A General Equilibrium Evolutionary Model with Generic Utility Functions and Generic Bell-Shaped Attractiveness Maps, Generating Fashion Cycle Dynamics

Abstract

We propose a discrete-time exchange economy evolutionary model, in which two groups of agents are possibly characterized by heterogeneous preference structures. With respect to the classical Walrasian framework, in our setting the definition of equilibrium, in addition to utility functions and endowments, depends also on population shares, which affect the market clearing conditions. We prove that, despite such difference with the standard framework, for every economy and for each population shares there exists at least one equilibrium and we show that, for all population shares, generically in the set of the economies, equilibria are finite and regular. We then introduce the dynamic law governing the evolution of the population shares, and we investigate the existence and the stability of the resulting stationary equilibria. More precisely, we assume that 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 described via a generic bell-shaped map, increasing for low visibility levels, but decreasing when the visibility of the group 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 may reproduce the recurrent dynamic behavior typical of the fashion cycle, presenting booms and busts in the agents' consumption choices, and in the groups' attractiveness and population shares. We illustrate the emergence of fashion cycle dynamics in the case of Stone-Geary utility functions, which generalize the Cobb-Douglas utility functions, and for different formulations of the attractiveness maps, already considered in the literature.

The First Fundamental Theorem of Welfare in a General Equilibrium Evolutionary Setting

“…1 This work completes the analysis of the discrete-time exchange economy evolutionary model in Naimzada and Pireddu (2019b), in which two groups of agents are characterized by possibly different preference structures, described within a class of utility functions. The reproduction level of a group is there related to its attractiveness degree, which depends on the social visibility level, determined by the consumption choices of the agents in that group.…”

“…For the brevity's sake, we will not report any equation describing the share updating rule, referring the interested reader e.g. to (2.4) in Naimzada and Pireddu (2018), or to (15) in Naimzada and Pireddu (2019b), where a more general formulation of the objects involved is proposed. We recall that the same general formulation of the evolutionary mechanism can be found also in (5) in Naimzada and Pireddu (2019b), where however, like in Naimzada and Pireddu (2018), just the Cobb-Douglas utility functions were considered.…”

“…to (2.4) in Naimzada and Pireddu (2018), or to (15) in Naimzada and Pireddu (2019b), where a more general formulation of the objects involved is proposed. We recall that the same general formulation of the evolutionary mechanism can be found also in (5) in Naimzada and Pireddu (2019b), where however, like in Naimzada and Pireddu (2018), just the Cobb-Douglas utility functions were considered. In all cases, if shares are constant, the equilibrium price determined through the market clearing condition is constant, too, and consequently also the equilibrium consumption choices are constant.…”

“…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. We recall that the setting in Naimzada and Pireddu (2019b) is a generalization of that in Naimzada and Pireddu (2019a), in which only Cobb-Douglas utility functions were considered. The context in Naimzada and Pireddu (2019a) is in turn an extension of the framework in Naimzada and Pireddu (2018), where just one particular formulation of the attractiveness was taken into account.…”

“…The remaider of the paper is organized as follows. In Section 2 we present the model, which coincides with the setting in Naimzada and Pireddu (2019b), as well as the needed definitions. In Section 3 we prove the first theorem of welfare economics, in relation to both the static and the dynamic notion of equilibrium, and we make some concluding comments.…”