Stéphane De Cara
email: stdecara@grignon.inra.fraccordsinternationaux

Gilles Rotillon
email: rotillon@u-paris10.fr

Multi-Greenhouse gas International Agreements

Keywords: Self-enforcing international environmental agreements, multiple pollutants, greenhouse gases, climate change, Kyoto Protocol Accords internationaux environnementaux, Multi-gaz, Stabilité, Gaz à effet de serre, Changements climatiques, Accord de Kyoto Self-enforcing international environmental agreements, multiple pollutants, greenhouse gases, climate change, Kyoto Protocol

We examine the influence of the multi-pollutant nature of climate change on the stability of international environmental agreements. We consider a $n$-player, two-stage game. Pollution results from two gases, which differ by their environmental impact and their abatement costs. Different kinds of agreements are examined: 'single-gas agreements' (one pollutant is neglected in international negotiations), 'comprehensive agreements' (the treaty encompasses the two pollutants), and 'gas-by-gas agreements' (the abatements in the two pollutants are set up separately in two different agreements). In each case, the outcome of the emission game is computed for any given partition of the set of countries. The stability conditions are used to determine the outcome of the first-stage game. In the case of homogeneous countries, we show that the size of stable agreements remains low (no more than two countries). We give necessary and sufficient conditions of existence of such agreements. Our main result shows that comprehensive agreements better resist free-riding incentives and thus are more likely to emerge than single-gas agreements.

Introduction

International environmental issues are still characterized by the lack of a real supranational body able to enforce Pareto-improving measures. As a consequence, any action should arise from an intergovernmental negotiation process. In the case of climate change, the public-bad nature of greenhouse gases emissions also implies that such decisions may be undermined by free-riding incentives. The reversal in the US policy regarding climate change highlights how difficult it can be to design efficient and effective international environmental agreements. So far, more than ten years after the Rio Convention and despite climate scientists' growing alarm (Intergovernmental Panel on Climate Change, 2001), international action is still merely restricted to a "global warning".

During the last decade, an important body of literature has addressed these issues from a game-theoretic perspective. [START_REF] Carraro | The International Dimension of Environmental Policy[END_REF] first argued that the "tragedy of the commons" could be somehow overcome thanks to a partial cooperation among countries. Their approach based on the use of cartel stability concepts shows that, even with a lack of international government, self-enforcing agreements may emerge as a stable equilibrium of a non-cooperative game. 1 Extensive surveys of the related literature can be found in Barrett (1997b), [START_REF] Carraro | The Structure of International Environmental Agreements[END_REF] and [START_REF] Finus | Game Theory and International Environmental Cooperation: A Suvey with an Application to the Kyoto Protocol[END_REF]. Nevertheless, these studies show that the size of such self-enforcing agreements remains generally low in terms of number of signatories. Therefore, such agreements are far from achieving the cooperative outcome.

In Kyoto, extending the set of gases in the targets definition from three to six gases has captured a lot of negotiators' efforts to reach an agreement. Likewise, deciding wether to include or not carbon sinks in emission targets has been one of the key-issues in the negotiation. Surprisingly, whereas the definition of the "basket of pollutants" to be included in the Kyoto Protocol has been at core of the recent talks, it has received little attention in economics studies, which analyze strategic dimensions of climate change negotiations. As stated by [START_REF] Hoel | Efficient International Agreements for Reducing Emissions of CO2[END_REF]: "Whatever type of international agreement is reached during the next decade, it will probably only cover CO 2 -not other climate gases. [. . . ] Although agreements encompassing all climate gases could be more efficient, practical considerations, will thus force governments, at least initially, to limit an agreement to CO 2 ."

In the immediate aftermath of the Rio Conference, evidence from the negotiation process appears to be in contradiction with Hoel's prediction. 2 Admittedly, as suggested by Hoel, the importance of energyrelated emissions and their straightforward link with economic growth may ease the monitoring of CO 2 emissions. But indeed, once quantitative targets are adopted, the wider the "portfolio" of pollutants is, the larger the opportunity is to find low-cost abatement options for a given level of the emissions target [START_REF] Houghton | A Multi-Gas Approach to Climate Policywith and without GWPs[END_REF]. Recent estimates unambiguously show the cost-effectiveness of a multigas approach [START_REF] Hayhoe | Costs of Multigreenhouse Gas Reduction Targets for the USA[END_REF][START_REF] Reilly | Multi-Gas Assessment of the Kyoto Protocol[END_REF][START_REF] Burniaux | A Multi-Gas Assessment of the Kyoto Protocol[END_REF]. This issue is also of great importance insofar as the choice of the set of pollutants strongly determines the sectors that should be involved in a regulation policy. This is particularly true for agriculture that accounts for most of the world emissions of nitrous oxide and methane and provides carbon storage both in soils and trees [START_REF] Babcock | Relative Efficiency of Sequestering Carbon in Agricultural Soils through Secon-Best Market-Based Instruments[END_REF][START_REF] De Cara | Emissions of Greenhouse Gases from Agriculture: the Heterogenity of Abatement Costs in France[END_REF][START_REF] Schneider | Agricultural Sector Analysis on Greenhouse Gas Emission Mitigation in the United States[END_REF]. 3 Both global environmental results and abatement costs are sensitive to the definition of the basket of pollutants. So must be the incentives to participate to an international agreement. The question examined in this paper is then to assess the influence of the set of pollutants on the formation of stable and self-enforcing agreements. We analyze the formation of international environmental agreements as the equilibrium of a two-stage non-cooperative game.

The model, the assumptions, and the different kinds of agreement we consider are defined in section 1. Section 2 is devoted to the analysis of the 'single-gas agreement' game, in which only one gas is included in the treaty, other possible sources of pollution being ignored. The outcome of the emission game is computed for any given size of the agreement. The use of the stability conditions makes the size of the agreement endogenous. It is shown that only low-sized agreements (two countries) are likely to emerge. For a commonly used class of payoff functions, we give necessary and sufficient conditions of the existence of a stable agreement. These results are then extended to 'comprehensive agreements', which encompass all the pollutants (section 3). 'Gas-by-gas agreements' are then considered as a generalization of the two former kinds of agreement in section 4. In this section, the environmental and economic results of each kind of agreement are compared and discussed. It is shown that comprehensive agreements not only lower abatement costs, but also better resist to free-riding incentives.

The model

Let consider a problem, in which n countries share a common resource, namely the atmosphere or the climate. The set of the n countries is denoted by I = {1, . . . , n}.

One environmental issue, two pollutants

The pollution is assumed to result from two gases, denoted by g 1 and g 2 . The abatement in gas g j j ∈ J = {1, 2} in country i ∈ I is denoted by q ij . These gases are considered to differ by their impact on the environment. An equivalence rule is assumed to have been established by the scientists and to have been accepted by the parties before the negotiations. As it is the case for GWP (for a given time horizon 4 ), the equivalence rule between the two gases is simply defined as a constant factor of conversion that allows to convert emissions of g 2 into g 1 equivalent. Hence, one mass unit of g 2 emitted now is assumed to be equivalent to β units of g 1 emitted now. 5 The total reduction in pollution is denoted by Q and is computed as follows:

Q = i∈I (q i1 + βq i2 ) = i∈I q i
where q i is the national total abatement in country i, expressed in terms of g 1 -equivalent.

Net benefits

Each country benefits from global reductions in the ambient pollution level. Following Barrett's assumption (1994), the gross benefit function is assumed to be increasing and concave in Q such that:

B i (Q) = b i n aQ - 1 2 Q 2 with b i , a > 0 ∀i ∈ I (1)
Abatement costs faced by each country depend uniquely on its own abatement and are increasing and convex in q ij . We express the function of abatement costs in terms of g j quantities so that:

C ij (q ij ) = 1 2 c ij q 2 ij with c ij > 0 ∀(i, j) ∈ I × {1, 2} (2)
Net benefit of country i thus depends on the vector of abatement decisions:

π i (q ij , q -i,j ) = B i (Q) -C i1 (q i1 ) -C i2 (q i2 ) (3) = b i n aQ - 1 2 Q 2 - 1 2 c i1 q 2 i1 - 1 2 c i2 q 2 i2 (4)
1.3. International environmental agreement(s) as a two-stage game

The generic game is a two-stage game. Countries decide non-cooperatively to join or not to join the environmental 'coalition(s)' in the first stage. This choice is assumed to be driven by the maximization of each country's own individual net benefit. The first stage game is a simultaneous open-membership game. 6 The second-stage game is the emission game. At this stage, countries set up their abatement levels. This choice is made jointly among the countries belonging to the agreement(s) and non-cooperatively for other countries. The partition of I is assumed to be common knowledge as decided in the first stage. One important feature of the formulation (3) is that the best-reply functions are non-orthogonal because of the quadratic setting of the benefit function. As noted by [START_REF] Botteon | Burden-Sharing and Coalition Stability in Environmental Negotiations with Asymmetric Countries[END_REF], this implies the presence of some leakage, due to the possibility for non-signatory countries to increase their emissions in reaction to signatories' decisions. This leakage tends to limit the size of stable agreements. 7 1.3.1. Second stage: Emission game Following the above quoted Hoel's assertion, single-gas agreements are first considered. In this case, countries are assumed to only consider one gas in the negotiation process. This kind of agreement thus leads to partition the set I into two types of countries: those which sign the environmental treaty on gas g j and those which do not. For an exogenous reason, gas g -j is not considered and abatement in this gas is set up individually by each country. 8 DEFINITION 1 (Single-gas j-agreement). A single-gas j-agreement is given by the partition of P(I) = {S j , {i} i∈I\S j } such that countries belonging to S j choose cooperatively their abatement in gas g j , while other countries behave like singletons. The abatement levels in gas g -j are set up non-cooperatively by all the countries. These decisions are assumed to occur simultaneously.

