Sustainable International Cooperation with Ancillary Benefits of Climate Policy

Abstract

This study aims to increase the long-term feasibility of international environmental agreements (IEAs) between asymmetric countries via a repeated game model by considering the effect of the ancillary benefits of climate policy. Generally, climate change mitigation generates not only primary public benefits, but also ancillary benefits. This study supposes that all countries have two-sided asymmetry: the public and ancillary benefits and cost parameters, which are high and low, respectively. The IEA model with a repeated game considers that a strategy dictates participating countries’ actions. Consequently, ancillary benefits affect the conditions under which participants cooperate in line with the strategy. Moreover, we find the minimum number of participating countries that needs to be satisfied before the agreement starts by considering the method for the selection of the countries that punish a deviator from the agreement between two types of countries so that our strategy is always effective. Additionally, the findings show that there is a possibility that ancillary benefits relax the condition of minimum participation. The results suggest that participating countries should recognize the effect of ancillary benefits when they negotiate on climate change mitigation.

This research did not receive any specific grants from public, commercial, or not-for-profit funding agencies.

Appendices

Appendix 1: Proof of Lemma 1

In a similar manner to Takashima (2018), we obtain the conditions for subgame perfection when the ancillary benefits are considered. We consider deviation by type i (=1, 2) countries.

First, let us consider a deviation by a type 1 country.

• (A) The incentive constraint for each country to play cooperate when no deviation occurs in any period. A participating country, j, receives b ₁(n ₁ + n ₂) + α ₁b ₁ − c ₁ in each period if no deviation occurs in the previous period. If country j deviates in period t and reverts to the strategy in period t + 1, it receives b ₁(n ₁ − 1) + b ₁n ₂ in period t and b ₁(n ₁ − θm) + b ₁(n ₂ − (1 − θ)m) + α ₁b ₁ − c ₁ in period t + 1. Thereafter, each country receives b ₁n ₁ + b ₁n ₂ + α ₁b ₁ − c ₁ from period t + 2 onward. Each country plays cooperate if

  $$ \begin{aligned}&\left(1+\delta \right)\left({b}_1\left({n}_1+{n}_2\right)+{\alpha}_1{b}_1-{c}_1\right)\ge {b}_1\left({n}_1-1\right)+{b}_1{n}_2\\&\quad +\,\delta \left({b}_1\left({n}_1-\theta m\right)+{b}_1\left({n}_2-\left(1-\theta \right)m\right)+{\alpha}_1{b}_1-{c}_1\right).\end{aligned} $$

  (1)

  By rearranging inequality (1), we obtain

  $$ m\ge \left({c}_1-{\alpha}_1{b}_1-{b}_1\right)/\delta {b}_1. $$

  Given that γ ₁ = c ₁/b ₁, we have

  $$ m\ge \left({\gamma}_1-{\alpha}_1-1\right)/\delta . $$

  (2)

  Similarly, if a type 2 country deviates, we obtain the lower bound for the number of punishing countries needed to maintain cooperation:

  $$ m\ge \left({\gamma}_2-{\alpha}_2-1\right)/\delta . $$

  (3)

  Inequalities (2) and (3) represent those conditions under which each signatory plays cooperate in every period, provided that the other signatories also cooperate. If the number of punishing countries m is less than the right-hand side of inequalities (2) and (3), a deviation occurs in period t to increase that country’s payoff.

