DRAFT for presentation at the AERE Conference, Seattle, June 2011
Collective Action and the Commons:
Are Cooperative Groundwater Institutions Stable in the
Presence of Environmental Externalities
Encarna Esteban Gracia and Ariel Dinar
Water Science and Policy Center
University of California, Riverside
Abstract
This paper focuses on economic impacts of cooperation between groundwater
users on aquifers management. Groundwater is a common pool resource with
several negative externalities associated with its use. Cooperation among users
is promoted as a means of achieving better management, internalizes the
damages of user’s activity and reduced extractions. This paper develops a
game theory framework to assess the value of cooperation in an aquifer that is
being over-exploited. The framework is applied to an aquifer that is divided into
three different sub-regions or sub-aquifers with different characteristics. The
water extractions in each sub-aquifer affect the groundwater extractions in the
other regions (extraction externality) and also the environment (environmental
externality). The model is tested numerically in one of the most important
aquifer in Spain, Eastern La Mancha aquifer. The results illustrate how
extraction externalities and environmental externalities interact in affecting the
likelihood of cooperation among the users.
Keywords:
cooperation,
non-cooperation,
externalities,
common-pool
resources, groundwater management, ecosystem, Shapley Value, Core, Game
Theory, public choice.
1. Introduction
The literature advocates public choice approaches for addressing common pool
resource (CPR) problems and their optimal management. CPRs are
characterized by open-access with associated congestion problems. Users of
CPR can access the resource and use all the quantity they want, but the
problem arises when the resource reaches a critical level of overexploitation,
which triggers congestion in the use of the resource. In the case of CPRs the
economic theory points out how markets by themselves are not efficient
instruments to manage the resource, and public intervention is required in terms
of property rights, quotas, taxes (Gordon 1954; Coase 1960; Ostrom 1990,
2010)
In recent decades, several approaches have been developed to analyze
different solutions to correct congestion/overexploitation of CPRs. The possible
interventions can be organized into three groups: the creation of property rights
(Coase 1960); the implementation of taxes and penalizations to incentive users
to reduce their consumption (Pigou 1920); and finally, the creation of institutions
and arrangements to incentivize users to internalize the congestion externality
and cooperate for preserving these resources (Gordon 1954; Ostrom 1990,
2010). There are also studies that apply game theory approaches to check the
feasibility of cooperation in CPRs and the way to achieve it (Negri 1989,
Polansky et al. 2006, Dufournaud and Harrington 1990, Tarui et al. 2008, Rubio
and Casino 2001, Provencher and Burt 1993, Madani and Dinar 2011).
Groundwater is an example of common pool resource1. Groundwater
resources are renewable resources that are used mainly for irrigation, with
‘geographically open-access’2. When there is no regulation users do not
internalize the external cost, leading to pumping rates that exceed recharge and
lead to congestion/overexploitation of the resource and related pollution issues
(Anderson et al. 1985, Tsur 1990, 1995, Dinar 1994, Dinar and Xepapadeas
1998, Roseta-Palma 2002, Yadav 1997).
While the literature provides options to deal with congestion externalities, it
does not address externalities that affect stability of the CPR management
1Gordon
studied the problem of CPRs in the case of fisheries harvesting. Milliman (1956)
identifies that similar inefficiencies are observed in the case of groundwater.
2 For only those who have physical access to the aquifer, which is different from the case of
fisheries, where everyone with access to a vessel can exploit.
2
arrangements in the presence of ecosystem damages that are different from
congestion externalities. Ecosystem externalities exist when the CPR is
connected to an ecosystem that can either be affected by the level of the CPR
or affect the quality of the CPR, and in addition may inflict indirectly on the
welfare of all users of the CPR and the society which they are part of.
In this paper we introduce a cooperative game theory framework for
governing groundwater resources when ecosystems are linked to the aquifers
and ecosystem damages are accounted for. We hypothesize that the stability of
the cooperative arrangements with and without the existence of externality cost
in the form of ecosystem damages differ due to differences in the overall
societal welfare. This calls for more aggressive public interventions in the case
of ecosystem damages from CPR congestion.
We develop a groundwater model to analyze the different users’ propensity
to cooperate and the efficiency of collective action in achieving a sustainable
groundwater management. We included an ecosystem health function that
connects the level of water in the aquifer and the value of the linked ecosystem.
The basis for cooperation in the case of existing ecosystem externalities is
smaller due to lesser incremental gains to the grand coalition. The higher the
ecosystem externality damage, the less the cooperative settlement will be
stable. In such case, we may expect more partial coalitions (partition game) and
especially when the aquifer is large and there exist many users. Thus, social
regulation is even more necessary in the case of an ecosystem externality.
The model analyzes the changes in net present value of social welfare and
farmers’ private benefits in the case of non cooperation and when cooperation
prevails. In the presence of ecosystem externalities the incremental gains for a
grand coalition may lead to creation of sub-coalitions that are preferred to the
grand coalition. The ‘status quo’ is defined in this paper where farmers
maximize the present value of their individual net present value of
income/benefits. In the ‘status quo’ farmers neither internalize the water
extractions affects on their neighbors’ income, nor the environmental externality.
