International Environmental Agreements: The Case of Costly Monetary Transfers

Abstract

Most existing international environmental agreements to resolve transboundary pollution problems appear constrained in the sense that either monetary transfers accompany uniform abatement standards (agreements based on a uniform standard with monetary transfers), or differentiated abatement standards are established, but without monetary transfers (agreements based on differentiated standards). For two asymmetric countries facing the challenge of a transboundary pollution problem, we compare the relative efficiency of these two second-best agreements. We study especially the role of the costs associated with transfer payments across countries in the choice of these agreements. To conduct this analysis, we use a negotiation game and the generalized Nash bargaining solution (Nash in Econometrica 21:128–140, 1953) as the equilibrium. For total welfare, our findings show that countries collectively prefer the uniform to the differentiated agreement if the cost of transfers is sufficiently low compared to the ratio for countries of the difference of the abatement costs between the two agreements. In the analysis of individual welfare, we also discuss the reluctance of one country to sign a specific type of agreement even if it is better off than in the case of non-cooperation.

Appendices

Appendix 1: Nash Equilibrium

1.1 Proof of Lemma 1

The objective of country 1 is to maximize its utility function with respect to its budget constraint, taking as given the level of abatement of country $2, a_{2}: \underset{a_{1}}{Max}[ B(a_{1}+a_{2})+\Omega _{1}-C(a_{1})]$.

The first-order condition of this program is the following: $B^{^{\prime }}(A)=C^{^{\prime }}(a_{1})$. Similarly, the first-order condition of the program for country 2 is the following: $\alpha B^{^{\prime }}(A)=\delta C^{^{\prime }}(a_{2})$.

1.2 Uniqueness of the Nash Equilibrium

The equilibrium is defined by:

$$\begin{aligned} B^{\prime }(a_{1}+a_{2})&= C^{\prime }(a_{1}) \quad \text {(1)} \\ \alpha B^{\prime }(a_{1}+a_{2})&= \delta C^{\prime }(a_{2}) \quad \text {(2)} \end{aligned}$$

Suppose there is another equilibrium:

$$\begin{aligned} B^{\prime }(\overline{a}_{1}+\overline{a}_{2})&= C^{\prime }(\overline{a} _{1}) \quad \text {(3)} \\ \alpha B^{\prime }(\overline{a}_{1}+\overline{a}_{2})&= \delta C^{\prime }( \overline{a}_{2})\quad \text {(4)} \end{aligned}$$

Assume that $a_{1}>\overline{a}_{1}$, so $B^{\prime }(a_{1}+a_{2})>B^{\prime }(\overline{a}_{1}+\overline{a}_{2})$ because of (1) and (3), and the convexity of $C(.)$. Consequently, $a_{2}>\overline{a}_{2}$ because of (2) and (4), and the convexity of $C(.)$. So $a_{1}+a_{2}>\overline{a}_{1}+ \overline{a}_{2}$, then $B^{\prime }(a_{1}+a_{2})<B^{\prime }(\overline{a} _{1}+\overline{a}_{2})$ because of the concavity of $B(.)$, which is contradictory to $a_{1}>\overline{a}_{1}$ and $B^{\prime }(a_{1}+a_{2})>B^{\prime }(\overline{a}_{1}+\overline{a}_{2})$.

Assume that $a_{1}<\overline{a}_{1}$, so $B^{\prime }(a_{1}+a_{2})<B^{\prime }(\overline{a}_{1}+\overline{a}_{2})$ because of (1) and (3), and the convexity of $C(.)$. Consequently, $a_{2}<\overline{a}_{2}$ because of (2) and (4), and the convexity of $C(.)$. So $a_{1}+a_{2}<\overline{a}_{1}+ \overline{a}_{2}$, then $B^{\prime }(a_{1}+a_{2})>B^{\prime }(\overline{a} _{1}+\overline{a}_{2})$ because of the concavity of $B(.)$, which is contradictory to $a_{1}<\overline{a}_{1}$ and $B^{\prime }(a_{1}+a_{2})<B^{\prime }(\overline{a}_{1}+\overline{a}_{2})$.

This proof shows that the Nash equilibrium is unique under the standard assumptions of a concave benefit function and a convex cost function.

Appendix 2: Existence and Uniqueness of the Cooperative Equilibria

1.1 Proof of Proposition 1a

The Pareto optimality results from the following program

$$\begin{aligned} \max _{\overline{a,}t}\text { }NB_{1}&=\max _{\overline{a,}t}\left[ B(2 \overline{a})+c_{1}\right] \\&s.t. \qquad \left\{ \begin{array}{l} \Omega _{1}=c_{1}+C(\overline{a})+(1+\lambda )t \\ \Omega _{2}=c_{2}+\delta C(\overline{a})-t \\ NB_{2}=\alpha B(2\overline{a})+c_{2}\ge \overset{\overline{-}}{NB_{2}} \end{array} \right. \end{aligned}$$

where $\overset{\overline{-}}{NB_{2}}$ is exogenous.

