Ahmad Naimzada

Nonlinear dynamics of a Cournot duopoly game with differentiated products

In this paper we investigate the dynamics of a Cournot duopoly game with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update the quantity produced in each period. As in Ahmed et al. (2015), the demand functions are derived from an underlying CES utility function. The present analysis reveals that a higher degree of product differentiation may destabilize the Nash equilibrium. Through local analysis we provide conditions for the stability of the market equilibrium and through global analysis we investigate some bifurcations which cause qualitative changes in the structure of the attractors and of their basins as some parameters are allowed to vary. Since a higher degree of product differentiation tends to reduce competition and may generate undesirable fluctuations, an implication of our findings is that a stronger competition could help in stabilizing the unique Nash equilibrium.

A Cournot duopoly game with heterogeneous players

We analyze a duopolistic Cournotian game with firms producing a homogeneous good, isoelastic demand function and linear total cost functions. In this economic setting, the traditional dynamic adjustment based on the classical best reply mechanism is very demanding in terms of rationality and information set. Therefore, in the competition we study, both the players adopt decisional mechanisms which are based on a reduced degree of rationality, being the agents supposed to have only limited informational and computational capabilities. We assume that the first player adopts a gradient rule mechanism, while the second one adjusts his output level according to a Local Monopolistic Approximation. We provide local stability conditions in terms of marginal costs ratio and complex dynamics are investigated. In particular, we show that two different routes to complicated dynamics are possible: a cascade of flip bifurcations leading to periodic cycles (and chaos) and the Neimark-Sacker bifurcation, which results in an attractive invariant closed curve.

Heterogeneous Fundamentalists and Imitative Processes

Developing a model with a switching mechanism, we show how complex dynamics can be generated even though heterogeneity arises among agents with the same trading rules (fundamentalists). We assume that there are two experts which are imitated by other operators. We show that (i) market instability and periodic, or even, chaotic price fluctuations can be generated; (ii) conditions exist under which an expert can drive another expert out of the market; (iii) two experts can survive when the dynamic system either generates a period doubling bifurcation around an attractor or when an homoclinic bifurcation leads to the merging of the two attractors (i.e. Dieci et al., 2001); (iv) a central role is played by the reaction to misalignment of both market makers and agents.

A multiscale time model with piecewise constant argument for a boundedly rational monopolist

We present a dynamic model for a boundedly rational monopolist who, in a partially known environment, follows a rule-of-thumb learning process. We assume that the production activity is continuously carried out and that the costly learning activity only occurs periodically at discrete time periods, so that the resulting dynamical model consists of a piecewise constant argument differential equation. Considering general demand, cost and agent’s reactivity functions, we show that the behavior of the differential model is governed by a nonlinear discrete difference equation. Differently from the classical model with smooth argument, unstable, complex dynamics can arise. The main novelty consists in showing that the occurrence of such dynamics is caused by the presence of multiple (discrete and continuous) time scales and depends on size of the time interval between two consecutive learning processes, in addition to the agent’s reactivity and the sensitivity of the marginal profit.

Two different routes to complex dynamics in an heterogeneous triopoly game

We study a triopoly game with heterogeneous players. The market is characterized by a nonlinear (isoelastic) demand function and three competitors. The main novelty is the double route to complex dynamics that we find and is quite rare in heterogeneous triopoly models. We show that the two routes have important implications for the economic interpretation of the dynamics emerging when the Cournot–Nash equilibrium becomes locally unstable. Moreover the model displays multistability of different attractors, requiring a global analysis of the dynamical system.

Introducing a price variation limiter mechanism into a behavioral financial market model.

