Campo de Montiel : Funcionamiento, Gestión, problemática y alternativas[END_REF] [START_REF] Sanz | El acuífero 23 de la Mancha Occidental y el acuífero 24 del Campo de Montiel : Funcionamiento, Gestión, problemática y alternativas[END_REF]). These conditions give rise to a very low ecient recharge rate of the aquifer of around 20% of precipitation. Moreover, drought episodes can be sustained over time, lasting several years, as happened in [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF]-1985and 1990-1996(cf. Olcina Cantos (2001) [START_REF] Cantos | Tipología de sequías en España[END_REF]). In the last decades, the WLM aquifer witnessed a critical decrease in water-table levels, due to the development of intensive irrigated agriculture, which increased groundwater extraction (3 thousand million m 3 over the past three decades according to López-Gunn (2012) [START_REF] López-Gunn | Groundwater governance and social capital[END_REF]), coupled with inecient management regimes (Esteban and Albiac (2011) [START_REF] Esteban | Groundwater and ecosystems damages: Questioning the Gisser-Sánchez eect[END_REF]). The degradation of water-table levels has caused damage to aquatic ecosystems as the "Mancha Húmeda" Biosphere reserve and also aects human uses downstream. In what follows, we rst estimate the ineciency, as dened in the previous section, in terms of stocks, and then in terms of welfare. We also discuss the protability of the dierent solutions until the date of the shock, that is gains obtained in [0, t a ]. Moreover, we complete this analysis by studying the problem when the intensity and the date of occurrence of the shock vary.

In Figure 1, we observe optimal solutions of stock G * (t) (on the left) and pumping rate g * (t) (on the right), in particular the pareto optimal (PO) (in green), the open loop In what follows, we use this shock as a baseline case. 4Focusing on the left-hand side of the gure, we note that G P O (t) > G OL (t) > G F B (t) for all t. This gives the ranking of eciency in terms of stocks. In particular at the steady state, the pareto optimal stock reaches a level of 76 711 Mm 3 , which is higher than levels obtained by the OL and FB solutions (of around 73 811 and 71 962 Mm 3 respectively). Thus, the dierence between the pareto optimal and the open-loop solutions is 2 899 Mm 3 whereas the dierence between the pareto optimal and the feedback solutions is 4 749 Mm 3 . We hence conrm theoretical results proved in Proposition 2: the cost and strategic eects in the steady state remain positive when a shock takes place. Moreover, we observe on the right-hand side that the pumping rate at the steady state is the same for the dierent solutions, with a value of approximately 181 Mm 3 /year, as demonstrated in Proposition 1.

We can also analyze the problem before the occurrence of the shock, between t=0 and t = t a = 20 years. On the right hand, we note that total extractions until the arrival of the shock, measured by the areas under the curves in [0, t a ], are higher in the feedback case (9 672 Mm 3 ) than in the OL (8 383 Mm 3 ) and PO (6 044 Mm 3 ) solutions. This means that the feedback strategy is the less conservative for the resource until t a = 20 years. In other words, the "impatience eect", that is the increase of extractions before the occurrence of the shock is most important in the feedback case, and least important in the pareto optimal solution.

Let us now calculate the ineciency in terms of welfare of the private solutions5 over the whole time horizon (in [0,∞)), as well as their protability before the occurrence of the shock (in [0,t a ]). Over the whole time horizon, the ineciency of the FB solution with regard to the PO solution is estimated at 37 478 thousands of Euros and the ineciency of the OL solution with respect to the PO solution at 14 393 thousands of Euros. This can be seen in Table 2, column 3. Maybe more surprisingly, in [0,t a ], the dierence of welfare between the PO and FB solutions is positive, but the dierence between the PO and OL solutions is negative, that is the OL strategy is more protable than the PO solution until the occurrence of the shock, (as can be seen in Table 2 column 2). However, this result is not related to the occurrence of the shock. To see why, let us illustrate the evolution of the instantaneous welfare (in euros per year), dened as the function F i (G, g i )e -ρt in equation [START_REF] Sanz | El acuífero 23 de la Mancha Occidental y el acuífero 24 del Campo de Montiel : Funcionamiento, Gestión, problemática y alternativas[END_REF], for the FB (in red), OL (in blue) and PO (in green) solutions in absence of any shock. Figure 2 displays this evolution for dierent time intervals and Table 3 provides the corresponding accumulative welfares (that is the area under the curve) of these zooms. We see in Table 3 that until the 5 th year of exploitation the most protable solution is the FB solution, that gives up his position to the OL solution before the end of the 10 th year. It is not until the 30 th year onward that the PO solution becomes the most protable. This gives interesting insights in how the ineciency of private solutions is distributed over time. simulate a shock of 210 Mm 3 /year, which is about 140 Mm 3 /year more intense than the shock described in the previous section (and illustrated in Figure 1), but takes place at the same date.

First of all, we analyze the problem at the steady state and quantify results presented in propositions 2 and 3. We note that the cost and the strategic eects remain positive. We also note that cost and strategic eects decrease with the more intense shock. Hence, we numerically conrm one of our main theoretical result in Proposition 3: the greater the value of the shock, the smaller the cost and strategic eects, dened as the dierences between steady state stock solutions, and then the smaller the ineciency from private solutions in terms of stocks. Moreover, we can observe that cost and strategic eects are half as high as that of the mid-intense shock of 70 Mm 3 /year, which is three times less intense. This means that cost and strategic eects do not vary proportionally with a change on the intensity of the shock. Thus we add to the previous result the nding that ineciency decreases with more intense shocks but less than proportionally. We conrm this in Table 4 where other simulations for shocks of dierent intensities are illustrated. Table 4: Dierences between stock values in millions of m 3 at the steady state. Secondly, in Table 5, we can observe that the total welfare in [0, ∞] decreases the more intense the shock (columns 3, 5, 7 and 9 rst three rows). This result is logical in the sense that after the occurrence of the shock, extractions decrease the higher the intensity of the shock due to the decrease on water availability.