The underlying concept of equilibrium that is used here is a Partial Agreement Nash Equilibrium with respect to a coalition S j (PANE wrt S j ). 9 The equilibrium of the second-stage game is given by solving the following problem:

P SG j            max (q ij ) i∈S j k∈S j π k (q k , q -k ) max q i,j π i (q i , q -i ) ∀i ∈ I\S j max q i,-j π i (q i , q -i ) ∀i ∈ I
which first-order conditions are given by the following 2n-system: 10

k∈S 1 B k (Q) = C i1 (q i1 ) ∀i ∈ S 1 (5) B i (Q) = C i1 (q i1 ) ∀i ∈ I\S 1 (6) βB i (Q) = C i2 (q i2 ) ∀i ∈ I (7)
Alternative to single-gas agreements may consist in an agreement that encompasses all the gases involved in the polluting process. This kind of agreements is refered hereafter as comprehensive agreements. Indeed, one can imagine two different settings in this case. The environmental agreement can specify targets (i) in aggregate level of abatement or (ii) in each gas separately. In the Kyoto Protocol, the first approach is retained. However, we show in appendix that these two approaches are equivalent in our framework. Therefore, in the remainder of the paper, we use the latter definition, which allows more straightforward comparison with other kinds of agreements.

DEFINITION 2 (Comprehensive agreement). A comprehensive agreement is given by the partition of P(I) = {S, {i} i∈I\S } such that countries belonging to S choose cooperatively their abatement in gas g 1 and in gas g 2 , while other countries behave like singletons. These decisions are assumed to take place simultaneously.

The corresponding problem is:

P C (1,2)      max (q i1 ,q i2 ) i∈S k∈S π k (q k , q -k ) max q i1 ,q i2 π i (q i , q -i ) ∀i ∈ I\S
That leads to the following first-order conditions:

k∈S B k (Q) = C i1 (q i1 ) ∀i ∈ S (8) β k∈S B k (Q) = C i2 (q i2 ) ∀i ∈ S (9) B i (Q) = C i1 (q i1 ) ∀i ∈ I\S (10) βB i (Q) = C i2 (q i2 ) ∀i ∈ I\S (11)
We now define a new kind of environmental agreement. The 'gas-bygas' agreement allows the presence of two co-existing agreements, each of them dealing separately with one pollutant. It is formally defined as follows:

DEFINITION 3 (Gas-by-gas agreement). A gas-by-gas agreement is given by a set S 1 of countries, which choose cooperatively their abatement in gas g 1 and a set S 2 of countries, which choose cooperatively their abatement in gas g 2 . Countries which do not belong to S j set noncooperatively their abatement in gas g j . These decisions are assumed to occur simalteneously.

The problem to be solved is thus the following:

P G                    max (q i1 ) i∈S 1 k∈S 1 π k (q k , q -k ) max q i1 π i (q i , q -i ) ∀i ∈ I\S 1 max (q i2 ) i∈S 2 k∈S 2 π k (q k , q -k ) max q i2 π i (q i , q -i ) ∀i ∈ I\S 2
which first-order conditions are:

k∈S 1 B k (Q) = C i1 (q i1 ) ∀i ∈ S 1 (12) β k∈S 2 B k (Q) = C i2 (q i2 ) ∀i ∈ S 2 (13) B i (Q) = C i1 (q i1 ) ∀i ∈ I\S 1 (14) βB i (Q) = C i2 (q i2 ) ∀i ∈ I\S 2 (15)
Using the specifications defined in equations ( 1) and ( 2), the firstorder conditions of P G become:

1 n (a -Q)γ S 1 ,i 1 = q i1 ∀i ∈ S 1 (16) β 1 n (a -Q)γ S 2 ,i 2 = q i2 ∀i ∈ S 2 (17) 1 n (a -Q)γ i,i 1 = q i1 ∀i ∈ I\S 1 (18) β 1 n (a -Q)γ i,i 2 = q i2 ∀i ∈ I\S 2 (19)
where:

γ S,i j = B S c ij = k∈S b k c ij ( 20
)
This problem is solved by summing aggregate emissions over the n countries (q 2i is weighted by β):

Q G = a A 1 (S 1 ) + β 2 A 2 (S 2 ) 1 + A 1 (S 1 ) + β 2 A 2 (S 2 ) ( 21
)
where

A j (S j ) = 1 n   k∈S j γ S j ,k j + k∈I\S j γ k,k j  
Indeed gas-by-gas agreements lead to partition of I into four types of countries: those which belong to both agreements (i ∈ S 1 ∩ S 2 , they are assigned a superscript ss ), those which belong to only one agreement (i ∈ S 1 \{S 1 ∩ S 2 } and i ∈ S 2 \{S 1 ∩ S 2 }, denoted respectively sn and ns ) and those which do not belong to any agreement (i ∈ I\{S 1 ∪ S 2 }, denoted nn ). Abatement levels depend on the partition P(I). The aggregate abatement levels are given by equations ( 22) to (25).

q ss i (P(I)) = a n γ S 1 ,i 1 + β 2 γ S 2 ,i 2 1 + A 1 (S 1 ) + A 2 (S 2 ) ( 22
)
q sn i (P(I)) = a n γ S 1 ,i 1 + β 2 γ i,i 2 1 + A 1 (S 1 ) + A 2 (S 2 ) ( 23
)
q ns i (P(I)) = a n γ i,i 1 + β 2 γ S 2 ,i 2 1 + A 1 (S 1 ) + A 2 (S 2 ) (24) q nn i (P(I)) = a n γ i,i 1 + β 2 γ i,i 2 1 + A 1 (S 1 ) + A 2 (S 2 ) (25)
For such a given partition of I, the net benefits are easily computable:

π ss i (P(I)) = a 2 b i 2n 2   n - n + B S 1 b i γ S 1 ,i 1 + β 2 B S 2 b i γ S 2 ,i 2 (1 + A 1 (S 1 ) + β 2 A 2 (S 2 )) 2   ( 26
)
π sn i (P(I)) = a 2 b i 2n 2   n - n + B S 1 b i γ S 1 ,i 1 + β 2 γ i,i 2 (1 + A 1 (S 1 ) + β 2 A 2 (S 2 )) 2   ( 27
)
π ns i (P(I)) = a 2 b i 2n 2   n - n + γ i,i 1 + β 2 B S 2 b i γ S 2 ,i 2 (1 + A 1 (S 1 ) + β 2 A 2 (S 2 )) 2   ( 28
)
π nn i (P(I)) = a 2 b i 2n 2 n - n + γ i,i 1 + β 2 γ i,i 2 (1 + A 1 (S 1 ) + β 2 A 2 (S 2 )) 2 (29)
The leakage is embodied in the factors A j (S j ) since a non-signatory's payoff is affected by the partition of I. The reaction functions of nonsignatories and signatories are thus clearly non-orthogonal. The payoff of any country, which signs an agreement on gas g j is increasing in the ratio of its own slope of marginal benefit (b i ) over the aggregate slope of marginal benefit within the coalition (B S J ).

Outcomes of the other kinds of agreement are obtained by imposing appropriate restrictions on the partition of the set of countries. The outcome from a single-gas j-agreement, is thus computed by using the previous equations ( 26)-( 29) and assuming that S -j = ∅. 11 Likewise, if the partition of I is assumed to be such that S 1 = S 2 = S, one can find the outcome of the comprehensive (1-2)-agreement.

First stage of the game: membership game

To find out sub-game perfect equilibrium of the two-stage game, we proceed by backward induction. Therefore, given the outcome of the emission game computed above, the equilibrium of the membership game consists in a partition of I such that unilateral deviation from a country is not individually profitable.

In the case of single-gas or comprehensive agreement, the choice that each country has to make in this first stage is binary, since the alternatives are lying between 'signing' or 'not signing' the treaty. Following [START_REF] Carraro | The International Dimension of Environmental Policy[END_REF], we use the internal and external stability concepts first developed by d [START_REF] Aspremont | On the Stability of Collusive Price Leadership[END_REF]. These conditions require that an agreement is stable if: (i) a signatory country is better off when remaining inside the treaty rather than leaving it and (ii) a non-signatory country is better off when remaining outside of the agreement rather than in joining it. Moreover, a stable agreement is required to be profitable, that is to say that a signatory should be better off when signing the treaty than the non-cooperative outcome. To be stable, a partition of I (P S (I) = {S, {i} i∈I\S }) is thus required to be such that:

π s i (P S ) ≥ π n i (P S\{i} ) ∀i ∈ S (30) π n k (P S ) ≥ π s k (P S∪{k} ) ∀k ∈ I\S (31) π s i (P S ) ≥ π n k (P ∅ ) ∀i ∈ S ( 32
)
where π s i (P S ) (resp. π n i (P S )) is the net benefit of a signatory (resp. non-signatory) country i in the situation P S .