• (B) The incentive constraint for n ₁ + n ₂ − m countries to play cooperate after a unilateral deviation in period t − 1. First, we consider deviation by a type 1 country. If countries play cooperate in period t, they first receive b ₁(n ₁ − θm) + b ₁(n ₂ − (1 − θ)m) + α ₁b ₁ − c ₁ and then b ₁(n ₁ + n ₂) + α ₁b ₁ − c ₁ from period t + 1 onward. If one country deviates in period t but cooperates in period t + 1, that country first receives b ₁(n ₁ − θm − 1) + b ₁(n ₂ − (1 − θ)m) and then b ₁(n ₁ − θm) + b ₁(n ₂ − (1 − θ)m) + α ₁b ₁ − c ₁ in period t + 1 as a result of punishment by m countries. Thereafter, each country receives b ₁(n ₁ + n ₂) + α ₁b ₁ − c ₁ from period t + 2 onward. Therefore, type 1 countries play cooperate after a unilateral deviation if

  $$ {\fontsize{9}{11}\selectfont{\begin{aligned}&{b}_1\left({n}_1-\theta m\right)+{b}_1\left({n}_2-\left(1-\theta \right)m\right)+{\alpha}_1{b}_1-{c}_1+\delta \left({b}_1\left({n}_1+{n}_2\right)+{\alpha}_1{b}_1-{c}_1\right)\\&\quad \ge {b}_1\left({n}_1-\theta m-1\right)+{b}_1\left({n}_2-\left(1-\theta \right)m\right)+\delta \left({b}_1\left({n}_1-\theta m\right)\right.\\&\qquad\qquad\qquad\qquad\,\,\quad\left.+\ {b}_1\left({n}_2-\left(1-\theta \right)m\right)+{\alpha}_1{b}_1-{c}_1\right).\vspace*{-12pt}\end{aligned}}} $$

  (4)

  By rearranging inequality (4), we obtain

  $$ m\ge \left({c}_1-{\alpha}_1{b}_1-{b}_1\right)/\delta {b}_1. $$

  Given that γ ₁ = c ₁/b ₁, we have

  $$ m\ge \left({\gamma}_1-{\alpha}_1-1\right)/\delta . $$

  (5)

  Condition (5) is obtained when a type 2 country deviates in period t.

  Second, we consider deviation by a type 2 country after a type 1 country’s unilateral deviation. Type 2 countries play cooperate after a type 1 country’s deviation if

  $$ {\fontsize{9}{11}\selectfont{\begin{aligned}&{b}_2\left({n}_1-\theta m\right)+{b}_2\left({n}_2-\left(1-\theta \right)m\right)+{\alpha}_2{b}_2-{c}_2+\delta \left({b}_2\left({n}_1+{n}_2\right)+{\alpha}_2{b}_2-{c}_2\right)\\&\quad \ge {b}_2\left({n}_1-\theta m\right)+{b}_2\left({n}_2-\left(1-\theta \right)m-1\right)+ \delta \left({b}_2\left({n}_1-\theta m\right)\right.\\&\qquad\qquad\qquad\quad\,\,\ \left.+\ {b}_2\left({n}_2-\left(1-\theta \right)m\right)+{\alpha}_2{b}_2-{c}_2\right).\kern0.5em \end{aligned}}}$$

  (6)

  By rearranging inequality (6), we obtain

  $$ m\ge \left({c}_2-{\alpha}_2{b}_2-{b}_2\right)/\delta {b}_2. \vspace*{-3pt}$$

  Given that γ ₂ = c ₂/b ₂, we have

  $$ m\ge \left({\gamma}_2-{\alpha}_2-1\right)/\delta . \vspace*{-3pt}$$

  (7)

  Inequalities (5) and (7) represent those conditions under which each signatory plays cooperate in every period, provided that the other signatories also play cooperate.

  If the number of punishing countries is less than or equal to the right-hand side of inequality (5) and (7), a deviator in period t − 1 increases its payoff by deviating in period t. That is, the deviator in period t − 1 deviates again in the next period.

• (C) The incentive constraint for m punishing countries to punish a deviation. First, we consider the payoff for a punishing country that fails to punish—that is, when it plays cooperate in period t after a deviation in t − 1. As the country defecting in period t will be punished in period t + 1, this defection leads to a loss in period t + 1.

