Investigaciones relativas al objeto de estudio

The formation of an ICA with costly enforcement is modelled as a four-stage cartel game. Let

= {1,2, … , } denote the set of players. Each player ∈ is faced with a membership choice at stage 1. Signatories, those who sign up to the agreement, form a coalition ⊆ . At stage 2, all signatories ∈ cooperatively set the abatement targets, denoted by , for each member in order to maximise joint welfare. Simultaneously, the overall monitoring expenditures denoted by = ∑∈ is chosen jointly and optimally by all members through balancing the expenditures of monitoring with benefits from increased compliance. We assume in our model that the aggregated monitoring expenditures are shared equally

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among coalition members, i.e. = , hence we focus on how to determine the optimal aggregated level of monitoring expenditure . At stage 3, with given abatement targets and the monitoring expenditure, coalition members and singletons choose abatement levels by maximising their individual welfare. The equilibrium solution determines the level of compliance. At the final stage, each signatory’s abatement is randomly monitored with a probability ∈ [0,1] depending on the monitoring expenditure and the size of the coalition

= | |. Thus the inspection probability can be represented as a function of the monitoring

“intensity” such that = ( ) with (0) = 0, > 0 and < 0. If emission reduction by any signatory is found to be less than its abatement commitment, a fine will be imposed.

We apply sub-game perfect equilibrium to solve the game, such that equilibria are obtained by backward induction.

Stage 4: Starting with the analysis of the final stage, notice that at this stage the set of signatories , the monitoring expenditure , and the signatories abatement and abatement targets are given. All signatories are monitored by an enforcement agency with probability ( ). A fine, denoted by , is imposed on the defector if non-compliance is detected. The fine is increasing in the level of defection, i.e. the shortfall of a country’s abatement compared to its target. Let ≡ (0, − ) denote the level of defection. The max operator ensures that overcompliance does not count as “negative defection”. We also introduce a parameter > 0 which reflects the severity of punishment. Here we assume that the fine imposed would reflect the severity of the offence; it could, for example, reflect the (global) damage from emissions. Note that, as Stranlund (2007) points out, different forms of penalty influence the choice of abatement in the way that the abatement depends on the punishment policy parameter f or abatement target or both. In our model we treat as an exogenous parameter that is set in pre-negotiations. Then a general way to write the fine is ( ; ) =

( , ; ). Fines are assumed to be weakly convex in the level of defection; there is no discount on punishment for more severe defections. Given the choices in previous stages, signatories’ expected fine is = ( ) ∙ ( , ; ).

Stage 3: At stage 3, countries choose their abatement levels strategically. Any player’s abatement cannot exceed the Business-as-usual (BAU) emissions level denoted by ̅ , such that ∈ [0, ̅ ]. Let and denote respectively abatement benefit and cost functions where the magnitude of abatement benefits depends on the global abatement denoted by = ∑_∈ .

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We can write = ( ) with ( ) > 0 and ( ) ≤ 0 . The cost is associated with individual abatement ( ) with ( ) > 0 and ( ) > 0 . At this stage, singleton players choose their abatement level to maximise their own payoffs by taking all others’

abatement as given. Under such an abatement denoted by , in equilibrium each non-signatory’s marginal abatement benefits are equal to its marginal costs:

( ) = , ∈ \ . (5.1) Each signatory decides on its abatement level to maximize its own expected payoff given the coalition’s monitoring expenditure and abatement targets decided at stage 2. The problem of signatory ∈ at this stage can be written as:

max ( ) = ( ) − ( ) − − ( ) ( , ). (5.2)

where = ( , … , ), denotes the abatement vector. Notice that an abatement target that is smaller than the Nash abatement level (in the absence of a coalition) denoted by would not be effective. Hence, even before we analyse the second stage we can assume targets ≥ . The enforcement mechanism cannot incentivise abatement levels beyond , hence, we can rule out overcompliance such that signatories’ abatement ∈ [0, ]. Note that we assume that fines, if collected, are distributed as a lump sum to all signatories. This assumption implies that the redistributed fine will not influence signatories’ strategic choice of abatement.

Taking the derivative of problem (5.2) with respect to , the first order condition for an interior solution is obtained as:

( ^∗) − ( ^∗) − ( ) ^{( , )}

∗= 0 ⇔ ( ^∗) − ( ^∗) = ( ) ^{( , )}

∗. (5.3) Eq. (5.3) shows how signatories’ abatement choice is influenced by the enforcement mechanism. Note that ^{( , )}

∗≤ 0, thus the equilibrium abatement of a signatory ∈ is not less than the non-cooperative Nash equilibrium level, where ( ^∗) = ( ^∗). For an interior solution, according to the equilibrium condition (5.3), a signatory chooses the optimal level of abatement by equalising marginal net gains of abatement with the marginal expected fine. Notice that, if the marginal expected fine is zero, no signatory will comply with the target and the non-cooperative abatement level will result. It is also interesting to note that Samuelson’s rule which determines the coalitional best level of abatement can be satisfied if

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− ( ) ^{( , )}

∗= ∑ _{∈ \{ }} . The intuition is as follows: if the marginal expected fine makes up for the externality of each signatory’s abatement, then the coalitional optimum will be achieved.

