Social norms for the stability of international enviromental agreements

Marta Biancardi; Lucia Maddalena; Giovanni Villani

doi:10.3934/dcdss.2022046

Article Contents

2022, Volume 15, Issue 12: 3541-3555. Doi: 10.3934/dcdss.2022046

This issue Previous Article Expected vs. real transaction costs in European option pricing Next Article Soliton solutions for quasilinear modified Schrödinger equations in applied sciences

Social norms for the stability of international enviromental agreements

1.
Department of Economics and Finance, University of Bari, Largo Abbazia S. Scolastica, 53 70124 Bari, Italy
2.
Department of Economics, Management and Territory, University of Foggia, Via Alberto da Zara 11, 71121 Foggia, Italy

^* Corresponding author: Lucia Maddalena
^* Corresponding author: Lucia Maddalena

Received: August 2021

Early access: March 2022

Published: December 2022

Abstract / Introduction Full Text(HTML) Figure(1) / Table(4) Related Papers Cited by

Abstract

This paper is devoted to study the stability of international environmental agreements (IEAs) in a pollution abatement context. Countries can decide to cooperate or to defect. Defector countries decide on their abatement levels by minimizing their own total cost whereas, signatory countries decide on their abatement levels by minimizing the aggregate of all cooperators.

In the model, all countries have the same environmental damage instead, respect to the non-environmental cost, we assume that each signatory country has to punish a non-signatory for its behaviour, at some cost to itself (see [17]). We propose two different cases in which we have that punishment is directly proportional to the level of pollution (see [6] or not (see [5]). Punishments can be in the form of trade sanctions or import tariffs, as a measure to encourage cooperation.

We model a differential game in order to determine both the optimal path of the abatement levels and stock pollutant as results of feedback Nash equilibria. Stability conditions, such as internal and external stability, are applied showing that diﬀerent answers about the size of a stable IEA can be obtained.

Keywords:

Mathematics Subject Classification: 91A10, 49L12.

Citation:

Full Text(HTML)

Figure 1. The solid line is the evolution of pollution s(t) in the the first model while, the dashodot line is the evolution in the second model. The gold surface is the evolution of s(t) in the first model while the violet in the second model.

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Table 1. Coalition Stability with $ p = 0.20 $

Coalition	$ \psi < 0 $	$ \psi=0 $	$ \psi=0 $
	$ \xi > 0 $	$ \xi > 0 $	$ \xi=0 $
	First Model
$ m=1 $
$ m=2 $			Stable
$ m=3 $	$ \xi\,\in\,[0.00; 6.94] $	$ \xi\,\in\,[0.00; 5.55] $	Stable
$ m=4 $	$ \xi\,\in\,[6.94; 12.69] $	$ \xi\,\in\,[5.55; 11.11] $	Never Stable
$ m=5 $	$ \xi\,\in\,[12.69; 18.11] $	$ \xi\,\in\,[11.11; 16.66] $	Never Stable
$ m=6 $	$ \xi\,\in\,[18.11; 23.39] $	$ \xi\,\in\,[16.66; 22.22] $	Never Stable
$ m=7 $	$ \xi\,\in\,[23.39; 28.59] $	$ \xi\,\in\,[22.22; 27.77] $	Never Stable
$ m=8 $	$ \xi\,\in\,[28.59; 33.75] $	$ \xi\,\in\,[27.77; 33.33] $	Never Stable
$ m=9 $	$ \xi\,\in\,[33.75; 38.88] $	$ \xi\,\in\,[33.33; 38.88] $	Never Stable
$ m=10 $	$ \xi >38.88 $	$ \xi >38.88 $	Never Stable
	Second Model
$ m=1 $
$ m=2 $			Stable
$ m=3 $	$ \xi\,\in\,[0.0000; 0.0005] $	$ \xi\,\in\,[0.0000; 0.0004] $	Stable
$ m=4 $	$ \xi\,\in\,[0.0005;0.0010] $	$ \xi\,\in\,[0.0004; 0.0009] $	Never Stable
$ m=5 $	$ \xi\,\in\,[0.0010; 0.0015] $	$ \xi\,\in\,[0.0009; 0.0014] $	Never Stable
$ m=6 $	$ \xi\,\in\,[0.0015; 0.0021] $	$ \xi\,\in\,[0.0014; 0.0019] $	Never Stable
$ m=7 $	$ \xi\,\in\,[0.0021; 0.0027] $	$ \xi\,\in\,[0.0019; 0.0026] $	Never Stable
$ m=8 $	$ \xi\,\in\,[0.0027; 0.0035] $	$ \xi\,\in\,[0.0026; 0.0034] $	Never Stable
$ m=9 $	$ \xi\,\in\,[0.0035; 0.0044] $	$ \xi\,\in\,[0.0034; 0.0044] $	Never Stable
$ m=10 $	$ \xi\,\in\,[0.0044; 2.4800] $	$ \xi\,\in\,[0.0044; 7.0100] $	Never Stable

