Climate change and complexity: Accounting for singularities in environmental dynamics

The complexity of climate change and environmental dynamics requires us to account for the possibility of singularities in the occurrence of future events. Abrupt and disruptive phenomena, often associated with extreme or catastrophic processes, provide clear examples of why singularities may be relevant for policymaking when designing appropriate control measures. However, the classical theory of differential equations and thus of dynamic optimization does not apply when environmental complexity implies that the source of the temporal variation in a state variable is singular with respect to the Lebesgue measure. In this paper we thus develop a theory of dynamic optimization capable to handling singularities (i) by defining a suitable differential equation (based on a new concept of derivative with respect to an abstract measure) describing singular dynamics, and (ii) by following the calculus of variations approach to establish first-order type necessary conditions for an optimization problem in which the dynamic constraint is characterized by singularity properties. We present an economic application in the context of climate change in which the social planner aims at minimizing the social cost of adaptation over a finite time-horizon in the case in which the cumulative damages are highly irregular and concentrated. We show how our theory can effectively be used to determine the optimal policy response in such a complex framework and thus how it can support policymakers to effectively account for the presence of eventual singularities in environmental dynamics.