In the case of gas-by-gas agreements, the stability conditions are considerably enriched, since each country faces four possible moves.

Single-gas agreements in the homogenous case

We first examine single-gas j-agreements. As noted by Hoel in the above quoted assertion, this type of agreement is likely to be less efficient than comprehensive agreements. The issue being addressed in this section consists in assessing the outcome of an agreement that neglects one source of pollution.

For the sake of simplicity, we focus on the homogenous case. Then, the following assumptions are made:

b i = b ∀i ∈ I (33) c ij = c j ∀(i, j) ∈ I × {1, 2} (34)
γ i,i j = γ j ∀(i, j) ∈ I × {1, 2} (35)
If |S| = s, we then have:

γ S,i j = sγ j Moreover, the following normalizations are used:

k = β 2 γ 2 γ 1 = β 2 c 1 c 2 and γ 1 = γ
k thus stands for the relative slope of the marginal abatement costs related to a supplementary emission unit of g 1 -equivalent.

The assumption of symmetric countries allows for great simplification of this problem. The first one is that an agreement can be fully depicted by its size. Hence, the discussion of the stability of a given agreement is summed up to a discussion on the size of the agreement. Another important simplification allowed by the assumption of homogeneity is that any burden-sharing rule within the agreement leads to the same results. 12 2.1. Single gas agreement: outcome of the emission game

In order to obtain the results of the second-stage game, equations ( 22) and ( 25) are used and combined with the above assumptions. Single-gas 1-agreements are first considered. Hence, S 2 is assumed to be empty. Since the countries are assumed to be ex ante identical, the partition of the set I is fully defined by the agreement size:

q s i (s 1 , 0) = aγ(s 1 + k) n + γ(s 1 (s 1 -1) + n(k + 1)) ∀i ∈ S 1 ( 36
)
q n i (s 1 , 0) = aγ(k + 1) n + γ(s 1 (s 1 -1) + n(k + 1)) ∀i ∈ I\S 1 (37)
Consitently to the intuition, signatories' abatement is higher than non-signatories' and the difference increases with respect to the number of signatories for low sizes of agreement. As mentioned above, the bestreply functions of the signatory and non-signatory countries are not orthogonal because of the non-linear setting of the benefit function. Hence, the abatement of a non-signatory country is decreasing with the size of the agreement.

The total abatement for a single-gas 1-agreement is thus:

Q(s 1 , 0) = aγ s 1 (s 1 -1) + n(k + 1) n + γ(s 1 (s 1 -1) + n(k + 1)) (38)
The signatory countries' abatement is first increasing with respect to the size of the agreement. Over a given threshold value of s 1 , the global reduction in the level of pollution is such that the abatement required from individual signatory is reduced by the entry of a new country, and, hence, q s i (s 1 , 0) is then decreasing with s 1 . For a given size of agreement, single-gas 1-agreement perform greater reduction in the total pollution than single-gas 2-agreement when the slope of marginal abatement cost for gas g 1 is lower than for gas g 2 expressed in the same unit (ie k < 1). The total abatement is the same in the two cases when k = s 1 (s 1 -1) s 2 (s 2 -1) . The comparison between the abatement achieved through the single-gas agreements is illustrated in figure 1.

Multi-Greenhouse Gas International Agreements

Abatement

Full cooperation

1 n I q i (s 1 , 0) q n i (s 1 , 0) s 2 Abatement q s i (s 1 , 0) q n i (0, s 2 ) 1 n I q i (0, s 2 )
Single-gas 1 agreement Single-gas 2 agreement q s i (0, s The net individual benefits are:

π s i (s 1 , 0) = a 2 b 2 1 n - n + γ(s 2 1 + k) (n + γ(s 1 (s 1 -1) + n(k + 1))) 2
(39)

π n i (s 1 , 0) = a 2 b 2 1 n - n + γ(1 + k) (n + γ(s 1 (s 1 -1) + n(k + 1))) 2 (40)
The net benefit of a non-signatory country is clearly increasing in the size of the agreement as the environmental quality increases with s 1 . This effect also holds for a signatory country, but it is partially offset by the increase in its abatement costs. Non-signatories are thus always better off than signatories. The net benefit are sketched for the two kinds of agreements in figure 2.

Single gas agreement: outcome of the membership game

Again, we first focus on single-gas 1-agreements and extend the results to single-gas 2-agreements. To address the stability issue, the following function, L(., 0), is used:

L(s 1 , 0) = π s i (s 1 , 0) -π n i (s 1 -1, 0) (41)
When L(s 1 , 0) is negative, at least one signatory country would be better off in exiting the treaty in the situation (s 1 , 0). When it is positive, one non-signatory country would be better off in joining the environmental coalition in the situation (s 1 -1, 0). Hence, the (s 1 , 0)agreement is stable when its size s 1 is such that L(s 1 , 0) ≥ 0 and L(s 1 + 1, 0) ≤ 0. PROPOSITION 1. Whatever the values of the parameters a, b, γ, k and n ≥ 3, the size of any stable single-gas agreement cannot be greater than two.

Proof. see appendix

Although a partial environmental coalition can emerge as an equilibrium of the membership game, incentives to defect are too large when the size of the agreement is greater than three. 13 We now characterize the conditions that allow an agreement of size two to be stable. As the stability function is negative for s j ≥ 3, the stable agreement, if it exists, should correspond to a situation for which L(2, 0) ≥ 0. Let denote the ratio of marginal abatement cost in each gas relatively to total marginal abatement cost by δ j (δ 1 = 1 k+1 and δ 2 = k k+1 ).

Multi-Greenhouse Gas International Agreements

1 n I π i (s 1 , 0) π s i (0, s 2 ) 1 n I π i (0, s 2 )
Single-gas 1 agreement Single-gas 2-agreement PROPOSITION 2. A single-gas j-agreement of size two is stable if and only if γ is sufficiently small (k and n ≥ 3 given), that is to say γ ≤ n k+1 / n -2(δ j + 1) + 2 (n -δ -j ) 2 -3δ j (n -1) . If this condition is not fulfilled, the non-cooperative outcome prevails.

π n i (s 1 , 0) π s i (s 1 , 0) π n i (0, s 2 )
Proof. see appendix COROLLARY 1. If γ(k + 1) ≤ 1 3 , a stable single-gas j-agreement of size two exists.

Proof. It is sufficient to see that lim

n→+∞ γ+ j = 1 3(k + 1)
and that

γ+ j ≥ 1 3(k + 1)
.

Notice that the threshold value of γ+ j depends on the weight of each gas in the marginal abatement costs expressed in g 1 -equivalent (δ 1 and δ 2 ). Depending on the slope of marginal abatement cost in gas g j relatively to g -j , stable single-gas j-agreement may exist while stable single-gas (-j)-agreement may not. However, since the difference between γ+ 1 and γ+ 2 decreases as n increases (the other parameters remaining constant), the sets of parameter values for which single-gas agreements exist tends to be the same when n is large.

The main result is that the maximum size of any stable singlegas agreement is two, and such an agreement holds when γ is small. This result may appear to be rather counter-intuitive: whatever gas at stake in the negotiation process, a stable agreement may emerge if and only if the slope of marginal environmental benefit is not too large relatively to the slope of marginal abatement cost. Indeed, when γ is large, the abatement level within the agreement is also large and so are the abatement costs faced by the signatories. Hence, the decrease in the pollution is not sufficient to offset the incentive to remain outside of the agreement.

Comprehensive agreements in the homogenous case

Let now consider the case of comprehensive agreements. This kind of agreement encompasses all the pollutants as the Kyoto Protocol does. That means that if a country decides to sign the agreement, he is required to reduce its emissions in all the gases. Reversely, if a signatory country decides to leave the agreement, no abatement in any gas is required. Again, we compute and analyse the outcome of the emission game and then examine the stability of such kind of agreements.

Comprehensive agreements: outcome of the emission game

The outcome of the comprehensive agreement is given by the equations ( 22)-( 25) and ( 26)-( 29) assuming that S 1 = S 2 = S and that |S| = s. The individual abatements are:

q s i (s, s) = aγs(k + 1) n + γ(s(s -1) + n)(k + 1) (42) q n i (s, s) = aγ(k + 1) n + γ(s(s -1) + n)(k + 1) (43)
The general interpretation of these results is not modified as compared to the comments made in the former section. Abatement levels from the signatory countries are s time bigger than from the non-signatory countries, with s given. As the size of the agreement increases, the nonsignatory countries' abatement decreases. For low sizes of agreement, the signatory countries' abatement is increasing with respect to s until s ≤ 1 + n γ(k+1) . Over this threshold, it is decreasing with respect to s. However, the total abatement remains strictly increasing with s, as the negative effect induced by the decrease in signatory countries' abatement is offset by the entry of a new member. These features are sketched on figure 3 (left).

