On time preference, rational addiction and utility satiation

A basic consumer problem with a unique good is considered, current consumption of this good influencing in a positive manner consumer intertemporal utility, while past consumption exerts a negative influence. Moreover, in the line of Fisher, a specification of preferences is retained so that the rate of time preference, assumes a long-run value - this means for a stationary consumption-path - that is non-monotonic as a function of consumption: impatience increases for low level of consumptions but decreases for higher ones. Such a framework allows for an integrated appraisal of addiction, satiation and the rate of time preference. It is shown that the emergence of an addiction phenomenon in the neighbourhood of an unsatiated long-run position exactly corresponds to letting the rate of time preference be an increasing function of past consumption habits. When addiction becomes sufficiently strong, the unsatiated stationary state becomes unstable and the satiated steady state becomes the only admissible stationary position.