Article Contents
Article Contents

Social norms for the stability of international enviromental agreements

• * Corresponding author: Lucia Maddalena
• 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.

Mathematics Subject Classification: 91A10, 49L12.

 Citation:

• 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.

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

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

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

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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