Acceptable reforms of agri-environmental policies Philippe Bontemsy
Gilles Rotillonz
Nadine Turpinx
March 20, 2008
Abstract We consider a model of regulation for nonpoint source water pollution where farmers are heterogenous along two dimensions, their ability to transform inputs into …nal production and the productive land they possess. Regulation takes place through non linear taxation/subsidization of agricultural production and land, in the presence of asymmetric information about ability to produce. We also introduce a political acceptability constraint implying that the regulation has to be optimally designed taking into account the distribution of incomes in the pre-reform situation. We show that the optimal reform essentially amounts to reallocate production towards less e¢ cient farmers, who bene…t from the reform at the expense of more e¢ cient producers. Importantly, incentive compatibility requirement puts some strong restrictions on the way production should be allocated in the agricultural sector and thereby, contrary to what happens under perfect information, it allows to get some clear-cut results on who is over-compensated or under-compensated by the optimal regulatory reform. Last, we calibrate the model using data on a french watershed (Don watershed). Simulations indicate that, on our exemple, satisfying a high degree of acceptability may not entail high welfare losses compared to lower degrees of political sustainability. A low size farm has a higher probability of being a reform winner than a large farm, even though the regulator has no speci…c redistribution concern towards small farms. JEL : D82, Q19. Key-words : Non Linear Taxation - Asymmetric Information - Non Point Source Pollution - Water Pollution.
This paper was written as a part of the project ‘Systems approach to environmentally acceptable farming’ (AgriBMPwater EVK1-CT-1999-00025) that has been funded by the European Union. The views expressed are purely those of the authors and may not in any circumstances be regarded as stating an o¢ cial position of the European Commission. Corresponding author: Gilles Rotillon. y Toulouse School of Economics (GREMAQ, INRA and IDEI). z EconomiX, Université de Paris-X Nanterre. 200, avenue de la République, 92000 Nanterre. Phone #: 01 40 97 78 19. Email:
[email protected]. x CEMAGREF, UMR 1273 Metafort, Clermont-Ferrand.
1
Acceptable reforms of agri-environmental policies 1
Introduction
The use of economic incentive instruments such as green taxes in environmental policy has become increasingly popular in many countries and in particular in Europe. However, new taxes are often unpopular in polluting sectors that may prefer traditional command and control regulation. When designing a green tax scheme, the regulator has thus to take into account the potential losses imposed to the polluters when comparing with the pre-reform (or status quo) situation. In other words, the design of green reforms is subject to a political acceptability constraint. In principle, a tax scheme can be designed such that no producer looses through a reimbursement rule using lump sum transfer. In such a case any polluting …rm would enter voluntarily in the environmental regulation scheme given that no losses are imposed compared to the status quo situation. Such a highly acceptable policy comes usually at an ine¢ ciency cost: because compensations are socially costly, the environmental regulation should not impose too much e¤ort in order to be acceptable by everybody. Alternatively, the public authority might impose the term of regulation without fearing about losses imposed to polluters. Here the strong commitment power possessed by the government allows to push forward a mandatory regulation. Of course, such a political power might be limited by laws protecting polluters’revenues through limited liability. Obviously, the resulting policy would be highly e¢ cient from an environmental perspective but would gain very low political support in the polluting industry. The concern about political support for reform is particularly deep in the agricultural sector in Europe (see e.g. Thurston, 2002). Rising interests about environmental quality depressed by highly intensive agriculture have given birth to a series of reforms relating the level of subsidy to some environmental e¤ort by farmers. Up to now, those policies have largely relied on voluntary adoptions by farmers and have gained little success. It has 2
been sometimes argued that o¤ered compensations were insu¢ cient to cover the compliance cost and that uncertainty about the durability of environmental voluntary programs have undermined their success. A large rate of participation is obviously needed to attain the required environmental protection goals, so that ensuring this rate in a voluntarily way might be excessively costly. This situation raises the issue of how to regulate agricultural pollution abatement while sustaining a given level of political support. In this paper, we consider the case of water pollution caused by the excessive use of fertilizers in agricultural activities. It is well known that regulating agricultural pollution is quite di¢ cult because of its nonpoint source feature: individual emissions cannot be measured and controlled easily (Shortle and Horan, 2001). To overcome this di¢ culty, we consider the regulation of nonpoint source water pollution by an environmental agency through non linear taxation/subsidization of agricultural production and land. These variables are typically more easily observable than individual emissions or even the amount of fertilizers spread over the …elds. The population of farmers we consider is heterogenous along two dimensions: their ability to transform inputs into …nal production and the available area they possess. Productive ability is assumed to be private information to the farmers while available area is an observable characteristic which may be part of the regulatory scheme. Hence, private information makes self-selection necessary. In addition, we assume that farmers’pro…ts are protected by limited liability such that the reform cannot yield negative pro…ts. We introduce political acceptability as part of the constraints that the regulator has to take into account. Following Lewis et al. (1989), the regulator has to satisfy a given proportion of farmers through intervention and a farmer is satis…ed if he does not loose from regulation compared to the status quo situation. When this proportion is zero, the resulting policy is such that the political acceptability constraint is non binding and hence it can be labelled as a mandatory policy, which however does not allow the agency to impose negative pro…ts to farmers because of limited liability constraints. On the contrary, when this proportion is one, the policy is voluntary by its very nature as any farmer is ready to accept the reform. 3
Bontems et al. (2005) analyze the mandatory case in details and show that asymmetric information yields to a larger range of less e¢ cient farmers that are induced to quit the sector and a lower production level for any type of farmers compared to what prevails when individual ability is perfectly known to the agency. Overall, the total level of pollution is reduced because …rst some of the less e¢ cient and most polluting farms are shut down and second because the remaining active farms are required to produce less intensively and therefore pollute less compared to the pre-reform situation. This paper is devoted to the case where the proportion of farmers who have to be satis…ed is strictly positive. An important feature of the model is that the identity of winners is endogenous and determined by the agency as part of the optimal policy. Two regulatory regimes are identi…ed. When the required level of acceptability is low, the regulatory pattern is similar to the mandatory one except that all farmers are now provided with a strictly positive level of pro…t which allows the agency to meet the acceptability constraint. When this level of acceptability is high, the regulatory pattern is quite di¤erent and entails pooling for a subset of farmers at the equilibrium, due to the presence of countervailing incentives. The regulatory regime depends on the observable farm area. We indicate the optimal rule of allocation of satis…ed farmers according to their characteristics. Importantly, an acceptable reform amounts to reallocate production towards less e¢ cient farmers who bene…t from the reform at the expense of more e¢ cient producers. Furthermore, it appears that the presence of asymmetric information imposes some restrictions on the policy pattern that yields clear predictions with regards to the (endogenous) identity of reform winners, contrary to the case where information is complete. In the last part of the paper, we calibrate the model using data from a French watershed (Don watershed) and we simulate the optimal policy under di¤erent scenarii concerning the level of political acceptability. For this, we elaborate a speci…c method to classify farmers along a one-dimensional type that represent their productive ability. We also use the results from a hydrological model calibrated on the watershed to estimate nitrogen emissions. Dam4
ages are estimated by computing the treatment cost necessary to meet water quality standards actually enforced in Europe (upper limit on nitrogen water concentration). We show that, on our particular application case, satisfying a high degree of acceptability does not entail high welfare losses compared to a low degree of acceptability. However, the environmental damages varies highly with the level of political support required. The paper is organized as follows. We brie‡y survey the related literature on mechanism design applied to agri-environmental policies in Section 2. Section 3 is devoted to assumptions and notations and analysis of regulation under complete information. Regulation when selfselection is necessary is analyzed in section 4 while section 5 is devoted to the empirical application to the Don watershed. Section 6 concludes. Most proofs are relegated into an appendix.
