UK Nonlinear News, August 2000

Attractors, bifurcations and chaos: nonlinear phenomena in economics

Tönu Puu

Reviewed by Alan Wilson

Springer-Verlag, Berlin and Heidelberg, (2000)

This book is essentially a new edition of the author's earlier book Nonlinear economic dynamics - but the author writes of this that it "has been so much changed that I felt it reasonable to give it a new title". The first chapter sets the scene and the following four focus on the core mathematics, taking in turn differential equations, partial differential equations, difference equations (or iterated maps) and bifurcation and catastrophe. Particular attention is paid to perturbation methods. The remaining eight chapters work through a range of economic examples.

The first chapter is in a way too short: it offers an historical perspective on dynamics in economics and explains the author's rationale in his choice of economic examples. He emphasizes the attachment of many economists to linear models with, in the dynamic case, "unique optimal trajectories being a cherished idea, as if society was a rocket and reaching the moon one single, unique as well as obvious, target". He commends scientists in other fields for understanding the importance of nonlinearities and multiple attractors and development paths. In Chapter 12, he acknowledges the importance of Brian Arthur's [see Arthur (1994) for a summary] recognition of the significance of positive returns to scale in economics (though not directly in those terms). It is the much-assumed law of diminishing returns which produces the convenience of unique solutions in most economic textbooks. Unfortunately for economists, the world is more complex!

Puu's treatment of the mathematics offers interesting insights. He illustrates key ideas by worked example - though they are more likely to be rooted in physics than economics. What he does achieve is an integration of algebraic and geometrical perspectives. The book is superbly illustrated and Puu is clearly one of those authors who can 'picture' his algebra.

In the economic chapters, he focuses on the representation of the classical models in the areas of monopoly (Joan Robinson), duopoly, oligopoly (Cournot), business cycles (Samuelson, Hicks, Harrod), international trade (Beckmann) and development (Harrod and Hotelling). The authors cited show the age of the models: Cournot, Nineteenth Century, the rest 1920s to 1950s. He deploys his functions based on truncated Taylor series to represent the models algebraically and in a framework in which different examples of dynamical behaviour can be explored. His basic method is to take the classical models and show by adding terms that multiple paths and bifurcations can be generated. As in the mathematics' chapters, he works through examples in detail so that, to an unusual extent, the student really can see the mathematics being worked out. Although his models often demonstrate chaotic behaviour, he does not really confront the argument put by Cohen and Stewart (1994), and by this reviewer (Wilson, 2000), that real social systems are unlikely to attain chaotic states.

Nonetheless, this is a rich book which will repay working through for many purposes. There are, perhaps, two fundamental weaknesses. First, the focus is almost entirely on economic models which can be described in terms of a relatively small number of variables. [Though he does cite Haaken's (1983) argument that "bifurcations in high dimensional systems are low dimensional.....[because].... the local stability of a systems in the neighbourhood of equilibrium is described by eigenvalues".] Secondly, when he introduces space, his focus is on continuous space representations - building on his earlier work with Martin Beckmann [e.g. Beckmann and Puu (1990)]. It would be argued by this author that the most interesting economic-geographic systems are large and complex and that the more interesting modelling questions arise from confronting these challenges. And secondly, that discrete space representations facilitate the mathematical modelling of such multi-region systems (Wilson, 2000). However, Puu deals with the micro foundations of such models and he is to applauded for the depth and intelligence he brings to bear on classical economic models.


Arthur, B. (1994) Increasing returns and path dependence in the economy, University of Michigan Press, Ann Arbor.
Cohen, J. and Stewart, I. (1994) The collapse of chaos: discovering simplicity in a complex world, Viking, London.
Haken, H. (1983) Advanced synergetics. Instability hierarchies of self-organising systems and devices, Springer-Verlag, Berlin and Heidelberg.M/DD>
Wilson, A. G. (2000) Complex spatial systems: the modelling foundations of urban and regional analysis, Prentice-Hall, Harlow

A listing of books reviewed in UK Nonlinear News is available.

UK Nonlinear News thanks Springer-Verlag for providing a review copy of this book.

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