If we use the first two constraints, the program becomes

$$\begin{aligned} \max _{\overline{a,}t}\text { }NB_{1}&=\max _{\overline{a,}t}\left[ B(2 \overline{a})+\Omega _{1}-C(\overline{a})-(1+\lambda )t\right] \\&\quad s.t. \qquad NB_{2}=\alpha B(2\overline{a})+\Omega _{2}-\delta C( \overline{a})+t\ge \overset{\overline{-}}{NB_{2}} \end{aligned}$$

The Lagrangian of this maximization problem is: $L=[B(2\overline{a})+\Omega _{1}-C(\overline{a})-(1+\lambda )t]+\rho [\alpha B(2\overline{a} )+\Omega _{2}-\delta C(\overline{a})+t-\overset{\overline{-}}{NB_{2}}]$, where $\rho$ is the multiplier associated with the constraint.

The first-order conditions (FOCs) with respect to $\overline{a}$ and $t$ give:

$$\begin{aligned} 2(\alpha (1+\lambda )+1)B^{\prime }(2\overline{a})=(1+\delta (1+\lambda ))C^{\prime }(\overline{a}) \end{aligned}$$

(11)

The total differential of $NB_{1}$ is equal to: $d(NB_{1})=[2B^{\prime }(2 \overline{a})-C^{\prime }(\overline{a})]d\overline{a}-(1+\lambda )dt$, and that of $NB_{2}$ is equal to: $d(NB_{2})=[2\alpha B^{\prime }(2\overline{a} )-\delta C^{\prime }(\overline{a})]d\overline{a}+dt$.

On the Pareto frontier, we have

$$\begin{aligned} d(NB_{1})&= \frac{C^{\prime }(\overline{a})(1+\lambda )(\delta -\alpha )}{ (\alpha (1+\lambda )+1)}d\overline{a}-(1+\lambda )dt \\ d(NB_{2})&= \frac{C^{\prime }(\overline{a})(\alpha -\delta )}{(\alpha (1+\lambda )+1)}d\overline{a}+dt \end{aligned}$$

So along the Pareto frontier $\dfrac{dNB_{1}}{dNB_{2}}=-(1+\lambda )$. Hence, the Pareto frontier is a decreasing line.

1.2 Proof of Proposition 1b

We consider two points on the Pareto frontier. Consider the case where $\delta C(\overline{a})= \Omega _{2}$ when the private consumption of country 2 and the transfers are both equal to 0. So the point $(NB_{1,\max },NB_{2,\min })$ defined by $NB_{1,\max }=B(2\overline{a} )+\Omega _{1}-C(\overline{a})$ and $NB_{2}=\alpha B(2\overline{a})$ with $\overline{a}$ defined by $\delta C(\overline{a})= \Omega _{2}$ is on the Pareto frontier. The point $(NB_{1,\min },NB_{2,\max })$ defined by $NB_{1,\min }=B(2\overline{a})$ and $NB_{2,\max }=\alpha B(2\overline{a} )+\Omega _{2}-\delta C(\overline{a})+t$ with $\overline{a}$ defined by $2\alpha B^{\prime }(2\overline{a})=\delta C^{\prime }(\overline{a})$ and $t$ defined by $(1+\lambda )t+C(\overline{a})=\Omega _{1}$ is also on the Pareto frontier. These two points are different so the Pareto set is not empty.

1.3 Proof of Proposition 1c

The social welfare function is strictly concave and the Pareto set is convex. So there exists a unique bargaining equilibrium for agreement UT.

1.4 Proof of Proposition 2a

The Pareto optimality results from the following program

$$\begin{aligned} \max _{a_{1},a_{2}}\text { }NB_{1}&=\max _{a_{1},a_{2}}\left[ B(a_{1}+a_{2})+c_{1}\right] \\&s.t. \qquad \left\{ \begin{array}{l} \Omega _{1}=c_{1}+C(a_{1}) \\ \Omega _{2}=c_{2}+\delta C(a_{2}) \\ NB_{2}=\alpha B(a_{1}+a_{2})+c_{2}\ge \overset{\overline{-}}{NB_{2}} \end{array} \right. \end{aligned}$$

where $\overset{\overline{-}}{NB_{2}}$ is exogenous.

If we use the first two constraints, the program becomes

$$\begin{aligned} \max _{a_{1},a_{2}}\text { }NB_{1}&=\max _{a_{1},a_{2}}\left[ B(a_{1}+a_{2})+\Omega _{1}-C(a_{1})\right] \\ s.t. \qquad NB_{2}&=\alpha B(a_{1}+a_{2})+\Omega _{2}-\delta C(a_{2})\ge \overset{\overline{-}}{NB_{2}} \end{aligned}$$

The Lagrangian of this maximization problem is: $L=[B(a_{1}+a_{2})+\Omega _{1}-C(a_{1})]+\rho [\alpha B(a_{1}+a_{2})+\Omega _{2}-\delta C(a_{2})-\overset{\overline{-}}{NB_{2}}]$, where $\rho$ is the multiplier associated with the constraint.

