CHAOS THEORY, NONLINEARITIES, AND ECONOMICS: A SPECULATIVE NOTE

by JOSHUA GANS*

Introduction’ Science has accomplished much in its pursuit of an understanding of

nature and the prediction of natural phenomena. Despite its considerable success, however, scientific endeavour has never quite achieved absolute perfection. There has always been some noise in the results of experiments and some errors in prediction. Usually, however, these have been dismissed as unavoidable and unpredictable randomness and therefore, provided they remained small, unimportant and worthy of being averaged out of focus. Even though success may be forthcoming with such approaches there comes a time when anomalies do become important and require some thought. For example, anomalies have been the spur of the relativity and quantum mechanics revolutions in the twentieth century. Today, scientists are once again taking a fresh look at anomalies and supposed randomness.

In an information ridden world where simple catchcries and T-shirt slogans are the most effective communication devices, the new “revolution” and approach in the natural sciences has been dubbed “chaos”. Chaos is an interesting and provocative name, for when one usually thinks of science one imagines order and control. Chaos theory is really no different in this respect for it addresses the idea that apparently random and complex phenomena could actually be determined by fundamental regularities. Thus, it asserts that what may seem like randomness and chance to the naked eye may actually be order in that it is representable mathematically. New developments in mathematics have allowed scientists in almost every field to turn back to the anomalies, previously dismissed as too random and complex for proper scientific analysis, and make practical use of them.

Economics has long been influenced by developments in the natural sciences. From physics, Classical and Neoclassical economics borrowed the concept of equilibrium and its complementary mathematics based on differential calculus in order to represent the perceived regularities and adjustment mechanisms of the market. And these tools were invaluable for

* Department of Economics, University of Queensland. 1. Thanks must be given to Professor Clem Tisdell, University of Queensland, and Dr Chee-

Wah Cheah. CIRCIT, who shared their thoughts on chaos theory with me. The views expressed in this paper, however, are solely mine.

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defining the characteristics of allocative efficiency in an economy with scarce resources.z

It would be an understatement to say that the Neoclassical approach has ever reached perfection even closely approximating the accomplishments of the natural sciences. Economic time series are characterised by random- ness around a trend. Even where good statistical fits have been found (such as with the aggregate consumption function), accurate predictions have not emerged. Such lack of success in explaining economic data has led to a school of economics which, using efficient market or rational expectations hypotheses, has attempted to explain randomness as being consistent with the behaviour of rational agents. The problem, however, with this form of analysis is that it is virtually impossible to formulate any policies to improve the performance of the economy and thus, it is not surprising that hardly any policies have emerged from this scho01.~

Here is where chaos theory has made its recent inroads into economics. The purpose of this paper is to consider the influence of chaos theory on economics and speculate upon its potential usefulness in policy formulation. The formal economics literature regarding chaos applications has grown considerably in the last decade. As will be argued in this paper, however, the influence of what is now known as chaos theory has existed in economics for some time, in some cases long before its emergence in the natural sciences. These applications, whilst often acknowledged, have not found their way into the textbooks, despite their simplicity.

What is chaos theory? Before turning to the economic applications of chaos theory it is

appropriate to outline, in general terms, just what chaos theory entails and discuss several concepts that fall within its scope.4 Chaos theory developed from a number of sources in very different branches of the natural sciences including meteorology, biology, chemistry and fluid mechanics. As such, it can probably best be described as a “family of models” (Schelling, 1978) whose common link was a mathematical structure which was nonlinear in n a t ~ r e . ~ This is in contrast to traditional science which has employed linear

2. The causal direction of this has not always been one way. Malthus had a significant influence on Darwin. More recently, some economists have attempted to go further, with mixed success-see Rugina (1989).

3. Blinder (1986), Thurow (1983) and Arrow (1978) provide excellent discussions regarding the policy imperceptiveness of the rational expectations school.

4. More detailed reviews of chaos theory that are not technical are provided by Hofstadter (1981), Prigogine and Stengers (1984). Davies (1987) and Gleick (1988).