The cooperative solution is reached when farmers introduce in their
maximization problem a function representing the environmental externality and
the other farmers’ consumption. In this case it is expected to obtain lower
private benefits and a decrease in farmers’ extractions, leading to a recovery in
3
the water table level. The social welfare is defined in the cooperative game, as
the farmers’ private benefits plus the incremental improvement in the
ecosystems due to the increase in the water level (private welfare plus external
benefits).
The model is applied to one of the most important aquifers in Spain-Eastern
La Mancha aquifer. The overexploitation of this aquifer during the last 30 years
is driving a significant depletion of the water table level. The case of this aquifer
is also important due to be one of the few examples worldwide of an aquifer
where cooperation between farmers is working successfully. This cooperation
began in the mid 1990s mainly because pressure from downstream users and
river basin authority to prohibit agriculture in this area.
A theoretical and empirical framework is developed in the paper. Section 2
presents the game theory model. Section 3 describes the study area of the
Eastern La Mancha aquifer and outlines the empirical model. Section 4
describes the results in the different scenarios and the effectiveness and
stability of possible coalitions between farmers. Finally, section 5 concludes.
2. A game theory model applied to groundwater management with
externalities
Assume a CPR shared by 𝑗 users, 𝑗 =1, …, 𝐽. For simplicity assume that each
user is confined to a certain land area with given characteristics that affects
both the performance of that user and also the performance of adjacent users
(extraction externality). Assume that the users/players are placed sequentially
such that player 𝑗 borders only with player 𝑗 − 1 and with player 𝑗 + 1(and that if
𝑗 = 1 it borders only with 𝑗 + 1, and if 𝑗 = 𝐽 it borders only with 𝑗 − 1). Assume
that each player 𝑗 uses the CPR for its own benefit. However, in doing so,
player 𝑗 inflicts negative extraction externalities on its neighbors 𝑗 − 1 and 𝑗 + 1.
For example, in the case of groundwater, such multidirectional externalities
could be in the form of a relative level of the water table. In addition, all players,
through their aggregate action impose an externality on an ecosystem that is
connected to the CPR and indirectly affects the players that are part of the
society that benefits from the ecosystem services.
Users of the CPR consider joint management options by agreeing on an
optimal management plan over a long-term planning horizon, which can be
4
translated into a net present value of benefits. Therefore, any cooperative
arrangement is referred to an allocation scheme of the present value of the sum
of net benefits over the planning horizon. For simplicity and without loss of
generality we will eliminate in the theoretical model the time variable and refer
to the annual level of decisions and payoffs.
Let 𝑋 be the maximum level of the CPR. Player 𝑗 uses 𝑥𝑗 ≤ 𝑋 such that
∑𝑗 𝑥𝑗 ≤ 𝑋 and produces 𝑦𝑗 = 𝑓𝑗 (𝑥𝑗 , ∑𝑖 𝑥𝑖 ; ∀𝑖 ≠ 𝑗 ) with
𝜕𝑦𝑖
𝜕𝑥𝑖
≤ 0. The expression
𝜕𝑦𝑖
𝜕𝑥𝑖
𝜕𝑦𝑗
𝜕𝑥𝑗
≥ 0,
𝜕 2 𝑦𝑗
𝜕𝑥𝑗 2
≤ 0, and
is the extraction externality. Let 𝑁 be the set of the
grand coalition players and 𝑆⊆𝑁 be the set of all possible partial coalitions. A
partial coalitions 𝑠 ⊆ 𝑆 may include 2, 3, …, 𝐽 − 1 members with the following
characteristic functional forms:
For a singleton coalitions {𝑗}:
𝑓𝑗 (𝑥𝑗 , 𝑥𝑗−1 , 𝑥𝑗+1 ) ∀ 𝐽 > 𝑗 > 1
𝑓𝑗 (𝑥𝑗 , 𝑥𝑗−1 ) ∀ 𝑗 = 1
𝑓𝑗 (𝑥𝑗 , 𝑥𝑗+1 ) ∀ 𝑗 = 𝐽
{𝑗}
𝑦𝑗 ={
(1)
For any coalition 𝑠 the characteristic function will depend on the number of
members in 𝑠 and on the location of 𝑆 in the CPR landscape.
For a two member coalition {𝑗, 𝑗 + 1} the characteristic function is:
{𝑗,𝑗+1}
𝑦2
𝑗+1
=𝑓𝑗 (∑𝑗
𝑥𝑗 ; 𝑥𝑗+2 ) ∀ 𝑗 = 1, 2, … , 𝐽 − 1
(2)
For a 3 player coalition:
{𝑗,𝑗+1,𝑗+2}
𝑦3
𝑗+2
=𝑓𝑗 (∑𝑗
𝑥𝑗 ; 𝑥𝑗+3 ) ∀ 𝑗 = 1, 2, … , 𝐽 − 2
(3)
.
.
.