The first-order conditions (FOCs) with respect to $a_{1}$ and $a_{2}$ are respectively: $B^{\prime }(a_{1}+a_{2})-C^{\prime }(a_{1})=-\rho \alpha B^{\prime }(a_{1}+a_{2})$, and $B^{\prime }(a_{1}+a_{2})=-\rho [\alpha B^{\prime }(a_{1}+a_{2})-\delta C^{\prime }(a_{2})]$.

The ratio of these FOCs gives

$$\begin{aligned} \frac{1}{B^{\prime }(a_{1}+a_{2})}=\frac{\alpha }{\delta C^{\prime }(a_{2})}+ \frac{1}{C^{\prime }(a_{1})} \end{aligned}$$

(12)

The total differential of $NB_{1}$ is equal to: $dNB_{1}=B^{\prime }(a_{1}+a_{2})(da_{1}+da_{2})-C^{\prime }(a_{1})da_{1}$, and that of $NB2$ is equal to: $dNB_{2}=\alpha B^{\prime }(a_{1}+a_{2})(da_{1}+da_{2})-\delta C^{\prime }(a_{2})da_{2}$.

On the Pareto frontier, we have

$$\begin{aligned} d(NB_{1})&= \frac{\delta C^{\prime }(a_{1})C^{\prime }(a_{2})}{\delta C^{\prime }(a_{2})+\alpha C^{\prime }(a_{1})}(da_{1}+da_{2})-C^{\prime }(a_{1})da_{1} \\ d(NB_{2})&= \frac{\alpha \delta C^{\prime }(a_{1})C^{\prime }(a_{2})}{ \delta C^{\prime }(a_{2})+\alpha C^{\prime }(a_{1})}(da_{1}+da_{2})-\delta C^{\prime }(a_{2})da_{2}\\ d(NB_{1})&= \frac{\delta C^{\prime }(a_{1})C^{\prime }(a_{2})da_{2}-\alpha C^{\prime }(a_{1})C^{\prime }(a_{1})da_{1}}{\delta C^{\prime }(a_{2})+\alpha C^{\prime }(a_{1})}\\ d(NB_{2})&= \frac{\alpha \delta C^{\prime }(a_{1})C^{\prime }(a_{2})da_{1}-\delta C^{\prime }(a_{2})\delta C^{\prime }(a_{2})da_{2}}{ \delta C^{\prime }(a_{2})+\alpha C^{\prime }(a_{1})} \end{aligned}$$

Along the Pareto frontier, we have $\dfrac{dNB_{1}}{dNB_{2}}=-\dfrac{1}{ \delta }\dfrac{C^{\prime }(a_{1})}{C^{\prime }(a_{2})}$. The Pareto frontier is differentiable, thus it is continuous.

1.5 Proof of Proposition 2b

We consider two points on this frontier: for $NB_{2\text { }}$, consider the case where $\delta C(a_{2})= \Omega _{2}$ when the private consumption is equal to 0. At that point, the maximum of $NB_{1\text { }}$is attainable for $B^{\prime }(a_{1}+a_{2})=C^{\prime }(a_{1})$. So the point $(NB_{1,\max },NB_{2,\min })$ defined by $NB_{1,\max }=B(a_{1}+\,a_{2})+\Omega _{1}-C(a_{1})$ and $NB_{2,\min }=\alpha B(a_{1}+a_{2})$ with $a_{2}$ defined by $\delta C(a_{2})= \Omega _{2}$ and $a_{1}$ defined by $B^{\prime }(a_{1}+a_{2})=C^{\prime }(a_{1})$, is on the Pareto frontier. Similarly, the point $(NB_{1,\min },NB_{2,\max })$ defined by $NB_{1,\min }=B(a+a_{2})$ and $NB_{2,\max }=\alpha B(a_{1}+a_{2})+\Omega _{2}-\delta C(a_{2})$ with $a_{1}$ defined by $C(a_{1})= \Omega _{1}$ and $a_{2}$ defined by $\alpha B^{\prime }(a_{1}+a_{2})=\delta C^{\prime }(a_{2})$ is on the Pareto frontier. So the Pareto set is not empty because these two points are different.

1.6 Proof of Proposition 2c

Assume that there exists an interval $[\underline{NB}_{2},\overline{NB_{2}}]$ on which the Pareto frontier is convex. Consider a point on this interval $\widetilde{NB}_{2}$ which is a convex combination of $\underline{NB_{2}}$ and $\overline{NB_{2}}$, that is, $\widetilde{NB}_{2}=\gamma \underline{ NB_{2}} +(1-\gamma ) \overline{NB_{2}}$. By definition of the local convexity of the Pareto frontier, the same convex combination of $NB_{1}$, denoted $\widetilde{NB}_{1}=\gamma \underline{NB_{1}}+(1-\gamma )\overline{ NB_{1}}$, is greater than the corresponding point $\widetilde{\widetilde{ NB_{1}}}$ of the Pareto frontier defined by

$$\begin{aligned} \widetilde{\widetilde{NB_{1}}}&= Max\text { }NB_{1} \nonumber \\ s.t.\quad NB_{2}&\ge \text { }\widetilde{NB}_{2}\text { } \end{aligned}$$