  Type 1 punishing countries implement the punishment if

  $$ {\fontsize{8.2}{10.5}\selectfont{\begin{array}{l}{b}_1\left({n}_1-\theta m\right)+{b}_1\left({n}_2-\left(1-\theta \right)m\right)\ge {b}_1\left({n}_1-\theta m+1\right)+{b}_1\left({n}_2-\left(1-\theta \right)m\right)+{\alpha}_1{b}_1-{c}_1,\end{array}}} $$

  or

  $$ {c}_1\ge {b}_1+{\alpha}_1{b}_1. \vspace*{-3pt}$$

  Type 2 punishing countries implement the punishment if

  $$ {\fontsize{8.2}{10.5}\selectfont{\begin{array}{l}{b}_2\left({n}_1-\theta m\right)+{b}_2\left({n}_2-\left(1-\theta \right)m\right)\ge {b}_2\left({n}_1-\theta m\right)+{b}_2\left({n}_2-\left(1-\theta \right)m+1\right)+{\alpha}_2{b}_2-{c}_2,\end{array}}} $$

  or

  $$ {c}_2\ge {b}_2+{\alpha}_2{b}_2.\vspace*{-3pt} $$

  From the assumption that c _i > b _i + α _ib _i (i = 1, 2), which denotes a solo cooperation is not profitable, the above inequalities always hold.

  We know that the definition of subgame perfection requires both types of countries not to deviate. Therefore, from inequalities (2), (3), (5), and (7), the condition for subgame perfection is

  $$ m\ge \left({\gamma}_1-{\alpha}_1-1\right)/\delta\ \mathrm{and}\ m\ge \left({\gamma}_2-{\alpha}_2-1\right)/\delta . $$

  This condition can be rewritten as

  $$ m\ge \left(\max \left\{{\gamma}_1-{\alpha}_1,{\gamma}_2-{\alpha}_2\right\}-1\right)/\delta . $$

Appendix 2: Proof of Lemma 2

Because of the assumption that a potential renegotiation can occur when the entire group consists of type 1 and type 2 punishing countries and when sub-groups consist solely of type 1 or type 2 punishing countries, the following potential renegotiations under these three scenarios must be considered.

We consider the type i punishing countries’ incentive for renegotiation after deviation by a type i (i = 1, 2) country.

• (A) Consider the case where type 1 and type 2 punishing countries (θm and (1 − θ)m) renegotiate in period t. Type 1 punishing countries receive b ₁(n ₁ − θm) + b ₁(n ₂ − (1 − θ)m) if they adopt the strategy, and b ₁(n ₁ + n ₂) + α ₁b ₁ − c ₁ if they do not punish by renegotiation. They receive b ₁n ₁ + b ₁n ₂ + α ₁b ₁ − c ₁ in each period irrespective of their action from period t + 1 onward. Therefore, renegotiation is deterred if

  $$ {b}_1\left({n}_1-\theta m\right)+{b}_1\left({n}_2-\left(1-\theta \right)m\right)\ge {b}_1\left({n}_1+{n}_2\right)+{\alpha}_1{b}_1-{c}_1. $$

  Assuming that γ ₁ = c ₁/b ₁, we have

  $$ m\le {\gamma}_1-{\alpha}_1. $$

  (8)

  For type 2 countries, renegotiation is avoided if

  $$ {b}_2\left({n}_1-\theta m\right)+{b}_2\left({n}_2-\left(1-\theta \right)m\right)\ge {b}_2\left({n}_1+{n}_2\right)+{\alpha}_2{b}_2-{c}_2. $$

  Assuming that γ ₂ = c ₂/b ₂, we have

  $$ m\le {\gamma}_2-{\alpha}_2. $$

  (9)

• (B) Consider that only type 1 punishing countries (θm) renegotiate in period t. Type 1 punishing countries will not renegotiate if

  $$ {\fontsize{9.5}{11.5}\selectfont{\begin{aligned}{b}_1\left({n}_1-\theta m\right)+{b}_1\left({n}_2-\left(1-\theta \right)m\right)\ge {b}_1{n}_1+{b}_1\left({n}_2-\left(1-\theta \right)m\right)+{\alpha}_1{b}_1-{c}_1.\end{aligned}}} $$