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Table 2. Coalition Stability with $ p = 0.30 $

Coalition	$ \psi > 0 $	$ \psi=0 $	$ \psi=0 $
	$ \xi > 0 $	$ \xi > 0 $	$ \xi=0 $
	First Model
$ m=1 $
$ m=2 $			Stable
$ m=3 $	$ \xi\,\in\,[0.00; 15.62] $	$ \xi\,\in\,[0.00; 12.50] $	Stable
$ m=4 $	$ \xi\,\in\,[15.62; 28.57] $	$ \xi\,\in\,[12.50; 25.00] $	Never Stable
$ m=5 $	$ \xi\,\in\,[28.57; 40.76] $	$ \xi\,\in\,[25.00; 37.50] $	Never Stable
$ m=6 $	$ \xi\,\in\,[40.76; 52.63] $	$ \xi\,\in\,[37.50; 50.00] $	Never Stable
$ m=7 $	$ \xi\,\in\,[52.63; 64.33] $	$ \xi\,\in\,[50.00; 62.50] $	Never Stable
$ m=8 $	$ \xi\,\in\,[64.33; 75.94] $	$ \xi\,\in\,[62.50; 75.00] $	Never Stable
$ m=9 $	$ \xi\,\in\,[75.94; 87.50] $	$ \xi\,\in\,[75.00; 87.50] $	Never Stable
$ m=10 $	Never Stable ($ \Omega <0 $)	Never Stable ($ \Omega <0 $)	Never Stable ($ \Omega <0 $)
	Second Model
$ m=1 $
$ m=2 $			Stable
$ m=3 $	$ \xi\,\in\,[0.0000; 0.0012] $	$ \xi\,\in\,[0.0000; 0.0010] $	Stable
$ m=4 $	$ \xi\,\in\,[0.0012;0.0024] $	$ \xi\,\in\,[0.0010; 0.0021] $	Never Stable
$ m=5 $	$ \xi\,\in\,[0.0024; 0.0038] $	$ \xi\,\in\,[0.0021; 0.0034] $	Never Stable
$ m=6 $	$ \xi\,\in\,[0.0038; 0.0053] $	$ \xi\,\in\,[0.0034; 0.0049] $	Never Stable
$ m=7 $	$ \xi\,\in\,[0.0053; 0.0073] $	$ \xi\,\in\,[0.0049; 0.0069] $	Never Stable
$ m=8 $	$ \xi\,\in\,[0.0073; 0.0099] $	$ \xi\,\in\,[0.0069; 0.0096] $	Never Stable
$ m=9 $	$ \xi\,\in\,[0.0099; 0.0137] $	$ \xi\,\in\,[0.0096; 0.0134] $	Never Stable
$ m=10 $	Never Stable ($ \Omega' <0 $)	Never Stable ($ \Omega' <0 $)	Never Stable

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Table 3. Coalition Stability with $ p = 0.40 $

