short-term analysis of human behavior, there is a deterministic relationship. That is, on a short-term scale, it is quite acceptable to consider the "relationship" variables as constants, and on the long-term one can neglect the "behavior" variables.
If the system is unstable, the problem becomes quite subtle. For example, even within the framework of short-term analysis, we cannot effectively treat the variables x as constants, because even if s is small enough, the dynamic behavior of the system can be strikingly different from the behavior of a system with zero s.
To conclude this section, I would like to quote from Hume (1748): “Ambition, greed, self-love, vanity, friendliness, generosity, patriotism - these feelings, mixed in various proportions and distributed in society, have been and still are from the beginning of the world the motive of all actions and the source of all enterprises that have ever been observed in the human community.
Application: Subordination principle for stochastic differential equations
In this appendix, we will discuss how the principle of subordination applies to stochastic differential equations. A general method has been given by Haken (1983) and Gardiner (1983). Since the problem is very complex, we would like to present two simple examples to demonstrate at least the main points. The examples are taken from Gardiner (1983, ch. 6).
As mentioned above, it often happens that a dynamical system is described by stochastic differential equations that have a wide spread of response times and whose behavior on a very short scale is of no interest. Now let's see how the principle of subordination can be applied to this type of equations.
Consider the Langevin equation describing the behavior of the "Brownian particle"
where T is the absolute temperature, k is the Boltzmann constant, m is the mass of the particle, and h(t) represents the effect of external shocks
with zero mean. Consider the situation when the friction coefficient b is not small and the mass m is very small.
The corresponding Fokker-Planck equation for the distribution function p (x, v, t ) has the form
Introducing the distribution function along the p *(x, t) coordinate as
and putting m → 0 we get the Fokker-Planck equation for p *(x , t )
This is a standard partial differential equation that is easily solved given the appropriate initial and boundary conditions.
Thus, we have excluded the fast variable ν from the equation, with respect to which we assume rapid convergence to the quantity
v (t) \u003d (2kT / b )l / 2 h (t).
We see that a large coefficient b in this expression leads to the fact that the variable υ will tend to a value that coincides with the value corresponding to the case of the constancy of the slow variable x. Therefore, the fast variable effectively obeys the slow one.
As another example, consider the deterministic equation from Sec.
There we showed how the principle of subordination can be applied to this system. The stochastic version of this system is given by the equations
where C and D are constants, and W 1 (t) and W 2 (t) are mutually independent. If the coefficient r 2 is large enough, we can replace y with the stationary solution of equation (9.A.5) expressed in terms of x and get
The Fokker-Planck equation for (9.A.4) and (9.A.5) has the form
where p is defined as
Just like in sect. 9.1, we want to get rid of y . Let's introduce
For a fixed x, z has zero mean. In terms of the variable z, we can write the Fokker-Planck equation
In order for when r 2 → 0 the expression
constituted the correct limit form, there must be such A that αβ /r 2 = r 1 A . For this limit to be discernible, it must not be drowned in noise, so as r 1 → 0 we must also have C 2 = 2r 1 B . Then
For L 1 0 to be independent of r 1 , we require that r 2 be independent of r 1 .
Therefore, the equality αβ /r 2 = r 1 A means that αβ must be proportional to r 1 . Gardiner considers various possibilities.
First, the case of quiet submission (in terms of Gardiner): α = a r 1 . In this case L 1 0 does not depend on r 1 , while L 0 2 and L 0 3 are proportional to r 1 . It can be shown that the usual elimination procedure leads to the equation
where p *(x, t) is the distribution along the coordinate. This corresponds to the adiabatic elimination of y, ignoring fluctuations in y, and simply substituting the deterministic value into the equation for x. Gardiner called this case "silent submission" because y is subordinate to x and does not contribute noise to the equation for x.
In the case of "noisy submission", when a and b are proportional to r 1 1/ 2,
the probability distribution is given by the equation
This case is called "noisy submission" because in the final equation, the subordinate variable makes noise on the slow variable by applying additive noise.
10 Synergy economy and its importance
The presence of analogies in the main provisions of various theories means that * there must be a more general theory that combines particular and unifies them with respect to these general properties.
P. A. Samuelsosh
We have considered the unstable behavior of various economic dynamic systems. It was shown that for economic evolutionary processes, linearity and stability are not universal, but rather limited. This arrangement of accents is different from those on which the traditional economy is built. For example, in the Fundamentals of Economic Analysis, Samuelson tried to identify precisely linearity and stability as basic properties in economic phenomena, because when using traditional static analysis and the correspondence principle, we can only deal with those systems in which small changes in parameters lead to small changes. characteristics. This book, in contrast to traditional dynamics, studies those properties of dissipative systems for which small shifts in parameters entail qualitative changes in dynamic behavior. We have shown that when a system becomes dynamically unstable, for example, due to a perturbation of the parameters, nonlinear terms become very important for elucidating the nature of its behavior. In this chapter, we will look at what this means in economics.