For a 𝐽 − 1 player coalition:
{𝑗,𝑗+1,…𝐽−1}
𝑦𝐽−1
=𝑓𝑗 (∑𝐽−1
𝑥𝑗 ; 𝑥𝐽 ) ∀ 𝑗 = 1, 2, … , 𝐽
𝑗
(4)
And for the grand coalition:
{𝑁}
𝑦𝐽 =𝑓𝑁 (∑𝐽𝑗 𝑥𝑗 ) ∀ 𝑗 = 1, 2, … , 𝐽
(5)
The environmental externality is a value function that connects between the
remaining level of the CPR, 𝑋 − ∑𝐽𝑗 𝑥𝑗 , and the value of the ecosystem, 𝐸. The
lower the remaining level of the CPR the lower is the value of the ecosystem:
𝐸 = 𝑔(𝑋 − ∑𝐽𝑗 𝑥𝑗 ), with:
5
𝜕𝐸
𝜕 ∑𝐽𝑗 𝑥𝑗
<0
We postulate that |
(6)
𝜕𝐸
𝐽
𝜕 ∑𝑗 𝑥𝑗
𝜕𝑦
| ≫ |∑𝑖 𝜕𝑥𝑖 |. Assume that a grand coalition solution
𝑖
for the aquifer produces a value of 𝑣(𝑁) that needs to be divided among the
players so that the following relationships hold. Assume that the allocation of
the cooperative gains are divided among the players such that player 𝑗 gets 𝛺𝑗 .
Each player has to be better off in the cooperative solution compared with
the status quo (individual rationality). Thus, 𝛺𝑗 ≥ 𝑦{𝑗} ∀𝑗 ∈ 𝑁. Each player has
also to be better of in the cooperative solution than in any partial coalition
arrangement, fulfilling also the group rationality condition ∑𝑗∈𝑠 𝛺𝑗 ≥ 𝑦{𝑠} ∀𝑠 ⊂
𝑆. And finally, the value of the entire set of players have to be fully allocated by
the grand coalition participants, namely, ∑𝑗∈𝑁 𝛺𝑗 = 𝑦{𝑁}. An allocation that fulfills
these three requirements is in the core of the game.
3. The Eastern La Mancha aquifer model
Eastern La Mancha aquifer is one of the biggest aquifer in Spain. This aquifer
extends over 7,200 km2, covering the regions of Albacete, Cuenca and
Valencia. The aquifer is part of the Jucar River Basin Authority, representing an
18 per cent of the total area of the Basin (see Figure 1). The area has a
Mediterranean weather characterized by important fluctuations in daily and
seasonal temperatures, with cold winters and hot summers. The precipitations
also register several seasonal fluctuations and vary between nearly 350
mm/year in the south to nearly 550 mm/year in the north.
This groundwater is mostly used to irrigation agriculture (80% of the total
water use), but also there is a slight consumption for urban (8% of total water
use) and industrial activities (12% of total water use). During the 1970s and
mainly due to the development of advanced and efficient irrigation technologies
a significant increase in groundwater extraction took place. From the seventies
the irrigated agriculture has extended to nearly 90,000 Hectares. During this
period the pressure on the aquifer has largely increased, generating a
significant over-exploitation. This over-exploitation is affecting, mostly in dry
periods, the whole Jucar River system and the ecosystems linked to the aquifer.
6
Figure 1. Eastern La Mancha aquifer location
Source: adapted from Martin de Santa Olalla et al. (2007)
This paper models the cooperation in Eastern La Mancha, where farmers
are currently cooperating to reduce their groundwater consumption (Esteban
and Albiac 2010a,b). The Eastern La Mancha aquifer is linked with Jucar River.
In some parts of the River the aquifer supplies water to the River but in other
parts the River supplies water to the aquifer. Because the decrease in the water
table level the contributions from the aquifer to the River decrease, increasing
the flow from the River to the aquifer. Because the political pressure of farmers
downstream, individuals in Albacete realized that if they do not reduce their
extractions the whole agriculture in the area could be jeopardized. Therefore,
cooperation was initiated in order to reduce the extractions to a sustainable
level (recharge equal extractions).
The paper assumes an aquifer area divided into three different sub-areas
characterized by different water table depths, total area, irrigated area and
cropping pattern, aquifer natural level3, and recharge. The model assumes that
the farmers’ activity affects the ecosystems linked to the aquifer and also
neighboring areas (environmental and extraction externalities respectively).
Farmers in each sub-area can act individually, ignoring both the ecosystem and
the effect of their extraction on their neighbors’ extractions. They can partially
cooperate with their closer neighbor, internalizing the environmental externality
and the extraction externality (in partial cooperation one region acts as ‘freerider’). The last option is full cooperation, where all farmers cooperate
internalizing the entire externalities.
3
Aquifer level when non human interventions have been made.
7
3.1. Study area: sub-aquifers definition
The aquifer is divided following the work of Castaño et al. (2010). Six different
sub-aquifers
can be defined: Northern Domain (ND) that is extended over
1,870 km2 and the irrigation surface is around 150 km 2; Central Domain (CD)
spans over 3,600 km2 with an irrigated area of 670 km2; El Salobral-Los Llanos
Domain (SLD) with an area of 400 km2 and 200 km2 of irrigation agriculture;
Moro-Nevazos Domain (MND) with 520 km2 and an irrigation area of 300 km2;
Pozocañada Domain (PCD) with 270 km2 of which 70 km2 are irrigated; finally
Montearagón-Carcelén Domain (MCD) with 600 km2 which is not irrigated4. For
simplicity in the model the last three areas located in the south of the aquifer
(SLD, MND, and PCD) are lumped as South Domain, SD. Figure 2 shows the
different parts of the aquifer.
Figure 2. Eastern La Mancha aquifer sub-areas
Source: adapted with some modifications from Castaño et al. 2010.
The main crops are classified as spring, summer, and summer-spring.