r 1 -r 2 0 30 70 210 [0, t a ] [0, ∞] [0, t a ] [0, ∞] [0, t a ] [0, ∞] [0, t a ] [0, ∞
However, we can also observe that the dierences of total welfare between solutions in [0, ∞] (columns 3, 5, 7 and 9, last three rows) decreases the more intense the shock. In other words, the ineciency in terms of welfare is maximal for the case without shock. Let us observe that the maximum of 41 265 (column 3 last row) exceeds by 3 787 thousands Euros the value of the baseline case. Hence, we numerically show that the greater the value of the shock, the smaller the dierences between solutions and then the smaller the ineciency, not only in terms of steady state stocks but also in terms of welfare. As we advanced in section 5.1, this result is explained by the fact that the less abundant the water to share in the future, the smaller the extend to which ineciencies from non-cooperative users develop. Now, we analyze extraction behavior before the occurrence of the shock (before t a ) for the various solutions illustrated in Figure 3. In [0, t a ], total extractions in the feedback case (9 757 Mm 3 ) remain higher than in the OL (8 810 Mm 3 ) and PO (6 474 Mm 3 ) cases. Moreover, in the three cases, we observe a more intense extraction behavior (see the righthand side of Figure 3) in comparison with the shock of mid-intensity of 70 Mm 3 /year (that was illustrated in Figure 1). In particular, total extractions increase by 85, 427 and 4307 Mm 3 for the FB, OL and PO cases respectively when the intensity of the shock increases by around 140 Mm 3 /year. We can conrm this result for dierent types of shocks by observing the left-hand side of Figure 4, that illustrates the total amount of extractions until t a with respect to the value of the shock, r 1 -r 2 . We note that the impatience eect increases the higher the shock but this increase is more important in the PO and OL cases than in the FB case. This could be interpreted as the fact that an increase in the "impatience eect" is less needed when considering the "strategic" externality, because extractions in the FB case are already very important in the beginning of the exercise. This is due to the fact that farmers have more information about the state of the result in the FB case and then they may react earlier to the shock. Finally, we can study the dierences in welfare between solutions in [0, t a ] (see Table 5, columns 2, 4, 6 and 8, last three rows): welfare obtained from PO-OL and OL-FB strategies vary in a non-monotonic way with respect to the value of the shock, reaching sometimes negative values. Again, the pareto optimum is not the most protable solution, if we analyze the problem in [0, t a ]. OL is performing better until the 20 th year of exploitation. We will see in next section that the ranking of solutions in [0, t a ] is driven by a change on t a . After the analysis and estimation of extraction behavior in the dierent cases according to the intensity of the shock, we study the dierent solutions with respect to the date of occurrence of the shock. In Figure 5, we observe optimal solutions of stock G * (t) (on the left) and pumping rate g * (t) (on the right), in particular the pareto optimal (in green), the open loop (in blue) and the feedback solutions (in red), for a shock of mid-intensity of 70 millions of cubic meters per year (Mm 3 /year), i.e. r 1 -r 2 = 70 Mm 3 /year occurring in the 5 th year of exploitation (t a = 5 years). In what follows, we compare this shock with the previous shock illustrated in Figure 1, which has the same intensity but takes place 15 years later.

In the long-run, steady state stocks do not depend on the date of occurrence of the shock as we note in the analytical solutions (equations ( 15), ( 16) and ( 17)). Individual welfare does change with the date of occurrence of the shock, as shown in Table 6. We note that total welfare in [0, ∞] increases the later the shock occurs for all information structures (see columns 3, 5, 7 and 9 of the rst three rows in Table 6). Logically, farmers are better o, if recharge rates are higher for longer times. The same result is obtained when we compute dierences between solutions (see Table 6, columns 3, 5, 7 and 9, of the last three rows). As an example, the loss in total welfare derived from private exploitation (feedback solution) with respect to optimal exploitation, is greater of 3 855 thousands of Euros when the shock occurs in t a = 20, instead of occurring earlier at t a = 5. Hence, the later the shock occurs, the greater the dierences in welfare between solutions and therefore the greater the ineciency. Farmers have more time to deploy their non-cooperative strategies and hence foster ineciencies.

Next, if we do again an analysis of protability in terms of stocks and welfare in [0, t a ], we conrm that until the arrival of the shock, total extractions are higher in the FB case (4 254 Mm 3 ) than in the OL (3 328 Mm 3 ) and PO (1 845 Mm 3 ) cases. Moreover, they are less important than in the previous shock that occurs at t a = 20. We note that this result is monotonic in time as shown in the right-hand side of Figure 4. In terms of welfare, we increases the later the shock occurs. This is linked to the fact that extractions are also more important when the shock takes place later for each information structure. However, if we analyze dierences between solutions in [0,t a ] (see last three rows of columns 2,4,6 and 8 in Table 6), these are negative when the shock occurs at an earlier date (t a = 5).

t a 0 5 20 50 [0, t a ] [0, ∞] [0, t a ] [0, ∞] [0, t a ] [0, ∞] [0, t a ] [0, ∞] PO 0
Until the 5th year of exploitation, the feedback solution seems to be the most protable strategy but it is not the optimal solution over the whole time horizon. To briey summarize this section, we have shown that in the long term (in [0, ∞)), ineciency from private exploitation can be reduced when water availability decreases.