2
Related literature
Regulating non point source pollution engendered by agricultural activities appears to be particularly a challenging task for regulators as individual emissions levels are typically dif…cult to monitor (Braden and Segerson, 1993). The related literature can be divided in two sub-…elds for the purpose of presentation (Xepapadeas, 1997, chapter 4). On the one hand, the design of collective incentives schemes in the multi-polluters context has been extensively analyzed starting from the pioneering work by Segerson (1988).1 On the other hand, some authors have suggested to rely on indirect individual regulation such as output/input based regulatory procedures (see Gri¢ n and Bromley, 1982, for an early analysis in the case of agricultural pollution). There, an important aspect is clearly the relationship of the regu1
In this approach, the set of polluters is considered like a team whose joint product is the level of pollution observed in the environmental media while individual contributions are private information (see Holmström 1982 for a seminal paper on moral hazard in teams). The regulatory agency monitors ambient pollution concentrations and implements contingent transfers to/from the polluters. For instance, taxes (subsidies) are charged (paid) to the agents when the ambient pollution concentration rises above (falls below) an exogenously determined target. The implementation of the incentive scheme yields the …rst-best outcome as a Nash equilibrium given that the pro…t, nonpoint emissions and fate and transport functions (and other essential informations) are common knowledge. This framework has been widely re…ned notably by incorporating risk aversion, asymmetric information on environmental and polluters’ characteristics and budget balance requirement (see Shortle and Horan, 2001, for a recent survey).
5
latory variables (inputs/outputs, including …eld management practices) with pollution ‡ows (Shortle et al. 1998). The indirect individual regulation approach is particularly useful in the case of large population of polluters where a collective incentive scheme based on some aggregate pollution measure is certainly di¢ cult to implement. From this perspective, in the context of nitrogen pollution, two important inputs are suitable candidates for regulation: the land allocated to polluting crops and the level of fertilizers used. For instance, Wu and Babcock (1996) develop a mechanism design approach to induce land-based nonpoint polluters to choose second-best input vectors depending on their land type. They consider farmers who are heterogeneous along one dimensional index of soil quality. Given the prior information on the distribution of the soil quality index, the second best policy consists in a menu of input vectors associated with transfer. Bontems et al. (2005) extend this approach by considering farmers heterogeneous in both their productive ability and the size of land available for production. In this setting, the regulation is based on the control of the land devoted to production and the production level. They show that a positive relationship between the size of land and the farmer’s ability may exacerbate adverse selection impacts. It should be noted that in this literature participation constraints are typically such that the income under regulation cannot be less than an exogenous level. Clearly, this excludes any attempt to take into account the pre-regulation distribution of incomes when designing the regulatory scheme. By contrast, Wu and Babcock (1999) among others have analyzed the case of voluntary mechanisms where an agent participates to the regulatory scheme if and only if his pro…t is greater than the pro…t under non-participation. In general, this opportunity pro…t from non participating depends on some individual characteristics that might be private information to agents. A closely related study by Lewis et al. (1989) introduces a political constraint to the regulatory scheme such that a given proportion of agents must be at least fully compensated for the cost of modifying productive decisions compared to the pre-reform ones. The moti6
vation of the regulation is that before the reform a pre-existing sub-optimal subsidy per unit implies that aggregate production is too large from a welfare perspective and hence Lewis et al. study how to design an optimal menu of production contracts that allow to reach the second best while simultaneously ful…lling the political constraint. In their setting agents are heterogenous along one dimension, namely the ability to produce which is private information to the agent. Our paper di¤ers from Lewis et al. following two aspects. First the motivation of the regulation is here to reduce pollution and second more importantly we account for additional heterogeneity of farmers according to emissions and size of the available land for production. This introduces some novelty in the sense that the regulator has not only to design the optimal regulation for a given class of farmers according to the land size but also to determine how to optimally allocate the weight of the political constraint according to the size of farms.
3
The model
3.1
The farmer’s behavior
Consider a watershed where a set of farmers produce output y (say milk) per land unit using a quantity s of land devoted to feed crops and a polluting input like fertilizers. Let us denote c(y; ) the production cost per land unit. We introduce heterogeneity among farmers through parameter
that belongs to the set
=
;
and that represents the farmer’s
ability to transform feed crops into the production of milk. Parameter
can be understood
as a function of several on-farm characteristics (management skills, soil quality, genetic value of the herd...). We assume that the cost of production c(:; ) is increasing convex in y. We also normalize the set of types by assuming that cost c is decreasing in . In addition, the marginal cost of producing milk is also decreasing with ability (cy < 0):2 In other words, a more e¢ cient farmer (higher type) is also associated with lower optimal rates of input use.3 2 3
We denote fx the …rst-order partial derivative of function f (x). In addition, we make the following technical assumptions: c yy < 0, c
7
y
> 0. These assumptions will
The land available for production that we consider is denoted by S which belongs to the set S = S; S . Land capacity S is distributed according to land density function h(S) with cumulative H(S). We also normalize the total land devoted to agriculture to unity. For each level of land capacity S, there is a continuum of farmers characterized by their ability, with mass unity. Farmers are distributed along this line segment according to ability density function f ( ; S) with cumulative function F ( ; S): We assume that f ( ; S) > 0 for every
and S: Note that we also assume that the set
this assumption does not presume that 1 F ( ;S) f ( ;S)
and
F ( ;S) f ( ;S)
does not depend on S.4 However,
and S are independent. Moreover, the risk ratios
are assumed to be strictly increasing in .5
Let us describe the private optimum of the pro…t-maximising farmer with ability
and
land capacity S, assuming there is no intervention by the environmental agency (status quo situation). It is given by solving the following program : max s;y
(s; y; )
(py
c(y; ))s s.t. s
S
where p is the (exogenous) product price.6 We assume that market and production conditions are such that all farms are active in the status quo situation. Hence the constraint s
S
is binding everywhere. Alternatively, we could have considered that the privately optimal cropped area s is interior, simply by adding a …xed cost k(s) for land in the pro…t function: (s; y; )
(py
c(y; ))s
k(s): This extension is straightforward and does not modify
drastically the features of the optimal policy derived in the paper. We denote y ( ) as the optimal production level of -type farmer.7 Given our assumptions help us in obtaining optimal policies that e¤ectively separate the di¤erent farmers. 4 This assumption can be relaxed but at a much higher cost of complexity for the analysis. In that case, it is possible to diminish incentives costs because private information is partially veri…able by the regulator (see Green and La¤ont 1986). 5 These regularity conditions are made in order to prevent the incidence of “pooling” in the optimal policy resulting from the probability function, that is policy in which the same allocation is selected for di¤erent values of . 6 Note that it is straightforward to extend the model by incorporating some ine¢ cient pre-reform production subsidy that would be added to the market price p. An optimal policy would remove this subsidy as it implies too much production from a welfare perspective. Nevertheless, the subsidy rate would in‡uence the shape of the regulatory reform as it is a key component of the status quo pro…t level. 7 Note that we omit to explicitly condition y ( ) on S in order to save on notations.
8
on c(:; :), it is easy to check that both the optimal production level y ( ) and the corresponding pro…t level
3.2
( ) are increasing in the farmer’s ability .
The Environmental Agency objective
Agricultural lands not only di¤er according to their productive e¢ ciency but also in the environmental impacts of production. We assume that pollution is represented by a pollution production function per hectare, denoted g(y; ), that estimates the emissions using simulation models, where g(:; :) is increasing with y and depends on . Importantly, we do not impose assumptions on the sign of g that may vary.8 As a consequence, a more productive farmer may or may not pollute less at the margin ceteris paribus. We denote E the total pollution level for the watershed: Z E=
S
S
Z
sg(y; )f ( ; S)d h(S)dS:
The problem of the regulatory (utilitarian) agency is to maximize a social welfare function W, written as the sum of taxpayers surplus, the farmers total surplus and the environmental damage D. We assume that the social cost of pollution D(E) depends on total pollution emitted and is increasing, convex. Note that we do not include any consideration to the consumer surplus in the welfare function. As the regulation is implemented on a small watershed in our empirical application, we expect that any variation in the agricultural production will have negligible impacts on total production sold on the market. As emphasized in the introduction, both the land e¤ectively used and the level of production are assumed to be observable and veri…able by anybody. Suppose that the regulator requests the -type farmer with land capacity S to produce output y( ) using land s( )
S
and o¤ers the monetary transfer t( ):9 The corresponding pro…t of the -type farmer can be written as follows: ( ) = (py( )
c(y( ); ))s( ) + t( )
8
(1)
In our empirical application, we have found that, for a given class of farms size, the level of pollution emissions per hectare tends to be the lowest for intermediate levels of dairy production per hectare. See section 5. 9 For the purpose of clarity, we omit to condition the allocation {s( ); y( ); t( )g on the land capacity S.