Spring crops are characterized by total revenue per hectare of 2,705 m 3/ha, the
revenue of summer crops is 6,269 m3/ha, and finally spring-summer crops
revenue is 7,415 m3/ha5 (Sanz et al. 2009, 2011).
3.2. Empirical model
The empirical model is an optimal control problem that combines economic,
agronomic and hydrologic variables. The farmers demand function is a fixed
coefficient crop production function where each crop is characterized by fixed
4
5
The sub-area MCD is not included in the model because there is no irrigation.
The main crop in spring is wheat, in summer is onion and corn, and alfalfa in summer-spring.
8
yields water requirements 𝑤𝑖 and a revenue per unit of land 𝑏𝑖 . Where the index
𝑖 = 𝑠𝑝, 𝑠𝑢, and 𝑠𝑠, represents the crops (spring, summer, and summer-spring
respectively).
Farmers decide on how much land to allocate to each crop, so the
groundwater consumption. The total number of hectares/year, per crop and
𝑗
area, is defined as ℎ𝑎𝑖 (𝑡), where the indexes represent the different aquifer
areas (𝑗 = 𝐶𝐷, 𝑁𝐷, 𝑆𝐷) and crop types. A Leontief crop production technology is
assumed, where the inverse demand is a discontinuous function. The demand
function depends on the available land area and the net income per land unit of
each crop. Farmers’ annual private revenues are the per hectare crop-level
revenue multiplied by the area per each crop:
𝑗
𝑗
𝐵 𝑗 [ℎ𝑎𝑖 (𝑡), 𝑡] = ∑ 𝑏𝑖 ∙ ℎ𝑎𝑖 (𝑡)
(6)
𝑖
𝑗
where 𝐵 𝑗 [ℎ𝑎𝑖 (𝑡), 𝑡] is the revenue function, and 𝑏𝑖 is the crop revenue per
hectare. Revenues are defined as the fix quantity of water requirement per crop
(𝑤𝑖 ) multiplied by crop prices (𝑝𝑖 ), so 𝑏𝑖 = 𝑤𝑖 ∙ 𝑝𝑖 .
The annual production costs function by each area is the water extraction
costs plus fixed costs per hectare6:
𝑗
𝑗
𝐶𝑗 [ℎ𝑎𝑖 (𝑡), 𝐻 𝑗 (𝑡), 𝑡] = 𝐶0 ∙ ∑ ℎ𝑎𝑖 (𝑡)
𝑖
𝑗
+ 𝐶1 ∙ (𝑆𝐿𝑗 − 𝐻 𝑗 (𝑡)) ∙ ∑ ℎ𝑎𝑖 (𝑡) ∙ 𝑤𝑖
(7)
𝑖
where 𝐶0 is the fixed pumping cost per hectare which is multiplied by the total
irrigated acreage, and 𝐶1 is the per unit of water constant marginal costs of
pumping. The term 𝐶1 is multiplied by the height of pumping, and by the total
water pumped to cover irrigated acreage. The pumping height is given by the
difference between the surface level of the aquifer (natural level) 𝑆𝐿𝑗 and the
actual water table level 𝐻 𝑗 (𝑡) in each sub-region. The water extractions per
hectare for spring, summer and summer-spring crops are given by parameter
𝑤𝑖 . The pumping costs per cubic meter are driven by the height of the water
table, which is the difference between the natural and the water table levels in
each sub-aquifer.
6
The model does not include other production costs (neither fix nor variable) as labor, other
input costs, amortization, or machinery costs.
9
The farmers’ annual private gross margin is defined as the difference
between the farmers’ annual revenue and the annual extraction cost (equation 6
minus equation 7). This is the result under the ‘status quo’, where farmers
maximize their private gross margin ignoring that their water use affects the
ecosystems and neighbors farmers.
𝑗
𝑗
𝑗
𝐺𝑀𝑗 [ℎ𝑎𝑖 (𝑡), 𝐻 𝑗 (𝑡), 𝑡] = 𝐵 𝑗 [ℎ𝑎𝑖 (𝑡), 𝑡] − 𝐶[ℎ𝑎𝑖 (𝑡), 𝐻 𝑗 (𝑡), 𝑡]
(8)
The damages to ecosystem are caused by the fall in the water table and the
progressive reduction in the flows that feed aquatic ecosystems. These
damages are assumed here to be linear in the meters of aquifer reduction,
which can be expressed also as the volume of the aquifer depletion. They are
computed as the product of two terms: the damage per cubic meter of aquifer
water depleted 𝛽, multiplied by the meters of aquifer depletion (𝑆𝐿𝑗 − 𝐻 𝑗 (𝑡)).
