International environmental agreements with external transfers

In this section, we turn our attention to external transfers which may be promising to encourage participations in IEAs. This transfer scheme requires free-riders, namely supporters, that do not abate by themselves to provide financial transfers to countries who later enter into the IEAs. The concept of external transfers originates from Carraro

and Siniscalco (1993) and is recently revisited by Ansink et al. (2019). Our study further applies it in the context of asymmetric countries.

To implement external transfers, climate agency, which one can refer to an international body (for example, Intergovernmental Panel on Climate Change) that is unable to give orders to sovereign states, should coordinate the negotiation process.

Such climate agency has three key roles: (1) to generate a support contract that countries decide to sign or not at the first stage of the game; (2) to propose transfer payment that supporters can accept or reject at the second stage; (3) to provide a conventional IEA contract that remaining countries can sign or not at the third stage. By this way, climate agency ensures the execution of external transfers and thus leads to the targeted IEA.

In this study, we assume that developed countries move first to choose whether or not to become supporters, which is more reasonable considering their great incentive to induce others’ abatements under strong asymmetry. This incentive comes from developed countries’ higher abatement benefits, higher abatement costs and abundant capital stock. On one hand, with the strict requirements for environmental quality, developed countries tend to take actions to reduce emissions, since they benefit much from global abatement. However, high costs drive them to buy others’ cheaper abatements rather than reduce on their own. On the other hand, based on equality and budget constraint, developed countries are capable and willing to financially transfer to developing ones. We can hardly expect a low-income country struggling to provide financial assistance to other rich countries.

The formation of IEAs with external transfers is modelled as the following four-stage game.

Stage 1: Developed countries independently and simultaneously choose to become supporters or not. Let S denote the set of supporters and 𝑠 ≡ |𝑆| the number of supporters.

Stage 2: The set of supporters 𝑆 decides whether or not to make the transfer payment proposed by the climate agency, to member countries. Note that supporters can reject to pay transfers if they find it to be unbeneficial.

Stage 3: The remaining countries choose independently to become a member or not. This leads to a partition of the set of countries into three subsets: supporters, members, and free-riders¹¹.

Stage 4: Members choose abatements to maximize the coalition payoff, while all other countries act as singletons to maximize individual payoffs.

Again, we solve the game backwards to derive SPNE. The solution for stage 4 is the abatement level of individual country, which can be derived as the same logic shown in Section 2.2.

In stage 3, every country except for supporters decides whether to become an IEA member. The solution for this stage is the number of developing and developed countries that are IEA members, denoted as (𝑚^∗, 𝑚^∗), that satisfies internal as well as external stabilities. Let 𝑇 (𝑠) denote the transfer to type i members which depends on the number of supporters and it seems intuitively reasonable to allocate equal transfers to member countries of the same type.

11 Changing the sequence of the above four-stage game so that an IEA without transfers has formed and then supporters decide whether or not to accept transfer proposal does not affect the final equilibrium.

Taking transfers into account, internal stability can be expressed as

𝜋 (𝑚^∗, 𝑚^∗) + ^{( )}_∗ ≥ 𝜋 (𝑚^∗ − 1, 𝑚^∗), (2.12) 𝜋 (𝑚^∗, 𝑚^∗) +𝑇 (𝑠)

𝑚^∗ ≥ 𝜋 (𝑚^∗, 𝑚^∗ − 1). (2.13) Correspondingly, external stability becomes

𝜋 (𝑚^∗, 𝑚^∗) > 𝜋 (𝑚^∗+ 1, 𝑚^∗) + 𝑇 (𝑠)

𝑚^∗ + 1, (2.14)

𝜋 (𝑚^∗, 𝑚^∗) > 𝜋 (𝑚^∗, 𝑚^∗+ 1) + 𝑇 (𝑠)

𝑚^∗ + 1 . (2.15) Obviously, using the following (2.16) and (2.17), it is easy to verify the minimum transfer 𝜏 that internally stabilizes a coalition (𝑚 , 𝑚 ). That is,

Turning to stage 2, the set of supporters S decides whether or not to pay the transfer 𝑇 (𝑠) as proposed by the climate agency to members. Recall that since transfers are not compulsory, supporters have rights to reject the proposal and if they reject, solution to this stage is 𝑇^∗(𝑠) = 0. If they accept, the transfer is 𝑇^∗(𝑠) = 𝑇 (𝑠) and consequently we invoke (2.16) to (2.19) to derive the self-enforcing coalitions. In what follows, we refer to 𝑚^∗ as a function of transfers: 𝑚^∗ 𝑇 (𝑠), 𝑇 (𝑠) .