The individual benefits are:

π s i (s, s) = a 2 b 2 1 n - n + γs 2 (k + 1) (n + γ(s(s -1) + n)(k + 1)) 2 (44) π n i (s, s) = a 2 b 2 1 n - n + γ(k + 1) (n + γ(s(s -1) + n)(k + 1)) 2 (45)
Figure 3 (right) shows the evolution of each country's payoff with respect to the size of the comprehensive agreement. Two effects make the non-signatories better off when s increases: (i) they benefit from the improvement in the environmental quality and (ii) in the same time, they face lower abatement cost. On the contrary signatories' abatement cost increases at least for low values of s and, hence, whatever s, nonsignatories are better off than signatories. In the simulation showed in figure 3, signatories' benefit is increasing, even for low values of s. This is ensured by the fact that γ is not too large. This point is important in the subsequent discussion on stability. q s i (s, s) q n i (s, s)

Multi-Greenhouse Gas International Agreements

1 n I π i (s, s) π n i (s, s) π s i (s, s) Net payoff

Comprehensive agreements: outcome of the membership game

The stability function in this case is defined as follows:

L(s, s) = π s (s, s) -π n (s -1, s -1) (46)
The interpretation of the stability function is the same as in the former section. The difference lies in the commitment of any signatory country to achieve abatement in both pollutants. PROPOSITION 3. Whatever the values of the parameters a, b, γ, k and n ≥ 3, the size of any stable comprehensive agreement cannot be greater than two.

Proof The general results regarding stability found in the former section also hold for comprehensive agreements. The stability of any comprehensive agreement involves low number of participating countries and is subject to the condition that γ is not too large. If not, the incentive to defect prevents any agreement of this kind from emerging as remaining outside of the agreement is an equilibrium strategy for all the countries.

Comparison between single-gas and comprehensive agreements

The comparison between the results above and those obtained in section 2 highlights the global and individual gains involved by the enlargement of the set of pollutants included in the treaty. It is obvious that a (s, s)-comprehensive agreement allows to achieve greater g 1 -equivalent abatement than any (s, 0) or (0, s) single-gas j-agreement.

As k tends to zero (resp. +∞), the difference between single-gas 1-(resp. 2-) agreements and comprehensive agreements decreases. Consequently, to assess the improvement in global welfare due to the enlarging of the set of pollutants encompassed in the treaty, it is necessary to consider the relative abatement costs for each gas expressed in g 1equivalent (k), and not only the relative environmental impact of each gas (β).

From the point of view of the non-signatories, enlarging the treaty is unambiguously profitable given that they remain oustide of the agreement and that the number of signatories remains constant. If so, the quality of the environment is improved while in the same time their abatement costs are lowered. However, as far as the signatories are concerned, the result is not as straightforward. Indeed, if extended in this way, the treaty would involve higher abatement for them and thus reduce their net benefit and increase the incentive to defect.

Nevertheless, corrollary 3 shows that, in this case, comprehensive agreements better resist to free-riding incentives, since the conditions that define a stable agreement are less restrictive (see also figure 4). COROLLARY 3. If two countries reach a single-gas j-agreement, then they can reach a comprehensive agreement. The reciprocal does not hold.

Proof. It is sufficient to notice that γ+ j ≤ γ+ for j ∈ J. Combined with propositions 2 et 4, these relations prove the corollary.

Gas-by-gas agreements

This section is devoted to the comparison of the results of gas-bygas agreements with other kinds of agreements. The space of strategy available for each country is extended to account for the possibility of cooperating on one gas and not on the other.

Graphical analysis of gas-by-gas agreements

Since four strategies are available to each country, any gas-by-gas is defined by the partition of

I into S 1 ∩ S 2 , S 1 \{S 1 ∩ S 2 }, S 2 \{S 1 ∩ S 2 } and I\{S 1 ∪ S 2 }.
In the homogenous case that we analyse, a gas-bygas agreement may be defined by the number s 1 of countries which cooperate on g 1 , the number s 2 of countries which cooperate on g 2 and by e as the number of countries which cooperate on both gases. Note however that the individual and global results of the emission game 0 0 within a gas-by-gas regime do not depend on the cardinal of S 1 ∩ S 2 , e (see the appendix). A graphical representation of the set of gas-by-gas agreements in {1, n} 2 is given in figure 5. PROPOSITION 5. The total abatement Q(s 1 , s 2 ) is increasing with respect to s 1 (resp. s 2 ) with s 2 (resp. s 1 ) given. The iso-abatement curves for (s 1 , s 2 ) ∈ I 2 are strictly decreasing at a decreasing rate. The slope of iso-abatement curves are independant of a, b, c and n are increasing in k.

1 3(k+1) k < 1 k > 1 γ+ 1 γ+ γ+ γ+
Proof. see appendix This is illustrated in figure 5. The lower is k, the steeper are the isoabatement curves. This rather intuitive result is due to the fact that a lower slope of marginal abatement cost in gas 1 comparatively to g 2 (in g 1 -equivalent) required less countries to cooperate on g 1 to reach the same environmental result. Corollary 4 is directly derived from these properties.

COROLLARY 4. Consider any (s 1 , s 2 ) gas-by-gas agreement. It exists Proof. Let s * be such that :

s * ∈ [s 1 , s 2 ] such that a (s * , s * ) agreement achieves the same level of global abatement. The closer to 0 (resp. to +∞) is k, the closer is s * to s 1 (resp. to s 2 ). ¡ ¡ ¢ £¢ ¢ £¢ ¤ £¤ ¤ £¤ ¥ ¥ ¦ ¦
Q(s 1 , s 2 ) = Q(s * , s * ) (47)
There is only one positive s * which satisfies that equation with (s 1 , s 2 ) given :

s * = 1 2 (Y + 1) (48)
with

Y = δ 1 (2s 1 -1) 2 + δ 2 (2s 2 -1) 2
PROPOSITION 6. For the same environmental result, the global benefit that is reached under the (s * , s * ) agreement is higher than under any environmentally equivalent (s 1 , s 2 ) gas-by-gas agreement.

Proof. see appendix

This is illustrated in figure 6 (right). Along each curve, the global benefit is constant. The difference between global benefit along two consecutive curve is also constant. It is easy to show that the slope of any iso-benefit curve is -1 k when it crosses the 45 degrees-line. On the example shown, the gradient of the global benefit is high when s 1 and s 2 are low and is decreasing as s j tends towards n. The maximum benefit yield through cooperation on g 2 (for s 2 = 100) requires about fifty countries to cooperate on g 1 and only fourty if the treaty encompasses the two gases.

Because of their cost-effectiveness perspective, comprehensive agreements do better than any other gas-by-gas agreements. However, the other side of the argument presented in proposition 6 is also noteworthy. Consider a comprehensive (s, s) agreement and operate a move along a iso-profit curve towards any (s 1 , s 2 ) agreement. The net global benefit is obviously the same, whereas the environmental quality is improved. Of course, this move requires the entry of non-signatories into S 1 and/or S 2 .

Membership game: an example

The results found in sections 2 and 3 suggest that stability requirement strongly restricts the feasible set of international agreements to a few participating countries. In the case of gas-by-gas agreements, stability conditions are complicated by the fact that four types of strategies are available to each country. Thus, even in the homogenouse case, since each country faces three alternative choices, twelve conditions are required to be fulfilled simultaneously for general partition of I (less if some strategies are not used by the countries).

To illustrate the possible second-stage outcomes, we discuss a numerical example. The parameters are such that the conditions exposed in corollaries 1 and 2 are fulfilled. Hence, the single-gas agreements defined by (s 1 , s 2 ) = (2, 0) and (s 1 , s 2 ) = (0, 2) are stable. The first yields higher global benefit and higher global abatement than the second since k < 1. The comprehensive (2, 2) agreement is also stable as it is not profitable for a signatory to defect (1450, 2 > 1447, 6) while it is also not profitable for a non-signatory to join the agreement (1470, 0 > 1461, 8). On this example, we also see that, starting from any single-gas j-agreemeent, both signatories and non-signatories would benefit from extending the treaty to gas g -j and negotiating a comprehensive agreement.

However, introducing gas-by agreements widens the set of possible outcomes. If a single-gas 1-agreement is signed, it is in non-signatories' interest to propose another agreement that deals uniquely with gas g 2 (1463.4 > 1462.6). In this case, the situation where two countries form a coalition on gas g 1 and two other countries cooperate on g 2 is stable, since no country faces incentives to change its decision in the membership game. Although this situation leads to the same global results than a comprehensive agreement, no country would accept in this case to sign a comprehensive agreement.

Concluding remarks

In the case of climate change, the difficulty to achieve full cooperation is strengthened by the number of pollutants involved in the polluting process. Our results partly show how the outcomes of a climate negotiation could be modified when the "basket of pollutants" is extended. This plays an important role in the economic and environmental results of the treaty, but also in the incentives to join the treaty or not.

From a cost-effectiveness perspective, it is unambiguously profitable to enlarge the set of pollutants included in an international agreement. Our results show that the superiority of comprehensive agreements also lies in the fact that such agreements better resist to incentive to defect than any single-gas agreement. Indeed, we have shown that for some set of parameters, self-enforcing comprehensive agreements may emerge as an equilibrium, while single-gas agreements may not. As soon as the decision to enlarge the "basket of pollutants" is made, arises the question of the equivalence rule between the pollutants. So far, the Global Warming Potential (GWP) index is used to convert emissions of different gases into CO 2 equivalent. Since this index relies only on scientific foundations, it does not give a proper account of the relative values of the different pollutants. From an economic perspective, this index is questionable for at least two reasons: (i) it does not account for the differences between uncertainties concerning the differences in the long-run impact of abatement in each gas 14 (ii) since no discount rate is used, the question of the inter-temporal choice of abatement in the different gases is neglected and this rule may mislead economic choices. 15 Our results stress the importance of this parameter on the set of stable agreements.