9
Note that if the regulator requests s( ) to be strictly inferior to S then set-aside is part of the regulatory proposal for a -type farmer with land capacity S. In particular, the regulator may …nd optimal to shut down some farmers by assigning them s( ) = 0. The regulatory’s objective can be expressed as Z
W
S
S
where
Z
Z
[ ( ) (1+ )t( )]f ( ; S)d h(S)dS D
S
S
Z
s( )g(y( ); )f ( ; S)d h(S)dS
!
is the (positive) marginal cost of public funds. Eliminating the transfer t(:) and
using (1), we obtain a reduced formulation of the regulator’s objective function: W=
Z
S
S
Z
[(1 + )(py( )
c(y( ); ))s( )
( )]f ( ; S)d h(S)dS D
Z
S
S
Z
s( )g(y( ); )f ( ; S)d h(S)dS
!
Finding the optimal policy amounts to …nd the land allocation s(:); the production level y(:) and the net pro…t (:) (or equivalently the transfer t(:)) that are feasible in a sense to be de…ned below and that maximize the welfare function de…ned above.
3.3
Incentive compatibility and acceptability constraints
Feasible allocations are …rst constrained by the information set of the regulator. Information asymmetry arises from the impossibility for the regulator to identify each farmer’s ability. Alternatively, one may assume that ability is observable but that institutional or political constraints prevent the regulator from perfectly discriminating farmers on that basis. Thus, a self-selecting policy remains the only option available to the regulatory agency. Following the Revelation Principle (Myerson, 1982) we can restrict our attention to the set of direct revelation mechanisms fs( ); y( ); t( )g
2
.
By choosing a particular contract among all o¤ered contracts, say a production level y(^), a forage area s(^) in exchange of a tax/subsidy t(^), a farmer implicitly announces to be of type ^. Hence, incentive compatibility requires to satisfy the following constraints: ( )
( ; ^); 10
8 ;8^
(IC)
where ( ; ^) = ps(^)y(^)
s(^)c(y(^); ) + t(^) and ( )
( ; ), which ensures that any
farmer …nds optimal to reveal his true type. The regulator’s problem is to design policies which are not only incentive compatible but also politically acceptable. Consequently, the regulator has to operate under the additional constraint that the policy has to be acceptable compared to the status quo policy, and can therefore be implemented and enforced. This allows the regulator to take into account the regulatory history in the agricultural sector before formulating the reform. Following Lewis, and al. (1989), suppose that only the farmers who bene…t from the policy will support it, and that the probability that a farmer will support the policy is strictly positive if only if his pro…t after the reform is greater than before, i.e. denote ( ( ) ( )
( )
( ). More precisely, we
( )) this probability and we assume for simplicity that (:) = 1 whenever
( ) and 0 either: The political implementation of the policy requires that the
following acceptability constraint should be taken into account: Z
S
S
Z
( ( )
( ))f ( ; S)d h(S)dS
(AC)
which indicates that the probability that the policy is accepted must be superior to a (exogenous) level
2 (0; 1].10 In addition, we assume that any farmer is free to leave the agricultural
sector following the reform. This implies that the pro…t obtained after the reform cannot be lower than the reservation pro…t. Assuming that the best outside option for any type of farmer yields a constant (not type dependant) pro…t, we normalize this reservation pro…t to 0 and we obtain the following individual rationality constraints: ( )
0;
8
(IR)
In this setting, the regulator is thus able to enforce its policy as long as it does not entail any net loss for farmers and provided that a proportion
of farmers gets after the reform at
least the income earned before. 10 The properties induced by the use of this criterion have been rarely studied in the economic literature. Apart from Lewis et al. (1989), a recent exception is Demange and Geo¤ard (2005) who analyzed the physician regulation.
11
Finally, we impose that the optimal production level after the reform cannot be greater than the level under the status quo situation (8 ; S; y( )
y ( )). Hence we forbid any
re-allocation of production across farmers such that some of them may be given incentives to produce more than their private optimum absent any regulation. First, we believe that this constitutes a reasonable property of the optimal policy as it should facilitate political support for the regulatory intervention given that all farmers have to make some e¤ort by reducing production. Second it helps us in deriving the optimal policy as will be clear below. Given the various constraints faced by the regulator, the program to be solved is as follows: max
s(:);y(:); (:)
W s.t. (IC), (IR), (AC), s( )
S and y( )
y ( ) 8 ; S.
Obviously, it is clear that the addition of the acceptability constraint to the regulator’s maximization program always induces a loss in welfare compared to the situation of a socalled mandatory policy which is obtained for the special case
= 0. It remains important to
ascertain the qualitative changes in the optimal regulatory policy when when
varies. Moreover,
= 1, it is also interesting to note that our model belongs to the class of adverse
selection models with type-dependant participation constraints (Jullien, 2000).11 However, in the general case where
2 (0; 1), the type-dependant participation constraint only applies
to an endogenous subset of types (the “winners” of the reform) which must be identi…ed as part of the optimal policy.
3.4
Regulation under complete information
Before analyzing the optimal self-selecting policy, we characterize in this section the optimal policy under complete information, ignoring incentive compatibility constraints (IC). The program to be solved is: max
s(:);y(:); (:)
W s.t. (IR), (AC), s( )
11
S and y( )
y ( ) 8 ; S.
In this class of adverse selection models, countervailing incentives are generally part of the optimal policy because of the interaction between the incentive compatibility constraints and the type-dependant participation constraints. See also Lewis and Sappington (1989) and Maggi and Rodriguez-Cläre (1995) for earlier analysis on countervailing incentives.
12
Ignoring the constraint y( )
y ( ) for the moment, we di¤erentiate the objective w.r.t.
y( ) and we obtain the following necessary conditions: (1 + )(p
cy (y P I ( ); )) = D0 E P I gy ((y P I ( ); ))
where P I denotes perfect information and E P I =
RS R
(2)
sP I ( )g(y P I ( ); )f ( ; S)d h(S)dS.
S
Equation (2) indicates that for an active farmer the optimal production level y P I ( ) is determined by equalizing social marginal bene…t and social marginal damage. Given the linearity of pro…t function with respect to the land input, the optimal production does not depend on the land capacity S. Moreover, it can be checked readily from equation (2) that the optimal regulation entails a decrease in production per hectare compared to the private optimum (i.e. y P I ( ) < y ( ) for any
so that the constraint on quantity is ful…lled).
Di¤erentiating the objective w.r.t. s( ) we get: @W = @s( )
( )f ( ; S):
where ( ) = (1 + )(py P I ( )
c(y P I ( ); ))
D0 E P I g(y P I ( ); )
denotes the social net marginal surplus of land. We assume that
( ) is increasing and that
( ) > 0 otherwise all farmers should be shut down.12 It follows that when exists a unique threshold type
PI
(S) such that for any S, whenever PI
not allowed to produce (8S, sP I ( ) = 0, 8
, the farmer is
(S)). On the contrary, the more e¢ cient
farmers produce using their total land capacity (8S, sP I ( ) = S, 8 > when
PI
( ) < 0 there
PI
(S)). Finally,
( ) > 0, then all farmers are allowed to produce at the optimum involving complete
information. To complete the description of the optimal policy under perfect information, it remains to characterize the redistribution policy (through the transfer) inside the population. Let us 12 Di¤erentiating ( ) w.r.t. and using (2) leads to 0 ( ) = (1 + )c (y P I ( ); ) D0 E P I g (y P I ( ); ): ( ) is increasing given that c < 0 and provided that g is negative or not too positive, which is guaranteed in our empirical application.
13
denote ~ P I ( ) = (py P I ( ) c(y P I ( ); )sP I ( ) as the pro…t gross of transfer under the reform. As the objective is decreasing in the rent ( ) left to any agent, it is clear that the regulator would like to bind all participation constraints at the optimum ( ( ) = 0, 8 ; 8S). Of course, such a policy is not politically acceptable as long as
is strictly positive. In the limit case
where all farmers must be compensated ( = 1), the regulator has to o¤er a positive transfer tP I ( ) =
( )
~ P I ( ) > 0 to any -type farmer (with land size S) which exactly annihilates
the negative impact of the reform for the farmer. In intermediate cases when 0 < determined by the shape of
< 1, the identity of compensated farmers is partly ~ P I ( ). Unfortunately, without additional assumptions,
( )
one cannot predict directly who is to be exactly compensated at the optimum, as
( )
~ P I ( ) is not necessarily monotonic. Indeed, we may get various situations depending on the model speci…cation. For instance, in …gure 1, the set of compensated farmers is composed of intermediary types.13 On the contrary in …gure 2, the set of compensated farmers is composed of the least and the most e¢ cient farmers. As we will see now, incomplete information puts more structure on the shape of the optimal pro…t due to the presence of incentive compatibility constraints and gives clearer predictions on the identity of compensated farmers.
