Fashion cycle dynamics in a model with endogenous discrete evolution of heterogeneous preferences

We propose a discrete-time exchange economy evolutionary model, in which two groups of agents are characterized by different preference structures. The reproduction level of a group is related to its attractiveness degree, which depends on the social visibility level, determined by the consumption choices of the agents in that group. The attractiveness of a group is initially increasing with its visibility level, but it becomes decreasing when its visibility exceeds a given threshold value, due to a congestion effect. Thanks to the combined action of the price mechanism and of the share updating rule, the model is able to reproduce the recurrent dynamic behavior typical of the fashion cycle, presenting booms and busts both in the agents' consumption choices and in the population shares. More precisely, we investigate the existence of equilibria and their stability, and we perform a qualitative bifurcation analysis on varying the parameter describing the group's heterogeneity degree. From a global viewpoint, we detect, among others, multistability phenomena in which the group coexistence is dynamic, either regular or irregular, and the fashion cycle occurs. The existence of complex dynamics is proven via the method of the turbulent maps, working with homoclinic orbits. Finally, we provide a social and economic interpretation of the main scenarios.