'Gas-by-gas' agreements -that is to say the possibility for agreements on each pollutant to coexist-could be a way to induce new countries to join the environmental agreement insofar as it could provide them with sufficient incentives. This result may be strenghtened by relaxing the assumption of homogeneity. Indeed, considering heterogenous countries regarding abatement cost in the different gases may ease the emergence of dissymetric gas-by-gas agreements. Further work is needed in this direction.

Change, 1995). Indeed, the 100-year GWP is often retained as the reference in international negotiations. However, this index varies in a quite wide range for different time horizons. The main reason for that is that the lifetime in the atmosphere differs widely from one gas to another.

5 Consider for instance that g 1 stands for CO 2 and g 2 for CH 4 . β is then simply the GWP of CH4 (β = 21 in this case).

6 Carraro (1998) reviews different membership rules for the first stage of the game. Chander and Tulkens's cooperative approach can be viewed as a unanimity membership rule. The assumption that the membership decisions are simultaneous is also different from the sequential game proposed by [START_REF] Bloch | Non-Cooperative Models of Coalition Formation in Games with Spillovers[END_REF].

7 Note, however, that in the original version of this model by [START_REF] Barrett | Self-Enforcing International Environmental Agreements[END_REF], the leakage plays an ambiguous role. Indeed, Barrett assumes a second-stage Stackleberg game, in which the signatory countries act jointly as the leader. This tends to increase the benefit of the signatories because they expect greater abatement from the non-signatories. Hence, without non-orthogonal best-reply functions, large stable agreements as found by Barrett are not possible (see also Barrett (1997b)).

8 Note that, even if not included in the treaty, each country takes into account the influence of gas g-j in the computation of its own net payoff.

9 This concept is formally defined in [START_REF] Chander | The Core of an Economy with Multilateral Environmental Externalities[END_REF].

10 Without loss of generality, these conditions are written for a single-gas 1-agreement. The results in the case of a single-gas 2-agreement are straightforward by using appropriate change in the meaning of β.

11 Indeed, due to the reference to Nash equilibrium, it is equivalent to consider that S j is empty or reduced to a singleton.

12 In the case of heterogenous countries, the burden-sharing rule within the agreement is not straightforward as soon as side payments between the signatories are allowed. Barrett (1997a) and [START_REF] Botteon | Burden-Sharing and Coalition Stability in Environmental Negotiations with Asymmetric Countries[END_REF] have examined this problem and compared different type of bargaining rule (Nash bargaining, Shapley value). The results are quite sensitive to the choice of this rule.

13 [START_REF] Barrett | Self-Enforcing International Environmental Agreements[END_REF] finds larger sizes of stable environmental coalition, which can be as large as the total number of countries. But once again, this result is due to the Stackelberg assumption that provides greater incentives to reach the agreement. However, even in this case, the environmental coalition is large only when the difference between the non-cooperative and full-cooperative global benefit are very small. As soon as the decisions in the first-stage game are simultaneous, the low size of the environmental coalition is robust to different specifications of the net benefit function [START_REF] De Cara | International Environmental Agreements : Stability, Transfers and Sequential Membership[END_REF].

14 This reason is often used as an argument not to include carbon sinks in the Kyoto targets. As a matter of fact, the environmental impact of storing carbon depends on the future use of wood and, to this respect, is not strictly equivalent to reduce emissions by the same amount [START_REF] Feng | Carbon Sequestration in Agriculture: Value and Implementation[END_REF]. 15 One may admit, that, in this case, the difficulty then lies in the choice of the appropriate discount rate. by a signatory country i is as follows: min q i1 ,q i2 C i1 (q i1 ) + C i2 (q i2 ) st q i1 + βq i2 ≥ qi Country i would thus tend to equate the marginal abatement costs in each gas (expressed in the same unit) and would choose q i1 and q i2 in order to make the abatement constraint binding. That is to say: βC i1 = C i2 and q i1 + βq i2 = qi . Thus, the abatement mix for a given target of abatement qi is the following:

q i1 = c i2 β 2 c i1 + c i2 qi ( 49
)
q i2 = βc i1 β 2 c i1 + c i2 qi (50)
The emission game is thus described by the following problem:

P C (1-eq)          max (q i ) i∈S k∈S
π k (q k , q -k ) st (49) and ( 50) max (q i1 ,q i2 ) π i (q i , q -i ) ∀i ∈ I\S that leads to first-order conditions that are equivalent to those for P C (1-2) .

Proof of proposition 1. Let first consider single-gas 1-agreements. Note that any coalition of size 3 leads to a negative stability function: L(3, 0) = -4a 2 b n(γ(n + 3) + (n -1)) + k(2γ(n(n + 1) -2) + n(n -1)) + k 2 (γn(n -1)) (n + γ(2 + n(k + 1))) 2 (n + γ(6 + n(k + 1))) 2 ≤ 0 (51) It is sufficient to see that the stability function is negative for s 1 ≥ 3. Since non-signatory payoff increases more rapidly with s 1 than signatory payoff, the stability function is negative for all s 1 ≥ 3. Hence, each signatory country faces a positive incentive to defect when s 1 ≥ 3. The same holds for a single-gas 2-agreement with appropriate change in the meaning of k.

Proof of proposition 2. We look for conditions on the parameters allowing the stability function L(2, 0) to be positive. We thus solve L(2, 0) = 0 with respect to γ. L(2, 0) can be rewritten as follows:

L(2, 0) = a 2 b (k + 1) ((k + 1) n (3n -4) -4) 2n 2 (1 + γ(k + 1)) 2 (n + γ (2 + n(k + 1))) 2 γ(γ + 1 -γ)(γ-γ - 1 ) (52)

The slope of an iso-abatement curve is given by :

ds 2 ds 1 = - ∂Q ∂s 1 (s 1 , s 2 ) ∂Q ∂s 2 (s 1 , s 2 ) = - 1 k 2s 1 -1 2s 2 -1 (57)
It is negative, decreasing with s 1 and increasing with k. On the 45 degrees-line the slope is equal to -1 k .

Proof of proposition 6. s * is defined as the solution of Q(s * , s * ) = Q(s 1 , s 2 ). Π(s * , s * ) -Π(s 1 , s 2 ) = 2ka 2 bγ(s 1 -s 2 ) 2 (s 1 + s 2 -1) ((Y -1)(s 1 + s 2 -1) + 2s 1 s 2 ) (k + 1)(2s

1 -1 + Y )(2s 2 -1 + Y )(γ((k + 1)n + (s 1 -s 2 1 + k(s 2 -s 2 2 ))) 2 + n) ≥ 0
This difference is obviously strictly positive for s 1 , s 2 ∈ I and s 1 = s 2 .

Figure 1 .

 1 Figure 1. Abatement levels in the case of single-gas j-agreements (parameter values: a = b = n = 100, c = 800, k = 0.5).

Figure 2 .

 2 Figure 2. Net benefits in the case of single-gas j-agreements (parameter values: a = b = n = 100, c = 800, k = 0.5).

Figure 3 .

 3 Figure 3. Abatement levels (left) and net benefits (right) in the case of comprehensive agreements (parameter values: a = b = n = 100, c = 800, k = 0.5).

Figure 4 .

 4 Figure 4. Comparison of γ values for which stable agreements exist

Figure 5 .

 5 Figure 5. Iso-abatement curves in the case of gas-by-gas agreements

Figure 6 .

 6 Figure 6. Iso-benefit curves for gas-by-gas agreements (parameter values: a = b = n = 100, c = 800, k = 0.5).

Table I .

 I Example of a gas-by-gas agreement payoffs (parameter values: a = b = n = 100, c = 800, k = 0.5).

					s 2
			1	2	3	4	100
	s1
		π ss	-	-	-	-	-
	1	π sn	-	-	-	-	-
		π ns	-	1448.5 1452.4 1459.3 4333.4
		π nn 1447.6 1455.1 1470.0 1492.1	-
		π ss	-	1450.2 1454.0 1461.0	-
	2	π sn 1449.3 1456.8 1471.6 1493.7	-
		π ns	-	1463.4 1467.2 1474.0 4333.9
		π nn 1462.6 1470.0 1484.7 1506.7 4908.0
		π ss	-	1457.9 1461.8 1468.7	-
	3	π sn 1457.1 1464.5 1479.2 1501.1	-
		π ns	-	1492.9 1496.6 1503.3 4334.8
		π nn 1492.1 1499.4 1514.0 1535.7 4908.1
		π ss	-	1471.7 1475.6 1482.5	-
	4	π sn 1470.9 1478.2 1492.8 1514.5	-
		π ns	-	1536.4 1540.0 1546.5 4336.1
		π nn 1535.7 1542.9 1557.2 1578.5 4908.3
		π ss	-	4633.0 4633.1 4633.2 4746.8
	100 π sn 4633.0 4633.1 4633.2 4633.4	-
		π ns	-	-	-	-	-
		π nn	-	-	-	-	-

The global warming potential index is computed and published by IPCC for different time horizons(20, 100 and

years) (Intergovernmental Panel on Climate

L(s, s) can be written as a polynomial expression of s, for which s = 1 is an obvious solution of L(s, s) = 0 because π s (1, 1) = π n (0, 0).