4
Regulation under incomplete information
The regulator has now to solve the following program: max
s(:);y(:); (:)
W s.t. (IC), (IR), (AC), s( )
S and y( )
y ( ) 8 ; S.
Standards arguments (see Guesnerie and La¤ont, 1984) indicate that incentive constraints can be replaced by the following set of constraints: 0
( )=
c (y( ); )s( )
(IC1)
13 We assume for simplicity a uniform distribution of for a given class of surface. Moreover, all farmers are supposed to be active at the perfect information optimum.
14
s0 ( )c (y( ); )
s( )c y (y( ); )y 0 ( )
0
(IC2)
Also, because the rate of growth of rents is positive ( 0 ( ) > 0), then participation constraints reduce to ( )
4.1
0:
Preliminary analysis
Denote y a ( ), sa ( ),
a(
) as the optimal second best allocation o¤ered to any -type farmer
for a given land capacity S. For each land capacity S, we know that the set of farmers allowed to produce will be an interval part on the interval [ ;
a
a
(S);
. This implies that the function
a(
) has a ‡at
(S)] ; corresponding to the constant payment for farmers that are
induced to quit the agricultural sector. Now, let us examine the constraint (AC). At least for some S, the continuous functions
a(
) and
( ) must clearly intersect or touch at least
once, otherwise (AC) would be violated or it would not bind, which is suboptimal. We will proceed in two steps. First, we characterize the optimal policy for a given class of surface S under the constraint that the proportion of farmers that gain from the reform is at least (S) which is given, i.e. Z
( ( )
( ))f ( ; S)d
(S).
Second, we determine the optimal pro…le for the proportions RS S (S)h(S)dS = .
(S) under the constraint
For that purpose, let us denote W S the welfare function (partially) restricted to the class
of surface S, assuming that the policy for any S 0 6= S is …xed to the optimal level (i.e. y a ( ), sa ( )): S
W =
Z
f(1 + )s( ) [py( )
c(y( ); )]
( )g f ( ; S)d
D(E)
where the total level of pollution E is rewritten as follows: E=
Z
S
S
Z
sa ( )g(y a ( ); )f ( ; S 0 )d h(S 0 )dS 0 + +
Z
S
S
15
Z
Z
s( )g(y( ); )f ( ; S)h(S)d sa ( )g(y a ( ); )f ( ; S 0 )d h(S 0 )dS 0 :
Actually, the following lemma indicates that, for a given S, the function once
) intersects
( ) from above at a point (S) if (S) is interior.
Lemma 1 For a given S, if a(
(i)
a(
) intersects
(ii) (S) satis…es
(S) 2 (0; 1), then
( ) once and from above at a point (S) 2
,
(S) = F ( (S); S).
Proof. see appendix A. As shown by Lemma 1, it is important to note that incentive compatibility constraints and the constraint 8 ; S; y( )
y ( ) together imply that the set of farmers who bene…t
from the reform is composed of the less e¢ cient producers whether they are active or not. Furthermore, the relative position of the threshold type
a
(S) and the intersection point (S)
is crucial for the pattern of the optimal policy. First the function the ‡at part of Then necessarily
a(
). This case occurs when ( (S)) =
on the increasing part of a
a(
a ( a (S)).
( ) intersects
(S) is "small" and we have
On the contrary, the function
a
a(
) on
(S) > (S).
( ) intersects
a(
)
): This occurs when the proportion (S) is "large" and we have
(S) < (S). We will successively analyze both cases.
4.2
The case when the proportion (S) is "small"
By a continuity argument, when (S) is "small", we expect that the optimal policy looks like the optimal mandatory one (i.e. for
(S) = 0). This policy is usually obtained in standard
regulation in the absence of political constraint (see e.g. Bontems et al. 2005). The optimal allocation of production is described in the following proposition.14 Proposition 1 For any S such that
(S) is "small",
(i) the set of farmers that are allowed to produce using their land capacity S is a 14
a
(S);
with
(S) > (S); and
We assume that y a ( ) < y ( ) for any farmers.
and that the optimal policy is separating for the set of active
16
(ii) the optimal level of production for any type ( ; S) active farmer is given by: p
cy (y a ( ); ) =
1 D0 (E a ) gy (y a ( ); ) 1+
where the total pollution level is E a = (iii) Moreover, for any
such that
<
RS R S
a
a
1+
(S) Sg(y
a(
c y (y a ( ); )
1
F ( ; S) f ( ; S)
); )f ( ; S)d h(S)dS:
(S), production is not allowed (sa ( ) = 0) and
all non active farmers get a constant payment
a(
)=
a ( a (S))
=
( (S)).
Proof. See appendix B. As indicated by proposition 1, when (S) is "small", the optimal policy is similar to the mandatory one (i.e. when
(S) = 0), except that the production levels and the range of
excluded farmers are qualitatively di¤erent as they depend on the particular value of
(S).
In particular, a positive distortion is added to the marginal damage when determining the optimal level of production for active farmers, which means that the agency has to distort production downward. Moreover, the minimum level of pro…t is equal to
( (S)) > 0 in
order to ful…ll the acceptability constraint and it is higher than what is obtained in the mandatory case. Although the production has a constant return to scale technology in the model, the size of farms plays nevertheless an important role through the correlation with parameter . Indeed, the incentive distortion brought by the optimal policy under asymmetric information depends on the risk ratio
1 F ( ;S) f ( ;S) .
For instance, if this risk ratio is increasing in S, then the downward
distortion due to adverse selection is exacerbated if the size S is larger. Consequently, a large size farm is thus induced to reduce production per acre more than a low size farm.15 And conversely, if the risk ratio is decreasing in S. Finally, note that if the optimal allocation of the weights (S) across the classes of surface is such that
(S) is "small" everywhere, then the optimal policy is totally identical to the
mandatory one, except for the basic level of pro…t given to all farmers. Let us index the 15 In the empirical application to the Don watershed, the estimated risk ratio is indeed non monotonic in S (see …gure 4 p.289 in Bontems et al. (2005)).
17
optimal levels of variables in the mandatory situation by : Formally, we would then have a
E a = E and (y a ( ); sa ( );
(S)) = (y ( ); s ( );
welfare level is lower (and decreasing in
(S)) for every S. However, the total
(S)) compared to the mandatory one because of
the social cost of giving up a strictly positive minimum pro…t to every farmer. To sum up, in that very particular case, the introduction of the acceptability constraint only implies to translate upwards the minimum pro…t level for all farmers and does not change anything else to the optimal policy nor to the optimal level of pollution.
4.3
The case when the proportion (S) is "large" a
We now turn to the opposite case where
(S) < (S) which occurs when
(S) is "large".
The optimal policy in that situation is described in the following proposition.16 Proposition 2 For any S, when
(S) is "large",
(i) the set of farmers that are allowed to produce using their land capacity S is an interval a
(S);
with
a
(S) < (S); and
(ii) the optimal level of production for any type ( ; S) farmer is continuous and de…ned implicitly by: p
cy (y a ( ); ) =
1 F ( ; S) D0 (E a ) gy (y a ( ); ) + c y (y a ( ); ) ; 8 2 [ a (S); 1+ 1+ f ( ; S)
y a ( ) = y(S); 8 2 [ 1 (S); p
cy (y a ( ); ) =
2 (S)] ,
1 D0 (E a ) gy (y a ( ); ) 1+
1+
c y (y a ( ); )
1
F ( ; S) ; 8 2 f ( ; S)
where the total pollution level is a
E =
Z
S
S
Z
Sg(y a ( ); )f ( ; S)d h(S)dS a
(S)
and where y a ( ) denotes the quantities that maximizes W S and with y(S) maximizing expected welfare over [ 1 (S);
2 (S)]
1 (S)) ,
with
16
Once again, we assume that y a ( ) < y ( ) for any of active farmers.