If we analyze the problem until the 5 th year of exploitation of the aquifer, the feedback adaptation behaviors perform best, as they react earliest to the shock. Indeed, this is logical as feedback solutions correspond to the strategies where most information is available. If we analyze the problem until the 20 th year of exploitation, open-loop strategies are most protable. Finally, over the whole time horizon, the pareto optimum is the most ecient solution.

6

Conclusions and discussion

We have extended the analysis of a regime shift made in de Frutos Cachorro et al. ( 2014) [START_REF] De Frutos Cachorro | Optimal adaptation strategies to face shocks on groundwater resources[END_REF] by taking into account dierent externalities which arise in the exploitation of a common groundwater resource, i.e. the dynamic and strategic interactions between users of the resource. Moreover, we have added to the existing literature which compares private and optimal exploitation of groundwater resources [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF] [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF], Rubio and Casino (2001) [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF]) the consideration of regime shifts following a decrease on the recharge rate of the aquifer.

We rstly show analytically that pumping cost and strategic eects decrease the greater the intensity of the shock. In order words, steady-state stock solutions get closer, and then ineciency private exploitation, dened as the dierence between the pareto optimal and the feedback solutions, is reduced when the resource becomes more scarce. This result can seem counter-intuitive at rst sight, but is in the same line with the Gisser and Sánchez eect (e.g. [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF]). In their paper, Gisser and Sánchez prove that if the storage capacity (area of the aquifer times storativity) of the aquifer goes to innity the dierence between private and optimal solutions goes to zero. In their model, the recharge of the aquifer and the storage capacity dene the evolution of the water table height, through an inverse relationship. In our context, instead of considering the problem of a big groundwater area we consider the problem of relling the aquifer with small quantities of water. Although pumping cost and strategic eects decrease with greater shocks, there might still be need for policy intervention. We show analytically how the optimal quota should be imposed in order to correspond to the pareto optimal extraction path.

Next, we apply our game for the particular case of the Western la Mancha (WLM) aquifer. In the last decades, the WLM aquifer witnessed a critical decrease in water-table levels, due to the development of intensive irrigated agriculture, coupled with inecient management regimes. Following the reasoning of Gisser and Sánchez in [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF], a welfare analysis is necessary to justify an intervention in the management system.

We conrm that private solutions are inecient compared to the pareto optimum, not only in terms of stocks but also in terms of welfare. In particular, in terms of stocks, the consideration of the strategic externality (feedback solution) exacerbates the overexploitation of the resource with respect to open loop strategies. These results are in agreement with the existing literature [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF] [START_REF] Negri | The Common Property resource as a Dierential Game[END_REF], [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF] [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF]). Moreover, the ineciency of private exploitation in terms of welfare decreases the higher the value of the shock and the earlier the shock occurs. This may be explained by the fact that there is less water to share in the future, and respectively, less time to compete for water and this is why the ineciency cannot develop as easily as in a case with greater water availability. Nevertheless, the policy implication of our analysis is that a regulation through a centralized management of the Western la Mancha aquifer is still justied, because the pareto optimal solution allows positive eciency gains, even though these gains are less important than in a case without shocks.

We also analyze how the ineciency of private solutions is distributed over time. As in de Frutos Cachorro [START_REF] De Frutos Cachorro | Optimal adaptation strategies to face shocks on groundwater resources[END_REF], we show that there is an "impatience eect", that is an increase in extractions before the occurrence of the shock. This "impatience eect" is more important in the FB case than in the PO and OL cases, increasing ineciency in the most plausible non-cooperative solution. However, the FB solution is less inuenced by an increase of the value of the shock than SO and OL solutions because farmers are already exploiting the resource stock more heavily in the beginning of the exercise than in the other solution cases. Indeed, farmers have extra information about the stock levels in the FB case, and then could adapt earlier to the shock. When the shock arrives at an earlier date, the FB (and then the OL solution) is the solution which entails the greatest gains in the short run, i.e. before the occurrence of the shock. This can explain why private farmers could be attracted by this solution. Nevertheless, the pareto optimum remains the most ecient solution over the entire time horizon. As we have mentioned previously, in order to bring the individual farmer to adopt the optimal solution path, licences could be imposed to bring the farmer from the FB solution path back to the PO solution path. Of course, those policies rely entirely on the individual farmer's faith to be compensated in future periods. Some possible extensions of the article should also be mentioned. Firstly, we can introduce uncertainty in the model, for example on the date of the shock, as realized in [START_REF] De Frutos Cachorro | Optimal adaptation strategies to face shocks on groundwater resources[END_REF], or on the intensity of the shock. Secondly, we could consider the date and the intensity of the shock as decision variables of the resource manager8 . To tackle this issue it would be necessary to redene the objective function taking into account the goal of water extraction. As the objective is dened currently, it is optimal to never implement the shock. With an objective that takes into account the trade-o between delivering water to farmers or to other users, the socially optimal date and intensity of the shock could be computed. Mathematically, such a problem could be solved in a Stackelberg game. We can also study as in [START_REF] Tsur | Endangered aquifers: groundwater management under threats of catastrophic events[END_REF] [START_REF] Tsur | Endangered aquifers: groundwater management under threats of catastrophic events[END_REF] an endogenous shock, that is an event whose occurrence is determined solely by the exploitation policy. Finally, it would be interesting to introduce heterogeneities between groups of farmers.