(13)

Moreover the concavity of the function $NB_{1}=B(a_{1}+a_{2})-C(a_{1})+ \Omega _{1}$ implies that $\widetilde{NB}_{1}=\gamma \underline{NB_{1}} +(1-\gamma )\overline{NB_{1}}<NB_{1}(\gamma \underline{a_{1}}+(1-\gamma ) \overline{a_{1}}, \gamma \underline{a_{2}}+(1-\gamma )\overline{a_{2}})= \overline{\overline{NB_{1}}}$. This level of utility is attainable because $\overline{\overline{a_{1}}}=\gamma \underline{a_{1}}+(1-\gamma ) \overline{a_{1}}$ and $\overline{\overline{a_{2}}} =\gamma \underline{a_{2} }+(1-\gamma )\overline{a_{2}}$ are convex combinations of possible abatements in the budget set. Moreover $\overline{\overline{NB_{2}}}(\gamma \underline{a_{1}}+(1-\gamma )\overline{a_{1}}, \gamma \underline{a_{2}} +(1-\gamma )\overline{a_{2}}) \ge \widetilde{NB}_{2}$ by the concavity of the function $NB_{2}$. So for the abatements $\overline{\overline{a_{1}}}$ and $\overline{\overline{a_{2}}}$ we have $\overline{\overline{NB_{2}}} \ge \widetilde{NB}_{2}$ and $\widetilde{\widetilde{NB_{1}}}<\widetilde{ NB}_{1}< \overline{\overline{NB_{1}}}$ which is contradictory with the construction of the Pareto frontier given by program 13, because we have found a level of utility $\overline{\overline{NB_{1}}}$ associated with a level $\overline{\overline{NB_{2}}}$ which is above the Pareto frontier. So the Pareto frontier cannot be convex locally. It is concave and the Pareto set is convex.

1.7 Proof of Proposition 2d

The social welfare function is strictly concave and the Pareto set is convex. So there exists a unique bargaining equilibrium for agreement D.

Appendix 3: Total Welfare

The goal of “Appendix”  is to exhibit simple examples which fulfill Conditions C1 and C3.

1.1 Positivity of Transfers in the UT Agreement: The Case with a Linear Cost, and $\alpha \simeq 0, \delta <1, \gamma =1/2$ (Condition C1)

Here, we assume a linear cost function $C(a)=ca$, where $c$ is a positive parameter. We show that when the benefits from global abatement of country 2 are low $(\alpha \simeq 0)$ and the abatement costs of country 2 are lower than that of country 1 $(\delta <1)$, then the transfers from country 1 to country 2 are positive.

For the uniform agreement with transfers from country 1 to country 2, the Nash bargaining problem with equal bargaining powers $(\gamma =1/2)$ leads to the following first-order condition with respect to transfer $(t)$:

$$\begin{aligned} \frac{\partial \overset{-}{V\text { }}}{\partial t} =0\Longleftrightarrow&-(1+\lambda )\left[ \alpha B(2\overset{-}{a})+\Omega _{2}-\delta c\overset{-}{a}+t-\overset{\wedge }{NB_{2}}\right] \\&+ \left[ B(2\overset{-}{a})+\Omega _{1}-c\overset{-}{a}-(1+\lambda )t-\overset{ \wedge }{NB_{1}}\right] =0 \\ \Longleftrightarrow&2(1+\lambda )t=B(2\overset{-}{a})(1-(1+\lambda )\alpha )-(1+\lambda )\Omega _{2}+\Omega _{1}+c\overset{-}{a}((1+\lambda )\delta -1) \\&+((1+\lambda )\overset{\wedge }{NB_{2}}-\overset{\wedge }{NB_{1}}) \end{aligned}$$

Concerning the term $((1+\lambda )\overset{\wedge }{NB_{2}}-\overset{\wedge }{NB_{1}})$, the welfare levels at the threat point are given by: $\overset{ \wedge }{NB_{1}}=B(B^{\prime -1}(c))+\Omega _{1}-cB^{\prime -1}(c)$ and $\overset{\wedge }{NB_{2}}=\alpha B(B^{\prime -1}(c))+\Omega _{2}$. That gives us:

$$\begin{aligned}&((1+\lambda )\overset{\wedge }{NB_{2}}-\overset{\wedge }{NB_{1}})=(1+\lambda )(\alpha B(B^{\prime -1}(c))+\Omega _{2}) -B(B^{\prime -1}(c))-\Omega _{1}+cB^{\prime -1}(c) \\&\Longleftrightarrow (1+\lambda )\Omega _{2}-\Omega _{1}+(1+\lambda )\alpha B(B^{\prime -1}(c) -B(B^{\prime -1}(c))+cB^{\prime -1}(c) \end{aligned}$$

We make the following simplifying assumption: $\alpha \simeq 0$. Then, the level of transfers becomes: $2(1+\lambda )t\simeq B(2\overset{-}{a})-c \overset{-}{a}(1-(1+\lambda )\delta )-B(B^{\prime -1}(c))+cB^{\prime -1}(c)$.