Acknowledgements

This research was initiated while the first author was at INRA ESR, UMR "Economie Publique" INA PG/INRA. A support from the French Ministry of the Environment (Socio-économie du changement climatique, subvention 98/140) is gratefully acknowledged. The authors thank Pierre-Alain Jayet, Carlo Carraro and Kathleen Schubert for their valuable comments on an earlier draft of this paper.

IPCC -Intergovernmental Panel on Climate Change; PANE wrt S -Partial Agreement Nash Equilibrium with respect to coalition S; UNFCCC -United Nations Framework Convention on Climate Change JEL codes: Q28

Notes

1 Numerous studies have extended these results, examining the influence of heterogeneity and burden-sharing rules among the signatories (Barrett, 1997a;[START_REF] Botteon | Burden-Sharing and Coalition Stability in Environmental Negotiations with Asymmetric Countries[END_REF], different equilibrium concepts [START_REF] Barrett | Self-Enforcing International Environmental Agreements[END_REF][START_REF] Ecchia | Coalition Formation in International Environmental Agreements and the Role of Institutions[END_REF][START_REF] Carraro | International Games on Climate Change Control[END_REF], and different specifications of the payoff functions [START_REF] Carraro | Policy Coordination for Sustainability : Commitments, Transfers and Band-Wagon Effects[END_REF][START_REF] Heal | Formation of International Environmental Agreements[END_REF][START_REF] Péreau | Partial and Global Cooperation with Unilateral Commitment in the Presence of Global Environmental Problems[END_REF][START_REF] De Cara | International Environmental Agreements : Stability, Transfers and Sequential Membership[END_REF]). An alternative approach, based on cooperative-game concepts has been proposed by Chander andTulkens (1992, 1997). These authors show that a supra-national agency can sustain Pareto-optimality through an appropriate transfer scheme among all countries. This transfer scheme presents the interesting property of lying into the core and thus resists to coalitional free-riding. However, this approach requires the existence of a supra-national body, which does not hold in reality (for a comparison of the two approaches see [START_REF] Tulkens | Cooperation versus Free Riding in International Environmental Affairs : Two Approaches[END_REF] 2 As soon as 1995, methane and nitrous oxide emissions are included in the targets first sketched in Rio. In Kyoto, the set of pollutants has been extended to include three other trace-gases (HFC, PFC and SF 6 ).

3 The controversy on accounting or not carbon storage in net emissions relies partly on the fact that the carbon sequestered may be released in the future, whereas reduction in emissions have a permanent effect [START_REF] Feng | Carbon Sequestration in Agriculture: Value and Implementation[END_REF].

Appendix

Alternative comprehensive agreements. An alternative type of comprehensive agreement is considered. Hence: DEFINITION 4 (Comprehensive (1-equivalent)-agreement). A comprehensive (1-equivalent)-agreement is given by the partition of P(I) = {S, {i} i∈I\S } such that countries belonging to S choose cooperatively their aggregate abatement {q i } i∈S expressed in terms of g 1 -equivalent, while other countries behave like singletons. These decisions are assumed to occur simultaneously.

Provided that each signatory country minimizes his total abatement cost subject to the aggregate abatement constraint, the problem faced where:

Only γ+ 1 is positive. Then, L(2, 0) ≥ 0 if and only if 0 ≤ γ ≤ γ+ 1 . Note that since π s (1, 0) = π n (0, 0) -the non-cooperative payoff-, this condition also ensures that the profitability condition is fullfilled.

Following the same reasonning for single gas 2-agreements, we find that a necessary and sufficient condition for L(0, 2) ≥ 0 is that 0 ≤ γ ≤ γ+ 2 with γ+ 2 defined as follows:

These thresholds define the value of γ, n and k for which a single-gas j-agreement of size two is stable.

Proof of proposition 3. The proof follows exactly the same argument as in proposition 1.

The negativity of L(s, s) for s ≥ 3 indicates that a signatory country faces a positive incentive to leave the agreement.

Proof of proposition 4. The reasonning 1 is the same as in proposition 2. We rewrite L(2, 2) as follows:

where γ+/-= n k + 1 / n -4 ± 2 n 2 -3(n -1) . γ+ is the only strictly positive root.

Proof of proposition 5. The proposition results from the study of the following partial dervatives:

∂Q ∂s 1

(s 1 , s 2 ) = aγn (2s 1 -1) (n + γ(s 1 (s 1 -1) + ks 2 (s 2 -1) + n(k + 1))) 2 ∂Q ∂s 2 (s 1 , s 2 ) = aγn (2s 2 -1) (n + γ(s 1 (s 1 -1) + ks 2 (s 2 -1) + n(k + 1))) 2


François Lieutier

Ye Hui

PINUS YUNNANENSIS NATURAL RESISTANCE TO A BARK BEETLE ASSOCIATED FUNGUS IS INCREASED BY A MILD WATER STRESS

come

Pinus yunnanensis resistance to a bark beetle associated

fungus is increased during a natural water stress

Two neighboring sites of an area forested with Yunnan pine (Pinus yunnanensis), located in Yunnan province (China) and clearly differing by their water availability, were compared for tree resistance to isolated and mass artificial inoculations with Leptographium yunnanense, a fungus associated with the bark beetle Tomicus sp.

During 18 months alternating dry and humid seasons in each site, trees' physiological parameters (soil water content at different depths, predawn needle water potential, needle water content) were followed in the two sites.

Simultaneously, the performances of the fungus after isolated inoculation (growth in the phloem, ability to induce the development of a phloem induced reaction) and its virulence after mass inoculations were periodically measured to appreciate tree resistance. The phloem induced reactions were more extended during the wet than during the dry season, and fungus grew further into the phloem during the humid season and in the wet site than during the dry season and in the dry site. Fungus virulence after mass inoculations was higher during the humid period and in the wet site than during the dry period and in the dry site. These results demonstrate that tree resistance to fungus inoculation and consequently to bark beetle attacks, is increased by a mild water stress. This is in agreement with the growth-differentiation balance hypothesis and with the idea that the fungus is favored by a large availability of water. In conclusion, drought is not a general explanation for Tomicus damage in Yunnan.

2 38 41 72 30 Fax 33 / 2 38 41 72 02

	Choice of Presentation: X Oral	 Poster
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	Symposium Number 10.2	(first Choice)	(Second Choice)
	See the Symposium Numbers in the Scientific Program of this circular.
	The Name of Presenting Author and the Address:
	First Name Francois	Middle Name	Last Name LIEUTIER
	Institution University of Orleans
	Mailing Address Laboratoire de Biologie des Ligneux et des Grandes Cultures,
	rue de Chartres, B.P. 6759, 45067 Orleans Cedex 2 FRANCE.
	Tel 33 /

E-mail francois.lieutier@univ-orleans.

fr 1 PINUS YUNNANENSIS RESISTANCE TO A BARK BEETLE ASSOCIATED FUNGUS IS INCREASED BY A NATURAL WATER STRESS.

  François LIEUTIER 1 , YE Hui 2 and Annie YART 3 1 Laboratoire de Biologie des Ligneux et des Grandes Cultures, Université d'Orléans, rue de Chartres, B.P. 6759, F-45067 Orléans Cedex 2, France. 2 Yunnan University, Institute of Ecology and Geobotany, 650091 Kunming, Yunnan, P.R. China. 3 INRA, Unité de Zoologie Forestière, Ardon, F-45160 Olivet, France. Email: francois.lieutier@univ-orleans.fr


C Benjamin

I Piot-Lepetit

Dynamic Microeconomic Modeling of French Beef Supply Response

The European Union beef market regulation is largely influenced by the Common Agricultural Policy (CAP). With the 1992 CAP reform, there was a partial shift by the EU from product price support to a more direct form of income support by way of direct payments. For beef there was a move to direct payments on intermediate products which was essentially a direct payment for the possession of various categories of animais and these were linked to a land resource base. The Agenda 2000 reform consists in a further price decrease associated with an increase in direct payments. This contribution focuses on the dynamics of beef supply response in the French beef sector. The objective of this paper is to provide a model of beef suppl y response within a dynamic microeconomic framework. The aim of this mode! is the understanding of cattleman decisions and to assess how the behaviour of beef producers is sensitive to changes in production prices and to changes in premiums. In particular, the paper studies how the beef suppl y response is modified when various exogenous variables like prices or policy instruments are changed.

