Alfredo Medio

Chaotic dynamics : theory and applications to economics

13. Chaotic Dynamics: Theory and Applications to Economics. By A. Medio. ISBN 0 521 39488 0. Cambridge University Press, Cambridge, 1995. 344 pp. £37.50 (hardbound), £15.75 (paperbound).

An explanation of international differences in education and workplace training

We develop a simple search equilibrium model of workplace training and education based on two features. First, investment in education improves job-related learning skills and reduces training costs burdened by firms. Second, firms with vacant skilled job slots can choose between recruitment from the market and training. Compared to Germany and Japan, the US has both a higher inflow rate into unemployment and a higher efficiency of the matching process. While the combined effort of these differences on the share of educated labor is ambiguous, the effect on the percentage of firms undertaking workplace training is to unambiguously reduce it.

Is a random walk the best exchange rate predictor

Abstract The paper discusses short-term exchange rate prediction, using the random walk hypothesis (RWH) as a benchmark to compare performances. After surveying some recent results in this field, the authors suggest a filtering-prediction method inspired by recent developments in nonlinear dynamical systems theory. The filtering of presumably noisy data is realized by means of a technique derived from Singular Spectrum Analysis (SSA) conveniently adapted to a nonlinear dynamics context. In particular, the authors develop a multichannel version of SSA. Filtered data are then used to perform an out-of-sample, short-term prediction, by means of a nonlinear (locally linear) method. This method is applied to exchange rate series of the major currencies and the predictions thus obtained are shown to outperform neatly those derived from the RWH. Finally, the application of a test recently developed by Mizrach confirms the statistical significance of the results.

On lags and chaos in economic dynamic models

Abstract This paper discusses the role of lags in dynamic economic models. Applications of discrete dynamical systems to economics are considered and the shortcomings of their (implicit) treatment of lags are criticized. An abstract, probabilistic view of lags is then provided, within which fixed delay lags are shown to be a special case. A basic equivalence is proved between: (i) a system with an indefinitely large number of agents, reacting to inputs with randomly gamma distributed, discrete lags, and (ii) a system with one single ‘representative‘ agent reacting to inputs with a continuous, multiple exponential lag. Finally, the paper analyses a single-loop feedback system cuopling a multiple exponential lag and a non-linearity of the one-hump type. The dynamic behavior of the system is studied by means of analytical and numerical methods, and the conditions for periodic and chaotic solutions are investigated.

Oscillations in optimal growth models

Abstract This paper discusses a model of optimal growth with non-zero discount rate. Most known results concern sufficient conditions for saddle-point ( SP ) property of the (unique) equilibrium point Here necessary and sufficient conditions for (local) SP are found which permits one to apply bifurcation theory. In particular, the paper considers bifurcation of periodic orbits from an equilibrium point by means of the Hopf theorem, thereby generalizing a result obtained by Benhabib and Nishimura for a special case. A nonconventional theory of the trade cycle may thus be based on very conventional assumptions. Finally, certain known ( SP ) stability conditions are discussed and related to the main result of the paper.

Continuous-time models of chaos in economics

Abstract In this paper, we shall discuss some applications of chaos theory to the study of continuous-time economic dynamic models, i.e., models represented by systems of ordinary differential equations. Two such applications will be considered. The first, discussed in Part I of the paper, is a continuous-time generalization of a class of non-linear, one-dimensional maps which encompasses the majority of existing economic models of chaotic dynamics. The second, discussed in Part II, is a model of inventory cycles of Keynesian inspiration, represented by a system of three differential equations, including a single ‘one-humped’ non-linearity. Two points will be given particular emphasis, namely: in Part I, the relationship between discrete- and continuous-time representation of economic phenomena; in both Parts I and II, the combined role of lags and non-linearity in generating chaotic output.

Discrete and continuous-time models of chaotic dynamics in economics

Abstract The paper discusses applications to economics of non-linear one-dimensional maps. The hypothesis implicit in a fixed delay lag is criticized and a continuous-time generalization of it is suggested. The resulting system of differential equations is investigated by means of analytical and numerical tools. A ‘symptomatology’ of chaos is developed and it is shown that the combination of non-linearity and exponential lags may indeed produce chaos in a continuous setting. It is also shown that the essential qualitative properties of the full flow can be captured by a ‘reconstructed 1-D map’.

Invariant Probability Distributions In Economic Models: A General Result

This paper discusses the asymptotic behavior of distributions of state variables of Markov processes generated by first-order stochastic difference equations. It studies the problem in a context that is general in the sense that (i) the evolution of the system takes place in a general state space (i.e., a space that is not necessarily finite or even countable); and (ii) the orbits of the unperturbed, deterministic component of the system converge to subsets of the state space which can be more complicated than a stationary state or a periodic orbit, that is, they can be aperiodic or chaotic. The main result of the paper consists of the proof that, under certain conditions on the deterministic attractor and the stochastic perturbations, the Markov process describing the dynamics of a perturbed deterministic system possesses a unique, invariant, and stochastically stable probability measure. Some simple economic applications are also discussed.

Introductory Notes on the Dynamics of Linear and Linearized Systems

In the following we provide terminology and concepts which are central to understanding the dynamical behavior of nonlinear systems. The first four sections are a necessarily very brief introduction to the dynamics of linear systems, in which we concentrate on those aspects most useful for acquiring a sense of the basic behaviours characterising systems of differential and difference equations. The last four sections introduce basic notions of stability, the linear approximation and the Hartman-Grobman Theorem, the use of the Centre Manifold Theorem, local bifurcation theory.*

IMPLICIT EQUILIBRIUM DYNAMICS

We discuss the problem known in economics as backward dynamics occurring in models of perfect foresight, intertemporal equilibrium described mathematically by implicit difference equations. In a previously published paper [ Journal of Economic Dynamics and Control 31 (2007), 1633–1671], we showed that by means of certain mathematical methods and results known as inverse limits theory it is possible to establish a correspondence between the backward dynamics of a noninvertible map and the forward dynamics of a related invertible map acting on an appropriately defined space of sequences, each of whose elements corresponds to an intertemporal equilibrium. We also proved the existence of different types of topological attractors for one-dimensional models of overlapping generations. In this paper, we provide an extension of those results, constructing a Lebesgue-like probability measure on spaces of infinite sequences that allows us to distinguish typical from exceptional dynamical behaviors in a measure–theoretical sense, thus proving that all the topological attractors considered in MR07 are also metric attractors. We incidentally also prove that the existence of chaos (in the Devaney–Touhey sense) backward in time implies (and is implied by) chaos forward in time.