Some reflections on past and future of nonlinear dynamics in economics and finance

This paper offers an overview of the literature on the economic and financial applications of theory of nonlinear dynamics, especially bifurcation theory. After a short introductory discussion of the first nonlinear dynamic models in social sciences and the economic relevance of the zoo of bifurcations and complicated dynamics that such models can generate, we present an overview of the literature on nonlinear dynamic models in the areas of underdevelopment, environmental poverty traps, the management of common goods, industrial organization and financial markets. The review of the literature is enriched by reflections and ideas for future research.     

Nonlinear dynamics in economic modelling

Decisions in Economics and Finance -     

Heterogeneous duopoly with isoelastic demand function

In this paper we analyze a duopolistic market with heterogeneous firms when the demand function is isoelastic (Puu, T., 1991. Chaos in duopoly pricing. Chaos, Solitons and Fractals 1, 573–581.). We consider the same heterogeneous firms of Zhang et al. (Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148.) introducing a nonlinearity in the demand function instead of the cost function. Stability conditions of the Nash equilibrium and complex dynamics are studied. In particular we show two different routes to complicated dynamics: a cascade of flip bifurcations leading to periodic cycles (and chaos) and the Neimark-Sacker bifurcation which originates an attractive invariant closed curve. Comparisons with respect to the Puu model and the model of Zhang et al. are performed.     

One-dimensional maps with two discontinuity points and three linear branches: mathematical lessons for understanding the dynamics of financial markets

We develop a simple financial market model with heterogeneous interacting speculators. The dynamics of our model is driven by a one-dimensional discontinuous piecewise linear map, having two discontinuity points and three linear branches. On the one hand, we study this map analytically and numerically to advance our knowledge about such dynamical systems. In particular, not much is known about discontinuous maps involving three branches. On the other hand, we seek to improve our understanding of the functioning of financial markets. We find, for instance, that such maps can generate complex bull and bear market dynamics.     

Dynamic properties of a Cournot–Bertrand duopoly game with differentiated products

In this paper we consider a Cournot–Bertrand duopoly model with linear demand and cost functions and with product differentiation. We propose a dynamic framework for the study of the stability properties of this kind of mixed oligopoly game, a rather neglected topic in the existing literature despite its relevance. In particular, in this paper we highlight the role of best response dynamics and of an adaptive adjustment mechanism for the stability of the equilibrium.     

Economics as a compartmental system: a simple macroeconomic example

The application in economics of methods and tools originally thought for other disciplines is not new. In this paper, we show how the compartmental approach, typical of Biology, Medicine, Chemistry, etc., can be used as a tool to represent the heterogeneity of the agents and introduce it into the dynamic models. We also show that in some cases, the subdivision of the agents into compartments can be useful for the government in order to obtain some results more efficiently. In this paper, the compartmental approach is applied to a macroeconomic model characterized by firms that may go bankruptcy whenever they are unable to repay their debts. Firms are grouped by size, and the consequences of these assumptions are analyzed, especially those regarding the policy implications.     

Endogenous lifetime,accidental bequests and economic growth

This paper introduces the concept of unintentional bequests in a closed economy à la Chakraborty (J Econ Theory 116:119–137, 2004) with overlapping generations. We show that scarce public investments in health can lead to poverty traps depending on the relative size of the output elasticity of capital. More importantly, the existence of unintentional bequests, rather than a market for annuities, means that health tax rates play a prominent role in determining the stability of the long-term equilibrium in rich economies. In fact, Neimark–Sacker bifurcations and endogenous fluctuations occur depending on the size of the public health system.     

Forward and backward dynamics in implicitly defined overlapping generations models

In dynamic economic models derived from optimization principles, the forward equilibrium dynamics may not be uniquely defined, while the backward dynamics is well defined. We derive properties of the global forward equilibrium paths based on properties of the backward dynamics. We propose the framework of iterated function systems (IFSs) to describe the set of forward equilibria and apply the IFS framework to a one- and a two-dimensional version of the overlapping generations (OLG)-model. We show that if the backward dynamics is chaotic and has a homoclinic orbit (a “snap-back repeller”), the set of forward equilibrium paths converges to a fractal attractor. Forward equilibria may be interpreted as sunspot equilibria, where a random sunspot sequence determines equilibrium selection at each date.     

Uncertainty about fundamental,pessimistic and overconfident traders: a piecewise-linear maps approach

Decisions in Economics and Finance - We analyze a financial market model with heterogeneous interacting agents where fundamentalists and chartists are considered. We assume that fundamentalists are...