The expression $\left[ B(2\overset{-}{a})-c\overset{-}{a}(1-(1+\lambda )\delta )\right]$ is superior or equal to $(NB_{1}-\Omega _{1})$ because $B(2\overset{-}{a})-c\overset{-}{a}(1-(1+\lambda )\delta )\ge B(2\overset{-}{ a})-c\overset{-}{a}-(1+\lambda )t$. Furthermore, the expression $\left[ -B(B^{\prime -1}(c))+cB^{\prime -1}(c)\right]$ is equal to $(-\overset{ \wedge }{NB_{1}}+\Omega _{1})$. We thus have,

$$\begin{aligned} 2(1+\lambda )t\simeq B(2\overset{-}{a})-c\overset{-}{a}(1-(1+\lambda )\delta )-B(B^{\prime -1}(c))+cB^{\prime -1}(c)\ge NB_{1}-\overset{\wedge }{NB_{1}} \end{aligned}$$

When country 1 signs a UT agreement, its utility is greater than that at the threat point, hence the transfers from country 1 to country 2 are positive.

1.2 The Conditions $a_{1}\le 1$ and $a_{2}\ge 0$ (Condition C1)

In order to have $a_{1}\le 1$ and $a_{2}\ge 0$, it is sufficient, for instance, that in the case with a linear cost function, we have $\alpha \simeq 0; (1+\lambda )\delta \simeq 1, (1+\lambda )\delta$ could be less than or greater than 1, in the neighborhood of 1; $c$ must be sufficiently large and the function $B^{\prime -1}(.)$ must be sufficiently small.

The second condition of Definition 1 gives us:

$$\begin{aligned} C(a_{1})=C(\overset{-}{a}^{*})+(1+\lambda )t^{*}\Longleftrightarrow ca_{1}=c\overset{-}{a}^{*}+(1+\lambda )t^{*}\Longleftrightarrow a_{1}=\overset{-}{a}^{*}+\frac{(1+\lambda )t^{*}}{c} \end{aligned}$$

Similarly, the first condition of Definition 1 gives:

$$\begin{aligned} a_{2}=2\overset{-}{a}^{*}-a_{1}\Longleftrightarrow a_{2}=2\overset{-}{a} ^{*}-\overset{-}{a}^{*}-\frac{(1+\lambda )t^{*}}{c} \Longleftrightarrow a_{2}=\overset{-}{a}^{*}-\frac{(1+\lambda )t^{*} }{c} \end{aligned}$$

One knows, from proof C1, that the first-order conditions related to the Nash bargaining solution with equal bargaining powers $(\gamma =1/2)$ in the UT agreement imply:

$$\begin{aligned}&(2B^{^{\prime }}(2\overset{-}{a})-c)\left[ \alpha B(2\overset{-}{a} )+\Omega _{2}-\delta c\overset{-}{a}+t-\overset{\wedge }{NB_{2}}\right] \\&+\, (2\alpha B^{^{\prime }}(2\overset{-}{a})-\delta c)(1+\lambda )\left[ \alpha B(2\overset{-}{a})+\Omega _{2}-\delta c\overset{-}{a}+t-\overset{ \wedge }{NB_{2}}\right] =0 \\&\Longleftrightarrow (2B^{^{\prime }}(2\overset{-}{a})-c)+(2\alpha B^{^{\prime }}(2\overset{-}{a})-\delta c)(1+\lambda )=0\Longleftrightarrow B^{^{\prime }}(2\overset{-}{a})=\frac{c(1+\delta (1+\lambda ))}{2(1+\alpha (1+\lambda ))} \end{aligned}$$

We make the following assumptions: $\delta (1+\lambda )\simeq 1$ and $\alpha \simeq 0$. Under these assumptions, the expression of the marginal benefit becomes $B^{^{\prime }}(2\overset{-}{a})\simeq c$. This implies the following optimal level of the uniform standard $\overset{-}{a}\simeq \frac{1 }{2}B^{^{\prime -1}}(c)$, which is positive.

The preceding proof also provides the expression of the transfers when $\alpha \simeq 0: 2(1+\lambda )t\simeq B(2\overset{-}{a})-c\overset{-}{a} (1-(1+\lambda )\delta )-B(B^{\prime -1}(c))+cB^{\prime -1}(c)$.

If we use the assumption $\delta (1+\lambda )\simeq 1$ and introduce the expression of the optimal uniform standard, we obtain: $(1+\lambda )t\simeq \frac{1}{2}cB^{\prime -1}(c)$.