Since supporters are identical, a natural assumption is that total transfers are paid by them evenly. So 𝑇^∗(𝑠) = 𝑇 (𝑠) if and only if

𝑠 ∙ 𝜋 𝑚^∗ 𝑇 (𝑠), 𝑇 (𝑠) , 𝑚^∗ 𝑇 (𝑠), 𝑇 (𝑠) − [𝑇 (𝑠) + 𝑇 (𝑠)]

≥ 𝑠 ∙ 𝜋 𝑚^∗(0,0), 𝑚^∗(0,0) .

(2.18) 𝜏 (𝑚 , 𝑚 ) = 𝑚 𝜋 (𝑚 − 1, 𝑚 ) − 𝜋 (𝑚 , 𝑚 ) , (2.16) 𝜏 (𝑚 , 𝑚 ) = 𝑚 𝜋 (𝑚 , 𝑚 − 1) − 𝜋 (𝑚 , 𝑚 ) . (2.17)

The left-hand side of inequality (2.18) is the payoff to the set of supporters when they accept the transfer proposal from climate agency and the right-hand side represents that when they refuse and the equilibrium of the no-transfer game is realized. In other words, if condition (2.18) holds, the best choice for supporters is to pay the transfers as proposed by the climate agency.

In stage 1, developed countries must decide whether to become supporters or not.

In the Nash equilibrium for this stage, no supporters withdraw from the supporter team and no other countries (free-riders and IEA members) deviate and become supporters.

The outcome of this stage is 𝑠^∗, which solves the following conditions for the set of supporters.

Internal stability

If a supporter country would deviate to become a free-rider, its payoff with 𝑠^∗− 1 supporters should be no higher than its supporter payoff with 𝑠^∗ supporters,

𝜋 𝑚^∗ 𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗) , 𝑚^∗ 𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗) −𝑇^∗(𝑠^∗) + 𝑇^∗(𝑠^∗) 𝑠^∗

≥ 𝜋 𝑚^∗ 𝑇^∗(𝑠^∗− 1), 𝑇^∗(𝑠^∗− 1) , 𝑚^∗ 𝑇^∗(𝑠^∗− 1), 𝑇^∗(𝑠^∗− 1) . (2.19)

If a supporter country would deviate to become an IEA member, its payoff with 𝑠^∗− 1 supporters should be no higher than its supporter payoff with 𝑠^∗ supporters

𝜋 𝑚^∗ 𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗) , 𝑚^∗ 𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗) −𝑇^∗(𝑠^∗) + 𝑇^∗(𝑠^∗) 𝑠^∗

≥ 𝜋 𝑚^∗ 𝑇^∗(𝑠^∗− 1), 𝑇^∗(𝑠^∗− 1) , 𝑚^∗ 𝑇^∗(𝑠^∗− 1), 𝑇^∗(𝑠^∗− 1)

+ 𝑇^∗(𝑠^∗− 1)

𝑚^∗ 𝑇^∗(𝑠^∗− 1), 𝑇^∗(𝑠^∗− 1) . (2.20)

External stability

For a non-supporting country that tends to become free-rider in the subgame, its supporter payoff with 𝑠^∗+ 1 supporters is lower than its free-rider payoff with 𝑠^∗ supporters,

𝜋 𝑚^∗ 𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗) , 𝑚^∗ 𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗)

> 𝜋 𝑚^∗ 𝑇^∗(𝑠^∗+ 1), 𝑇^∗(𝑠^∗+ 1) , 𝑚^∗ 𝑇^∗(𝑠^∗+ 1), 𝑇^∗(𝑠^∗+ 1)

−𝑇^∗(𝑠^∗+ 1) + 𝑇^∗(𝑠^∗+ 1)

𝑠^∗+ 1 . (2.21)

For a non-supporting country that tends to become an IEA member in the subgame, its supporter payoff with 𝑠^∗+ 1 supporters is lower than its IEA member payoff with 𝑠^∗ supporters.