A Resolution for the open-loop case

We are going to solve the open-loop case proceeding rstly between t a and ∞. The Hamiltonian of this problem is:

H i = F i (G, g i ) + π i (r 2 -(1 -α) M i=1 g i ) = a b g i - 1 2bθ i g 2 i -(z -cG)g i + π i (r 2 -(1 -α) M i=1 g i ).
(21) Applying the maximum principle and assuming interior solutions, we have the usual rst order conditions:

∂H i ∂g i = 0 ⇒ a b -z + cG - 1 bθ i g i -π i (1 -α) = 0, ( 22
) πi = - ∂H i ∂G + ρπ i ⇒ πi = -cg i + ρπ i . (23)
The equilibrium of the open-loop game is obtained by solving M strategies which verify the conditions ( 22) and (23) (i=1..M), i.e. a linear system of 2M equations. To simplify the analytical resolution of the problem, we assume that players are symmetric, θ i = 1 M , g = g i and π = π i . From (22), we nd the optimal rate of extraction as a function of the resource stock and the shadow price:

g = 1 M (a -zb + cbG -πb(1 -α)). (24)
Substituting (24) in the second part of equation ( 12) and adjoint variable (23), we have the following dynamic system:

Ġ = r 2 -(1 -α)(a -zb) -cb(1 -α)G + πb(1 -α) 2 , ( 25
) π = 1 M (-c(a -zb) -c 2 bG + cb(1 -α) + ρM )π, (26)
which allows us to nd the roots of the characteristic polynom:

β 1,2 = ρM + c(1 -α)b(1 -M ) 2M ± ρ 2 M 2 + cb(1 -α)(-2M (1 -α)cb + c(1 -α)b(1 + M 2 ) + 2ρM (1 + M )) 2M . (27)
From equations ( 24), ( 25) and ( 26), with Ġ = 0 and π = 0, we nd the steady state of the problem:

g OL ∞ = r 2 (1 -α)M , (28)
π OL ∞ = cr 2 M ρ(1 -α) , (29)
G OL ∞ = r 2 cb(1 -α) + r 2 M ρ - a cb + z c . (30)
Assuming parameters are positives, g ∞ and π ∞ (equations ( 28) and ( 29)) have always positives values. Moreover, in what follows, we assume parameters such as the value of G ∞ (equation (30)) is also positive.

Finally, we nd optimal extraction path with β 2 , the negative root:

G OL+ (t) = e β 2 (t-ta) (G ta -G OL ∞ ) + G OL ∞ , (31)
g OL+ (t) = r 2 (1 -α)M - β 2 (1 -α)M e β 2 (t-ta) (G ta -G OL ∞ ), (32)
π OL+ (t) = e β 2 (t-ta) (π ta -π OL ∞ ) + π OL ∞ , (33)
and,

π ta = a b(1 -α) - z -cG ta (1 -α) - 1 b(1 -α) 2 (r 2 -β 2 (G ta -G OL ∞ )), (34)
which is obtained from equations ( 22) and (25).

In a second step, we will solve the problem between 0 and t a . In this period, the Hamiltonian of the problem is described by:

H i = F i (G, g i ) + π i (r 1 -(1 -α) M i=1 g i ) = a b g i - 1 2bθ i g 2 i -(z -cG)g i + π i (r 1 -(1 -α) M i=1 g i ).
(35) We use the same principle of resolution than previously. We have rst order conditions (equations ( 21), ( 22), (23) with r 2 = r 1 ) by applying the maximum principle. Moreover, we assume that players are symmetric.

In a nite horizon problem, we write solutions as described below:

G OL-(t) = C 1 e β 1 t + C 2 e β 2 t + C 3 , (36)
π OL-(t) = D 1 e β 1 t + D 2 e β 2 t + D 3 . (37)
Substituting G OL-(t) and π OL-(t) (equations ( 36) and (37)) in rst order conditions (22), (23), and taking into account boundary conditions G(0) = G 0 and π(t a ) = π OL+ (t a ), we obtain a system of 6 equations with 6 unknowns (C i , D i with i=1,2,3). We nd the following solutions to the system: 9

C 1 = -(1 -α)b(-ρM cr 1 + ρM c(1 -α) 2 bπ ta -(1 -α)c 2 br 1 ) -c 2 e β2ta ρM (1 -α)bG 0 D 1 +ce β2ta ρM (r 1 + (1 -α)zb -(1 -α)a) + cr 1 M β 2 -(1 -α)ρM 2 π ta (β 2 -ρ) + c 2 e β2ta b(1 -α)r 1 D 1 +ce β2ta ρM (r 1 + (1 -α)zb -(1 -α)a) + cr 1 M β 2 -(1 -α)ρM 2 π ta (β 2 -ρ)) D 1 , (38)
9 Solution of D3 is not detailed here, but they are available from authors request.

29

with

D 1 = ρM (-e β1ta β 1 (ρM +cb(1-α)-M β 2 )+(1-α)cb(-(1-α)cb(e β2ta -e β1ta
)+M e β1ta (β 2 -ρ))),

C 2 = -(-ρM -cb(1 -α) + β 2 M )(-c 2 b 2 (1 -α) 2 r 1 -ρM e β1ta cb(1 -α)G 0 (β 1 + cb(1 -α)) D 2 +cb 2 (1 -α) 3 ρM π ta + e β1ta (β 1 + cb(1 -α))(ρM (r 1 -(a -zb)(1 -α)) + cb(1 -α)r 1 ) D 2 , (39)
with

D 2 = cb(1 -α)ρM ((β 2 -ρM -cb(1 -α) + M β 2 )), +cb(1 -α)(e β2ta cb(1 -α) + e β1ta (-cb(1 -α)(β 2 -ρ)M )), C 3 = ρM (r 1 -(a -zb)(1 -α)) + cb(1 -α)r 1 cb(1 -α)ρM , (40)
and π ta described in equation (34). Finally, taking into account that G(t) is a continuous function (G OL-(t a ) = G OL+ (t a )), we nd optimal solutions for the open loop game, that is G OL (t), g OL (t) and π OL (t).

B Resolution of the feedback case

Now, for the feedback case, we solve as previously rst the problem between t a and ∞.

