Anna Agliari

The dynamics of a triopoly Cournot game

This paper reconsiders the Cournot oligopoly (noncooperative) game with iso-elastic demand and constant marginal costs, one of the rare cases where the reaction functions can be derived in closed form. It focuses the case of three competitors, and so also extends the critical line method for non-invertible maps to the study of critical surfaces in 3D. By this method the various bifurcations of the attractors and their basins are studied. As a special case the restriction of the map to an invariant plane when two of the three firms are identical is focused.

Global Bifurcations In Duopoly When The Cournot Point Is Destabilized Via A Subcritical Neimark Bifurcation

An adaptive oligopoly model, where the demand function is isoelastic and the competitors operate under constant marginal costs, is considered. The Cournot equilibrium point then loses stability through a subcritical Neimark bifurcation. The present paper focuses some global bifurcations, which precede the Neimark bifurcation, and produce other attractors which coexist with the still attractive Cournot fixed point.

Some global bifurcations related to the appearance of closed invariant curves

In this paper, we consider a two-dimensional map (a duopoly game) in which the fixed point is destabilized via a subcritical Neimark-Hopf (N-H) bifurcation. Our aim is to investigate, via numerical examples, some global bifurcations associated with the appearance of repelling closed invariant curves involved in the Neimark-Hopf bifurcations. We shall see that the mechanism is not unique, and that it may be related to homoclinic connections of a saddle cycle, that is to a closed invariant curve formed by the merging of a branch of the stable set of the saddle with a branch of the unstable set of the same saddle. This will be shown by analyzing the bifurcations arising inside a periodicity tongue, i.e., a region of the parameter space in which an attracting cycle exists.

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.

Global dynamics in a non-linear model of the equity ratio

A model for firms’ financial conditions is proposed, which ultimately reduces to a two-dimensional non-invertible map in the variables mean and variance of the equity ratio. The possible dynamics of the model and the global behaviour are investigated. We describe the mechanism of bifurcations leading to fractalization of the basins and/or fractalization of their boundaries, showing how a locally stable attractor may be almost globally unstable. Multistability is also investigated. Two, three or four co-existing attractors have been found and we describe the mechanism of bifurcations leading their basins to become chaotically intermingled, and thus to unpredictability of the asymptotic state in a wide region. The knowledge of such regimes, besides those associated with simple dynamics, may be of help for the operators. While the use of the technical tools we propose to study the global dynamics and bifurcations may be of help for further investigations. ” 2000 Elsevier Science Ltd. All rights reserved. In this paper we present a model of fluctuating growth in which firms’ financial conditions play a crucial role. Our analysis starts from the distribution of firms according to their equity ratio, that is the ratio of the equity base or net worth to the capital stock, a proxy of financial robustness. We identify two dynamic laws for the mean and the variance of this distribution. The motion over time of the average equity ratio is the engine of growth and fluctuations. The dynamic pattern of the dispersion of the distribution, captured by the variance, however, interacts with evolution of the average equity ratio. Given the non-linear nature of the map which describes the laws of motion of the mean and the variance of the equity ratio, a wide range of dynamic patterns are possible. Fixed points or periodic orbits, attracting closed invariant curves and thin annular chaotic areas wide chaotic areas or explosions may occur. For quite plausible values of the parameters which characterize the map, the dynamics of the equity ratio can be regular or chaotic, and the motion of capital and output can be characterized as a process of fluctuating growth, although, as we shall see, often very sensitive to small perturbations. The goal of the present paper is to show how the global properties (deriving from the structure of the basins, their boundaries and the critical curves of non-invertible two-dimensional maps) may be used to understand the dynamic behaviour of the model, especially when analytical results are not accessible, as in our case, where not only the equilibrium values, but also the number of existing fixed points, cannot be explicitly known.

Original article: Endogenous cycles in discontinuous growth models

In this paper we consider a discontinuous one-dimensional piecewise linear model describing a neoclassical growth model. These kind of maps are widely used in the applied context. We determine the analytical expressions of border collision bifurcation curves, responsible for the observed dynamics, which consists of attracting cycles of any period and of quasiperiodic trajectories in exceptional cases.

Some Methods for the Global Analysis of Closed Invariant Curves in Two-Dimensional Maps

It is well known that models of nonlinear oscillators applied to the study of the business cycle can be formulated both as continuos or discrete time dynamic models (see e.g. [23], [33], [34]). However, economic time is often discontinuous (discrete) because decisions in economics cannot be continuously revised. For this reason discrete-time dynamical systems, represented by difference equations or, more properly, by the iterated application of maps, are often a more suitable tool for modelling dynamic economic processes. So, it is useful to study the peculiarities of discrete dynamical systems and their possible applications to the study of self sustained oscillations. This is the main goal of this Chapter, where we describe, on the light of some recent results about local and global properties of iterated maps of the plane, some particular routes to the creation/destruction of closed invariant curves, along which self sustained oscillations occur.

A Cournot duopoly with bounded inverse demand function

In 1991 Puu proposed a Cournot duopoly model based on an iso-elastic demand function and constant marginal costs for the competitors. These are probably the simplest economic assumptions under which an oligopoly game easily leads to complex dynamics, as suggested by Rand in 1978. In fact a period-doubling cascade towards chaos was observed by Puu. Further, the extension of the model to cases with three competitors leads to more complex dynamics, such as Neimark-Hopf bifurcations of subcritical types, and multistability, i.e. coexistence of the Cournot equilibrium point with an attracting invariant closed curve. See Agliari et. al. 2000 and Puu 2000.

Homoclinic and Heteroclinic Bifurcations in an Overlapping Generations Model with Credit Market Imperfection

We investigate Matsuyamas (Econometrica, 72, pp. 853-84, 2004) model modi- fied only to include endogenous and forward looking labor supply decision. Young agents supply one unit of labor endowment elastically to a competitive labor market. While, old agents of ex-ante identical individuals are divided in equi- librium into depositors and entrepreneurs. Depositors lend funds in the form of interest bearing loans, while entrepreneurs borrow funds in the competitive credit market. We emphasize the interaction between credit and labor markets and show the possibility of occurrence of multiple steady states, local and global indeterminacy, and endogenous fluctuations. When young agents become optimistic about the future deposit rate then they decide to work harder and invest more. Countercyclical borrowing constraint will help agents to fulfill their initial optimistic expectations, because the next period credit volume and deposit rate can increase simultaneously. By conducting global bifurcation analysis, we show that credit cycles can occur through a self- fulfilling expectation mechanism. History-versus-expectations considerations can exist and escape from underdevelopment as well as fall into poverty can to be a self-fulfilling prophecy.

On the Fractal Structure of Basin Boundaries in Two-Dimensional Noninvertible Maps

In this paper we give an example of transition to fractal basin boundary in a two-dimensional map coming from the applicative context, in which the hard-fractal structure can be rigorously proved. That is, not only via numerical examples, although theoretically guided, as often occurs in maps coming from the applications, but also via analytical tools. The proposed example connects the two-dimensional maps of the real plane to the well-known complex map.