Symmetry-breaking bifurcations and representative firm in dynamic duopoly games

In this paper, we investigate the question of whether the assumption of the ‘representative agent’, often made in economic modeling, is innocuous or whether it may be misleading under certain circumstances. In order to obtain some insight into this question, two dynamic Cournot duopoly games are considered, whose dynamics are represented by discrete-time dynamical systems. For each of these models, the dynamical behavior of the duopoly system with identical producers is compared to that with quasi-identical ones, in order to study the effects of small heterogeneities between the players. In the case of identical players, such dynamical systems become symmetric, and this implies that synchronized dynamics can be obtained, governed by a simpler one-dimensional model whose dynamics summarizes the common behavior of the two identical players. In both the examples, we show that a negligible difference between the parameters that characterize the two producers can give dynamic evolutions that are qualitatively different from that of the symmetric game, i.e. a breaking of the symmetry can cause a noticeable effect. The presence of such bifurcations suggests that economic systems with quasi-identical agents may evolve quite differently from systems with truly identical agents. This contrasts with the assumption, very common in the economic literature, that small heterogeneities of agents do not matter too much.