The manifestation of non-linear effects can be discovered in a wide variety of examples, from sociology, population dynamics, economics and ecology. In each case mathematical models can be built that have the potential for a wide range of behavior from stability, gradual growth, persistent oscillations, self-organization, rigidity to change, infinite sensitivity to externalities, all the way to chaotic and unpredictable swings. Of course mathematical models are far from the real world but the possibility that a well behaved system could, at some point, engage in a radically different, and uncontrollable, form of behavior gives food for thought. Moreover, as more and more examples are found in the real world of qualitative changes in behavior, of chaos, sensitivity or rigidity, it becomes important to take them into account wherever policies are being made and the implications of actions contemplated.
Take an obvious example where non-linear effects occur. There has been much debate about the greenhouse effect. Suppose, therefore, we ask what will be the effect of increasing carbon dioxide on plant growth? The whole question of the effects global warming, increased humidity and carbon dioxide on vegetation is a highly complex issue. Not only will growth rates change but the whole balance of a region will be modified, with some species being favoured over others. For example, what may be good conditions for the growth of a certain crop may be even better for weeds and predators. In turn, the effects of these changing vegetation patterns will feed back into the atmosphere, both directly – in terms of the amount of carbon dioxide that is fixed by plant-life – but also indirectly, for as the mixes and yields of different vegetation changes so too will the economics and even the lifestyles of a given region. As the economy and social structure of a region changes so too does its energy demands, which results in different amounts of carbon dioxide being released into the atmosphere. Moreover, there will be a variety of lags in the various feedback loops of such a system, so that attempts to control variations in one part of a cycle may have the effect of magnifying another. Even the attempt to isolate a single variable in this whole complex system becomes incredibly complex. A single variable will exhibit the whole range of behaviors from extreme sensitivity to extreme stability as well as limit cycles, bifurcation points, large oscillations and possibly even chaotic behavior. Yet this system, by itself, is part of a much wider system that is embedded in global and local politics, attitudes towards agriculture and population density. Each of these elements is, in turn, dependent upon yet other factors which even include religious and ethical values – of key importance in population growth and attitudes to the environments.
This single example shows how complex a system may be. It shows that a given problem may be sensitive to a wide range of externalities, each of which is linked to a variety of other factors. No single policy, no rigid plan is capable of meeting the subtleties and range of possibilities within natural and social systems. Clearly a whole new philosophy is demanded.
Economics is currently under the scrutiny of experts in the field of non-linear dynamics and a variety of analyses of short and long term stock market trends have been made. There are deep questions to be answered about the very definition of economic systems and about the meaning of their variables, such as value. Chaos theory and non-linear dynamics have been added to those voices that are questioning the whole basis of economic theory.
The concept of money, for example, is highly complex and analysts are questioning the idea of economic equilibrium and of an intrinsically stable market. As Richard Day of the University of Southern California puts it: “An economic world in which money enters in a nontrivial way can be highly complex in its behavior in theory, just as in reality”. Day himself has shown that even the simple models, in which expenditure and income lag behind each other, can give rise to chaotic fluctuations.
The Systems Dynamics Group at MIT have a variety of models in which a human economist or policy maker can “drive” the computer model. In one of these, a seasonal variation in the demand for beer is passed on to the main supplier and its distributors. When a human operator attempts to smooth out fluctuations the system tends to move towards ever more uncontrolled oscillations. (In essence a non-linear iteration is dominating the system.)
Dr. Ping Chen of Univ. Texas at Austin has made an extensive analysis of monetary data from the Federal Reserve and argues that economics contains inherently chaotic behavior. Even if external shocks could be totally eliminated the economy will not run smoothly, for wild fluctuations are inherent in its very dynamics and recessions and downturns are often independent of external shocks. One analyst has suggested that the 10 October 1987 crash arouse out of the non-linear dynamics of the market and not through a combination of external causes.
R.H. Day has made an analysis of a variety of situations, such as investment in competing new technologies which, over time, shows a shift from steady expansion and economic health into one of financial crisis. Day’s agricultural model. in which a variety of factors such as market, prices, supply, investment in new buildings etc., are included, shows an initial set-up period, followed by a five year cycle. In time the system’s internal dynamics change again and move into a period of irregular oscillations. During the former period the economics of the situation are relatively stable but in the latter period they are highly sensitive to any new trend.
In a variety of analyses of different industries and technologies, quite distinct regions of behavior have been discovered, some of these are quite stable, other chaotic, some oscillate violently, or are extremely sensitive to an external trend or perturbation.
iii Order in Chaos
The notion that it is the inherent non-linear dynamics of the market that produce fluctuations rather than a combination of externalities suggests to some analysts that it may be possible to carry out “micro forecasting”. ( A more careful analysis may indicate that it is impossible to separate out endogenous from exogenous causes. ) Some claim to see the characteristics of deterministic chaos – i.e. strange attractors – within economic data. If this is true it suggests that while economic fluctuations are unpredictable they will always lie within certain bounds.
In addition, there are suggestions that a degree of self-similarity holds. Self-similarity is associated with fractal structures and would suggest that a certain range of behavior patterns repeat at various scales of time, from years, months, days and even hours. If this is true then micropredictions will take into account that a random fluctuation will fall within a particular range. A number of investment houses are currently developing sophisticated computer models to investigate this chaotic behavior.
Other analysts are looking for “co-operative effects”, for example, the manifestation of decision processes that are made in a collective way. Often the behavior of a crowd is simpler to predict than that of an individual. So where people respond to the news and other externalities in a collective way it may give rise to predictable results. Or the market itself may exhibit a degree of self organization.
Economics is only one factor in which public policies are concerned. The above brief overview suggests a variety of ways in which non-linearities may be effecting the market. If this is true then it would mean that many economic policies are ineffective and are attempting to change what is inherent in the dynamics of the market itself. Major swings may not be the result of externalities and oscillations may be purely chaotic.