A dynamic model of adaptive consumers with endogenous unimodal preferences

Abstract

We consider a discrete-time dynamic model to describe the repeated choices of adaptive consumers that at each time period adjust their request for a given good according to the discrepancies between the observed price and the fair price perceived on the basis of their utility function. Moreover, consumers’ preferences are endogenously modified on the basis of past consumption experience. The model considered is derived from the one proposed in D’Orlando and Rodano (2006), with a different assumption about the way the utility function changes according to the past consumption. In fact, in D’Orlando and Rodano (2006) the consumption preferences increase whenever past consumption increases, whereas in the model proposed in this paper a saturation effect is introduced, so that the same assumption holds for low and moderate past consumption, whereas current consumption decreases if the quantity consumed in the previous time period was too high. This leads to a unimodal preference function instead of an increasing one, which implies that the two-dimensional map, whose iteration represents the time evolution of the consumers’ choices, is transformed from an invertible map to a noninvertible one. Hence different global dynamic properties are obtained that influence the structure of the attractors and basins of attraction. These global dynamic features, described by the method of critical curves, interact with the property that the map is also characterized by the presence of a denominator that can vanish, giving rise to different kinds of singularities denoted as focal points and prefocal curves in Bischi et al. (1999), Bischi et al. (2003), Bischi et al. (2005), that strongly influence the structure of the basins of attraction. We describe the structure of the basins of attraction, and the contact bifurcations that change the qualitative properties of their global structure, by using geometric and numerical methods guided by the combined study of critical and prefocal curves.