Let us return to the study of the levels of differentiated standards $a_{1}$ and $a_{2}$.

$$\begin{aligned} a_{1}&= \overset{-}{a}^{*}+\frac{(1+\lambda )t^{*}}{c} \Longleftrightarrow a_{1}=\frac{1}{2}B^{^{\prime -1}}(c)+\frac{1}{2} B^{\prime -1}(c)\Longleftrightarrow a_{1}=B^{^{\prime -1}}(c)\le 1 \\&\text {if }(c)\text { is sufficiently large and }B^{\prime -1}(.)\text { is sufficiently small.} \\ a_{2}&= \overset{-}{a}^{*}-\frac{(1+\lambda )t^{*}}{c} \Longleftrightarrow a_{2}=\frac{1}{2}B^{^{\prime -1}}(c)-\frac{1}{2} B^{\prime -1}(c)\Longleftrightarrow a_{2}=0 \end{aligned}$$

1.3 Proof of Proposition 3

The maximum value attainable by the social welfare function in the UT agreement is:

$$\begin{aligned} \overset{-}{V}^{*}=\left[ B(2\overset{-}{a}^{*})-C(\overset{-}{a} ^{*})-(1+\lambda )t^{*}-\overset{\wedge }{NB_{1}}\right] ^{\gamma }\times \left[ \alpha B(2\overset{-}{a}^{*})-\delta C(\overset{-}{a} ^{*})+t^{*}-\overset{\wedge }{NB_{2}}\right] ^{1-\gamma } \end{aligned}$$

(14)

This function, if $\alpha$ is strictly positive, can be written as follows:

$$\begin{aligned} \frac{\overset{-}{V}^{*}}{\alpha ^{1-\gamma }}\!=\!\left[ B(2\overset{-}{a} ^{*})\!-\!C(\overset{-}{a}^{*})\!-\!(1+\lambda )t^{*}\!-\!\overset{\wedge }{ NB_{1}}\right] ^{\gamma }\times \left[ B(2\overset{-}{a}^{*})\!-\!\frac{ \delta }{\alpha }C(\overset{-}{a}^{*})\!+\!\frac{t^{*}}{\alpha }\!-\!\frac{ \overset{\wedge }{NB_{2}}}{\alpha }\right] ^{1-\gamma } \end{aligned}$$

(15)

Now we can write the social welfare function with particular values of the differentiated standards:

$$\begin{aligned} \frac{V(a_{1},a_{2})}{\alpha ^{1-\gamma }}=\left[ B(a_{1}+a_{2})-C(a_{1})- \overset{\wedge }{NB_{1}}\right] ^{\gamma }\times \left[ B(a_{1}+a_{2})- \frac{\delta }{\alpha }C(a_{2})-\frac{\overset{\wedge }{NB_{2}}}{\alpha } \right] ^{1-\gamma } \end{aligned}$$

(16)

Notice that the first terms in the brackets in Eqs. 15 and are the same because of Definition 1. Hence, the condition of superiority of the differentiated social welfare function over the uniform becomes:

$$\begin{aligned} \left[ B(a_{1}+a_{2})-\frac{\delta }{\alpha }C(a_{2})-\frac{\overset{\wedge }{NB_{2}}}{\alpha }\right] ^{1-\gamma }&> \left[ B(2\overset{-}{a}^{*})- \frac{\delta }{\alpha }C(\overset{-}{a}^{*})+\frac{t^{*}}{\alpha }- \frac{\overset{\wedge }{NB_{2}}}{\alpha }\right] ^{1-\gamma } \end{aligned}$$

(17)

$$\begin{aligned} \Longleftrightarrow \left[ -\delta C(a_{2})\right]&> \left[ -\delta C(\overset{-}{a}^{*})+t^{*}\right] \end{aligned}$$

(18)

because $B(a_{1}+a_{2})=B(2\overset{-}{a}^{*})$ by Definition 1, $\alpha$ and $(1-\gamma )$ are positive. From Definition 1, we have $t^{*}=\frac{C(a_{1})-C(\overset{-}{a}^{*})}{1+\lambda }$. Introducing this expression of transfers into Equation 18, we obtain:

$$\begin{aligned}&\left[ -\delta C(a_{2})\right] > \left[ -\delta C(\overset{-}{a}^{*})+ \frac{C(a_{1})-C(\overset{-}{a}^{*})}{1+\lambda }\right] \end{aligned}$$

(19)

$$\begin{aligned}&\Longleftrightarrow \delta (1+\lambda )\left[ C(\overset{-}{a}^{*})-C(a_{2})\right] > \left[ C(a_{1})-C(\overset{-}{a}^{*})\right] \end{aligned}$$

(20)

From Condition C1 and the first condition of Definition 1, we have $a_{2}<\overset{-}{a}^{*}<a_{1}$. This implies the positivity of the terms in brackets in Condition 2, say $\left[ C(\overset{-}{a }^{*})-C(a_{2})\right] >0$ and $\left[ C(a_{1})-C(\overset{-}{a}^{*}) \right] >0$.

1.4 Superiority of $a_{1}^{*}$ over $a_{2}^{*}$ (Condition C3)

Our objective here is to show that $a_{1}^{*}>a_{2}^{*}$, when the countries have the same bargaining powers $\gamma =1/2$, and a linear cost function $C(a)=ca$, where $c$ is a positive parameter.