The annual ecosystem damages are given by the following expression:
𝐷 𝑗 [ 𝐻 𝑗 (𝑡), 𝑡] = 𝛽 ∙ [𝑆𝐿𝑗 − 𝐻 𝑗 (𝑡)]
(9)
The farmers’ problem can be formulated as the maximization of the private
gross margin. The dynamic problem is as follows:
𝑇
𝑗
𝑀𝑎𝑥 𝐹𝑃 = ∑
𝑡=0
1
∙ [𝐵 𝑗 [ℎ𝑎 𝑗 (𝑡), 𝑡] − 𝐶[ℎ𝑎 𝑗 (𝑡), 𝐻 𝑗 (𝑡), 𝑡] ]
(1 + 𝑟)𝑡
𝑗
𝑠. 𝑡. 𝐻̇ 𝑗 =
𝑅 𝑗 + (𝛼 − 1) ∙ ∑𝑖 ℎ𝑎𝑖 ∙ 𝑤𝑖
𝐴𝑆 𝑖
𝐻 𝑗 (0) = 𝐻0𝑗
(10)
where 𝑀𝑎𝑥 𝐹𝑃𝑗 maximizes the farmers’ private gross margin over the planning
period subject to the hydrological behavior of the water flowing into the aquifer,
and the water table level initial value7. r is the discount rate, 𝐻̇ 𝑗 is the change in
the water table level over time 𝑡 (𝐻̇ 𝑗 = 𝐻 𝑗 (𝑡 + 1) − 𝐻 𝑗 (𝑡)). 𝑅 𝑗 is the natural
recharge of the aquifer8. The parameter 𝛼 is the return flow coefficient (the
water that returns to the aquifer due to water leaching and percolation). 𝐴𝑆𝑗 is
the aquifer area multiplied by the aquifer storativity coefficient. The total water
𝑗
extractions are represented by equation ∑𝑖 ℎ𝑎𝑖 ∙ 𝑤𝑖 . The total water extractions
Water stock at the first period (𝑡 = 0).
This model assumes a fix recharge during the time. This is a simplification of the model
because the recharge depends on stochastic elements like weather, soil, type of crops,
irrigation technology, etc. The recharge is different in each sub-region.
7
8
10
multiplied by (1 − 𝛼) is the total water from extractions that return to the aquifer
(with 𝛼 representing the irrigation technology efficiency).
When farmer 𝑗 internalizes the entire externalities (environmental and
extraction) of their activity the problem becomes:
𝑇
𝑗
𝑀𝑎𝑥 𝐹𝑃𝐸 = ∑
𝑡=0
1
∙ [𝐵 𝑗 [ℎ𝑎 𝑗 (𝑡), 𝑡] − 𝐶[ℎ𝑎 𝑗 (𝑡), 𝐻 𝑗 (𝑡), 𝑡] − 𝐷 𝑗 [ 𝐻 𝑗 (𝑡), 𝑡] ]
(1 + 𝑟)𝑡
𝑗
𝑠. 𝑡. 𝐻̇ 𝑖 =
𝑅 𝑗 + (𝛼 − 1) ∙ ∑𝑖 ℎ𝑎𝑖 ∙ 𝑤𝑖
𝐴𝑆 𝑖
𝑗−1
+ [𝐻0
𝑗−1
−𝜎
𝑗−1
𝑅 𝑗−1 + (𝛼 − 1) ∙ ∑𝑖 ℎ𝑎𝑖
∙
𝐴𝑆𝑗−1
∙ 𝑤𝑖
]
𝐻 𝑗 (0) = 𝐻0𝑗
(11)
where the 𝑗 − 1 represents the neighbor sub-aquifer that is being affected by
farmers in region 𝑗 consumption. The parameter 𝐻0𝑗−1 is the initial water table
level in region 𝑗 − 1, 𝜎 𝑗−1 shows the impact of the neighbor groundwater
consumption in region 𝑗. This value is calculating with the total water
consumption in region 𝑗 − 1, the weight is calculating dividing the total
extractions in region 𝑗 − 1 by the aquifer area, 𝜎
𝑗−1
𝑅 𝑗−1 +(𝛼−1)∙∑𝑖 ℎ𝑎𝑖
𝐴𝑆 𝑗−1
∙𝑤𝑖
𝑗−1
𝑗−1
=
∑𝑖 ℎ𝑎𝑖
∙𝑤𝑖
𝐴𝑆 𝑗−1
, finally
represents the extractions in region 𝑗 − 19 (Gura and
Maschler 2008:173-189).
4. Numerical application to the Eastern La Mancha aquifer
Different scenarios have been run by a period horizon of 50 years using the
program GAMS with the solver CONOPT. Farmers’ gross income, social
welfare, and environmental damages are reported. The parameters used in the
simulations are shown is Table 1.
Two main scenarios are simulated: in the first one farmers do not internalize
the environmental externality that their activity generates to the ecosystems; in
the second scenario, farmers internalize the environmental externality
introducing the ecosystem damage function into their optimization problem. An
additional scenario with a sensitivity analysis has been simulated in order to
9
In the case of region CD which is being affected by both neighboring ND and SD regions, the
externality should include both 𝑗 + 1 and 𝑗 − 1.
11
show the effect of different economic values of ecosystem in the coalitions’
stability.