5. Chaos theory shares some common characteristics with another “family of models“ in mathematics, catastrophe theory. Developed by Rene Thom, catastrophe theory “is a branch of topology which addresses discontinuous changes in natural phenomenon, and classifies them into distinct types” (Davies. 1987, p. 104). Chaos theory makes considerable use of bifurcation processes which Thom developed in detail as a part of his work; hence the parallels often drawn between the two theories.

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techniques (primarily, differential calculus) to yield neat mathematical solutions and highly accurate predictions, A person was placed on the moon using linear techniques. This approach was very convenient for dealing with behaviour of averages that had a smooth and continuous nature. But not all of nature was linear. In fact, according to the pioneering chaos theorists, much of what now should be significant to scientists was fundamentally nonlinear and subject to forces that were irregular and complex.

Mathematically, a system is nonlinear if its basic mathematical law cannot be represented as a straight line in any dimensional space. In such a system the whole is greater than the additive sum of its parts. This being the case nonlinear systems are far more difficult to analyse than their linear counterparts. Change within the system does not occur proportionally since its components interrelate in a complex way. In contrast to linear systems they “cannot be reduced or analysed in terms of simple subunits acting together. The resulting properties can often be unexpected, complicated and mathematically intractable” (Davies, 1987, p. 25).6 Indeed, given the practical difficulties of working with nonlinear systems, it is hardly surprising that it was not until the information technology revolution that choas theory was able to emerge at all. But when these systems were experimented with, on a mathematical level, several important implications become apparent.

It is beyond the scope of this paper to consider in detail the mathematics of nonlinear systems, that task has been comprehensively treated e1sewhere.l Here some of the basic properties of nonlinear systems will be used as springboards for discussing the impact chaos has had on economics.

Nonlinear economic dynamics The idea that a simple equation, algorithm or process can generate

complex results has been the cornerstone of chaos theory. The name “chaos” derives from this. To convey this feature in simple terms one of the simplest nonlinear equations known as the logistic curve has been used. This curve has its origins in biology where the reasonable hypothesis was made by May that the population of a species in the coming year was dependent upon current numbers. This created a feedback loop which in its simplest specifications took the following form:

x(t + 1) = ax(t)(l - x(t)) where x is the population in a given time period, and a is a fixed parameter. For low levels of a (i.e. 0 < a < l), the population declines to zero. At slightly higher levels of a (i.e. 1 <a < 21, the population converges smoothly to a steady or equilibrium level. Raising the parameter beyond this yields dampened oscillations (2 < a < 3), a two period limit cycle (3 < a < 3.449), a four period

6. It must be emphasised, however, that the holistic approach should not be adhered to in a metaphysical sense, only in a practical one (see Simon, 1981; Hofstadter and Dennett, 1981, for a detailed discussion of this aspect of complex systems).

7. See Eckmann and Ruelle (1985).

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limit cycle, an eight period cycle and so on until apparent chaos sets in at values of a above 3.5699 (although a three period cycle is apparent for a short time after this value). In the chaotic range it is possible that the time path generated would pass known tests of randomness, even though we know that the path is completely determined by the simple logistic equation above. The mathematics and more detailed discussions of this phenomenon are comprehensively reviewed in Baumol and Benhabib (1989), Savit (1988), and Gabisch (1987). For the purposes of this paper the point to be emphasised is that chaotic time paths are a potential property of all nonlinear systems.

The idea that complex dynamic time paths could be caused by simple deterministic equations has had an immediate effect on economics.8 After all, economic instability and fluctuations in prices and quantities have been the focus of much econometric work since the War. The current trend, as was mentioned above, is to treat economic instability as being caused by exogenous, stochastic factors. This is not a new trend. W. Stanley Jevons once pinned the blame for economic fluctuations on sunspots. Nonlinear dynamics of the chaotic variety has brought “the quest for motivating factors of business cycles . . . to more earthbound considerations” (Heilbroner, 1980, p. 260). This is because chaos implies that what may previously be con- sidered as random may in fact have a simple underlying explanation. As such, it is likely that economic fluctuations are created by endogenous mechanisms. This is certainly not a new idea. The models of Keynes, Hicks, Harrod, Wicksell, Kaldor, Hayek, and Schumpeter (to name but a few) all considered causes for instability which were endogenous to the economy.B In most of these cases the instability was created by a nonlinear feedback process as was exemplified by the importance Keynes placed on expectations formation?* Some, however, considered linear feedback (e.g. Samuebon’s multiplier-accelerator model (1939); and Frisch, 1933). This latter group of models proved to be mathematically tractable but it was hard to find the generated smooth cycles in reality. Certainly, chaotic time paths were not a feature of the model unless some random exogenous shock was applied repeatedly. The result was that in practice these linear models seemed to be more exogenous than endogenous.

