History-dependent optimization with applications to circular and sustainable economics

We propose a general history-dependent framework (GHDF), an intertemporal optimization framework in which the instantaneous reward function depends on the memory of (eventually all) the previous decisions. By considering a very general history-formation process, through the introduction of a memory function, we provide a wide embracing framework, keeping it being both tractable and interpretable within many diverse contexts and allowing to enrich the analysis by treating at the same time several kinds of history dependencies. An easily implementable history-dependent framework (EIHDF) is provided, a version of our general history-dependent framework (GHDF) for which the primer (i.e. instantaneous reward function, feasible set and history-formation process) are defined in a recursive way so that it is easily implementable while still as general as needed to be widely applicable. Taking into account the fact that the environment keeps in memory our activities and decisions, we further provide a general sustainable framework (GSF) which introduces a basis for future analysis in environmental and sustainable issues and encompasses many existing models in the environmental literature (including circular economy models). It is designed in a very amenable and flexible manner so that it can be adapted to many contexts and one can easily remove or add different effects that will be needed to be addressed. The mathematical results (including existence of a solution and dynamic programming tools) are derived as an application of our general history-dependent (GHDF) framework and can be directly used. As examples, we address the model by Morhaim and Ulus [37] and all its history-dependent and habit formation applications, as well as many environmental models. These include optimal management of natural resources, circular economy (CE) models (with or without recycling habits) and circular and cumulative causation (CCC) models, all of which being particular cases of our framework.