The problem of player i is:

max g i (.) ∞ ta F i (G, g i ) e -ρ(t-ta) dt, (41)
with F i (G, g i ) (equation ( 8)), constrained by the dynamics

Ġ = -(1 -α)(g i + M j=1, j =i a j G + b j ) + r, (42)
with r = r 2 and conditions [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF] and G(t a ) = G ta . For each player i (i = 1..M ) the optimal value of the resource, V i (G), have to verify the Hamilton-Jacobi-Bellman equation:

ρV i (G) = max g i (F i (G, g i ) -V i G (G)(r 2 -(1 -α)(g i + M j=1, j =i g j ))), i = 1..M, (43)
with

V i (G) = AG 2 + BG + C, (44)
g j = a j G + b j . (45)
Assuming that players are symmetric, we rewrite the equation (43) as follows:

ρV i (G) = max g i (F i (G, g i ) -V i G (G)(r 2 -(1 -α)(g i + (M -1)g j ))), i = 1..M, (46)
Next, solving the problem on the right hand-side of (46), we nd the optimal pumping rate of player i, g * i :

g * i = a * i G + b * i . (47)
with,

a * i = b(c -(1 -α)2A) M , (48)
Next, we solve the problem between 0 and t a . The value function of the problem of player i, V i (t, G)10 veries the Hamilton-Jacobi-Bellman equation:

ρV i (t, G) -V i t (t, G) = max g i (F i (G, g i ) -V i G (G)(r 1 -(1 -α)(g i + M j=1, j =i g j ))), i = 1..M, (53)
with

V i (t, G) = A(t)G 2 + B(t)G + C(t), (54)
g j (t) = a j (t)G + b j (t), (55)
and the transversality condition,

V i (t a , G ta ) = V + (G ta ). (56)
The value of V + (G ta ) is obtained from the rst step of resolution of the problem, and is described in equation (51). To solve the second step of the problem, we are going to use a similar process as previously. The challenge here lies in the fact that strategies of players depend on the stock of the resource G and on functions a j (t) and b j (t) in an independent way. So, the resolution of the problem is largely numerical.

Once again, assuming players are symmetric, we rewrite (53) as

ρV i (t, G)-V i t (t, G) = max g i (F i (G, g i )-V i G (G)(r 1 -(1-α)(g i +(M -1)g j )), i = 1..M, (57)
and we solve the right part of equation (57). We then nd the expression (47) that is the optimal pumping rate of player i, g * i (t), with a i = a i (t) and b i = b i (t), described in equations ( 48), (49) and A = A(t), B = B(t), which are now functions that depend on t. Now, substituting g * i (t) in the right part of equation ( 53), and equalizing the right and left parts of the equation, we have to solve a system of 3 dierential equations in A(t), B(t) and C(t), which are coecients of the value function V (t, G), between t = 0 and t = t a , (see equation ( 54)), taking into account boundary conditions:

A(t a ) = A * , B(t a ) = B * , C(t a ) = C * ,
derived from the transversality condition (56):

V -(G ta , t a ) = A(t a )G 2 ta + B(t a )G ta + C(t a ) = V + (G ta ).
At this stage, we obtain A * (t) and B * (t) by a numerical approximation method. Substituting A * (t) and B * (t) in expression g * i (t), we nd the optimal values of coecients b * (t) and a * (t) of the pumping rate. Next, we substitute these values in the equation of motion (42) with r = r 1 . Finally, we obtain the numerical solution of the feedback problem between 0 and t a , that is G F B-(t) and g F B-(t), where the initial condition G(0) = G 0 , is given.

C Resolution of the pareto optimum

To solve this problem, we separate it into two parts and proceed by backward induction. First, we solve the maximization between t a and ∞. The problem of the social planner is to nd φ(G ta ),

φ(G ta ) = max g i (.) ∞ ta M i=1 F i (G, g i ) e -ρ(t-ta) dt, (58)
with F i (G, g i ) (equation ( 8)), constrained by equation ( 9) with r = r 2 and conditions [START_REF] Provencher | The Externalities Associated with the Common Property Exploitation of Groundwater[END_REF] and G(t a ) = G ta . The Hamiltonian of this problem is given by:

H = M i=1 ( a b g i - 1 2bθ i g 2 i -(z -cG)g i ) + λ(-(1 -α) M i=1 g i + r 2 ),
where λ is the adjoint variable. Applying the maximum principle and assuming interior solutions, we have the usual rst order conditions:

∂H ∂g i = 0 ⇒ a b - 1 bθ i g i -(z -cG) -λ(1 -α) = 0, i = 1..M, (59) λ
= - ∂H ∂G + ρλ ⇒ λ = -c M i=1 g i + ρλ, i = 1..M. (60)
We assume that players are symmetric in order to simplify the analytical resolution of the problem. Thus, θ i = 1 M and g = g i . From (59), we nd the optimal extraction volume as a function of the resource stock and the shadow price:

g = 1 M (a -zb + cbG -λb(1 -α)). (61)
Substituting (61) in the second part of equation of motion [START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF] and equation of adjoint variable (60), we have the following dynamic system: (64)

Ġ = r 2 -(1 -α)(a -zb) -cb(1 -α)G + λb(1 -α) 2 , (62)
From equations (61), ( 62) and (63), with Ġ = 0 and λ = 0, we nd the steady state of the pareto optimum problem:

g P O ∞ = r 2 (1 -α)M , (65)
λ P O ∞ = cr 2 ρ(1 -α) , (66)
G P O ∞ = r 2 cb(1 -α) + r 2 ρ - a cb + z c . (67)
Since we assume that all parameters are positive, g ∞ and λ ∞ (equations ( 65) and (66)) are always positive. Moreover, in what follows, we consider parameters such that G ∞ (equation ( 67)) is positive.

