On History-Dependent Optimization Models : A Unified Framework to Analyze Models with Habits, Satiation and Optimal Growth

In this paper, we provide a framework for history-dependent utility models. Habit formation is a particular case of history-dependence and there is not yet any consensus on the way to formalize habits. Our framework allows to study not only habit models that are either additive or multiplicative or neither but also satiation models and other specifications. Such history-dependent modellings have their motivations and applications in many areas among which decision theory, finance, behavioral and environmental economics. We further consider discrete time infinite horizon dynamic optimization programs in which the instantaneous utility is history-dependent. In this unified setting one can deal with the usual optimal growth models with or without habit formation and with or without satiation effects. The way that the history-dependence is formalized allows to use dynamic programming tools. We show that the value function is the unique fixed point of the Bellman operator