It is unnecessary to emphasise the implications for economic analysis and policy when instability is postulated to be endogenous rather than exogenous. If the variables of the economy move according to simple deterministic trends then their future values are potentially predictable. Moreover, if endogenous instability is causing a loss of production, employment or other forms of inefficiency then there is a potential role of government policy to stabilise the economy (Brock and Malliaris, 1989, p. 305). It must be noted, however,

8. This statement must be taken in a qualified sense for there were economists who explored

9. For a good review of these theories see Zarnowitz (1985).

nonlinear feedback mechanism which gives rise to chaos.

nonlinear systems well before the current wave [e.g. Goodwin, 1982).

10. Savit (1988) and Baumol and Benhabib (1989) also argue that expectations may form a

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that for both prediction and policy, deterministic chaos only represents a potential benefit. Whether the potential, however, translates into the actual will be discussed later.

Spurred on by the potential benefits of endogenous models, chaos theory has stimulated a new literature in nonlinear economic dynamics in both theory and applied econometrics.” Research in theory has centred around developing models with plausible parameters that lead to chaotic results. For example, Day (1982). has modified a simple Neoclassical growth model to yield a chaotic time path due to nonlinearities in production and savings functions!* Other theoretical developments have concentrated on ex- pectations formation as a cause of chaos. This is reflected in the literature surrounding sunspot equilibria which expresses the idea that economic fundamentals move in a deterministic fashion while economic agents believe that prices and quantities are affected by irrelevant random factors (e.g. sunspots). These expectations are postulated to be self-fulfilling and hence, one gets sunspot equilibria and an inherent unpredictability in the market. Such results are very ironic since they closely reflect what Keynes said in a non-mathematical way, that self-fulfilling expectations lead to unpre- dictability. Some bottlenecks are evident, however.

. . . to our knowledge, no one has constructed examples of economic models that generate chaotic paths where the parameters of the tastes and technology are constrained by the range of parameter values established by empirical studies. (Brock and Mallieris, 1989. p. 318)

The initial theoretical results are very interesting but to date they are still very abstract and there will probably be a lengthy time-lag before they can be flung “upon the floodtide of practical politics” (Edgeworth in Heilbroner, 1980, p. 173).

Chaos theory has also induced applied econometricians to develop tests

11. Reviews of the potential that chaos theory holds for nonlinear economic dynamics are given by Day (1982). Gabisch (1987), Savit (1988). and Baumol and Benhabib (1989). Grandmont (1986) and Brock and Mallairis (1989, Chpt 10) give a more technical discussion of chaos theory’s potential in econometrics. An entire issue of the Journal of Economic Theory was devoted to the subject (1986, vol. 40, no. 11. An article by Michaels (1989) considers implications for organisation development and Routh (1989) considers the more philosophical and methodological issues surrounding the chaos approach. Perhaps one of the most significant contributions to the literature is the proceedings of a inter- disciplinary nature organised by Kenneth J. Arrow and Philip Anderson (a Nobel prize winning physicist)-see Anderson, Arrow and Pines, 1988. This work contains contri- butions from both economists and natural scientists discussing the potential for a better understanding of the economy as a result of nonlinear analysis.

12. Another example of a nonlinear,model yielding a chaotic time path in growth theory is contained in Gans (1989a). Baumol and Wolff (1983) also find evidence of chaos in a simple model of R & D.