Introduction

The European Union (EU) plays an important role on the international exchanges of beef both in imports and exports. Its main export areas are the East Europe, the Near East and the North Africa, while its main import areas are countries from the MERCOSUR and from the ACP areas with which preferential exchanges exist (Lomé agreement). However, since the early 1980s, the EU beef market is structurally over target and, thus, is largely dependent of trade with countries outside the EU. Until 1995, exports were made with the help of subsidies. However, due to the 1993 GATT agreements, this system must be progressively removed. A new challenge for the EU is to find a way of opening up new export outlets without subsidies. Moreover, the EU beef market regulation is largely influenced by the Comrnon Agricultural Policy (CAP). One of the objectives identified by the EU is an increase of competitiveness internally and externally. The 1992 reform was an attempt to improve market balance by reducing intervention prices and by increasing direct payments with an introduction of a ceiling on the number of animais eligible for support. These changes represented a switch in the nature of the support provided rather than a reduction in its amount. For cereals a payment per tonne was converted to a land based area aid using historical yields. For beef there was a move to direct payments on intermediate products which was essentially based on the possession of various categories of animais with a maximal amount of livestock units per forage areas. The main premiums for beef producers are the suckler cow premium and the special premiums for male animais. However, particularly because of the disruption caused by consumer concems over BSE, the EU beef market remains over target and the Commission projections indicated a further accumulation of stocks up to 2005 if the 1992 agricultural policy was maintained. Hence, the CAP reform within the Agenda 2000 consists in a further price decrease by 20 % over 3 years associated with an increase in direct payments. From 200/01 and over 3 years, the

DOCUMENTATION ÉCONOMIE R U RALE RENNES
1111111 11111 111111111111111 IIIII IIII IIII * 0 0 8 9 8 0 * first special beef premium will be increased of 55 % while the increase of the second one is of 38% as for the suckler cow premium increase. This new price reduction is defended as a way of opening up new export outlets without subsidies and rebalancing internai consumption to the benefit of beef. However, the income support mechanism remains partly based on product price and mainly on possession of animais. This contribution focuses on the dynamics of beef suppl y response in the European beef sector. The objective of this paper is to provide a model of beef suppl y response within a dynamic microeconomic framework. The aim of this model is the understanding of cattleman decisions and the assessment of how the behaviour of beef producers is sensitive to changes in production prices and to changes in premiums. In particular, the paper studies how the choice of keeping or slaughtering an animal (the beef supply response) is modified when previous exogenous variables are changed. This is done by considering the cattle herd as a capital good. Changes in the capital stock represent investment (or disinvestment) decisions that are influenced by market prices and compensatory payments. Furthermore, as a cattle become older, its capital value is expected to change, implying that investment decisions in the cattle herd are different for each age category. On this basis, a dynamic mode! of the cattle's herd size and age structure is developed by considering explicitly the influence of market prices on culling and replacement decisions for each age category in the cattle herd. In particular, biological information conceming the dynamics of the cattle population is exploited in the specification of the mode!. It is shown to play a crucial role in the economic adjustments of beef production to changes in relative prices. The biological structure of the cattle and the fact that cattleman makes his production decisions on the basis of expected prices are explicitly introduced. Thus, within this modelling, dynamic relationships are derived from optimising behaviour of producers over time and not simply from ad hoc considerations. The approach taken in this paper to analysis beef supply is as follows. In section 2, the particular features of cattle raising are discussed and some simplifying assumptions are introduced. The dynamic microeconomic mode!, based on the behaviour of individuals whose objective is to maximise expected profits over their entire work-life horizon and allowing both different types of animais on farm and the existence of premiums, is presented. In section 3, this profit fonction is maximised by dynamic programming methods to obtain the relevant behavioural relations which determine the econometric specification. The resulting empirical equations implied by the mode! are described. Section 4 then discusses the farm level panel dataset for France constructed for the period 1995-1997, the estimation methods and reports the estimation results. Section 5 concludes.

The theoretical framework

Basic assurnptions

In this section, we give a brief description of the options open to the cattleman during each period in which optimising decisions must be taken. Sorne specific assumptions depend on mode! specification, data availability and estimation needs. To simplify, we do not consider either the market for feed or the milk sector explicitly. We distinguish among different categories of animals, because they enter as different decision variables in the maximisation problem: heifers, suckling cows, male calves and male cattle. The distinction between these different categories of animais allows us to integrate various subsidies provided by the Common Agricultural Policy (CAP). We assume that the cattleman does not sell female calves. All female calves are grown to heifers, while male calves can be sold or grown to male cattle. Given data availability, it is necessary to assume that, after twelve months of age, calves are considered heifers or male cattle. The capital stock in this model is given by the reproductive herd. They are animais used to produce beef cattle and consist of suckling cows and heifers that have reached the breeding age. A heifer can be used for reproductive purposes only after eighteen months of age. Because of data limitations, however, we assume that heifers can reproduce only after two years of age, and are actually bred at least once when reaching this age. There are no male animais other than male cattle. Bulls are not explicitly considered in the model, since in the beef sector nearly all male calves grow to male cattle. The production of young animais is assumed to be proportional to the total number of animais kept in the stock of capital, or reproductive herd. This assumption implies that the number of calves bom is determined by the choice of the number of cows in the production capital stock. Hence, if Cr is the number of calves barn at period t, and Kr is the reproductive herd at the beginning of period t,

c 1 = FC,+ MC 1 =ÀK 1 for À= ÀF +ÀM <1 (1)
where FC 1 = ÀFK 1 and MC 1 = ÀM K 1 represent females and males calves, respectively.

Assume that there is no calf mortality. The simplification does not change the final results, since the mortality rate is very small and can be neglected in the maximisation of the profit function.

The following constraints summarise the stock-flow relations implied by above assumptions on cattle inventory management. The first constraint describes the evolution of the capital stock. Cows may be kept in the reproductive herd or sold for slaughter. The variation of the actual stock of suckling cows derives from the addition of newly bred heifers 0 1 (gross investment) and from the sale of cows for slaughter VK 1 (disinvestment).

Kt+ I -Kt = Ôt -VK 1 (2)
The second equation depicts the evolution of the stock of heifers Hr. In each period, the initial stock of heifers and the age distribution of this stock are given from the producer's point of view. Hence, the optimal decision about heifers concerns sale of heifers for slaughter, or selection of heifers for the stock of capital (reproductive herd) and breeding heifers. Breeding increases the stock of cows, since once bred, heifers are cows by definition. Thus, the variation of the actual stock of heifers derives from the addition of newly female calves FCr, which have reached the age of twelve months, from the selection ofheifers to reproductive purpose 0 1 , and from the sale of heifers for slaughter VH,.

Hr+1-Ht = FCt -<>t -VHt

(3) Similar decisions must be taken with respect to the stock of male animais. However, the decision concerning male cattle clearly does not affect future production of calves. For male cattle, the cattleman has only two possibilities: sale for slaughter or placement on feed to be sold in the future. Thus, the variation of the actual stock of old male cattle derives from the addition in the stock of male calves 11 1 , which have reached the age of twelve months, and from the sale of some of them for slaughter

VB 1 • Bt+1 -Bt =rit-VBt (4)
The cattleman can sale male calf now for slaughter or it can enter the male cattle herd. Thus, the allocation of the production of male calves (MC,) derives from the selection of young male cattle for the replacement of the male cattle herd 11 1 , and from the sale of some of them for slaughter VMC 1 •

(5)

Modeling representative cattleman behaviour

The cattleman is assumed to maximise profits, not only for this period, but over his entire production horizon. Thus, the producer tries to maximise the expected net present value of his enterprise. First, a profit function is constructed using years as the unit observation period. This profit fonction is quadratic in its arguments, a fact that guarantees a unique maximum and that satisfies the condition for the existence of certainty equivalents. The function is maximised by dynamic programming methods to obtain the relevant behavioural relations, which determine the econometric specification. The cattleman's revenue cornes from selling animais for slaughter, at any point in time, the premiums granted according to the number of animais 1 , while his costs consists of various production and investment costs (maintenance and ageing costs of the cattle stock kept on farm, cost of breeding/producing calves, feeding costs). Assume that the cattleman maximises his profits not only for this period, but for the whole period for which he is in business. Assume further that we know that the cattleman will be in business T periods and then retire. In each period, he must make decisions based on actual facts and on facts that he does not yet know for the future periods.

The features of cattle herd management described above translate into the different choice options for each decision variable. Male are animais that, from the decision-making point of view, can either be sold now or in the future. Heifers can also be sold now, or can be bred and thereby be transformed into cows (added to the capital stock), and in turn used to produce new calves. The reproductive capacity of heifers makes them different, and this difference is expressed in the profit function.

Keeping an animal one more period entails a cost and yields the benefit of being able to sell it in the future. If the cattleman acts rationally, he will decide how many cattle to keep in such a way as to equate marginal expected cost and marginal expected benefit, both discounted to the present period. Given constant unit costs, the function to be maximised by the cattleman may be approximated by the following quadratic expression:

rr(VKt, VHi,Ot, VMCt, VBi, rit I Kt ,Ht, Bi ,t) The terms of 1r may be interpreted as follows:

= Max(PKiVKt + PHiVHt + PMtVMCt + PsiVBt +SKtKt +SMtrit +sBtrit-1 -½q(K 1 -VKi +o 1 ) 2 -½b2(Ki-VKi/ _ _!__!J3(Ct) 2 2 1 2 1 2 -2 b4(Ht -VH 1 -oi + FCi) -2 b5(Ht-VHt-8t) -½b6(Bt -VBi + rit l -
I See the annex I for more details on the premiums granded under the Common Agricultural Policy.

