Natural cycles and pollution

In this paper, we study a competitive economy where a pollution externality, coming from production, impairs the renewable resource affecting the consumption demand in turn. A proportional tax, levied on the production level, is introduced to finance public depollution expenditures. In the long run, two steady states can coexist, the one with a lower resource level, the other with a higher level. Interestingly, a higher green tax rate reduces the natural resource in the low steady state, giving rise to a Green Paradox (Sinn, 2008). Moreover, the green tax can be welfare-improving in the higher steady state but never in the lower one. Therefore, in the second one, it is better to reduce the green tax rate as much as possible. Conversely, the optimal tax rate is positive and unique in the steady state with more natural resource. In the short run, the two steady states can collide and disappear through a saddle-node bifurcation. Since consumption and natural resource are substitutable goods, a limit cycle can arise around the higher stationary state. To the contrary, this kind of cycles never occurs around the lower steady state, no matter the resource effect on consumption. Finally, focusing on the variety of bifurcations of codimension two, we find a Bogdanov-Takens loop.