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which reveal whether economic time series are generated by deterministic chaos or stochastic processes. This is definitely not an easy task since it is a characteristic of chaotic time series that the variables look random and pass usual tests of randomness. “A host of examples shows that traditional linear statistical methods are often useless in this respect” (Grandmont and Malgrange, 1986, p. 6). In the natural sciences, the method used to locate chaos has been to represent the data in various forms of phase space l3 and by using sufficient amounts of data to observe regularities or bounds in the data. One example of this was contained in a paper by Ruelle and Takens (1971) which told of the discovery of a chaotic regularity known as a “strange attractor” in an experiment used to generate fluid turbulence. The attractor indicated that the time path of the model seemed to be attracted to certain areas and did not venture into others. Thus, bounds to fluid behaviour under turbulent conditions were observed. To take another example, if data were generated by the logistic equation discussed above then, when mapped onto the dimensions of itself in the current and next time period, the variable would yield a hill-shaped curve. The curve itself rises with the parameter a. This phenomenon is outlined in an excellent review by Baumol and Benhabib (1989)!‘

The problem, however, with such an approach in econometrics, is the same that has plagued the subject since its beginnings. Economic phenomena are rarely reducible to laboratory conditions where the number of degrees of freedom are limited. Thus, even if a time series were actually generated by a simple equation such as the logistic curve, how do we know that the parameter will remain fixed over time? It too is probably affected by other forces in the economy and these may also be nonlinear. Because of the number of free variables in the economy it is hard to believe that the nonlinear equation defining the economy will be anything but complex. That factor makes it virtually impossible to discover regularities hidden in the data even if they actually exist.

This is not to say that it is impossible that an economic time series might be generated by a simple nonlinear system of a low dimension (e.g. a single equation). In fact, if it was possible to come up with a measure for the complexity or dimension of the forces underlying the data, we could reasonably assert that if the dimension were low, the time series would more than likely be generated by deterministic chaos than stochastic processes. “Unfortunately, it is not easy to come up with a notion of dimension that is easy to calculate and give reliable results” (Brock and Malliaris, 1989,

13. Phase [or state] space describes a graph which portrays dynamic behaviour along the dimensions of free variables in the system without time as a dimension.

14. This method of revealing regularities in complex behaviour is similar to the effect of discovering which keys of a typewriter correspond to letters by observing a typist type “the quick brown fox jumps over the lazy dog”.

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p. 303)!5 Nonetheless, this is a possible direction for applied work, but in reality econometricians doubt that there are any models generated in a purely deterministic sense. There is always some noise present (Brock and Malliaris, 1989, p. 305). Also, intuitively, it is difficult to believe that low dimensional chaos really exists. Xke the example of asset prices. “There are many participants in a financial market with many complex sets of human relationships, motivations, and reactions” (Savit, 1988, p. 272). It could hardly be expected that these would converge to a simple model of any sort. Thus, it is not surprising that the results to date have been inconclusive!6 The butterfly effect

Meteorologists have long had the dream of being able to predict the weather perfectly. Lorenz (1963) was no exception and thus, when he first tested a simple model of the global weather system on his computer he had reason to be encouraged by the results. His simple, three differential equation, nonlinear model yielded a very complex time path not unlike the way the world weather pattern actually behaved. There was a faint hope that perfect predictability was only a hand’s breadth away. But the hand proved to be invisible when Lorenz discovered that his nonlinear model was extremely sensitive to only the slightest change in initial conditions. The resulting effect suggested that a butterfly flapping its wings off Hong Kong could cause hurricanes in the Gulf of Mexico! Small changes could translate into very large ones!’

15.

16.

17.

Paul Davies (1987. pp. 76-77) surveys attempts to come up with measures for complexity. These include measuring systems by the number of equations needed to specify them, a concept called “logical depth’ based on heuristics, forms of interaction, and information context. However, “organization and complexity, in spite of their powerful intuitive meanings, lack generally agreed definitions in the rigorous mathematical sense” (Davies, 1987, p. 77). It is interesting to speculate on what deterministic chaos means for the rational expectations hypothesis. According to that hypothesis agents should he able to learn the underlying laws of movements in economic variables from observation of the data and hence not make systematic prediction ermrs. But what if the underlying model is generated by a single chaotic system? The question is, of course, empirical. A test for experimental economists could be as follows: program a computer with the simple logistic equation described above. Inform the participants of this equation save for the value of the fixed parameter. Then allow the subjects to put data into the computer and observe the response. If the rational expectations hypothesis holds under conditions of deterministic chaos then, eventually, the subject should be able to discover the value of the parameter. It is left for the reader to imagine the plausibility of the rational expectations hypothesis given this simple test of its validity at the micro-micro level. Another fictitious example which may help conceptualise this effect in the time travel paradox of one person travelling back in time and whilst only making a small change in the context of history causes a large change in world events. For instance, in the Star Pek episode, “The City on the Edge of Forever”, Dr McCoy manages to allow Hitler to win World War I1 by saving an American woman’s life.