Finally, we have the optimal extraction paths from t = t a , with ρ 2 , the negative root:

G P O+ (t) = e ρ 2 (t-ta) (G ta -G ∞ ) + G ∞ , (68)
λ P O+ (t) = e ρ 2 (t-ta) (λ ta -λ ∞ ) + λ ∞ , (69)

g P O+ (t) = r 2 (1 -α)M - ρ 2 (1 -α)M (G ta -G ∞ )e ρ 2 (t-ta) , (70)
with,

λ ta = a b(1 -α) + -z + cG ta (1 -α) - r 2 b(1 -α) 2 + 1 b(1 -α) 2 ρ 2 (G ta -G ∞ ), G
and ρ 1 , ρ 2 described in equation ( 64). This constitutes a system of 6 equations and 6 unknowns, which we can solve to nd optimal solutions for the problem for the rst period, between 0 and t a . We nd optimal values of A i , B i (i = 1..3)13 : We have obtained: (85) and

A i = C i 1 + C i 2 (2υG *
G F B ∞ = r 2 M (1 -α)a 1 - b 1 a 1 , (84)
W = ρ 2 M 2 b 2 + 4M 2 ρ(1 -α)c b + 4(M -1) 2 (1 -α) 2 c 2 > 0.
Stability condition implies that 2A < c 1-α and then a 1 > 0. To obtain G F B ∞ > 0 from (84) for all r 2 we must impose b 1 < 0.

Remind also that

G P O ∞ = r 2 cb(1 -α) + r 2 ρ - a cb + z c , G OL ∞ = r 2 cb(1 -α) + r 2 M ρ - a cb + z c .
To have G P O ∞ > 0 and G OL ∞ > 0 for all r 2 we must impose -a + bz > 0.

As we assume that all the parameters are positive, it is easy to verify that lim

r 2 →0 G F B ∞ = lim r 2 →0
G OL ∞ = lim . This is immediate from equations ( 15) and ( 16) for the pareto optimal and the openloop cases. Assuming -a + bz > 0, it is enough to prove that the derivatives of expressions described in these equations with regard to r 2 are bigger than 0.

In the feedback case, replacing expressions of equation ( 85) in (84) and dierentiating with respect to r 2 , it is clear that

∂G F B ∞ ∂r 2 = 1 b(1 -α) [c -2A(1 -α)] > 0,
due to the stability condition.

D.3

Proof of proposition 2:

Sign of G F B ∞ -G OL ∞ and G OL ∞ -G P O ∞ .
Firstly, from equation (19), It is evident that

G OL ∞ -G P O ∞ < 0 .
Next, replacing expressions of equation ( 85) in (20), we can see that (86)

G F B ∞ -G OL ∞ = -

Figure 1 :

 1 Figure 1: Solutions of G * (t) (left-hand side) in millions of cubic meters and g * (t) (righthand side) in millions of cubic meters per year : the pareto optimum (in green), the open-loop (in blue) and the feedback (in red) cases, when r 1 -r 2 = 70 and t a = 20 years.

Figure 2 :Figure 3 :

 23 Figure 2: Evolution of F i (G, g i ) = F i (G, g i )e -ρt (in Euros per year) obtained by the PO (in green), OL (in blue) and FB (in red) strategies for the problem without shock (r 1 = r 2 = 360). Zoom in [0, 5] (left-top side), [0, 10] (right-top side), [0, 20] (left-bottom side) and [0, 30] (right-bottom side).

Figure 4 :

 4 Figure 4: Total amount of extractions until t a in millions of Mm 3 with respect to the value of the shock r 1 -r 2 (left-hand side) and the date of occurrence (t a ) (right-hand side), for the FB, OL and PO cases.

5. 2 . 2

 22 Variation of the date of the shock

Figure 5 :

 5 Figure 5: Solutions of G * (t) (left-hand side) in millions of cubic meters and g * (t) (righthand side) in millions of cubic meters per year : the pareto optimum (in green), the open-loop (in blue) and the feedback cases (in red), when r 1 -r 2 = 70 and t a = 5 years.

  λ = -c(a -zb) -c 2 bG + (cb(1 -α) + ρ)λ,(63)which allows us to nd the roots of the characteristic polynom:ρ 1,2 = ρ ± ρ 2 + 4cb(1 -α)ρ 2 .

  ta unknown. Substituting (68) and (70) in problem (58), we can compute the scrap value, φ(G ta ) 11 :φ(G ta ) = + κG ta + ιG 2 ta , with12 (71)κ = -r 2 ρ(4cb(1 -α) + ρ) + ρ 2 (1 -α)(a -zb) + 4cb(1 -α) 2 ρ(a -zb) ρ(1 -α) 2 b(η + ρ + 4cb(1 -α)) + (2cb(1 -α) + ρ)r 2 η -ρ(1 -α)η(a -zb)) ρ(1 -α) 2 b(η + ρ + 4cb(1 -α)) ,

  ι = -c(-4cb(1 -α) -ρ + η) (1 -α)(η + ρ + 4cb(1 -α)) , and(73)η = √ ρ ρ + 4bc(1 -α).