Since Eq. (5.3) is a condition for an interior solution ^∗≤ , it is also possible that

( ) − ( ) > ( , )

. (5.4)

In this case we have a corner solution, where ^∗= and full compliance is achieved. This happens when the punishment for a small deviation from the target is sufficiently severe.

From Eq. (5.3) we can see that the higher the monitoring probability , the higher is the abatement ^∗ (and the compliance level). This indicates an implicit relationship between equilibrium abatement of signatories and monitoring probability, that is:

∗

> 0, ∈ . (5.5)

We can now write the optimal abatement as the reduced form ^∗= ^∗( , ( ) ). Since the detection probability is a function of with > 0, we can write:

∗ , ( ) = ^∗( , , ), ℎ

∗

> 0, ∈ . (5.6)

Hence, by raising the monitoring expenditure , the inspection probability and the equilibrium abatement ^∗ increase. However, the relationship between ^∗ and is not straightforward, as it depends on the form of the fine function and will be discussed in the next section.

Stage 2: Now we move to the second stage. At this stage coalition members jointly determine abatement targets and monitoring payment . Because monitoring is costly and increasing the target is costless, it is always better to increase the target and to lower monitoring efforts while maintaining the expected fine. Unless there is an upper bound of the target, we cannot obtain a solution. However it is reasonable to assume that the target is bounded by the BAU emissions ̅, i.e. ≤ ̅ . Note that coalitional payoffs are decreasing in monitoring expenditures. Since we assume fines collected from defectors will be paid back to

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signatories, the collected fines are welfare neutral from the perspective of the coalition. The coalition solves the following problem: where is the Lagrangian multiplier for the target. By taking the partial derivatives with respect to , and , we obtain the following first order conditions:

= _∗ It is clear from (5.7d) that if the constraint is non-binding, then = 0 and Samuelson’s rule applies according to (5.7b). However, we prove in the Appendix that the optimal target is always chosen as a corner solution, i.e. ^∗= ̅ . Then the first order conditions (5.7a-d) can be reduced to the following:

In Eq. (5.8) the term ^∗ gives the marginal incentive to abate. Since we assume that overcompliance does not pay, additional monitoring will not increase abatement beyond the target level. Eq. (5.8) indicates that in equilibrium, the coalitional marginal net gains of the

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increased abatement due an increase in monitoring efforts is equal to the marginal costs of monitoring which are unity by assumption.

Eq. (5.9) leads to interesting insights. First, if > 0, then ∑_{∈ ∗}− ∗ and ^∗ must both be positive. This is true because it is never optimal for a coalition to induce abatement higher than the coalitional best level and we can rule out that ∑_{∈ ∗}− ∗< 0. Hence, >

0 implies that the level of abatement that is optimal to enforce falls short of the coalitional best (Samuelson) level of abatement. Furthermore, from Eq. (5.9) the coalitional best abatement can be induced by the optimal target if ∑_{∈ ∗}− ∗= 0. The shadow value indicates the marginal gain if the constraint on the target could be relaxed. Second, note that, from Eq. (5.9) it is also possible that increasing the target is not effective for inducing higher abatement. In that case ^∗= 0. This is the case when the fine function is linear.

From above analysis of Eq. (5.9), it can be concluded that, if the fine function is convex and therefore ^∗> 0, and under the binding target ^∗= ̅ , signatories are incentivised to choose an abatement level that is equal to or lower than the coalitional first best level obtained from the Samuelson’s rule, i.e. ∑∈ _∗≥ ∗. Particularly, the target is set optimally so that signatories could be induced to choose the coalitional best abatement level. As argued before this will be achieved if

− ^{( , )}

∗= ∑ _{∈ \{ }} . (5.10) We can also see from (5.10), that the target cannot play a role in incentivising signatories’

abatement when the marginal fine is constant. Some implications can be gained from equilibrium condition (5.9). Firstly, under the optimal enforcement mechanism, the target and monitoring are used as two instruments to increase signatories’ compliance level. Since setting higher target is costless while increasing the monitoring probability is not, it is always better to choose the maximum target level. Secondly, setting higher targets can increase the abatement, but it is not optimal for a coalition to abate more than the level obtained from the Samuelson condition. Under the Samuelson condition each coalition member makes the abatement choice that internalises the externality imposed on all other signatories.

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Stage 1: At the initial stage all countries make decisions on the membership by evaluating their payoffs of being a signatory or a singleton. Countries evaluate payoffs depending on the anticipated strategic decisions on the optimal enforcement policy and the abatement. We assume the sub-game perfect equilibrium in our model is unique under the optimal enforcement policy. Therefore, for each coalition structure we introduce a valuation function

( ) to represent each player’s payoff under a coalition . By applying the solution concept of cartel stability (d'Aspremont et al., 1983) to represent the Nash equilibrium at this stage, a stable coalition is defined as:

(a) internal stability: ( ) ≥ ( \{ }), ∀ ∈ (5.11) (b) external stability: ( ) ≥ ( ∪ { }). ∀ ∈ \ (5.12)