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Although the butterfly effect may cause you some concern the next time you wave goodbye to someone, the significant problem it revealed was that the scientist using a nonlinear model would only be able to forecast accurately if the necessary data existed at a level of absolute perfection. And this turned out to be a universal problem of all chaotic nonlinear models. For the meteorologist this meant that any imperfection in data, even if the correct dynamic laws of the world weather system were known, would drastically limit predictive power-often to uselessness. But more than meteorology would be affected by this. Even the prediction of planetary orbits (traditionally, a benchmark for determinism) would become less accurate as time goes on!8 The errors in dynamics grow exponentially with time. It is somewhat paradoxical that chaos theory, which promised so much in predictability, also came to deny that potential in actuality.

This poses a significant problem for the practical application of chaos to economics. If applied econometricians at first believed that they would be able to use deterministic chaos to predict fluctuating economic variables, they now realise that this dream has been considerably dashed. Economic data are notoriously unreliable. Therefore, even if economic variables were found to be determined by a low dimensional (simple) chaotic system, economists could never know with enough accuracy what present conditions were in order to predict the future perfe~tly!~ In contrast to conventional wisdom, neat enough is not good enough.20 Moreover, normal practices of smoothing and deseasonalising time series may cause an important loss of information (Savit, 1988, p. 289; Allen, 1988, p. 108). Even the slight errors allowed by techniques of linear approximation are likely to pose problems if the fundamentals are nonlinear. Consequently, the outlook for economic forecasting is bleak."

18. Practically, we could keep updating the data and keep calculations ahead of the action (Davies. 1987, p. 54). Nonetheless, pure determinism may exist and we cannot use it.

19. The problems of noise compound this problem even further. 20. An interesting twist on the problems of analysis given the sensitivities of nonlinear systems

is given by Savit. He argues that "[iln the market place, the study of the system disturbs the system itself. . . . Market prices in a nonlinear system can be very sensitive to small changes in the environment.. . . This interference by an observer on the phenomenon observed, while not usually important in macroscopic physical systems, is reminiscent of analogous considerations in quantum mechanics which governs the microscopic physical world" (Savit, 1988, p. 290).

21. The prospects of the fruitful application of chaos theory in economics are further reduced for the butterfly effect may, according to an argument by Sims (1984). wipe out the possibility of finding low dimensional chaos in financial markets. "[Tlhe financial logic leads us to believe that low dimensional chaotic deterministic generators for stock prices and returns over daily to weekly time periods should be extremely unlikely. After all. if a predictable structure is present, arbitraging on the part of traders should destroy it provided it is easy to be recognised by traders over very short time intervals" (Brock and Malliaris. 1989. p. 324). Of course, this logic assumes a rational expectations or efficient markets hypothesis.

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The butterfly effect also hints at difficulties for policy makers. In order to exert some influence over the economic system it is important that the policy maker have some conception of the difference between the actual and desired policy goal and the path which must be traversed to close that gap. In a complex nonlinear environment, however, the butterfly effect suggests that the accuracy of knowledge required to make certain policy decisions is impractical. Whatever action we take we cannot know for sure whether this would or has improved the situation. How should the policy maker react to this dilemma?

Similar problems have confronted policy makers in the past. The butterfly effect dilemma is very similar to the difficulty exposed when the rational expectations school suggested that no government policy could operate effectively in a systematic fashion because agents would come to adjust for anticipated changes. Their approach, which emphasised the intrinsic randomness of economic events, also implies a dearth of effective policy decisions (Thurow, 1983). At least, not ones with which we could be sure of their results. The key words in this argument, however, are sureness and certainty. The butterfly effect and other similar reactions are based on the idea that we can never be sure of the results of our policies. In the past, this idea provoked inaction on the part of politicians who would not take a risk. Keynes was incensed by the arguments against an active government policy in 1930s’ Britain.