The first-order conditions in the D agreement, when $\gamma =1/2$, are the following:

$$\begin{aligned} \frac{\partial V}{\partial a_{1}}\!=\!0&\Longleftrightarrow \frac{\left[ B^{^{\prime }}(a_{1}\!+\!a_{2})\!-\!c\right] }{\left[ B(a_{1}\!+\!a_{2})\!+\!\Omega _{1}-ca_{1}\!-\!\overset{\wedge }{NB}_{1}\right] }\!+\!\frac{\left[ \alpha B^{^{\prime }}(a_{1}\!+\!a_{2})\right] }{\left[ \alpha B(a_{1}\!+\!a_{2})\!+\!\Omega _{2}\!-\!\delta ca_{2}\!-\!\overset{\wedge }{NB}_{2}\right] }\!=\!0 \\ \frac{\partial V}{\partial a_{2}}\!=\!0&\Longleftrightarrow \frac{\left[ B^{^{\prime }}(a_{1}\!+\!a_{2})\right] }{\left[ B(a_{1}\!+\!a_{2})\!+\!\Omega _{1}\!-\!ca_{1}\!-\!\overset{\wedge }{NB}_{1}\right] }\!+\!\frac{\left[ \alpha B^{^{\prime }}(a_{1}\!+\!a_{2})-\delta c\right] }{\left[ \alpha B(a_{1}\!+\!a_{2})\!+\!\Omega _{2}\!-\!\delta ca_{2}\!-\!\overset{\wedge }{NB}_{2}\right] }\!=\!0 \end{aligned}$$

The ratio of these first-order conditions gives us (assuming that we do not divide by 0):

$$\begin{aligned} \begin{array}{cc} &{}\displaystyle \frac{B^{^{\prime }}(a_{1}+a_{2})-c}{B^{^{\prime }}(a_{1}+a_{2})} = \frac{\alpha B^{^{\prime }}(a_{1}+a_{2})}{\alpha B^{^{\prime }}(a_{1}+a_{2})-\delta c} \\ &{} \Longleftrightarrow B^{^{\prime }}(a_{1}+a_{2}) = \frac{\delta c}{\delta +\alpha }\Longleftrightarrow a_{1}+a_{2}=B^{^{\prime }-1}(\frac{\delta c}{\delta +\alpha }) \end{array} \end{aligned}$$

(21)

If we replace $B^{^{\prime }}(a_{1}+a_{2})=\frac{\delta c}{\delta +\alpha }$ in Equation 21, we obtain the following relation: $U_{2}=\delta U_{1}$ where $U_{1}\!=\!\left[ B(a_{1}\!+\!a_{2})\!+\!\Omega _{1}\!-\!ca_{1}\!-\!\overset{\wedge }{NB} _{1}\right]$ and $U_{2}\!=\!\left[ \alpha B(a_{1}\!{+}\!a_{2})\!+\!\Omega _{2}\!-\!\delta ca_{2}\!-\!\overset{\wedge }{N\!B}_{2}\right]$.

We thus have:

$$\begin{aligned} U_{2}\!=\delta U_{1}&\Longleftrightarrow \left[ \alpha B(a_{1}\!+a_{2})\!+\Omega _{2}\!-\delta ca_{2}\!-\overset{\wedge }{NB}_{2}\right] \!=\delta \left[ B(a_{1}\!+a_{2})\!+\Omega _{1}\!-ca_{1}\!-\overset{\wedge }{NB}_{1}\right] \\&\Longleftrightarrow (\delta -\alpha )B(a_{1}+a_{2})+\overset{\wedge }{NB} _{2}-\delta \overset{\wedge }{NB}_{1}=\delta c(a_{1}-a_{2}) \end{aligned}$$

We are interested in whether the term $(\delta -\alpha )B(a_{1}+a_{2})+ \overset{\wedge }{NB}_{2}-\delta \overset{\wedge }{NB}_{1}$ is positive. We replace the expressions of the utility levels at the threat point $\overset{ \wedge }{NB}_{1}$ and $\overset{\wedge }{NB}_{2}$ which were calculated above, and obtain:

$$\begin{aligned} (\delta -\alpha )\left[ B(B^{^{\prime }-1}(\frac{\delta c}{\delta +\alpha } ))-B(B^{\prime -1}(c))\right] +\delta cB^{\prime -1}(c)=\delta c(a_{1}-a_{2}) \end{aligned}$$

We have $B^{^{\prime }-1}(\frac{\delta c}{\delta +\alpha })>B^{\prime -1}(c)$ because $\frac{\delta c}{\delta +\alpha }<c$ and $B^{^{\prime }-1}(.)$ is a decreasing function. We can conclude then that $a_{1}^{*}>a_{2}^{*}$ if $\delta >\alpha$.