  Assuming that γ ₁ = c ₁/b ₁, we have:

  $$ m\le \left({\gamma}_1-{\alpha}_1\right)/\theta . $$

  (10)

• (C) Consider that only type 2 punishing countries ((1 − θ)m) renegotiate in period t. Type 2 punishing countries will not renegotiate if

  $$ {b}_2\left({n}_1-\theta m\right)+{b}_2\left({n}_2-\left(1-\theta \right)m\right)\ge {b}_2\left({n}_1-\theta m\right)+{b}_2{n}_2+{\alpha}_2{b}_2-{c}_2, $$

  or

  $$ m\le \left({\gamma}_2-{\alpha}_2\right)/\left(1-\theta \right). $$

  (11)

  To prevent a punishing country’s renegotiation, (8), (9), (10), and (11) are necessary and sufficient. These four conditions are summarized as

  $$ m\le \min \left\{{\gamma}_1-{\alpha}_1,{\gamma}_2-{\alpha}_2,\left({\gamma}_1-{\alpha}_1\right)/\theta, \left({\gamma}_2-{\alpha}_2\right)/\left(1-\theta \right)\right\}. $$

  This is equivalent with

  $$ m\le \min \left\{{\gamma}_1-{\alpha}_1,{\gamma}_2-{\alpha}_2\right\}, $$

  since θ ∈ [0, 1].

Appendix 3: Proof of Proposition 2

We show the requirements for the minimum number of participants required for agreements to be sustained as a WRP equilibrium using the Flexible Penance strategy.

• (a) Case γ ₂ − α ₂ < γ ₁ − α ₁

  Let m ^∗ be an integer where γ ₁ − α ₁ − 1 < m ^∗ ≤ γ ₁ − α ₁. If m ^∗ ≤ (γ ₂ − α ₂)/(1 − θ), renegotiation by type 2 punishing countries is always prevented. Rearranging this equation, we obtain the lowest number of type 1 punishing countries θm ^∗ required to deter renegotiations by type 2 punishing countries:

  $$ \theta {m}^{\ast}\ge {m}^{\ast }-\left({\gamma}_2-{\alpha}_2\right). $$

  (12)

  Condition (12) represents the lowest number of type 1 punishing countries required to deter renegotiation by type 2 punishing countries. Therefore, the number of type 1 participants must be larger than θm ^∗. The minimum number of type 1 participants, n ₁, can be written as:

  $$ {n}_1>{m}^{\ast }-\left({\gamma}_2-{\alpha}_2\right). $$

  (13)

  On the assumption that n ₁ + n ₂ > m, the total number of participants must be greater than m ^∗. Therefore, (13) and n ₂ > m ^∗ − n ₁ need to be satisfied.

• (b) Case γ ₁ − α ₁ < γ ₂ − α ₂

  Let m ^∗∗ be an integer where γ ₂ − α ₂ − 1 < m ^∗∗ ≤ γ ₂ − α ₂. If m ^∗∗ ≤ (γ ₁ − α ₁)/θ, then only deviation of type 1 punishing countries is prevented. Rearranging this, we obtain the lowest number of type 1 punishing countries (1 − θ)m ^∗∗ required to deter their renegotiation:

  $$ \left(1-\theta \right){m}^{\ast \ast}\ge {m}^{\ast \ast }-\left({\gamma}_1-{\alpha}_1\right). $$

  (14)

  Similar to case (a), (14) allows us to obtain the lowest number of type 2 participants n ₂:

  $$ {n}_2>{m}^{\ast \ast }-\left({\gamma}_1-{\alpha}_1\right). $$

  (15)

  On the assumption that n ₁ + n ₂ > m, the total number of participants must be greater than m ^∗∗. Therefore, (15) and n ₁ > m ^∗∗ − n ₂ need to be satisfied.