  a ) = B 1 e ρ 1 ta + B 2 e ρ 2 ta + B 3 = 2ιG ta + κ,

r 1 + b( 1 - 1 - 1 ,D 1 = 1 = 2 + c 2 b 2 ( 1 2 ,D 2 =

 11111122122 (t a ) + τ ), i=1,2, with,C 1 1 = b(1 -α)(ρc(r 1 + (1 -α)e ρ 2 ta (a -zb) -r 1 e ρ 2 ta ) -ρ 2 cr 1 ) D α)((1 -α)c 2 (r 1 b + e ρ 2 ta ρbG 0 -r 1 be ρ 2 ta )) α) 2 ρ(ρ 2 -ρ -cb(1 -α)) D ρ((ρ2-ρ)e ρ 1 ta cb(1-α)+c 2 b 2 (1-α) 2 (e ρ 2 ta -e ρ 1 ta )-cb(1-α), ρ 1 e ρ 1 ta +ρ 1 e ρ 1 ta (ρ 2 -ρ)), C 2 -(cb(1 -α) + ρ -ρ 2 )(ρ 1 e ρ 1 ta ρ((1 -α)(a -zb) -r 1 ) D -α) 2 (G 0 ρ + r 1 -r 1 e ρ 1 ta ) D 2 , +cb(1 -α)((G 0 ρ -r 1 )e ρ 1 ta ρ 1 -ρr 1 e ρ 1 ta ) + cb(1 -α) 2 e ρ 1 ta ρ(a -1 -α) + ρ -ρ 2 )(cb 2 (1 -α) 3 ρ) D cb(1 -α)ρ((ρ 2 -ρ)cb(1 -α)e ρ 1 ta + c 2 b 2 (1 -α) 2 (e ρ 2 ta -e ρ 1 ta )) +cb(1 -α)ρ(-cb(1 -α)e ρ 1 ta ρ 1 -ρe ρ 1 ta ρ 1 + e ρ 1 ta ρ 1 ρ 2 ),and,A 3 = -r 1 ρ -ρa + ρzb + ραa -ραzb + cr 1 b -cr 1 bα bρ(α -1)c .

  the continuity of the function of the variable state, i.e. G P O-(t a ) = G P O+ (t a ), we obtain optimal solution of the stock G P O (t) and water pumping g P O (t) for the pareto optimum problem. D Steady states and proofs of propositions D.1 Conditions for positive steady-state values.

  with

a 1 =

 1 b(c -(1 -α)2A) M , b 1 = a -zb -b(1 -α)B M , A = M [2 c(1 -α) -ρ/b] + √ W 4(2M -1)(1 -α) 2 , B = -2A [M r 2 -M (a -bz)(1 -α)] -c(a -bz) (2M -1)(1 -α) 2 b2A -M ρ -M cb(1 -α) ,

2 c b r 2 ( 1 -

 21 α)(M -1)

Table 1 :

 1 Values of parameters of the Western La Mancha aquifer.

	Parameters	Description	Units	Value
	b	Water demand slope	(Million Cubic Meters/Year) 2 Euros -1	0.097
	a	Water demand intercept Million Cubic Meters/Year	4403.3
	z	Pumping costs intercept Euros/Million Cubic Meters	266 000
	c	Pumping costs slope	Euros/(Million Cubic Meters) 2	3.162
	r	Natural recharge	Million Cubic Meters/Year	360
	G 0	Stock level (in volume) Million Cubic Meters	80960
	H 0	Current water table	Meters	640
	S L	Surface elevation	Meters	665
	A	Aquifer area	Square Kilometers	5500
	S	Storativity coecient	unitless	0.023
	ρ	Social discount rate	Year -1	0.05
	α	Return ow coecient	unitless	0.2
	M	Number of players	unitless	2

Table 2 :

 2 Welfare (and dierences of welfare) obtained by the PO, OL and FB strategies (respectively between solutions) in thousands of Euros, when r 1 -r 2 = 70 and for the date of occurrence t a = 20.

	6
			[0, t a ]	[0, ∞]
		PO	111 462 146 658
		OL	114 886 132 265
		FB	101 039 109 180
		PO-OL -3 424 14 393
		OL-FB 13 847 23 085
		PO-FB 10 423 37 478
		[0, 5]	[0, 10]	[0, 20]	[0, 30]
	PO	45 997 75 471 109 655 127 758
	OL	66 880 93 282 114 894 124 670
	FB	70 257 89 395 101 100 105 741
	PO-OL -20 883 -17 811 -5 239	3 088
	OL-FB -3 377	3 887	13 794 18 929
	PO-FB -24 260 -13 924 8 555	22 017

Table 3 :

 3 Welfare (and dierences of welfare) obtained by the PO, OL and FB strategies (resp. between solutions) in thousands of Euros, for the problem without shock (r 1 = r 2 = 360) in [0, t] for t= 5, 10, 20 and 30.In what follows, we analyze how previous results change according to the intensity and date of occurrence of the shock.

	F´PO(t)

Table 5 :

 5

	]

Welfare (and dierences of welfare) obtained by the PO, OL and FB strategies (respectively between solutions) (in thousands of Euros) for dierent values of the shock r 1 -r 2 and for the date of occurrence t a = 20.

Table 6 :

 6 Welfare (and dierences of welfare) obtained by the PO, OL and FB strategies (respectively between solutions) (in thousands of Euros) for a shock of value r 1 -r 2 = 70 Mm 3 and dierent dates of occurrence.

			133 375 48 912 138 348 111 462 146 658 144 321 152 021
	OL	0	120 991 68 017 125 350 114 886 132 265 133 402 137 086
	FB	0	101 058 70 268 104 725 101 039 109 180 109 876 111 594
	PO-OL	0	12 384 -19 105 12 998	-3424	14 393 11 219 14 935
	OL-FB	0	19 933 -2 251 20 625 13 847 23 085 23 526 25 492
	PO-FB	0	32 317 -21 356 33 623 10 423 37 478 34 745 40 427

can see in

Table 6 (rst three rows, columns 2, 4, 6 and 8) that welfare obtained in [0,t a ]

On the other hand, we call non-adaptation behaviour, when farmers do not have information about the shock until it happens and then change their extraction decisions just from the date of occurrence.

