Fashion cycle dynamics induced by agents' heterogeneity for generic bell-shaped attractiveness functions

ABSTRACT

We extend the exchange economy evolutionary discrete-time model with heterogeneous agents introduced in ‘Fashion cycle dynamics in a model with endogenous discrete evolution of heterogeneous preferences’, by A. Naimzada and M. Pireddu, appeared in Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 055907 (2018), by considering generic bell-shaped attractiveness functions for the two groups of agents, increasing for low visibility levels, but decreasing when the visibility of the group exceeds a given threshold value, due to a congestion effect. The reproduction level of a group is related to its attractiveness degree, which depends on its social visibility level, determined by the consumption choices of the agents in that group. We investigate the asymptotic heterogeneity among agents, finding that the global dynamics may differ according to the chosen functional form for the attractiveness. Nonetheless, the model is still able to reproduce the recurrent dynamic behaviour typical of the fashion cycle.

KEYWORDS:

• Heterogeneous agents
• evolution
• fashion cycle
• bifurcations
• multistability
• complex dynamics

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 37N40
• 39A60
• 91B55

Acknowledgments

The authors thank the anonymous Referees for the helpful and valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This occurs for instance with the parameter configuration we will deal with in Section 3 when considering $\mu =2$ rather than $\mu =6.5,$ i.e. for $vx=0.9,wx=0.2,vy=0.15,wy=2,\beta =0.1,\alpha =0.9,\mu =2,\hat{V}=0.8$ and letting $\sigma >0$ free to vary. On the other hand, for the attractiveness formulation in (20(20) ${\mathcal{A}}_{i,t}^{″}=f_2\left(\sigma d^2\right(V_{i,t},\hat{V}\left)\right)=1-\sigma (V_{i,t}-\hat{V}{)}^2,$(20) ) it holds that $\left(\mathrm{\partial }{\mathcal{A}}_{\alpha }\right(a)/\mathrm{\partial }\left(\sigma d_{\alpha }^2\right))\equiv -1$  and thus, in the case of a parabolic map, $\left(\mathrm{\partial }{\mathcal{A}}_{\alpha }\left(a\right)/\mathrm{\partial }\left(\sigma d_{\alpha }^2\right)\right){|}_{a=a^{\ast }}$ and $\left(\mathrm{\partial }{\mathcal{A}}_{\alpha }\left(a\right)/\mathrm{\partial }\left(\sigma d_{\alpha }^2\right)\right){|}_{a=a_j^{\ast }}$ do not depend on σ, and no issues arise.

2 We here consider polarized values for $v_x$ and $v_y,$ as both of them are positive but one is much larger than the other. Such case approximates those frameworks in which visibility and attractiveness are produced by the consumption of a single good. However, our results hold true also in the case of more balanced values for $v_x$ and $v_y.$