# Stable International Environmental Agreements: Large Coalitions that Achieve Little

## Abstract

**:**

## 1. Introduction

## 2. Analysis

- A coalition of size K is externally unstable if a non-signatory has an incentive to enter the coalition, i.e., if it is better off being a member of a coalition of size K + 1 than being a non-member of the existing coalition of size K, ${w}_{s}^{K+1}>{w}_{n}^{K}$.
- A coalition of size K + 1 is internally unstable if a signatory country is better off leaving this coalition and being a non-signatory in a world with a coalition of size K, i.e., ${w}_{n}^{K}>{w}_{s}^{K+1}$.

**Proposition.**

**Proof.**

^{–66}times its globally optimal abatement. It is not surprising that the global welfare derived from such little abatement is also minuscule compared to the welfare the world would achieve with the grand coalition.

## 3. Interpretation

## 4. Modifications of the Model

## 5. Summary and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Barrett, S. Self-enforcing international environmental agreements. Oxf. Econ. Pap.
**1994**, 46, 878–894. [Google Scholar] [CrossRef] - Carraro, C.; Siniscalco, D. Strategies for the international protection of the environment. J. Public Econ.
**1993**, 52, 309–328. [Google Scholar] [CrossRef] - Finus, M. Game Theory and International Environmental Cooperation; Edward Elgar: Cheltenham, UK, 2001. [Google Scholar]
- Karp, L.; Simon, L. Participation games and international environmental agreements: A non-parametric model. J. Environ. Econ. Manag.
**2013**, 65, 326–344. [Google Scholar] [CrossRef] - Diamantoudi, E.; Sartzetakis, E. Stable international environmental agreements: An analytical approach. J. Public Econ. Theory
**2006**, 8, 247–263. [Google Scholar] [CrossRef] - Chander, P.; Tulkens, H. A core-theoretic solution to for the design of cooperative agreements on transfrontier pollution. Int. Tax Public Financ.
**1995**, 2, 279–293. [Google Scholar] [CrossRef] - Chander, P.; Tulkens, H. The core of an economy with multilateral environmental externalities. Intern. J. Game Theory
**1997**, 26, 379–401. [Google Scholar] [CrossRef] - Currarini, S.; Marini, M. A sequential approach to the characteristic function and the core in games with externalities. In Advances in Economic Design; Koray, S., Sertel, M.R., Eds.; Springer: Berlin, Germany, 2003; pp. 233–250. [Google Scholar]
- Marini, M. The sequential core of an economy with environmental externalities. Environ. Sci.
**2013**, 1, 79–82. [Google Scholar] [CrossRef] [Green Version] - Carraro, C.; Siniscalco, D. R&D cooperation and the stability of international environmental agreements. In Environmental Negotiations: Strategic Policy Issues. Cheltenham; Carraro, C., Ed.; Edward Elgar: Cheltenham, UK, 1997. [Google Scholar]
- Helm, C.; Schmidt, R. Climate cooperation with technology investments and border carbon adjustment. Eur. Econ. Rev.
**2015**, 75, 112–130. [Google Scholar] [CrossRef] [Green Version] - Katsoulacos, Y. R&D spillovers, cooperation, subsidies and international agreements. In International Environmental Negotiations: Strategic Policy Issues; Carraro, C., Ed.; Edward Elgar: Cheltenham, UK, 1997; pp. 97–109. [Google Scholar]
- Pathways to a Low-Carbon Economy: Version 2 of the Global Greenhouse Gas Abatement Cost Curve; McKinsey & Company: New York, NY, USA, 2009; Available online: https://www.mckinsey.com/business-functions/sustainability/our-insights/pathways-to-a-low-carbon-economy (accessed on 22 October 2019).