Table 1. Aquifer parameters
Parameters
Description
Value (units)
Intercept of the pumping function cost
340 (€/ha)
𝐶0
Increase
in
costs
of
pumping
a
cubic
meter,
per
meter
𝐶1
0.0025 (€/m3∙m)
of decline in water table
Return flow coefficient
Social discount rate
Value of water for ecosystems (damage of depletion)*
Water requirements of spring crops
Water requirements of summer crops
Water requirements of spring-summer crops
Crop revenue of spring crops
Crop revenue of summer crops
Crop revenue of spring-summer crops
Water table natural level above sea level ND
Water table natural level above sea level CD
Water table natural level above sea level SD
Current water table elevation above sea level in ND
Current water table elevation above sea level in CD
Current water table elevation above sea level in SD
Recharge without irrigation return flows ND
Recharge without irrigation return flows CD
Recharge without irrigation return flows SD
Area of the aquifer times storativity in ND
Area of the aquifer times storativity in CD
Area of the aquifer times storativity in SD
𝛼
𝑟
𝛽
𝑤𝑠𝑝
𝑤𝑠𝑢
𝑤𝑠𝑠
𝑏𝑠𝑝
𝑏𝑠𝑢
𝑏𝑠𝑠
𝑆𝐿𝑁𝐷
𝑆𝐿𝐶𝐷
𝑆𝐿𝑆𝐷
𝐻0𝑁𝐷
𝐻0𝐶𝐷
𝐻0𝑆𝐷
𝑅 𝑁𝐷
𝑅 𝐶𝐷
𝑅 𝑆𝐷
𝐴𝑆 𝑁𝐷
𝐴𝑆 𝐶𝐷
𝐴𝑆 𝑆𝐷
Source: Sanz et al. (2009, 2011) and Judez et al. (2000, 2002)10
0.2
0.05
50,000 (€/m)
2,705 (m3/ha)
6,259 (m3/ha)
7,415 (m3/ha)
544 (€/ha)
2,500 (€/ha)
1,300 (€/ha)
680 (m)
660 (m)
670 (m)
650 (m)
635 (m)
640 (m)
75 (hm3)
150 (hm3)
95 (hm3)
59.84 (km2)
115.6 (km2)
38.08 (km2)
By each scenario different simulations have been run: i) ‘status quo’, where
farmers maximize their private gross income without internalizing any
externality, either the extraction externality; ii) partial cooperation, farmers
cooperate in groups of two11 (with the closer neighbor) internalizing both the
10
The economic value of the ecosystems has been approximated using the studies by Judez et
al. (2000, 2002). These authors calculate the economic value of the wetland “Tablas de
Daimiel” (National Park in Spain). This ecosystem is being affected by the decrease in the water
table level of the Western La Mancha aquifer. Similarly to what happen in “Tablas de Daimiel”,
several wetlands are also being affected in the case of Eastern La Mancha. The depletion of the
aquifer is affecting the survival of some ecosystems in this area. Due to the lack of information
about the economic value of the wetlands in this region we are using as an approximation the
value calculated to “Tablas de Daimiel”. A sensitivity analysis is made in the empirical
application in order to check the results with different economic values of ecosystems.
11 The third region, not into the coalition, acts as ‘free-rider’ without internalizing any externality,
neither environmental nor extraction.
12
environmental and the extraction externalities; iii) full cooperation, coalition
between the three regions where the entire environmental and extraction
externalities are internalized.
4.1. Analysis of results
In the first baseline (status quo) scenario, farmers do not internalize the effect of
their activity on the ecosystems that are linked with the aquifer. In the ‘status
quo’ non externality is internalized and each region acts separately. In partial
cooperation and full cooperation, the extraction externality is taken into account
but the environmental externality is ignored by all coalitions. The results of this
scenario are presented in Table 2.
Table 2. Baseline simulation: ‘status quo’, partial cooperation and full
cooperation when farmers do not internalize the environmental externality
‘Status quo’
Variable
ND
CD
SD
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
40.09
1,801.73
1,759.54
Partial cooperation ND and CD
ND+CD
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
54.25
753.70
699.30
94.34
2,556.61
2,462.41
Partial cooperation CD and SD
ND
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
* Data in million Euros
54.25
753.70
699.30
48.69
2,562.71
2,512.59
SD
48.69
2,562.71
2,512.59
CD+SD
88.71
4,375.53
4,288.27
Full cooperation
ND+CD+SD
142.96
5,130.41
4,991.15
The results suggest that partial or full cooperation achieves highest levels of
gross margin. The best results are achieved in the simulation of full cooperation,
demonstrating that when farmers decide to cooperate, all increase their private
benefits. In a similar way the social welfare increases when cooperation
between farmers takes place. The highest social welfare is reached in full
cooperation and the lowest is reached in the ‘status quo’. Partial cooperation
13
between CD and SD is always preferred than partial cooperation between ND
and CD (in terms of social welfare, gross margin, and environmental damage).
Table 3 presents the results when farmers internalize the environmental
externality of their activity. The model assumes that for any type of cooperation,
farmers internalize the environmental externality, while in the ‘status quo’
farmers continue ignoring this damage12. When partial cooperation exists the
region that is no cooperating does not internalize any of the externalities.
Table 3. Results of the ‘status quo’, partial cooperation and full cooperation
when farmers internalize the environmental externality**
‘Status quo’
Variable
ND
CD
SD
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
91.14
2,488.78
2,580.09
Partial cooperation CD and SD
ND
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
40.09
1,801.73
1,759.54
48.69
2,562.71
2,512.59
Partial cooperation ND and CD
ND+CD
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
54.25
753.70
699.30
54.25
753.70
699.30
SD
48.69
2,562.71
2,512.59
CD+SD
85.27
4,313.33
4,400.14
Full cooperation
ND+CD+SD
Environmental damage (€/year)*
137.55
Gross margin (€/year)*
4,982.48
Social welfare (€/year)*
5,123.90
* Data in million Euros
** The value of the environmental externality is 50,000€ per meter of aquifer depletion or
replenishment.