𝜋 𝑚^∗ 𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗) , 𝑚^∗ 𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗) +𝑇^∗(𝑠^∗) 𝑚^∗

> 𝜋 𝑚^∗ 𝑇^∗(𝑠^∗+ 1), 𝑇^∗(𝑠^∗+ 1) , 𝑚^∗ 𝑇^∗(𝑠^∗+ 1), 𝑇^∗(𝑠^∗+ 1) −𝑇^∗(𝑠^∗+ 1) + 𝑇^∗(𝑠^∗+ 1)

𝑠^∗+ 1 . (2.22)

Now, we prove the existence of SPNE with supporters, which can be found as long as 𝑛 is large enough. As mentioned before, there are lots of SPNEs but we focus on the result without free-riders. A later description will suggest this seemingly harsh result to be easily achievable under strong asymmetry.

Proposition 2.3. For an integer 𝑠^∗, if it holds that

[𝑛 + (𝑛 − 𝑠^∗)𝛾](𝑛 𝛽 + 𝑛 − 𝑠^∗) − [4𝑛 − 3𝑛 + 2𝑛 (𝑛 − 𝑠^∗)𝛾]𝛽 − 2𝑛 (𝑛 − 𝑠^∗)(1 + 𝛾)𝛽 − (𝑛 − 𝑠^∗)[4𝛾(𝑛 − 𝑠^∗) − 3𝛾 + 2𝑛 ] 2[𝑛 + 𝛾(𝑛 − 𝑠^∗)](𝑛 𝛽 + 𝑛 − 𝑠^∗) − 2𝑛 𝛽 − 2𝛾(𝑛 + 6 − 𝑠^∗)

≤ 𝑠^∗≤ 𝑛 − 4 (2.23) then there exists a SPNE with 𝑠 = 𝑠^∗ ,

𝑇^∗(𝑠) = 𝜏 (𝑛 , 𝑛 − 𝑠) 𝑖𝑓 𝑠 = 𝑠^∗ 0 𝑖𝑓 𝑠 ≠ 𝑠^∗, 𝑇^∗(𝑠) = 𝜏 (𝑛 , 𝑛 − 𝑠) 𝑖𝑓 𝑠 = 𝑠^∗

0 𝑖𝑓 𝑠 ≠ 𝑠^∗,

(2.24)

and 𝑚^∗(𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗)), 𝑚^∗(𝑇^∗(𝑠^∗), 𝑇^∗(𝑠^∗)) = (𝑛 , 𝑛 − 𝑠^∗).

Proof: See Appendix A.3. ∎

Proposition 2.3 illustrates that, if the number of supporters 𝑠^∗ satisfies (2.23), and if the supporters pay transfers as descried in eq. (2.24), a self-enforcing agreement including all developing countries and developed countries except for those supporters can be induced. Given eq. (2.24), transfers will be paid only when desired number of developed countries, which is proposed by the international climate agency, choose to become supporters. We use such a transfer scheme to discourage deviation in the first stage. Once any deviation happens, it will lead to the no-transfer equilibrium and gives

each country its payoff when no transfers is allowed. Other transfer schemes, for example, replacing 0 with other values when 𝑠 ≠ 𝑠^∗, may also achieve the goal.

We proceed to analyze the existence of supporters and the effectiveness of external transfers when strong asymmetry exists. This leads to our next proposition.

Proposition 2.4. With the option of external transfers, a self-enforcing coalition includes all developing countries and at least four developed countries if asymmetry is strong, as long as the number of developed countries exceeds eight.

Proof. See Appendix A.4. ∎

Other coalitions including all developing countries and more than four developed countries may also be self-enforcing, based on the parameter set. However, if the asymmetry is not strong, a more demanding condition for 𝑛 is needed for the existence of 𝑠^∗.

Obviously, paying external transfers will become easier if the number of supporters is large since the burden of each supporter is lower. Henceforth, it is often possible to support a large stable IEA if there is a sufficient number of countries that are able to act as supporters. This is also illustrated in Proposition 2.4.