1 | Diamantoudi/Sartzetakis [5] find that the maximum coalition size in this case is reduced to 4 if the game is specified in terms of emissions instead of abatement and if a non-negativity constraint on emissions is imposed. |

2 | This can be shown by means of Figure 2. Although the marginal abatement cost function is strictly concave, the considerations carry over to piecewise linear functions. In the case of indifference whether to join or not to join, the areas $b+d$ and $e$ are equally large. A fifth country is willing to join if the benefit from additional abatement of the other countries, $g+h$, is at least as large as its own net cost (abatement cost minus benefit from own abatement), $b+d+f+h$. In the linear model of Karp/Simon [4], $b+d=e.$ Thus the condition for the fifth country to enter is that $g\ge e+f$. In general, the Kth country is willing to join if at least $\frac{1}{2}\left({A}_{S}^{K}-{A}_{S}^{K-1}\right)=\left(K-2\right)\left({A}_{S}^{K-1}-{A}_{S}^{K-2}\right)$. This implies ${A}_{S}^{K}=\left(2K-3\right){A}_{S}^{K-1}-\left(2K-4\right){A}_{S}^{K-2}$. For large $K$, ${A}_{S}^{K}/{A}_{S}^{K-1}={A}_{S}^{K-1}/{A}_{S}^{K-2}$. Using this in the previous equation gives ${A}_{S}^{K}/{A}_{S}^{K-1}=2K-4$. |

3 | If the MAC curve becomes linear at ${A}_{i}={A}_{S}^{{K}^{*}}$, we have $\left({A}_{S}^{N}-{A}_{S}^{{K}^{*}}\right)MA{C}^{\prime}=N-{K}^{*}$, where $MA{C}^{\prime}=\alpha \gamma {\left({A}_{S}^{{K}^{*}}\right)}^{\alpha -1}$ is the derivative of the marginal abatement cost. Using ${A}_{s}^{{K}^{*}}={\left(\frac{{K}^{*}-1}{\gamma}\right)}^{1/\alpha}$ and rearranging terms yields the result. |

4 | S-shaped cost curves are common in economics, representing initially increasing and later decreasing returns to scale. An S-shaped marginal cost curve like the one proposed here implies slightly decreasing (almost constant) returns to scale if A is small and rapidly decreasing returns to scale if A is large, i.e., almost linear relationship between the abatement capital invested and the abated emissions until a threshold and extreme increases in investments required after the threshold to achieve further emission reductions. |

**Table 1.**Critical coalition sizes, stable coalitions, and their properties for different values of $\alpha $.

$\mathit{\alpha}$ | 2 | 1 | 0.5 | 0.2 | 0.1 | 0.01 |

$\overline{\mathit{K}}$ | 1.8 | 2 | 2.37 | 3.32 | 4.69 | 22.17 |

${\mathit{K}}^{*}$ | 2 | 2 and 3 | 3 | 4 | 5 | 23 |

abatement increment,${\left(\frac{{\mathit{K}}^{*}-\mathbf{1}}{{\mathit{K}}^{*}-\mathbf{2}}\right)}^{\mathbf{1}/\mathit{\alpha}}$ | – | 2 | 4 | 7.59 | 17.76 | 130.39 |

share of GC abatement${\left(\frac{{\mathit{K}}^{*}-\mathbf{1}}{\mathit{N}-\mathbf{1}}\right)}^{\mathbf{1}/\mathit{\alpha}}$ | 0.10 | 0.02 and 0.03 | 0.004 | 2.5*10^{–8} | 1.2*10^{–14} | 4.8*10^{–66} |

share of GC global welfare | 0.03 | 0.0004 and 0.0009 | 0.0004 | 0.6*10^{–8} | 0.6*10^{–14} | 8.7*10^{–65} |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rauscher, M.
Stable International Environmental Agreements: Large Coalitions that Achieve Little. *Games* **2019**, *10*, 47.
https://doi.org/10.3390/g10040047

**AMA Style**

Rauscher M.
Stable International Environmental Agreements: Large Coalitions that Achieve Little. *Games*. 2019; 10(4):47.
https://doi.org/10.3390/g10040047

**Chicago/Turabian Style**

Rauscher, Michael.
2019. "Stable International Environmental Agreements: Large Coalitions that Achieve Little" *Games* 10, no. 4: 47.
https://doi.org/10.3390/g10040047