When the environmental externality is internalized the social welfare
increases, in all the cooperative simulations, compares with the baseline
scenario. On the other hand, the gross margin decreases, also in the three
cooperation simulations, due to the increase in the farmers’ internalization
costs. The highest social welfare is achieved when full cooperation exists. In the
case of partial cooperation, the best social welfare is reached when CD and SD
regions cooperate. The results are different when we analyze the gross margin.
The highest gross margin is reached in the ‘status quo’ due to non
12
The results of the ‘status quo’ are equal to the ones obtained in the baseline scenario.
14
internalization of any externality by the farmers. But when cooperation exists,
partial cooperation yields better payoffs than full cooperation, and cooperation
between regions ND and CD is superior to that between CD and SD (more
efficient in social welfare terms)13.
The last scenario illustrates a sensitivity analysis using different economic
values for the environmental externality. The values used are half and double of
the ecosystem value assigned in the previous scenario (Table 3). The results
are shown in Table 4.
The results in the sensitivity analysis corroborate the ones obtained in the
previous scenario. In the case of a small value of ecosystem damages (25,000€
per meter of depletion) the highest value of social welfare is achieved by the full
cooperation. Partial cooperation between CD and SD regions has also a higher
social welfare than the coalition between ND and CD regions. But in the case of
the gross margin, the highest value is achieved with the coalition between CD
and ND. The gross margin in this case is even higher than the ‘status quo’ one.
With a small value of ecosystem damage partial cooperation is preferred over
individual coalitions in the Status Quo.
In the case of a higher value of environmental damages (100,000€ per meter
of depletion) the results replicate the ones obtained in the previous scenario.
The highest social welfare is reached under full cooperation and the lowest in
the ‘status quo’. The social welfare in the coalition between CD and SD regions
is higher than the one in the coalition between ND and CD. The highest gross
margin is in the ‘status quo’ one, followed by the coalition between CD and SD.
The lowest one is achieved in the full cooperation because all farmers are
internalizing the environmental externality.
In partial cooperation simulations the ‘free-rider’ region does not internalize the environmental
externality. This fact generates a higher gross margin in partial cooperation compare with full
cooperation. So, the total gross margin is higher with partial cooperation rather than with full
cooperation.
13
15
Table 4. Sensitivity analysis: results of the ‘status quo’, partial cooperation and
full cooperation with different values of environmental damages**
SENSITIVITY ANALYSIS: VALUE ECOSYSTEMS 25,000 €/m
‘Status quo’
Variable
ND
CD
SD
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
20.04
1,801.73
1,779.58
Partial cooperation ND and CD
ND+CD
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
27.13
753.70
726.42
46.46
2,540.74
2,587.31
27.13
753.70
726.42
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
44.51
4,374.29
4,420.35
70.28
5,110.25
5,184.71
SENSITIVITY ANALYSIS: VALUE ECOSYSTEMS 100,000 €/m
‘Status quo’
Variable
ND
CD
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
80.18
1,801.73
1,719.45
175.42
2,376.47
2,552.07
Partial cooperation CD and SD
ND
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
108.51
753.70
645.04
Partial cooperation ND and CD
ND+CD
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
Variable
CD+SD
Full cooperation
ND+CD+SD
Variable
Variable
SD
24.34
2,562.71
2,536.93
Partial cooperation CD and SD
ND
Environmental damage (€/year)*
Gross margin (€/year)*
Social welfare (€/year)*
24.34
2,562.71
2,536.93
108.51
753.70
645.04
SD
97.38
2,562.71
2,463.89
SD
97.38
2,562.71
2,463.89
CD+SD
163.45
4,232.58
4,397.75
Full cooperation
ND+CD+SD
Environmental damage (€/year)*
264.53
Gross margin (€/year)*
4,863.44
Social welfare (€/year)*
5,132.23
* Data in million Euros
** The value of the environmental externality is per meter of aquifer depletion or replenishment.
16
4.2. Game theory allocations
We apply a couple of game theory allocation solution concepts to the aquifer
management problem, referring mainly to the results in Tables 2, 3, and 4. Two
cooperative game theory allocation schemes, the Nash-Harsanyi solution and
the Shapley Value are applied. Once we calculate the resulting allocations of
the full cooperation payoff among the three regions, we test for their inclusion in
the Core, which is one of the conditions for stability, and also apply the
Loheman Power Index, which is a proxy for stability of an allocation scheme (for
more explanation see Dinar et al. 2007:122-131). The Shapley Value (𝛺𝑗 ) is
calculated, based on the incremental contributions of the players to the grand
coalition. The Nash-Harsanyi allocation (𝛩𝑗 ) is calculated, by maximizing the
grand coalition’s members incremental payoff, based on an equal incremental
allocation of the total cooperative payoff (compared with the status quo) to all
players.
We start with calculations of the Nash-Harsanyi allocation, which is
presented in Table 5.