1.5 Proof of Proposition 4

The condition of superiority of the social welfare function in the UT agreement over the one in the D agreement $V^{*}(a_{1}^{*},a_{2}^{*})$ is:

$$\begin{aligned} \frac{\overset{-}{V}(\overset{-}{a},t)}{\alpha ^{1-\gamma }} \!&= \!\left[ B(2 \overset{-}{a})\!-\!C(\overset{-}{a})\!-\!(1\!+\!\lambda )t\!-\!\overset{\wedge }{NB_{1}} \right] ^{\gamma }\times \left[ B(2\overset{-}{a})\!-\!\frac{\delta }{\alpha }C( \overset{-}{a})\!+\!\frac{t}{\alpha }\!-\!\frac{\overset{\wedge }{NB_{2}}}{\alpha } \right] ^{1-\gamma } \\ \frac{V^{*}(a_{1}^{*},a_{2}^{*})}{\alpha ^{1-\gamma }} \!&= \!\left[ B(a_{1}^{*}\!+\!a_{2}^{*})\!-\!C(a_{1}^{*})\!-\!\overset{\wedge }{NB_{1}} \right] ^{\gamma }\times \left[ B(a_{1}^{*}\!+\!a_{2}^{*})\!-\!\frac{\delta }{\alpha }C(a_{2}^{*})\!-\!\frac{\overset{\wedge }{NB_{2}}}{\alpha }\right] ^{1-\gamma }\nonumber \end{aligned}$$

(22)

Notice that the first terms in the brackets in Eq. 22 are the same because of Definition 2. Hence, the condition of superiority of the uniform social welfare function over the differentiated function becomes:

$$\begin{aligned} \left[ -\delta C(\overset{-}{a})+t\right] >\left[ -\delta C(a_{2}^{*}) \right] \end{aligned}$$

(23)

Given Definition 2, we have $t=\frac{C(a_{1}^{*})-C(\overset{-}{a})}{ (1+\lambda )}$. By introducing this expression of the transfer in Eq. 23, we obtain the result. Condition C3 and the first condition of Definition 2 imply the positivity of the terms in brackets in Condition C4, that is, $\left[ C(\overset{-}{a})-C(a_{2}^{*}) \right] >0$ and $\left[ C(a_{1}^{*})-C(\overset{-}{a})\right] >0$.

Appendix 4: Individual Welfare

1.1 Proof of Proposition 5

We begin the proof with a remark. Based on Proposition 2, we know that the social welfare in the UT agreement $\overline{V}^{*}$ is greater than that in the D agreement $V^{*}$, if the condition $\delta (1+\lambda ) [ C(\overset{-}{a})-C(a_{2}^{*})] <[ C(a_{1}^{*})-C( \overset{-}{a})]$ (Condition C4) is verified, and if $a_{1}^{*}$ is superior to $a_{2}^{*}$ (Condition C3). This condition implies the relation $(1+\lambda )\delta \dfrac{C^{\prime }(a_{2})}{C^{\prime }(a_{1})}<1$ if $a_{1}^{*}$ is superior to $a_{2}^{*}$, because increasing marginal abatement costs lead to $\frac{[ C(a_{1}^{*})-C( \overset{-}{a})] }{a_{1}^{*}-\overline{a}}<C^{\prime }(a_{1}^{*})$ and $\frac{[ C(\overset{-}{a})-C(a_{2}^{*})] }{\overline{ a}-a_{2}^{*}}>C^{\prime }(a_{2}^{*})$.

We use the first-order conditions of the programs associated with the UT and D agreements. We obtain:

$$\begin{aligned}&\text {(UT) }\gamma (1+\lambda )(NB_{2}^{U}-\overset{\wedge }{NB_{2}} )=(1-\gamma )(NB_{1}^{U}-\overset{\wedge }{NB_{1}}) \end{aligned}$$

(24)

$$\begin{aligned}&\text {(D) }\gamma C^{\prime }(a_{1}^{*})(NB_{2}^{D}-\overset{\wedge }{ NB_{2}})=(1-\gamma )\delta C^{\prime }(a_{2}^{*})(NB_{1}^{D}-\overset{ \wedge }{NB_{1}}) \end{aligned}$$

(25)

Proposition 2 implies that $\overline{V}^{*}$ is greater than $V^{*}$ under Conditions C3 and C4. So we have $\overline{V}^{*}=(\frac{\gamma (1+\lambda )}{(1-\gamma )})^{\gamma }(NB_{2}^{U}-\overset{\wedge }{NB_{2}})> (\frac{\gamma C^{\prime }(a_{1}^{*})}{(1-\gamma )\delta C^{\prime }(a_{2}^{*})})^{\gamma }(NB_{2}^{D}-\overset{\wedge }{NB_{2}})=V^{*}$. As $(1+\lambda )\delta \dfrac{C^{\prime }(a_{2}^{*})}{C^{\prime }(a_{1}^{*})}<1, NB_{2}^{U}$ is greater than $NB_{2}^{D}$. Moreover $(NB_{1}^{U}-\overset{ \wedge }{NB_{1}})>(NB_{1}^{D}-\overset{\wedge }{NB_{1}})$ if and only if $\dfrac{\gamma (1+\lambda )}{(1-\gamma )}(NB_{2}^{U}-\overset{ \wedge }{NB_{2}})>\dfrac{\gamma C^{\prime }(a_{1}^{*})}{(1-\gamma )\delta C^{\prime }(a_{2}^{*})}(NB_{2}^{D}-\overset{\wedge }{NB_{2}})$.