10.1 Synergetic economy and its relation to synergetics
We're not just looking for truth, we're looking for fascinating and illuminating truth, we're looking for theories that solve big problems. Finally, we need as deep theories as possible.
Karl R. Popper (1972)
Synergy economy belongs to the area economic theory. It concerns the temporal and spatial processes of economic evolution. In particular, Synergetic Economics deals with unstable non-linear systems and focuses on non-linear phenomena in economic evolution, such as structural changes, bifurcations and chaos.
I have considered many variations of the title of this book, such as The New Foundations of Economic Analysis, The New Evolutionary Economics, Chaotic Economics, and Synergistic Economics. To reflect the new approach to economic dynamics, the book was titled Synergistic Economics. The choice was made under the influence of Haken's synergy.
Haken defined synergetics as a general theory of the dynamic behavior of systems that have special properties. Synergetics deals with the cooperative interaction of many subsystems, which macroscopically manifests itself as self-organization. Synergetics focuses on critical points at which the system changes the nature of its macroscopic behavior and may experience non-equilibrium phase transitions between oscillations, spatial structures and chaos. The area of interests of synergetics is not limited only to transitions between equilibria and quasi-equilibrium attractors, similar to limit cycles. Synergetics tries to cover other transitions that do not have a specific final form. Thus, we can also consider the synergetic economy as part of synergetics as a whole.
It should be emphasized that although we are building a synergistic economy based on general synergetics, the fundamental ideas of economic evolution presented in this book were also strongly influenced by the work of Prigogine and others (see, for example, Nicolis and Prigogine, 1977, Prigozhin, 1980). , Prigozhin and Stengers, 1984, Yanch, 1980).
10.2 Relation of the synergistic economy to the traditional theory of economic dynamics
The path of knowledge runs through conjectures and refutations, from old problems to new ones.
Karl R. Popper (1972)
Before considering the significance of the synergistic economy for various economic problems, we will discuss the relationship between the synergistic and traditional economy. Since synergetic economics deals with economic evolution, it is part of the theory of economic dynamics. Many theories fall under this concept - both the theory of business cycles and the theory of economic growth, and many analytical methods such as the correspondence principle. All these theories and methods form the content of the traditional theory of economic dynamics. Synergetic economics, on the other hand, is an extension of the traditional theory of economic dynamics in the sense that the results of the latter can be explained within the framework of this new theory, moreover, it tries to explain other economic phenomena that the traditional theory ignores. From the perspective of a synergistic economy; the theories that make up the traditional theory of economic dynamics are not universal, but only special cases. And although we cannot say that the synergetic economy solves all the problems of economic evolution, we can conclude that this new theory allows the dynamic economy to explain and even predict some dynamic economic processes that cannot be explained using traditional theories and methods. . Synergy economics offers an encouraging new direction for explaining complex economic phenomena.
The opinion has been established that for understanding economic phenomena, the approaches of traditional economics, for example, Arrow-Debray's system of competitive equilibrium, are quite suitable starting points. Traditional economics offered science some fundamental economic mechanisms, such as competition, cooperation, and the rational behavior of economic objects.
The synergistic economy is based on somewhat different concepts. The concepts of rational behavior, sustainability and balance, which play a fundamental role in the development of the traditional economy, do not lose their importance here. However, the synergistic economy shifts the focus to concepts such as unsustainability, which are not covered by the traditional economy. Synergy economy sources of complexity
economic evolution finds itself in instability
and non-linearity more than stability and linearity (or proximity to linearity), as is typical of traditional economics.
The main subject of the traditional theory of economic dynamics is the theory of business cycles. This theory is also of great importance for the synergetic economy. However, this is more than just a revival of interest in the formal theory of endogenous cycles, which has indeed grown in last years. We show that many economic mechanisms can generate oscillations. Business cycles can be the result of a non-linear interaction between various economic and political factors. They can arise not only in a competitive, but also in a planned economy.
The traditional theory of business cycles deals mainly with the regular (periodic) change of variables. Within the framework of the traditional theory of economic dynamics, there is no theory that would satisfactorily explain, using endogenous mechanisms, the irregularity of the dynamics of real economic data. Until the advent of modern nonlinear dynamical theory, chaos remained something incomprehensible. The very concept of chaos for the dynamic theory of economics is completely new. Synergetic economics offers some analytical methods to study the endogenous chaos of economic systems. It shows that chaos lies in the nature of any evolutionary economic system. The fact that chaos exists means that accurate economic predictions are almost impossible.
Synergetic economics has given a new understanding of how economic evolution is influenced by stochastic processes. It has been shown that if a dynamical system is stable, the effect of zero-mean noise on economic analysis can be neglected - such a simplification will not affect the qualitative conclusions of the analysis. So the point of view on small fluctuations that prevails in traditional economics is correct only if the system is known to be stable. However, if the system is unstable, noise impact analysis becomes very difficult. Small fluctuations can cause significant changes in the behavior of a dynamical system.