Table 5. Nash-Harsanyi allocation of the cooperative payoffs
Variable
No
internalization
of externality
Internalization of
externality
(50,000 €/m)
Internalization of
externality
(25,000 €/m)
Internalization of
externality, value of
externality 100,000
Gross Margin
ND+CD+SD
NDCDSD
Difference
Incremental gains
above status
Allocation
of
cooperation payoff
Core conditions
5116
5130
14
5116
4982
-134
5116
5110
-6
5116
4863
253
4.65
N/A
NA
84.33
{757.65, 1805.65,
2566.65}
In the Core
N/A
N/A
Not in the Core
Not in the Core
{837.33, 1885.33,
2646.33}
Not in the Core
Social Welfare
ND+CD+SD
NDCDSD
Difference
Incremental gains
above status
N-H Allocation of
cooperation payoff
Core conditions
4970
4991
21
4970
5123
153
5041
5184
143
4827
5132
305
7
51
47.66
101.66
{706, 1766,
2519}
In the Core
{750, 1810, 2563}
{773.66, 1826.66,
2583.66}
In the Core
{746.66, 1820.66,
2564.66}
In the Core
In the Core
As can be seen from Table 5 allocation of the gross margin doesn’t allow
always having incremental benefits, especially when the value of the
environmental externality is relatively low.
In addition, the grand coalition
values are always higher in the case of the social welfare calculations
17
compared with the case of the gross margin calculations. In the case of using
the social welfare, the cooperation benefits are the highest when the value of
the environmental externalities is the lowest.
In the case the Shapley Value we report the cooperation allocations and in
addition we calculate measures of stability of the solution. Table 6 provides the
results of such analyses.
Shapley value calculations suggest that in the case of the gross margin the
Shapley Value did not yield results that are in the Core for scenarios of the two
lower values of the environmental externality (25,000€ and 50,000€). All the
rest of the scenarios Sahpley Value allocations are in the core. In addition, the
Loehman Power Index can be presented in terms of Coefficient of Variation
(CV) of the set of individual Power Indexes with higher values of CV indicating
less stability. The results in Table 5 suggest that the allocation solutions for the
Social welfare scenarios are more stable than those for the gross margin. In
addition, stability of the Shapley solution decreases as the value of the
environmental externality increases.
Table 6. Shapley Value Allocations and Stability measures
Variable
No internalization
of externality
Internalization of
externality, value of
externality 50,000 E
Internalization of
externality, value of
externality 25,000 E
Internalization of
externality, value of
externality 100,000 E
Gross Margin
Shapley Allocation
of
cooperative
payoff
Core conditions
CV of
Loehman
Power
Index/Stability
{754, 1808, 2568}
{714, 1737, 2531}
{745, 1799, 2567 }
{683, 1665, 2515}
In the Core
Not in the Core
Not in the Core
In the Core
0.71
N/A
N/A
0.55
Social Welfare
Shapley Allocation
of
cooperative
payoff
Core conditions
CV of
Loehman
Power
Index/Stability
{701, 1770, 2521}
{727, 1852, 2544}
{784, 1817, 2583}
{691, 1873, 2568}
In the Core
In the Core
In the Core
In the Core
0.63
0.71
0.22
0.53
5. Summary and Conclusions
Common pool resources (CPRs) are goods with the characteristics of ‘openaccess’ and non rivalry in the consumption until congestion affects the resource
availability, leading to scarcity. Additionally, negative externalities may be
involved in the use of such resources. A typical example of CPR is groundwater
18
used for irrigation purposes. Due to the significant development of irrigated
agriculture worldwide and the exponential growth of world population during the
last century the pressure on water resources has largely increased. This
pressure creates several problems of water contamination and water scarcity.
This paper focuses on the study of groundwater management, which
traditionally was unregulated, leading to problems of overexploitation. Due to
the overexploitation, two main externalities can be identified, environmental
externalities (impacts on ecosystems linked to the aquifer) and extractions
externalities (impacts on other groundwater users). The paper develops a
model of an aquifer, which is divided into three different regions; each region
extracts water for irrigation purposes, generating both externalities. The model
analyzes the propensity for cooperation between regions under scenarios of
internalizing and not internalizing the different externalities. An empirically
application of the model is carried out in one of the most important aquifers in
Spain, the Eastern La Mancha aquifer.
The results of the analysis suggest that when environmental externalities are
not accounted for by players, the grand coalition is the preferred result. Farmers
and society are always better of when there is cooperation (partial or full) than
when farmers acts individually. But, when the environmental externalities are
accounted for the results may be less obvious. When regions/plyers internalize
the environmental externality the incentives to cooperate decrease; the higher
the environmental damage the lower the incentives to cooperate. Even though
in terms of social welfare the preferred option is the full cooperation, the Core
becomes smaller. These results are supported by some game theory allocation
solutions (Nash-Harsanyi and Shapley value).
Additionally, some stability measures have been calculated (Loheman
Power Index). These measures suggest that social welfare considerations lead
to more stable than gross margin ones. The stability of the possible coalitions
decreases as the value of the ecosystems’ damage increases. Without
environmental externalities famers have incentives to cooperate due to an
increase in their private gross margin, and the society welfare increases as well.
However, as soon as the environmental externalities are introduced the farmers’
gross margin incentives associated with cooperation disappear.
19
Acknowledgements
This paper was prepared while the first author was a post doc researcher at the
Water Science and Policy Center, University of California, Riverside. The
authors appreciate the assistance of David Sanz (Instituto de Desarrollo
Regional, University of Castilla la Mancha, Spain) for his valuable help with the
agronomic data and hydrological information.
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