In the present paper, we consider a nonlinear financial market model in which, in order to decrease the complexity of the dynamics and to achieve price stabilization, we introduce a price variation limiter mechanism, which in each period bounds the price variation so that the current price is forced to belong to a certain interval determined by the price realization in the previous period. More precisely, we introduce such mechanism into a financial market model in which the price dynamics are described by a sigmoidal price adjustment mechanism characterized by the presence of two asymptotes that bound the price variation and thus the dynamics. We show that the presence of our asymptotes prevents divergence and negativity issues. Moreover, we prove that the basins of attraction are complicated only under suitable conditions on the parameters and that chaos arises just when the price limiters are loose enough. On the other hand, for some suitable parameter configurations, we detect multistability phenomena characterized by the presence of up to three coexisting attractors.

Effects of Size, Composition, and Evolutionary Pressure in Heterogeneous Cournot Oligopolies with Best Response Decisional Mechanisms

We study heterogeneous Cournot oligopolies of variable sizes and compositions, in which the firms have different degrees of rationality, being either rational firms with perfect foresight or naive best response firms with static expectations. Each oligopoly can be described using its size and composition, that is, the fraction of firms that are rational. We take into account two frameworks, one in which the decisional rules are exogenously assigned and the other in which the firms may change their heuristics. We consider a switching mechanism based on a logit rule, where the switching propensity is regulated by a parameter which represents the evolutionary pressure. In the fixed fractions setting, we prove that, in general, the composition has a stabilizing effect, while increasing the oligopoly size leads to instability. However, we show that, for particular parameters settings, stability is not affected by the composition or the firms number. Similarly, in the evolutionary fractions setting, we analytically prove that when marginal costs are identical, increasing the evolutionary pressure has a destabilizing effect. Nevertheless, focusing on particular examples with different marginal costs we are able to show that evolutionary pressure may also have a stabilizing or a neutral role.

Complex Dynamics in an Evolutionary General Equilibrium Model

We propose an exchange economy evolutionary model with discrete time, in which there are two utility-maximizing groups of agents which differ in the preference structure. Assuming an evolutionary mechanism based on the relative utility values realized by the two kinds of agents, we analytically and numerically investigate the existence of equilibria, their stability, and possible phenomena of coexistence between groups, mainly in terms of the heterogeneity degree in the preference structure. We find that our system has two trivial equilibria, at which just one of the two groups is present, and possibly a nontrivial equilibrium, characterized by the coexistence of the two groups of agents. Such nontrivial equilibrium may be stable, attracting all trajectories, or unstable. In the latter case, interesting, periodic, or chaotic, dynamics arise. We prove that the nontrivial equilibrium emerges via a transcritical bifurcation and loses stability via a flip bifurcation, after which the coexistence between groups is oscillatory in nature, presenting a regular or irregular behavior. In order to better investigate the role of the heterogeneity degree parameter, we perform a bifurcation analysis considering different scenarios, characterized by a balanced or unbalanced endowment distribution of the two goods.

Adaptive decision dynamics: Bifurcations, multistability and chaos

Abstract In this paper we propose a model describing the dynamical process of decision and opinion formation of two economic homogeneous interacting and boundedly rational agents. The decisional process represented in our model is given by an adaptive adjustment mechanism in which two agents take into account the difference between their own opinion and the opinion of the other agent. The smaller that difference, the larger the weight given to the comparison of the opinions. By means of an auxiliary variable describing the distance between the opinions, we obtain a one-dimensional dynamical system for which we investigate, via analytical and numerical tools, the stability of the unique steady state, its bifurcations, as well as the existence of a globally absorbing interval and of chaotic dynamics. We also investigate multistability phenomena, i.e., the presence of coexisting attractors. Finally, we relax the assumption of homogeneity between agents and we show that there is a strong correspondence between the dynamic behaviors in the scenarios with and without homogeneity.

Multiple Attractors and Nonlinear Dynamics in an Overlapping Generations Model with Environment

This paper develops a one-sector productive overlapping generations model with environment where a CES technology is assumed. Relying on numerical and geometrical approaches, various dynamic properties of the proposed model are explored: the existence of the phenomenon of multistability or the coexistence of different attractors was demonstrated. Finally, we describe a nontypical global bifurcation which determines the appearance of an attracting cycle.