18
a
(S)
1 (S)
(S)
2 (S)
.
and that the optimal policy is separating for the set
2 (S);
,
(iii) Moreover, for any
such that
active farmers get a constant payment
a
<
(S), production is not allowed and all non
a(
)=
a ( a (S))
> 0.
Proof. See appendix C. Figure 3 depicts the optimal output pro…le described in Proposition 2. As indicated in the proposition, the output pro…le is generally not e¢ cient. In particular, holding the total level of pollution constant, we …nd that, compared to the perfect information situation, ine¢ ciently high levels of production are induced for low ability types (for
2 [ a (S);
1 (S)))
while ine¢ ciently low levels of production are induced for highly productive farmers (for 2
2 (S);
).
The intuition goes as follows. We knows that the political acceptability constraint imposes that
a(
(S)) =
( (S)). For all types such that
its type at the rate of
> (S), the farmer’s pro…t increases with
Sc > 0. The agency will thus tend to induce lower output on
this range in order to limit the rate of growth of informational rents. On the contrary, for < (S), the farmer’s pro…t decreases with its type at the rate of Sc < 0 and consequently, the agency tends to limit rents by inducing high levels of output. The pooling interval on [ 1 (S);
2 (S)],
where all types of farmers are required to produce at the same constant level
y(S) appears because around (S) the two tendencies exhibited above would require that output would decrease with ability which contradicts incentive compatibility (IC2). Thus, the agency has to induce the same level of production y(S) in the interval [ 1 (S);
2 (S)]
around (S). Also the farmers in this range receive the same transfer level. To sum up, the optimal policy involves an intermediary range of types producing the same level of production which maximize expected welfare while the farmers with lower ability are required to overproduce and the farmers with higher ability are required to under-produce. Compared to the situation where political acceptability is not required (mandatory policy), it is clear that political participation induces the agency to reorganize production in a way that bene…t to the farmers with low ability. This is because essentially the revenue under
19
status quo increases more rapidly than the revenue after intervention. Indeed for an active farmer, under the status quo we have requires that
0(
)=
0
( ) =
Sc (y ( ); ) and incentive compatibility
Sc (y( ); ): Moreover, we have assumed that y( )
y ( ) and c is
decreasing with y. Hence the revenue is more equally distributed in the population, so that the low-types farmers bene…t at the expense of high-types farmers. Subsidies given to the low-types farmers in order to induce them to leave the sector are …nanced through implicit taxation of high-types farmers. However, in our model, it is clear that the total level of pollution always decreases after intervention. Indeed, given that for any active farmer we have y( )
y ( ) and given that the range of active farmers is smaller after regulation, it
follows that total pollution is lower.
4.4
The optimal allocation of proportions (S)
In this subsection, we indicate the rule that determines the optimal allocation of proportions (S) across the surface classes. The regulator seeks to solve the following maximization program: max W = (:)
WS = =
Z
S
S
s.t. Z Z
W S h(S)dS
f(1 + )sa ( ) [py a ( )
c(y a ( ); )]
a
( )
D(E a )g f ( ; S)d
S
(S)h(S)dS
S
Solving this program yields to the following result. Proposition 3 The optimal rule for the allocation of proportions classes is such that the marginal (negative) impact of
(S) across the surface
(S) on expected welfare should be
equalized across the di¤ erent classes of surface. Proof. We introduce a state variable (S) de…ned by _ (S) = (S)h(S) with the conditions (S) = 0 and (S) =
. Let (S) denote the associated co-state variable. The lagrangian 20
can be written as follows: L = =
Z
S
S
Z
S
W h(S)dS +
S
S
Z
S
(S) [ (S)h(S)
S
S
W + (S) (S) h(S)dS +
Z
_ (S)] dS
S
_ (S) (S)dS
(S) (S) + (S) (S)
S
and pointwise maximization leads to the following necessary conditions (for an interior optimum): @L = _ (S) = 0 ) (S) = @ dW S @L = + @ (S) d (S)
8S
=0
We thus have equality across classes of surface for the marginal impact of (S) on expected welfare. Using the envelop theorem and Lemma 1, it is now easy to prove that this marginal impact is negative: dW S d (S)
=
=
@ (S) ( (S)) + Sc (y; (S)) @ (S) i h 0 a0 ( (S)) ( (S)) < 0: f ( (S); S) 0
Recall that (S) is the marginal type in the class S, such that for any -type farmer gains from the reform while for any
(3)
(S), the
> (S), the -type farmer looses from
the reform. Determining the optimal allocation of proportions
(S) is hence equivalent to
determining the optimal marginal type (S) for each class S. In equation (3), the term between brackets represents the di¤erence between the rates of the (S)-type farmer rent’s growth in the status quo and the optimal regulation situations. This term is evaluated from the regulator’s perspective through the marginal social cost of public funds ( ). Moreover, f ( (S); S) represents the relative number of (S)-type farmers. Thus, the welfare impact of increasing the proportion of winners (S) at the margin can be interpreted as the marginal
21
welfare cost in terms of socially costly rents that must be provided in order to compensate the marginal farmer with ability (S). Finally, note that, as we will see below in the simulations, the hierarchy between classes S in terms of the optimal number of winners depends in particular on the level
of the political
constraint.
5
Empirical application to the Don watershed
The Don watershed (71 706 ha) is a typical breeding area in the western France, near Brittany: most of the farms breed cattle, with a large share of grassland area (50 % of the Don total area), corn and cereals being cropped for both grain and forage. The weather is typically oceanic, groundwater is scarse and the water supply for the population comes from dams and in-river pumping stations. At the outlet of the Don watershed there are two pumping stations for drinking water, supplying around 150,000 people. There, the nitrate concentration regularly reached or exceeded the EU guidelines of 50 mg/l from the mid-nineties and several recovery programs have been implemented on a voluntary basis, unfortunately with few improvements of the water quality. In the following, we …rst describe the estimation procedure for farms’types, cost function and nitrogen emissions. We then present the main results that can be obtained by simulating the optimal policy. Instead of making use of the …rst-order conditions of the regulator problem, we actually solve numerically this program directly using the Gams software (Brooke et al., 1998) and the Conopt optimization procedure (Drud, 1994).
5.1
Farms’types, cost functions and nitrogen emissions
The model parameters come from several surveys on the Don watershed: farm systems, farming practices, soil functioning, water and nitrogen transfers on this watershed have been monitored from 1998 to 2003. Data describing on-farm practices have been collected from a sample of 10 % of the farms after a strati…cation of the whole population on some production
22
system criteria (see Turpin et al., 2005, for details). This survey highlights the importance of heterogeneity among the dairy farms (Turpin et al., 2006) that however could be reduced to two non-independant characteristics, their available area (owned or rented) and their (productive) ability to transform this area into …nal production. The critical point in simulations is the assessment of the private information parameter (i.e. productive ability), which is necessary for calibrating the model. We proceeded in two steps. First, some experts in the local extension services, who have been involved in the survey, designed together a classi…cation of the farmers along one dimensional type that represents their productive ability. While designing this classi…cation they explained why and on which criteria they considered that a given farm had a higher (smaller) ability than the ones classed just below (above). Then, we combined these variables and designed a function that links them to our one-dimensional parameter . The technical variables that captured the di¤erence between the farms are: the productivity of the grassland area, that depicts grass availability and the quality of the grazing management (both a¤ect the food supply for cows), the dairy yield, as a proxy for the genetic quality of the herd and because high-yielding cows are more sensitive to quality management of feeding than low-yielding ones, the balance in the animal diet between energy and protein supply (the availability of energy is crucial to the e¢ ciency of N use by the animals but this variable is also a clue for the management capacity of the farmer), the distance between a theoretical concentrate intake, assessed with high quality forages, and on-farm observations that measures the actual concentrate intake.17 Agri-environmental issues often focus on soil quality. In dairy production systems, the soil quality is di¢ cult to measure, mostly because grass intake by grazing cows is usually 17
This concentrate intake has an important impact on the pro…tability and on the nutrient balance for dairy farms.