Coalition	$ \psi < 0 $	$ \psi=0 $	$ \psi=0 $
	$ \xi > 0 $	$ \xi > 0 $	$ \xi=0 $
	First Model
$ m=1 $
$ m=2 $			Stable
$ m=3 $	$ \xi\,\in\,[0.00; 27.77] $	$ \xi\,\in\,[0.00; 22.22] $	Stable
$ m=4 $	$ \xi\,\in\,[27.77; 50.79] $	$ \xi\,\in\,[22.22; 44.44] $	Never Stable
$ m=5 $	$ \xi\,\in\,[50.79; 72.46] $	$ \xi\,\in\,[44.44; 66.66] $	Never Stable
$ m=6 $	$ \xi\,\in\,[ 72.46; 93.56] $	$ \xi\,\in\,[66.66; 88.88] $	Never Stable
$ m=7 $	$ \xi\,\in\,[93.56; 114.37] $	$ \xi\,\in\,[88.88; 111.11] $	Never Stable
$ m=8 $	$ \xi\,\in\,[114.37; 135.02] $	$ \xi\,\in\,[111.11; 133.33] $	Never Stable
$ m=9 $	Never Stable ($ \Omega > 0 $)	Never Stable ($ \Omega > 0 $)	Never Stable
$ m=10 $	Never Stable ($ \Omega > 0 $)	Never Stable ($ \Omega > 0 $)	Never Stable ($ \Omega > 0 $)
	Second Model
$ m=1 $
$ m=2 $			Stable
$ m=3 $	$ \xi\,\in\,[0.0000; 0.0023] $	$ \xi\,\in\,[0.0000; 0.0018] $	Stable
$ m=4 $	$ \xi\,\in\,[0.0023; 0.0040] $	$ \xi\,\in\,[0.0018; 0.0040] $	Never Stable
$ m=5 $	$ \xi\,\in\,[0.0040; 0.0073] $	$ \xi\,\in\,[0.0040; 0.0066] $	Never Stable
$ m=6 $	$ \xi\,\in\,[0.0073; 0.0100] $	$ \xi\,\in\,[0.0066; 0.0100] $	Never Stable
$ m=7 $	$ \xi\,\in\,[0.0100; 0.0150] $	$ \xi\,\in\,[0.0100; 0.0147] $	Never Stable
$ m=8 $	$ \xi\,\in\,[0.0150; 0.0230] $	$ \xi\,\in\,[0.0147; 0.0224] $	Never Stable
$ m=9 $	Never Stable ($ \Omega' > 0 $)	Never Stable ($ \Omega' > 0 $)	Never Stable
$ m=10 $	Never Stable ($ \Omega' > 0 $)	Never Stable ($ \Omega' > 0 $)	Never Stable

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Table 4. Coalition Stability with $ p = 0.50 $

Coalition	$ \psi < 0 $	$ \psi=0 $	$ \psi=0 $
	$ \xi > 0 $	$ \xi > 0 $	$ \xi=0 $
	First Model
$ m=1 $
$ m=2 $			Stable
$ m=3 $	$ \xi\,\in\,[0.00; 43.40] $	$ \xi\,\in\, [0.00; 34.72] $	Stable
$ m=4 $	$ \xi\,\in\, [43.40; 79.36] $	$ \xi\,\in\, [34.72; 69.44] $	Never Stable
$ m=5 $	$ \xi\,\in\, [79.36; 113.22] $	$ \xi\,\in\, [ 69.44 ;104.16] $	Never Stable
$ m=6 $	$ \xi\,\in\, [113.22; 146.19] $	$ \xi\,\in\, [104.16; 138.88] $	Never Stable
$ m=7 $	$ \xi\,\in\, [146.19; 178.71] $	$ \xi\,\in\, [138.88; 173.61] $	Never Stable
$ m=8 $	Never Stable ($ \Omega > 0 $)	Never Stable ($ \Omega > 0 $)	Never Stable
$ m=9 $	Never Stable ($ \Omega > 0 $)	Never Stable ($ \Omega > 0 $)	Never Stable
$ m=10 $	Never Stable ($ \Omega > 0 $)	Never Stable ($ \Omega > 0 $)	Never Stable
	Second Model
$ m=1 $
$ m=2 $			Stable
$ m=3 $	$ \xi\,\in\, [0.0000; 0.0038] $	$ \xi\,\in\, [0.0000; 0.0030] $	Stable
$ m=4 $	$ \xi\,\in\, [0.0038; 0.0077] $	$ \xi\,\in\, [0.0030; 0.0066] $	Never Stable
$ m=5 $	$ \xi\,\in\, [0.0077; 0.0127] $	$ \xi\,\in\, [0.0066;0.0113 ] $	Never Stable
$ m=6 $	$ \xi\,\in\, [0.0127; 0.0197] $	$ \xi\,\in\, [0.0113; 0.0179] $	Never Stable
$ m=7 $	$ \xi\,\in\, [ 0.0197; 0.0317] $	$ \xi\,\in\, [ 0.0179; 0.0288] $	Never Stable
$ m=8 $	Never Stable ($ \Omega' > 0 $)	Never Stable ($ \Omega' > 0 $)	Never Stable
$ m=9 $	Never Stable ($ \Omega' > 0 $)	Never Stable ($ \Omega' > 0 $)	Never Stable
$ m=10 $	Never Stable ($ \Omega' > 0 $)	Never Stable ($ \Omega' > 0 $)	Never Stable

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