It is not an accident that the Conservative government have landed US in the mess where we find ourselves. It is the natural outcome of their philosophy:

You must not press on with telephones or electricity, because this will raise

You must not hasten with roads or housing, because this will use up

You must not try to employ everyone, because this will cause inflation. You must not invest, because how can you know that it will pay? You must not do anything, because this will only mean that you can’t do

Safety first! The policy of maintaining a million unemployed has now been

We will not promise more than we can perform. We, therefore, promise

the rate of interest.

opportunities for employment which we may need in later years.

something else.

pursued for eight years without disaster. Why risk a change?

nothing. This is what we are being fed with. (Keynes and Henderson, 1929, P. 124)

Chaos theory restates the obvious that we can never be absolutely sure about the potential outcome of our policy decisions. What, however, applies for government also applies for private business. The individual economic agent cannot be sure whether actions taken will result in a higher welfare level. This does not prevent decisions being made. Without decisions to produce we would not have production. So in answer to the butterfly effect dilemma, policy makers should probably act much as they have done in the past- assessing decisions on the basis of the information available, accepting uncertainties and acknowledging the risks involved.

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The role of chance The above discussion would seem to indicate that chaos theory has little

of practical value to offer economics. Certainly, randomness is fundamental to complex nonlinear systems. The unknown butterfly is everpresent but it does highlight the significance of chance and randomness in determining the present and future. The anomalies and the irregular assume a new importance for their presence suggests potential irreversible events that could take place. In this sense, there is a bridge between chance and determinism in a way economists have not generally incorporated in their outlook.

Some economists, however, have recognised that chance could play an important role in determining the economic environment, in particular the often assumed constant organisations, institutions, technology, and tastes of society. A little known exposition of such ideas by a well-known economist is Arrow’s The Limits of Organization. In that book, Arrow outlines the factors that influence the actual structure of organisations. Starting at the level of the individual, Arrow argues that in order to perform a productive function an economic agent must typically learn an information code, a technical language for instance. This not only is a considerable investment on the individual’s part but it is an irreversible one. There is a great incentive to continue to use a code once it is learnt for the simple difficulties of the costs of changing weighed against the benefits of using the present code. Banslated up to the organisational level this practical irreversibility implies “that the actual structure and behaviour of an organization may depend heavily upon random events, in other words on history“ (Arrow, 1974, p. 49). “It follows that organizations, once created, have distinct identities, because the costs of changing the code are those of unanticipated obsolescence” (p. 5QZ2 No small wonder then that the organisations and institutions of the economy display a diversity not at all predicted by the usual economic theory based on optimising agents. In Arrow’s formulation the fact that organi- sations and institutions are established at different times is enough to account for the qualitative heterogeneity of the economic world.

More examples like this are present in the past literature and in recent times some of these ideas have begun to be considered again. In the last few years, Stiglitz has written a number of papers expanding upon the idea that technical progress may result in a local rather than a global shift in the production fun~tion.’~ Over time this can give the production function a nonlinearity, or even a kink in extreme scenarios. In fact, when applied to

~~

22. This practical irreversibility can be likened to a cassette player which plays forward but cannot rewind. Thus, if you miss a part you must wait until the cassette runs its course before you can correct your mistake. A more detailed discussion of self-reinforcement and lock-in is contained in Arthur (1988).

23. This idea stems from an early paper by Atkinson and Stiglitz (1969) which has been expanded upon in Stiglitz (1987, 1989).

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learning by doing and learning to learn phenomena Stiglitz’s conclusions bear considerable resemblance to Arrow’s. Allen (1987) has also expressed similar ideas into what irreversibility and chance mean for the economics of innovation. Game theory is a potential candidate for similar analysis (Baumol and Benhabib, 1989, p. 77). In fact, Schelling (1978) anticipated many chaotic phenomena in his analysis of interactive human behaviour. Along another route, Thurow (1981, p. 125) considers that critical limits in the ecological environment may pose problems for economic regulation of pollution based on marginal approaches. Also, as mentioned earlier, Keynes found that chance events of experience played an important role in financial market behaviour (1936, Chpt. 12). In other places, Keynes went further in the importance he gave to forces low on our priorities of attention.

