Designed to be stable: international environmental agreements revisited

Abstract

In a three-stage game, we revisit the non-cooperative coalition approaches into international environmental agreements by tackling a fundamental design flaw in these approaches. We show how a treaty can effectively remove the free-riding problem from its roots by farsightedly choosing its members’ emissions. We prove that under this approach, the grand coalition is a self-enforcing equilibrium. We will argue how the modified timing of the coalition game suggested in this article is more realistic and consistent with real-world practices. Another advantage of the farsighted rule is its simplicity and applicability to all coalition game settings, regardless of whether agents are homogeneous or heterogeneous.

A Proof of Proposition (4.1)

Since the premise of this article is that the grand coalition is unstable under the Stackelberg solution, that means $w_{k-1,St}^{n}>w_{k,St}^{m}=w_{k}^{m}$ ($w_{k-1,C}^{n}>w_{k,C}^{m}=w_{k}^{m}$). So, to curb the free-riding incentives the agency should choose $e_{k-1}^{m}$ to lower the welfare of the free-rider, which also means we readily have $w_{k-1}^{n}\le w_{k-1,St}^{n}$ ($w_{k-1}^{n}\le w_{k-1,C}^{n}$). Moreover, by the negative externality assumption, we have $\frac{\partial w_{s}^{n}}{\partial e_{s}^{m}}<0$, therefore, $e_{k-1}^{m}\ge e_{k-1,St}^{m}$ (and given the fact that $e_{k-1,St}^{m}\ge e_{k-1,C}^{m}$, then $e_{k-1}^{m}\ge e_{k-1,C}^{m}$). In addition, by the nature of our assumptions, the members’ and non-members’ emissions are strategically substitute, therefore, $e_{k-1}^{n}\le e_{k-1,St}^{n}$ (and $e_{k-1}^{n}\le e_{k-1,C}^{n}$). Consequently, $w_{k-1}^{m}\le w_{k-1,St}^{m}$. The welfare comparison for the farsighted and Cournot members depends on how much the increase in members’ emissions compensates for the decrease in non-members’ emissions.

Mathematically, for the farsighted rule and a treaty of size $k-1$ we solve the following Lagrangian:

$$\begin{aligned} {\mathscr {L}}=(k-1)\left\{ B\left( e_{k-1}^{m}\right) -D((k-1)e_{k-1}^{m}+e_{k-1}^{n})\right\} +\lambda (w_{k}^{m}-w_{k-1}^{n}), \end{aligned}$$

(10)

with $B'-(k-1)D'[1+g']-\lambda \frac{\partial w_{k-1}^{n}}{\partial e_{k-1}^{m}}=0$ as the first order condition, where the Lagrange multiplier $\lambda$ is strictly positive given the binding constraint. Therefore, at the solution: (i) automatically, we must have $w_{k-1}^{n}\le w_{k-1,St}^{n}$ and $w_{k-1}^{m}\le w_{k-1,St}^{m}$ (adding a constraint to an optimization problem cannot be welfare improving); and (ii) the treaty’s net marginal benefit of a member’s emission must be negative at the constraint optimum, i.e., $B'-(k-1)D'[1+g']<0$. The latter condition, paired with the premise of eliminating the free-riding incentives in a negative externality context, i.e., $\frac{\partial w_{s}^{n}}{\partial e_{s}^{m}}\le 0$, yields in $e_{k-1}^{m}\ge e_{k-1,St}^{m}$, and since $e_{k-1,St}^{m}\ge e_{k-1,C}^{m}$ (for a formal proof see Finus et al. 2021), then $e_{k-1}^{m}\ge e_{k-1,C}^{m}$, then we also readily have $e_{k-1}^{n}\le e_{k-1,St}^{n}$, and $e_{k-1}^{n}\le e_{k-1,C}^{n}$.

A similar argument is applied to other coalition sizes.