• PKtVK 1 + PHiVHi + PMtVMC 1 + PBtVB 1 is the total revenue from selling vanous cattle categories: suckler cows, heifers, male calves, male cattle.

• sK 1 K 1 ++sMtTJt +S BtTJt-I is the revenue from premiums. The former premium is granted per cows each year while the latter is granted only twice in the male cattle life. In our model, we assume that a male calf which enter in the cattle herd is grown until the age of two years and that the special beef premium is obtained when a male enters in the cattle herd at the age of twelve month and when he reaches the age of two years, i.e., after one year in the cattle herd2.

• _J_ b 1 (K, -VK, + 8 1 )2 is the maintenance cost of the capital stock.

2 • _ J_b 4 (H 1 -VH 1 -8 1 +FC 1 )2 and _ J_b 6 ( B 1 -VB 1 + ri 1 ) 2 is the feeding cost of respectively, 2 2
heifers and male cattle. Feeding costs for heifers are assumed to differ from those for male cattle for the following reasons: first, if heifers are kept in part for possible future breeding, there is no need to feed them as much as male cattle; second, the capacity of heifers for transforming feed into weight is less than the capacity of male cattle.

• _J_b 3 (C 1 )

2 is the production and maintenance cost of calves. an -2b7(B,-VB,) IS te cost O O mg respectively the capital stock , the heifers and the male cattle due to ageing. The ageing costs are also assumed to be different for cows, heifers, and male cattle. It is the additional cost involved in keeping the animal one more period, resulting from the need for more feed and the higher probability of death, etc., as the animal becomes older. The principal element of the aging cost, however, is the loss in value at sale. Animais sold for slaughter are classified by weight, sex, and age. Older animals are generally worth less, ceteris paribus. The productivity and the expected benefits and costs differ according to age.

2 1 2 1 2 d I 2 h f h Id' • -2b2(K1 -VK1)
• ris the one period discount rate,

• E is the expectation operator,

• and rr(VK,+ 1 ,VH,+ 1 ,8,+ 1 ,VMC,+ 1 ,VB,+ 1 ,ri,+ 1 / K 1 + 1 , H,+ 1 , B 1 + 1 , t + I) the profit function for the next period t+ 1.

Empirical implementation

The method used is dynamic programming, which is a recursive maximisation procedure starting from the last period T, i.e., the period after which the cattleman will retire. In the usual fashion, having obtained the solution for this period, we then solve for the next to last period; and so on, as many times as necessary for determination of the general solution until period t, for any t (Howard, 1966, 317-320).

The mode! represented by equations ( 2) and ( 3) is equivalent to maximising

n(VKr, VHr,8 1 , VMC 1 ,VB 1 , Tlt / K 1 , H 1 , Bpt) = 2>T-t[Tr1 -P1,1 (K1 -Kt-1 +VKt-1 -81-1) -P2,1 (H, -Ht-1 +VHt-1 +81-1-FC1-1) -P3,1(B1 -B1-1 +VB1-1-TJ1-1)] (7)
where the Lagrange multipliers P 1 , 1 , P 2 , 1 and P 3 , 1 are the shadow prices of an animal in stock for cows, heifers and male cattle . Expression ( 7) is maximised with respect to VK 1 , VH,, ô, , VMC 1 , VB1 and 77 1 •

The first conditions derives above can be solved for each endogenous variable in terms of past, current and future price levels, and lagged values of the capital stock. Following Nerlove and Fomari (1998), (ECRF), Omitting the expectations operator for simplicity, we obtain the Expectationally Conditional Reduced Form (ECRF) of the dynamic maximisation model that expresses endogenous variables in terms of exogenous and predetermined ones, conditional on expectations of future levels of exogenous variables:

(8) (9) (12) (13) (14) (15) (16)
The only unambiguous effect concems the impact of headage premiums. The sukler cow premiurns should have always a positive effect on the sales and the nurnber of animais. The effect of anticipated prices cannot be predicted in general, while the impact of observed prices is in general indeterminate, depending on the value of the estimated shadow costs.

First econometric analysis

In the following section we describe the econometric estimation techniques, the data used in the econometric analysis, and lastly discuss the results from the estimation.

Price and premium expectation formulation

For each anticipated price, we use the following formulation * P t+l = <XJPt-2 +a2Pr-l +a3pt * where p t+l is the anticipated price that each producer makes in t for the following time period t+I , Pt -i is the current price in t-i ,and a 1 ,a 2 ,a 3 are the coefficients to be estimated.

For each anticipated premium, we use naive expectations since each producer knows the value of premiums for each year:

* s t+I = St
The data for France are drawn from the national Farru Business Survey for the 1995-1997 years. These is annual national survey collected by agencies of the government. The sample of farms are chosen so as to be representative of national agriculture in the country. In general, each survey farm remains in the survey for 5 or 6 years. Hence, a balanced panel of 343 farrns can be constructed for the period.

Results for suckler cows

Equations ( 8) to ( 16) have been estimated by OLS. Although the ECRF expressions determine the optimal suppl y fonctions and gross investment behaviour of the producer, in terms of his anticipations of future price levels and predetermined variables, their evident complexity and the presence of many cross-and within equations restrictions present formidable obstacles to empirical investigation. The large number of leads and lags in the price and stock variables, the appearance of the same parameters in different equations, the large number and non-linear nature of cross-equation restrictions on parameters, create difficulties in direct estimation. Hence, in a first step we estimate the unrestricted model.

For the number of the cow, ail the coefficients in the regression which are statically significant have the predicted sign. We can note the negative effect of the price of heifers showing the substitution between cows and heifers. The suckler cow premium has an important positive effect. For the equation explaining the sales of cows as expected from the model, the price of the cows and the suckler cow premium have a positive and significant effect.

The last table provides a comparison between the effect of a decrease in the current price level and an increase in the current premium for the suckler cows. For each variable, there exists a positive effect of the premium which is larger than the effect of slaughter price.

Concluding remarks

This paper has provided a dynamic optimisation model to describe the behaviour of a representative cattleman maximising his expected profits over an infinite time horizon. It allows us to derive a reduced form which defines each endogenous variables (sales of different animal categories and investment in the herd) by current, past and future exogenous variables as slaughter prices or headage premiums. The model is estimated on a balanced panel data set of 353 farms for the French cattle sector for the period 1995-1997.

Results from the estimation show a positive effect of the premium which is larger than the effect of slaughter price conceming the choices of the number of cows maintained in the herd and the sales of cows during the current period.

The disaggregation by animal categories is helpful in assessing supply response to changes in market conditions or in agricultural policy. This work can be extended in two directions. Firstly, the various equations of the reduced form can be estimated simultaneously. Secondly, it is possible to simulate different reforms of the Common Agricultural Policy.

  PKt for suckling cows, PH1 for heifers, PMt for male calves, p 81 for male cattle and the headage premiums are respectively for the suckler cows sx 1 , and for the male cattle SM 1 , and ss1-

											(6)
			½lry(Bt -VBi)	2
	+ rEi {rr(VKt+ I, VHi+I,ôt+I, VMCt+I, VBt+I, rit+I / Kt+I, Ht+t,Bt+J ,t + 1)})
	subject to the constraints (1) to (5).
	Where	the	subscript	t	refers	to	the	current	period,	and	where
	rr(VK,, VH,, o,, VMC,, VB,, ri, I K ,, H ,, B,,t) is the expected present value of profits. The slaughter
	prices are

Table 1 .

 1 Estimates for the number of cows, the sales of cows and the number of heifers added to the herd The t ratios are reported in brackets under the coefficients.

		Dependent variables
		Number	of Cows sales VK,	Heifers added to the
		cows K 1		herd
	Interce 5.141	-3.518	-4.037
	ot	(2.54)	(1.63)	(-2.11)
	0.238*10:J (-0.10) ~ -~ ~ --0.346* 10• 4 ~ (-1.86) -0.176* 10 4 ~ (-0.09) 0.224*10-4 ~ (-1.86) -0.647*10•~ ~ (-2.06) ~ -x / -0.760*10-3 ~ (44.87) x i 0.87	o.550* 1 o•J (2.55) ---0.137* 10 4 (-0.09) 0.715*10• 4 (0.47) 0.307*10• 3 (1.99) -0.442*10• 3 (-1.75) --0.152*10• 3 (11.16) 0.36	0.543*10• 4 (0.19) 0.594*10-3 (2.06) 0.22 1 *10• 3 (0.81) -0.103*10• 3 (0.66) 0.112* 10-3 (0.71) 0.186*10-;! (1.18) 0.181 *10-4 (0.05) 0.186* 10• 2 (1.18) -o. 750* 10• 4 (0.25) 0.145*10• 3 ( 10.23) 0.34
	DW	1.83	1.89	1.96

Table 2 .

 2 Elasticities of suckler cow variables relative to slaughter price and premium.

	Variables	Slaughter price for cows	Suckler cow premium
	Number of cows	-0,00026	0,959
	Sales of cows	0,313	1,007
	Heifers added to the herd	0,029	0,925
	Elasticities are evaluated for the average point of the sample

ln fact, during the studying period, the special premium was obtained between 10 and 22 months and a fier 23 months.