The great events of history are often due to secular changes in the growth of population and other fundamental economic causes, which, escaping by their gradual character the notice of contemporary observers, are attributed to the follies of statesmen or the fanaticism of atheists. (1920, p. 8) These examples from the literature were for the most part written before

the emergence of chaos theory in the natural sciences and seem to have anticipated many of its conclusions.*~ Nonetheless, these have existed themselves as irregularities on the fringe of economics and apart from the textbooks. But the blurring of the complexity of the economic environment may have lost some of its edge in explanation. Chaos theory leads to an understanding and appreciation of the importance of irregularity or chance and suggests that economics may benefit from a focus away from the usual practice of looking only at the behaviour of averages.

Conclusion Reactions concerning the promise chaos theory holds for economics have

been mixed. Some show qualified optimism suggesting that chaotic “insight should help construct better (and perhaps simpler) explanations for these kinds of events (large qualitative changes in market behaviour), and ultimately improve our ability to anticipate, if not precisely predict such large changes in market behaviour” (Savit, 1988, p. 287). Others are more pessimistic but believe that the research may be worth pursuing regardless (Brock and Malliaris, 1989; Day, 1982; Boldrin, 1988). Still others are heartened by the co-operation and interdisciplinary flavour of the research taking place (Routh, 1989, p. 52; Anderson et a]., 1988).

~

24. The notion of irreversibility has just begun to gain considerable recognition in the natural sciences. The analyses of “arrows in time” are grounded in thermodynamics (Hawking, 1988); increasing complexity in evolution (e.g. dissipative structures of Prigogine and Stengers, 1984); and the collapse of the wave function in quantum mechanics (Flood and Lockwood. 1986, Chpt. 7; Davies, 1987). In economics, such considerations have led to conclusions that the concept of equilibrium is an outdated analytical tool (Kaldor. 1972).

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The discussion in this paper suggests that chaos theory may manifest itself in economics in a rather different way than has been speculated by many. The butterfly effect indicates that the hopes chaos theory initially generated were probably too high. But this does not mean that it represents a Pandora’s Box. Quite the contrary. The role chaotic phenomena such as the butterfly effect accord to irreversibility, chance, and irregularity alludes to a change in direction for economic analysis from averages and the neat mathematical solutions of linear equilibrium.

This is not a reason to abandon modelling, but rather the opposite! Without the model, we would not be able to “order” the system to an extent sufficient to realize that something ”inexplicable” was occurring. With it, we can be aware of the emergence of some new mechanism or factor, and we can then search for the best manner in which to include it. That is to say that the model we have of a particular situation will probably always require modification because the real world is itself evolving. (Allen, 1988, p. 117)

Similar sentiments have been expressed by Herbert Simon (1981) who believed that computer simulation offer the empirical data of tomorrow for analysis of complex systems (see also, Hofstadter, 1981).

As has been pointed out, such ideas are not new to economics. None- theless, the textbooks and conventional analysis have shunned their existence. Chaos theory indicates that this may have been unwise and thus signals a re-emergence of such thought in economics. The words of May (whose work in biology we have encountered already) are particularly apt in his plea for a change towards new thinking.

I would urge that people be introduced to the equation y = 4ax(l- x) early in their mathematical education. This equation can be studied phenomenologically by iterating it on a calculator, or even by hand, Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student’s intuition about nonlinear systems.

Not only in research but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties. (May in Hofstadter, 1981, p. 29-the notation of the equation was altered to make it more compatible with that of this paper.)

With a recognition of nonlinear processes, history gains a new importance in economics, as it does elsewhere. “When you think about a variable, the evolution of it must be influenced by whatever other variables it’s interacting with. Somehow their mark must be there” (Farmer, in Gleick, 1988, p. 266.) Bygones are not forever bygones. Questions and issues turn from what is, to how it came to be. With a mathematical basis for the analysis of the latter question, it may be that this will be chaos theory’s lasting mark on economic thought.

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