23
unobserved. In this application, the soil quality a¤ects the descriptive variables cited by the local experts: fertile and easy-to-manage soils enable high production for the grasslands areas, provide good quality forages and thus enable high-yielded cows, balanced diets and the use of low amounts of supplementing concentrates. The distributions h(S) and f ( ; S) have been estimated using a discrete approximation of Beta density functions. Farms are split into ten di¤erent classes of surface, while ability falls into hundred di¤erent classes for each S (Figure 4). The parameterization of the model considers explicitly the density of farms over the watershed. Once the parameter
is designed, the survey data allow for an estimation of the cost
function using the speci…cation described in Table 1. parameters for the estimation of the cost function: c(y; ) = w#( ):( ( )y 2 + ( )y + ( )) + !( )y + % with , and having the quadratic form: ( ) = 1 2 + 2 + parameters for ( ) parameters for ( ) parameters for ( ) 6 * 2.47 10 -0.00237 * 215.0 * 1 1 1 6 -9.12 10 * 0.00796 * 293.8 * 2 2 2 6 * 7.91 10 -0.0206 * 121.1 * 3 3 3 The other parameters (#, ! and %) have been calibrated so that p = cy in the observed situation * signi…cant at 0.05 level.
3
Table 1: Estimated parameters for the cost function, adapted from Bontems et al. (2005)
Individual emissions have been estimated here using the SWAT -Soil and Water Assessment Tool- model. SWAT is a semi-distributed watershed model with a GIS interface (DiLuzio et al., 2002) that outlines the sub-watersheds and stream networks from a digital elevation model and calculates daily water balances from meteorological, soil and land-use data. SWAT simulates each sub-basin separately according to the soil water budget equation taking into account daily amounts of precipitation, runo¤, riverbed transmission losses, percolation from the soil pro…le, and evapotranspiration. We introduce in the model the agricultural practices on a …eld base for the surveyed farms and the model was calibrated on 24
the whole watershed for water and N loads. The type-dependent emissions have then been determined from the individual estimations, using a quadratic function and a constrained maximum likelihood method (see Table 2). parameters for the estimation of the emission function : g(y; ) = 1 2 + 2 + 3 y 2 + 1 2 + 2 + 3 y + {1 2 + {2 + {3 parameters for ( ) parameters for ( ) parameters for {( ) 5 7.668 10 0.0120 { 29.379 * 1 1 1 5 -1 10 * -0.0524 * {2 -12.98 * 2 2 5 7.668 10 0.0139 * { 17.28 * 3 3 3 constrained value (to have non negative emissions). * signi…cant at 0.05 level. Table 2: parameters estimation for emission function, adapted from Bontems et al. (2005)
For the surveyed farms, there is no clear relationship between the supply of milk and the nitrogen quantity emitted to the river. Indeed, the farmers can provide the same amount of milk with very di¤erent practices on their …elds (including the crop rotation, the share of cropped forages on the animal food, the amount of inorganic fertilizers used), and thus the pollution they emit can be also very di¤erent. In our application watershed, the farms with medium level of
have the lower emission
rates: low -farms emit large amount of nitrogen because they are less e¢ cient in production and thus require higher amounts of fertilizers for relatively low productive (but highly polluting) grasslands and the farms with the higher
values reach such high levels of productivity
per hectare that they overload the natural absorption capacity of their …elds and pollute more (see Figure 5). Last, the damage cost function we introduce in our model integrates the cost of fresh water treatment only. Other amenities could have been included in this function, such as the costs associated with the decrease of biodiversity or recreation activities, the increase of eutrophication or the degradation of the landscape. However, due to the lack of data, we do not include them in the simulations. 25
5.2 5.2.1
Results obtained from simulations Dairy production
Figures 6 and 7 depict optimal production levels for each of the ten area classes for
= 50%.
Figure 6 depicts the production level for one class of area and Figure 7 focusses on the production decrease, expressed here as a proportion of the private optimum production level for all the classes S. Let us …rst examine the production level pattern, that is similar to the one derived theoretically in Figure 3. All farmers are active in each class S and this remains a property of the optimal reform for all classes for almost all the acceptability levels in the context of our simulations (as soon as
6%). The production constraint y a ( )
y ( ) is binding for the lowest types in
each classes. This follows from the regulator’s desire to limit the rents distributed to the winners of the reform by increasing their production at the maximal level. On the contrary, all other higher types are required to reduce their production. In particular, a range of intermediate types are induced to produce at the same level because of the con‡ict between the acceptability (AC) and incentive (IC) constraints, as highlighted in the theoretical part. At the right side of this pooling zone, all the types reduce their production level and, in each class S, this decrease is smaller for the more e¢ cient farmers. For the highest types, a pooling phenomenon appears because the second order condition, IC2, that imply the monotony of y, is binding: the regulator would want to decrease the production level of the high types, that contradicts incentive compatibility constraints and production levels remain the same for all types. Figure 7 depicts the production pattern, as a proportion of the private optimum level, for all the surface classes. The pooling phenomenons depicted above appear as quick falls in this Figure. In the same class, the production decrease (for types between the two pooling zones) is lower the more e¢ cient farmers, except for the larger farms. This pattern suggests that, although the regulator reorganizes production in a way that bene…ts to the farmers with low ability, the proportion of production decrease is more important for low medium type farmers 26
than for high medium types ones. Low medium type farmers decrease their production level in a greater proportion (but are less taxed) than high medium type ones who decrease less their production level but are more taxed. Last, the simulations provide information about the proportion of winners for each class S (see Figure 7 for the case where
= 50%). In …gure 7, the smaller farms (depicted in dotted
lines) experience the larger proportion of winners: the types range for which the farms produce at their private optimal level is large, we have
a
(S) comprised between 0.6 and 0.7, which
means that nearly 70% of the farmers in these class bene…t from the regulation (remind that we have
a
(S) < (S)). Middle size farms experience a smaller proportion of winners, and
this proportion is not monotonous with S. The proportion of production decrease is not monotonous with S either. 5.2.2
Farmers’pro…t versus emissions
Although the farmers with high types decrease less their dairy yield, the optimal policy leads to a higher decrease of emissions per hectare for these farmers (see Figure 8 for the case where =50%). The decrease of total emissions induced by regulation ranges from 47% when = 0% (mandatory policy) to 2% when
= 100%. This decrease is associated with a
reduction in dairy production at the watershed scale, ranging from 13.5 % ( =0%) to 0.3 % ( =100%). Figure 9 illustrates the relationship between the level of emissions, the size of pro…ts and the size of farms. Whatever the size of farms, the di¤erence between the dashed curve (status quo) and the thick curve (optimal regulation) represents the reduction of pollution induced by the reform, for a given level of pro…t. As the lowest active types who get the lowest pro…ts are not required to decrease production as shown above, there is no pollution reduction either, contrary to what happens for the highest levels of pro…t. More interestingly, the simulations suggest that a small farm that earns little will pollute less than a medium size farm earning the same amount. And conversely, a very pro…table small farm pollutes more than a medium
27
size farm with average pro…t. This situation implies that there is a real trade-o¤ between sustaining farmers’pro…ts and reducing pollution across the di¤erent classes of farms. 5.2.3
Welfare and its components
In our case study, the optimal proportion (S) (weighted by the density h(S)) is increasing in for any class S. In others words, the proportion of winners for a given class increases in the required level of acceptability. However, the ranking of classes according to the proportion of winners changes when
increases. For example, class S3; which includes areas that range
from 30 to 35 ha, is actually the class with the highest proportion of winners (weighted by the density h(S)) only when
is lower than 38% and higher than 98%. In between, either
class S2 or S4 are the ones with the greatest number of winners, depending on the value of
: On the contrary, the classes with large size farms (e.g. S8, S9 and S10) possess the
lowest proportions of reform winners. Hence, a low size farm has a higher probability of being (over-)compensated than a large size farm. It is important to understand that this feature of the policy is not driven by a willingness from the regulator to protect small farms at the expense of large farms, but it comes instead from the lower social cost of rents left to small farms. Let us denote P( ) the relative variation of total pro…t with respect to the status quo situation: P( ) =
RS R S
a(
RS R )f ( ; S)d h(S)dS ( )f ( ; S)d h(S)dS S RS R ( )f ( ; S)d h(S)dS S
Similarly, let us de…ne W( ), M( ), D( ) as the relative variations of respectively the welfare, the aggregate production and the estimation of social damage. Figure 10 depicts the patterns of these indicators with respect to the weight
of the political constraint. As
expected, relative damage D( ) increases with the weight of the political constraint meaning that less and less incentives for extensive farming are given. This is re‡ected in the pattern of the (relative) aggregate production curve (M( )). As farmers are giving up less production and as they are more compensated when
raises, their total pro…ts increase (see the curve 28
P( )). Recall that we noted earlier that, as soon as under the optimal policy. Indeed, when
reaches 6%, all farmers remain active
is su¢ ciently large, it becomes too costly for the
agency to induce some farmers to quit as they have to be at least compensated. Indeed, it appears far less costly to keep them active with a production level close to their laissez-faire optimum. Last, relative welfare W( ) is actually rapidly decreasing so that the reform only brings a modest welfare gain for high values of
. Moreover, when
is greater than 98%
then the damage reduction is no longer important enough to ensure a welfare gain compared to the status quo situation.
