A Review of Tipping Points and Precaution using HJB equations

Abstract

This paper analyzes three models in environmental economics with the property that tipping can occur in the ecological part of the model: the pollution control model where tipping suddenly shifts up the damage, the fishery model where tipping suddenly lowers the carrying capacity, and the Ramsey growth model where tipping suddenly decreases the total factor productivity. The question is whether the possibility of tipping gives rise to precaution in the sense of lower production, lower harvest, or higher accumulation of capital, respectively. The paper shows that precaution always results in the pollution control model but in the other two models, it depends on the elasticity of intertemporal substitution. In these models, the tipping point is uncertain, and the analysis employs a hazard rate that can be constant or state dependent. The paper shows that the Hamilton-Jacobi-Bellman equations are the appropriate framework to implement this hazard rate. The results are mostly known, but this paper puts the results together in a transparent way, using the HJB equations consistently. In this way, it pays tribute to this technique in optimal control that proves to be useful when tipping points can occur. Extensions to differential games are discussed at the end of each section.