Proposition 3 When r 2 decreases (respectively increases), the cost and strategic eects decrease (respectively increase).Proposition 3 exposes one of our main results: it shows that pumping strategies, at the steady state, derived from private and optimal solutions get closer if the aquifer recharge decreases. This result might be counter-intuitive at rst sight as the occurrence of a shock reduces the ineciency of the dierent non-cooperative solutions. It may be explained by the fact that a reduction on the recharge rate, implies a reduction on the water available to share in the future. Thus, ineciency cannot develop to the same extent as in a case with greater water availability.We remind that Rubio and Casino found the same expressions (19) and (20) in[START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF] in their model without shock. They show rst that the dierence between solutions declines as the discount rate and/or the number of farmers increases. They also conrm that the same result is obtained when the storage capacity of the aquifer 2 increases (GSE eect). In this paper, we add to Rubio and Casino's result the importance of a recharge rate variation.In order to correct this ineciency, instruments such as licences or quotas can be used that bring the resource user to respect pareto optimal extraction paths. Let's consider[START_REF] Esteban | Groundwater and ecosystems damages: Questioning the Gisser-Sánchez eect[END_REF] In[START_REF] Rubio | Competitive versus ecient extraction of a common property resource: The groundwater case[END_REF], the storage capacity of the aquifer corresponds to the expression G/H, where G is the volume of water and H is the water-table height of the aquifer.

We remind that ineciency in terms of welfare is dened as the dierence between gains obtained from the pareto optimal solution and the dierent private equilibria over the whole time horizon.

This choice is motivated by the fact that the decrease of the water-table obtained in ta = 20 can be compared to the estimated drop of 3 000 Mm 3 over the last 30 years reported by López-Gunn.

Solutions of welfare in Tables5 and

are associated to individual strategies.

This does not mean that there is an optimal time in order to implement policy instruments. Policy measures should be implemented over the whole planning horizon, as shown in Proposition 4, in order to reach the PO path.

These numbers are obtained by deducting total extractions before ta for a shock of 210 Mm 3 /year and a shock of 70 Mm 3 /year, more specically, by computing the dierences 9 757-9 672,

810-8 383 and 6 474-6 044 for the feedback, open loop and respectively, pareto optimal solutions.

We thank an anonymous referee for indicating this possibility.

We remind that in this type of problem with a nite horizon planning, the value function has to be described as a function that depends on G and t independently.

We nd that the expression φ(ta, Gta) does not have the independent term ta. In what follows, we write the scrap value function, φ(Gta).

We do not detail expression of because it is not necessary for the resolution of the problem, but it is available from the authors upon request

We do not provide detailed solutions of Bi (i = 1..3) because the equations are too long and they are not necessary for the proofs, however, they are available from the authors upon request.

and A and B in equation ( 85), appendix D. Substituting now g * i on the right hand-side of (46) and equalizing the left and hand sides of the equation, we obtain optimal values of coecients A, B and C of V i with regards to variables a j and b j , the coecients of the pumping rate function of player j (j = i). Moreover, replacing A * and B * (see formulas in (85)) in equations ( 48) and (49), and taking into account that players are symmetric, that is, g 1 (t)=g i (t) for any t = t a ..∞ (and then a 1 =a i and b 1 =b i for any i, (i = 1..M ), we nd optimal values of coecients of the pumping rate function of the individual player, a * 1 and b * 1 . Finally, substituting a * 1 and b * 1 in (47) and replacing g * 1 = g * i for any i, (i = 1..M ) in the dynamics of the aquifer (42) with r = r 2 , we may solve the dierential equation (42), constrained by condition G(t a ) = G ta . Thus, we obtain optimal solutions of the feedback problem, G F B+ (t), g F B+ (t) and the optimal value function V * (G) in [t a ,∞):

and

with,

and b * 1 , a * 1 described in equations ( 48) and (49).

We now turn to the second part of the problem, between 0 and t a , considering the optimal solution for the rst part. The problem for the social planner is now:

and the transversality condition:

with φ(ta, G ta ) described by equation ( 71). The Hamiltonian can be written as:

where λ is the adjoint variable. Applying the maximum principle and assuming interior solutions, we have the usual rst order conditions (59) and (60). From this and equation of the motion of the state (76), we have the system of dierential equations:

We know that the solutions of the nite problem are now, of the shape:

with,

D.4

Proof of proposition 3:

and

.

Finally, if we dierentiate equations ( 19) and (86) with respect to r 2 , it is immediate that

< 0

and

Proof of proposition 4:

We have to prove that g = g P O is a solution of the feedback problem constraint by g ≤

We solve as previously rst the problem between t a and ∞. Considering the symmetric case, V P O+ (G) and g P O+ are the solutions of the Hamilton-Jacobi-Bellman equation:

with

and r = r 2 . Likewise, V F B+ (G) and g F B+ (G), constraint by a quota ḡ(G) are the solution of the Hamilton-Jacobi-Bellman equation of player i:

with

and r = r 2 .

The objective is to nd ḡ(G) such that g F B+ = g P O+ .

We propose that V F B+ (G) = V P O+ /M and g F B+ = g P O+ . By rewriting (89), with

V F B+ (G) = V P O+ /M and g F B+ = g P O+ , we obtain

We want to prove that ḡ = g P O+ is the optimal quota of the above problem, i.e., there exists µ > 0 such that (91) holds. Using equations ( 87) and (91), we obtain:

(M -1) ∂F (g P O+ , G) ∂g -M µ = 0, ∀G, µ > 0.

Thus, µ can be dened by: µ = (M -1) ∂F (g P O+ ,G) ∂g M .

From (87), we know that ∂F (g P O+ , G) ∂g > 0, then µ > 0, and ḡ = g P O+ .

In a similar way, using the Hamilton-Jacobi-Bellman equation in a nite time horizon with a scrap value, we can show that g = g P O-is a solution of the feedback problem with quota restriction ḡ = g P O-between 0 and t a .