6
Conclusion
Incentive policies that promote the adoption by farmers of costly but pollution-decreasing practices are supported by an increasing number of both farmers and environmentalists. These policies have already been tested in many areas in Europe with agri-environmental schemes. In spite of design, enforcement and implementation di¢ culties, EU Member States have to ensure a programme of measures to mitigate water pollution within the Water Framework Directive (2000/60/EC) and they need to select among the set of potential measures, the most cost-e¤ective ones. Most of current agri-environmental schemes consider however uniform instruments and face low commitment, mostly because heterogenous farmers are not uniformly scattered within a watershed which, in general, is not physically uniform either. We present here the regulation of nonpoint source water pollution through non linear taxation/subsidization of commodity output supply, associated with the satisfaction of a given proportion of farmers following intervention. An important feature of this scheme is that the identity of winners is endogenous. The optimal reform actually amounts to reallocate production towards ine¢ cient farmers who bene…t from the reform at the expense of more e¢ cient producers The conceptual framework presented here helps delineating the optimal policy pattern. The empirical parameterization of the model explores the multiple product framework in the
29
context of farms’heterogeneity and provides additional insights. First, there is a strong heterogeneity in the relationship between the dairy supply and the amount of nitrogen emitted in the river for the population of surveyed farms. This heterogeneity in the degree of connection between commodity outputs and polluting emissions is often stressed in the literature but seldom measured. The second insight, that should be the subject of future research, deals with the proportion of winners that shifts the policy from one pattern to the other. Assessing this proportion in one case study only limits the trust of more general conclusion that could have been drawn on this topic. Last, the empirical application highlights the fact that there exist win-win opportunities for costless provision of water quality in agricultural watersheds when considering the farms’ heterogeneity, as for most cases in our application the welfare raises compared to the status quo situation.
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[5] DiLuzio M., Srinivasan R., Arnold J.G. and Neitsch, S.L., 2002, “ArcView Interface for SWAT2000 User’s Manual”. GSWRL Report 02-03, BRC Report 02-07, Texas Water Resources Institute TR-193y, College Station, TX. [6] Drud A.S. (1994), “CONOPT - a large scale GRG code", ORSA Journal on computing, 6:207-216. [7] Gouriéroux C. and Montfort A., (1995), Statistics and Econometrics models, Cambridge University Press, 562 p. [8] Green J. and J.J. La¤ont (1986), “Partially veri…able information and mechanism design”, Review of Economic Studies, 103:447-456. [9] Gri¢ n R.C. and Bromley D.W. (1982), “Agricultural run-o¤ as a nonpoint externality: a theoretical development”, American Journal of Agricultural Economics, 64:547-542. [10] Guesnerie R. and JJ. La¤ont (1984) “A complete solution to a class of principal-agent problems with an application to the control of a self-managed …rm”, Journal of Public Economics, 25, 329-369. [11] Holmstrom B. (1982), “Moral Hazard in Teams”, Bell Journal of Economics, 13:324-340. [12] Jullien B. (2000), “Participation constraints in adverse selection models”, Journal of Economic Theory, 93:1-47. [13] Lewis T.R., Feenstra R. and R. Ware (1989) “Eliminating price supports”, Journal of Public Economics, 40, 159-185. [14] Lewis T.R. and D. Sappington (1989), “Countervailing incentives in agency problems”, Journal of Economic Theory, 49:294-313. [15] Maggi G.and Rodriguez-Clare A., (1995), “On Countervailing Incentives”, Journal of Economic Theory, 66:238-263.
31
[16] Myerson R., (1982), “Optimal Coordination Mechanisms in Generalized Principal-Agent Problems”, Journal of Mathematical Economics, 10, 67-81. [17] Segerson K., (1988), “Uncertainty and incentives for nonpoint pollution control”, Journal of Environmental Economics and Management, 15:87-98. [18] Shortle J.S., Horan R.D. and D.A. Abler (1998) “Research issues in nonpoint pollution control”, Environmental and Resource Economics, 11, 571-585. [19] Shortle J.S. and Horan R.D. (2001), “The economics of nonpoint pollution control”, Journal of Economic Surveys, 15, 255-289. [20] Thurston J., How to reform the CAP: A guide to the politics of European agriculture, The Foreign Policy Centre, 2002, London. [21] Turpin N., Laplana R., Strauss P, Kaljonen M., Zahm F., Bégué V, 2006, Assessing the cost, e¤ectiveness and acceptability of best management farming practices: a pluridisciplinary approach, IJARGE, special issue on "features of environmental sustainability in agriculture: where do we stand ?", 5, 2/3, 272-288. [22] Turpin, N., Bontems, P., Rotillon, G., Barlund, I., Kaljonen, M., Tattari, S., Feichtinger, F., Strauss, P., Haverkamp, R., Garnier, M., Porto, A. L., Benigni G., Leone, A., Ripa M.N., Eklo O.M., Romstad E., Bordenave, P., Bioteau, T., Birgand, F., Laplana, R., Lescot, J.-M., Piet, L., and Zahm, F., 2005, AgriBMPWater: systems approach to environmentally acceptable farming, Environmental Modelling and Software, 20/2 (Policies and Tools for Sustainable Water Management in the European Union - Edited by R.A. Letcher and C. Giupponi), 187-196. [23] Wu J. and B.A. Babcock (1996) “Contract design for the purchase of environmental goods from agriculture”, American Journal of Agricultural Economics, 78, 935-945.
32
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33
Appendix A
Proof of lemma 1 0
Proof of part (i): Using the envelop theorem, we get before any regulation As c
y
< 0 and y a ( )
But we have also sa ( )
a(
the increasing part of
c (y ( ); )
S; so using equation (IC1) we obtain: a0
0 (S) > 0, then
) intersects a(
0
( )
( )
( ) once and from above, either on the ‡at part or on
).
Proof of part (ii): given (i), for the constraint
R
binding at the optimum, (S) must satisfy (S) =
B
Sc (y ( ); ).
y ( ), we have: c (y a ( ); )
If
( )=
R
( ( ) (S)
( ))f ( ; S)d
(S) to be
f ( ; S)d = F ( (S); S).
Proof of proposition 1
Assume that 0 < (S) < ( ( )=
a
(S). Then, integrating (IC1), we get: R
( (S))
a ( (S)) for (S) c (y(u); u)s(u)du for a (S)
>
a
(S)
:
(4)
Using (4), we rewrite the objective W S as follows: W
S
=
Z
[(1 + )s( )(py( ) a (S) " Z Z ( (S)) a
(S)
c(y( ); )] f ( ; S)d #
Z
a
(S) h
c (y(u); u)s(u)du f ( ; S)d a
(S)
i ( (S)) f ( ; S)d D(E)
Integrating by parts, we obtain: WS =
Z
(1 + )s( )(py( ) a
c(y( ); )) + s( )c (y( ); )
(S)
( (S))
Z
f ( ; S)d
D(E)
34
1
F ( ; S) f ( ; S)d (5) f ( ; S)
It remains to maximize W S w.r.t. s(:) and y(:). It is then clear that the optimal allocation (s(:); y(:)) follows the same rule as the one under a mandatory policy. We now check that equation IC2 is satis…ed. For any type-( ; S) farmer who is allowed to produce, equation IC2 can be written as: Sc y (y( ); )y 0 ( )
0
Derivating the …rst-order condition for y with respect to , and dropping any argument, we get: (1 + )(cyy y 0 + c y )
D0 (E)gyy y 0
D0 (E)g 1
+
y
F
(c
f
yy y
0
c
y
+c
y)
+ c
y
d d
1
F f
=0
This equation can be written as: 0
(1 + )c
y =
y
+ D0 (E)g
1 F f
y
D0 (E)gyy +
(1 + )cyy
1 F f
We have already made the following technical assumptions: c d d
1 F f
< 0. Thus, if in addition g
y
d yd
c
c
1 F f
yy
< 0; c
y
yy
< 0; c
y
> 0 and
< 0 and gyy > 0, then we have y 0 ( ) > 0, 8 and the
second order conditions are ful…lled.
