Self-enforcing agreements under unequal nationally determined contributions

Abstract

The Paris Agreement builds on intended nationally determined contributions (INDCs) submitted by most participating nations. The INDCs vary across nations, since national circumstances differ, including national incomes and damages. The INDCs follow a bottom-up approach, whereby nations submit their plans of actions in a Nash-like form (i.e., taking the other nations’ plans as given). We build a model with normal goods and an unequal world income distribution to consider the endogenous formation and stability of an international environmental agreement (IEA) under the bottom-up approach. Nations provide carbon abatement and produce R&D efforts that promote improvements in environmental efficiency of their outputs. Nations share R&D efforts and enjoy R&D spillovers if they join an IEA. Non-members do not enjoy R&D spillovers. Global carbon abatement rises as the IEA expands in size due to the R&D spillovers. We show that the Grand Coalition is stable under a nearly perfectly equitable income distribution, where all nations make positive carbon abatement and R&D contributions. We also consider a more realistic world income distribution, in which some nations lack sufficient income to provide carbon abatement and R&D. In this case, the stable coalition contains all (wealthier) nations that make positive abatement and R&D contributions. For a very unequal world income distribution, not even a bilateral IEA, with the two richest nations in the world, is stable. The stable IEAs provide too little carbon abatement relative to the first best.

Appendix

Proof of Proposition 2

We only need to show that the Grand Coalition is internally stable for $\beta >0$. For $\beta \in ( {0,1} ]$ and $m=n$, the coalition N is internally stable if and only if $u( {n,N} )\ge u( {n-1,N/{\{ n \}}} )$ or

$$\begin{aligned}&\frac{2W+n+( {n-1} ) \beta ( {2W-n} )}{2[ {1+n( {1+( {n-1} ) \beta } )} ]}\nonumber \\&\quad \ge \frac{2W+n+( {n-2} )\beta ( {2( {W-w_n } )-( {n-1} )} )}{2[ {n+1+( {n-1} )( {n-2} ) \beta } ]}. \end{aligned}$$

(A.1)

After some algebraic manipulations, condition (A.1) simplifies to

$$\begin{aligned}&( {n-2} )[ {n+1+n( {n-1} )\beta } ]w_n +1\nonumber \\&\quad -[ {n+( {n-1} )( {n-2} )\beta -3} ]W-n( {2n-1} )\ge 0. \end{aligned}$$

(A.2)

Combining conditions (5.28), evaluated at $m=n$, with the left-hand side of condition (A.2) yields

$$\begin{aligned}&( {n-2} )[ {n+1+n( {n-1} )\beta } ]w_n +1\nonumber \\&\quad -[ {n+( {n-1} )( {n-2} )\beta -3} ]W-n( {2n-1} ) \nonumber \\&>( {n-2} )[ {1+n( {n-1} )\beta } ]W-[ {n+( {n-1} )( {n-2} )\beta -3} ]W+1\nonumber \\&\quad +\frac{( {n-2} )( {2n+1} )}{2}-n( {2n-1} ). \end{aligned}$$

(A.3)

The right-hand side of condition (A.3) simplifies to

$$\begin{aligned} \left[ {1+\left( {n-1} \right) ^{2}\left( {n-2} \right) \beta } \right] W-\frac{n\left( {2n+1} \right) }{2}>0. \end{aligned}$$

(A.4)

The left-hand side of condition (A.4) is greater than zero because $W>nw_n >{n\left( {2n+1} \right) }/2$. Condition (A.4) implies that

$$\begin{aligned}&( {n-2} )[ {n+1+n( {n-1} )\beta } ]w_n +1\nonumber \\&\quad -[ {n+( {n-1} )( {n-2} )\beta -3} ]W-n( {2n-1} )>0. \end{aligned}$$

(A.5)

Inequality (A.5), in turn, proves that $u\left( {n,N} \right) >u\left( {n-1,N/{\left\{ n \right\} }} \right)$. $\square$

Proof of Proposition 3

Since the stand-alone structure is by default internally stable and we have already shown that the Grand Coalition is internally stable in Proposition 2, we need to show that profile-stable coalitions with at least two and at most $n-1$ nations are internally stable.

Let ${M^{*}}/{\{ m \}}\equiv \{ {1,\ldots ,m-1} \}$ and $W^{{M^{*}}/{\{ m \}}}\equiv \sum \nolimits _{i=1}^{m-1} {w_i }$ in what follows. For $\beta \in ( {0,1} ]$ and $n-1\ge m\ge 2$, a coalition $M^{*}=\{ {1,\ldots ,m} \}$ is internally stable if and only if

$$\begin{aligned}&\frac{2W+n+( {m-1} )\beta ( {2W^{M^{*}}-m} )}{2[ {n+1+m( {m-1} )\beta } ]}\nonumber \\&\quad \ge \frac{2W+n+( {m-2} )\beta ( {2W^{{M^{*}}/{\{ m \}}}-( {m-1} )} )}{2[ {n+1+( {m-1} )( {m-2} )\beta } ]}, \end{aligned}$$

(A.6)

Since $W^{{M^{*}}/{\{ m \}}}=W^{M^{*}}-w_m$, inequality (A.6) implies

$$\begin{aligned}&( {m-2} )[ {n+1+m( {m-1} )\beta } ]w_m -( {m-1} )( {2W+2n+1} )\nonumber \\&\quad -[ {( {m-1} )( {m-2} )\beta -( {n+1} )} ]W^{M^{*}}\ge 0. \end{aligned}$$

(A.7)

Since $w_m >w_n$, conditions (A.7) imply that for $\beta \in ( {0,1} ]$ and $n-1\ge m\ge 2$,

$$\begin{aligned} w_m >\frac{2[ {W+( {m-1} )\beta W^{M^{*}}} ]+2n+1}{2[ {n+1+m( {m-1} )\beta } ]}. \end{aligned}$$

(A.8)

Given condition (A.8), we can affirm that the left-hand side of condition (A.7) is greater than

$$\begin{aligned}&( {m-2} )[ {W+( {m-1} )\beta W^{M^{*}}} ]\nonumber \\&\quad +\frac{( {m-2} )( {2n+1} )}{2}-( {m-1} )( {2W+2n+1} )\nonumber \\&\quad -[ {( {m-1} )( {m-2} )\beta -( {n+1} )} ]W^{M^{*}}. \end{aligned}$$

(A.9)

The expression (A.9) simplifies to

$$\begin{aligned} \left( {nW^{M^{*}}-mW} \right) +\left( {W^{M^{*}}-\frac{m\left( {2n+1} \right) }{2}} \right) >0. \end{aligned}$$

(A.10)

The left-hand side of condition (A.7) is positive because $nW^{M^{*}}>mW$ and ${W^{M^{*}}}/m>w_n >{( {2n+1} )}/2$. Hence, the left-hand side of condition (A.7) is greater than zero, which implies that $u( {m,M^{*}} )>u({m-1,{M^{*}}/{\{m\}}})$ for $\beta \in ( {0,1} ]$ and $n-1\ge m\ge 2$. This, in turn, implies that the Grand Coalition is the only coalition structure that satisfies profile stability, internal stability and external stability. $\square$