C
Proof of proposition 2
Given Lemma 1, we can integrate (IC1), and we get: ( R (S) ( (S)) + c (y(u); u)s(u)du for R ( )= ( (S)) (S) c (y(u); u)s(u)du for
(S) > (S)
(6)
Using the de…nition of pro…ts given by (6), we get for W S : S
W =
Z
[(1 + )s( )(py( ) " Z
c(y( ); )] f ( ; S)d
(S)
( (S)) + Z
(S)
"
Z
#
(S)
c (y(u); u)s(u)du f ( ; S)d
( (S))
35
Z
#
c (y(u); u)s(u)du f ( ; S)d (S)
D(E)
Integrating W S by parts, we get : W
S
Z
=
+
h
(1 + )s( )(py( )
Z
Z
i ( (S)) f ( ; S)d
c(y( ); ))
(S)
s( )c (y( ); ) s( )c (y( ); )
F ( ; S) f ( ; S)
1
(S)
(7)
f ( ; S)d
F ( ; S) f ( ; S)
f ( ; S)d
D (E)
Before derivating (7), note that the expression of the derivatives will depend on whether is greater or lower than (S).
C.1 For
Analysis for
< (S)
< (S), derivating (7) with respect to y( ), we obtain the following …rst-order condition
for an (interior) optimum: @W S = 0 ) (1 + ) [p @y( )
cy (y a ( ); )]
gy (y a ( ); )D0 (E a )
cy (y a ( ); )
F ( ; S) = 0 (8) f ( ; S)
Similarly, with respect to s(:), we obtain the following expression for the social marginal surplus of land : @W S = (1+ ) [py a ( ) @s( )
c(y a ( ); )] f ( ; S)
c (y a ( ); )F ( ; S) g(y a ( ); )D0 (E a ) f ( ; S)
Note that, sign(@W S =@s( )) = sign(B( )) where B( ) = (1 + ) [py a ( )
c(y a ( ); )]
c (y a ( ); )
F ( ; S) f ( ; S)
D0 (E a )g(y a ( ); )
(9)
Using (8) and dropping any argument, we get from (9) : B0 = Note that when c
(1 + )c
c
F f
c
d d
F f
D0 (E a )g :
is small with respect to the other terms, B 0 ( ) > 0. Assuming B( ) < 0
and B( ) > 0, there exists a unique threshold type 36
a
(S) such that for every
<
a
(S),
@W S =@s( ) < 0 and consequently sa ( ) = 0 so that all su¢ ciently ine¢ cient farmers are shut down. Moreover, for any
>
a
(S); @W S =@s( ) > 0 and consequently sa ( ) = S. The
threshold type separating active from inactive farmers is given by: (1 + )(py a ( a (S))
c(y a ( a (S));
D0 (E a ) g(y a ( a (S));
a
(S)))f ( a (S); S)
a
(S))f ( a (S); S) = c (y a ( a (S));
a
(S))F ( a (S); S)
This equation means that the regulator equalizes the marginal surplus (net of damage) of keeping active the
a
(S)-type farmers with the corresponding incentive cost for all smaller
type farmers (in proportion F ( a (S); S)).
C.2 For
> (S)
Analysis for
> (S), derivating (7) w.r.t y( ) yields to: (1 + )(p
cy (y a ( ); ) = gy (y a ( ); )D0 (E a )
cy (y a ( ); )
1
F ( ; S) f ( ; S)
(10)
Once again, the optimal production level follows the same rule as for the optimal mandatory policy and for the optimal policy when (S) is "small".
C.3
Pooling around (S)
From (8) and (10), it is clear that y a ( ) decreases discontinuously at lim
! (S)
y a ( ) > lim
! (S)+
= (S) (that is,
y a ( )). Thus, it is not possible to implement y a ( ) without
violating incentive compatibility constraints (more precisely (IC2)). The optimal policy involves pooling with y a ( ) = y(S) in a neighborhood [ 1 (S);
2 (S)]
around (S) (Lewis and
Sappington, 1989). It remains to determine
1 (S); 2 (S)
and y(S) . First, it is easy to show that y a ( ) is
continuous (see Lewis and Sappington (1989)). Then, rewriting the pro…t function for any -type farmer, we get: 8 < ( (S)) + R a ( )= R : ( (S))
R (S) c (y a (u); u)sa (u)du + 1 (S) c (y(S); u)sa (u)du pour R 2 (S) c (y(S); u)sa (u)du c (y a (u); u)sa (u)du pour 2 (S) 1 (S)
(S)
37
(S) > (S)
Denote AW (s( ); y( ); ) = (1 + )s( )(py( ) R
c(y( ); )
( )
D(E): Then W S =
[AW (sa ( ); y a ( ); )] f ( ; S)d : Di¤erentiating W S with respect to y(S), we get: dW S = dy(S)
Z
2 (S) 1 (S)
@AW (sa ( ); y(S); ) f ( ; S)d = 0: @y
This equation together with the continuity of the production schedule gives implicitly and y(S).
38
1 (S); 2 (S)
Euros
6
( )
~P I ( )
-
mass Figure 1: When compensated farmers are of intermediary types.
39
-
Euros 6 ( )
~P I ( )
-
-
-
mass 3
Y
Figure 2: When compensated farmers are of low and high types.
ya 6
y(S)
0
a
-
(S)
1 (S)
2 (S)
Figure 3: Optimal production level y a as a function of
40
when (S) is large.
0,16 available area 0,14 10 < S < 20
0,12
20 < S < 30 0,10
30 < S < 35 35 < S < 40
0,08
40 < S < 45 45 < S < 50
0,06
50 < S < 60 0,04
60 < S < 70 70 < S < 80
0,02
80 < S
0,00 0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
Figure 4: Estimated density for each available area class (S) for the farms on the Don watershed - adapted from Bontems et al. (2005).
41
emissions (kg N-NO3/ha) 120 100 80 60 40 20 dairy yield (litres/ha) 0 0
2 000
4 000
6 000
8 000
10 000
12 000
Figure 5: Dairy yield and nitrogen emissions for the surveyed farms on the Don watershed (here are plotted farms with 35 to 40 ha of available area).
42
y (milk yield l/ha)
12 000
y°
11 000 10 000 9 000 8 000 small farms (25 ha) 7 000 very large farms (95 ha)
6 000
ya medium size farms (47.5 ha)
5 000 4 000 3 000
θ
2 000 0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
Figure 6: Simulated dairy yield for three farm size (small, medium and large) for the optimal regulation ( = 50%).
43
1,0
dairy yield y as a percent of baseline y° 100
95
90 0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
Figure 7: Simulated dairy yield y as a percent of the baseline y for each farm in the Don watershed grouped per size class. Acceptability = 50%. The smaller farms are in dotted lines, the larger one in bold lines.
44
1,0
N emitted (kg N/ha)
y (milk yield l/ha)
140
12 000 y°
120
10 000
100 8 000 80 6 000
ya 60
4 000 40
g0
50 mg NO3/l
2 000
20 ga
0 0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
Figure 8: Dairy yield and nitrogen emissions for medium size farms in the optimal regulation ( = 50%).
45
θ
0 1,0
emissions (t N) 7 medium size farms (42.5 ha) 6
5
4 small farms (25 ha)
3
2
1 profit (1000 euros) 0 0
20
40
60
80
100
120
Figure 9: Emissions of nitrogen (tons) as a function of pro…t (1000 euros) at the farm level for the status quo situation (dashed curve) and for the optimal policy for = 50% (thick curve).
46
140
% 20 15 10
W
5 0 -5 0
10
-10
20
30
40
50
60
70
80
90
M
-15 -20
P
-25 -30
D
-35 -40 -45 -50 -55 -60 -65
Figure 10: Relative variation with respect to of damage ( D), farmers’pro…t ( ), milk production ( y), transfert from farmers to the regulatory agency ( t) and social welfare ( W). The mandatory regulation is not depicted here.
47
100
α