It should be noted that the emphasis on instability can also be found in the writings of Karl Marx, Keynes, Schumpeter and other economists, although these economists find different sources of instability. "Vision" of the move economic development in a synergistic economy is very similar to Schumpeter's vision of the course of development. Schumpeter's innovation impulses (shocks) can be viewed as a "supply of energy" leading to qualitative changes in the system: an economy without innovation is forced to remain in stagnation (stable equilibrium), and innovation impulses can
can lead to chaos. However, this does not mean that everything that a synergetic economy can "offer" is already contained in the works of Schumpeter - after all, even in the case when people's views on the same problems coincide, the explanation of the ongoing processes may be different, and this difference may cause a different "level of understanding". The conclusions of the synergetic economy can be verified using real economic data. Mathematics plays a significant role in the synergetic economy. Mathematics helps us express exactly what we mean by instability, cyclical development, chaos, etc. You will not find any of this in the works of the authors mentioned above.
Various authors often emphasize the role of inaccurate information and irrationality in economic analysis. For example, by demonstrating the complexities of moving along a chaotic trajectory, Simon gave definitions of bounded rationality and a satisfactory level of production. He showed that, due to the complexity of calculating the optimal strategy, the subjects of the economy will not find the optimal path, and instead will choose a satisfactory level of production as the goal. The possibility of chaotic behavior may give another direction to the interpretation of Simon's bounded rationality.
Synergy economics emphasizes the interaction of different variables and different levels of the system. Although the importance of such interactions is recognized and " system analysis”, there this approach did little to understand the processes of social evolution. System analysis obviously presupposes stability. In this respect, it is still within the framework of the traditional economy.
The introduction of nonlinearity and instability into economics can lead to new discussions. For example, it becomes more difficult to answer the delicate question of which theory of economics is more true. The existence of chaos also affects the way in which economic theory can be tested. The classic way to test a theory is to formulate a theoretical prediction, which is then tested against experimental data. If the phenomena are chaotic, long-term predictions are essentially impossible, so that the procedure for testing a theory becomes very difficult. Moreover, the synergetic economy can play a significantly negative role in the development of econometrics. If it is proved that the theory is not capable of making any accurate predictions, one may decide that the development of finer models and more accurate parameter estimates has become redundant. It also seems that the impact of the concept of chaos can adversely affect not only econometrics, but the entire economy in
Synergetic Economics
Time and Change in Nonlinear Economics
Berlin Heidelberg New York London
Paris Tokyo Hong Kong Barcelona
V.- B. Zang
Synergistic
ECONOMY
Time and Change in Nonlinear Economics
Translation from English by N. V. Ostrovskaya, edited by
V. V. Lebedeva and V. N. Razzhevaikin
MOSCOW "MIR" 1999
BBC 16.22.9 З27
V.- B. Zang
Z27 Synergetic economy. Time and change in non-linear economic theory: Per. from English. - M.: Mir 1999. -335 p., ill.
ISBN 5-03-003304-1
The Chinese economist's book was written during his tenure at the Swedish Institute for Advanced Study and was published in 1991 in the famous Springer Literature Series on Synergetics, edited by Hermann Haken. The book uses the modern mathematical apparatus of nonlinear analysis for problems of macroeconomic dynamics.
It will be useful for specialists in the field of macroeconomics, applied mathematicians, graduate students and students of economic universities.
BBC 16.22.9
The publication was supported by the Russian Foundation fundamental research project
Editorial Board of Literature in Mathematical Sciences
Originally published in English under the title:
"Synergetic Economics" by Wei-Bin Zhang.
Copyright © Springer-Verlag Berlin Heidelberg
1991. All Rights Reserved. © translation into Russian, "Mir",
ISBN 5-03003304-1 (Russian)
ISBN 0-387-52904
In troduction .......................................................... ................................................. ............................................... |
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Time and changes in economic theory .............................................. ...................................... |
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Economic evolution. Introduction ................................................ ...................... |
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Equilibrium theories in economic analysis....................................................... ............ |
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Dynamic theories in economics .............................................................. ............................... |
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The Samuelson correspondence principle and its limitations .............................................................. |
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Instability in economic analysis .............................................................. ................ |
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ELEMENTS OF MATHEMATICAL THEORETICAL AND DYNAMIC SYSTEMS.... ....... |
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Dynamics and balance ............................................................... ............................................... |
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Classification of differential systems of the second order .............................................. |
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The principle of stability in linear approximation .............................................................. |
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Lyapunov's direct method ............................................... ............................................... |
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Structural stability .................................................................. ................................................ |
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Conservative systems .................................................................. ......................................... |
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Theory of bifurcations.................................................... ................................................. . |
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Singularity theory .................................................................. .................................................. |
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Catastrophe theory .................................................................. ................................................. ..... |
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Appendix: Some remarks on the theory of bifurcations.................................................... ... |
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Sets of equilibria and structural changes in economic systems .............................................. |
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Catastrophe theory and comparative static analysis .............................................. |
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Modeling regional dynamics .................................................................. ................. |
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Some examples of structural changes .................................................................. ........... |
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Business cycles in the Kaldor model .............................................. ............. |
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Resource management................................................ ............................... |
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Dynamic choice of mode of transport and bifurcation .......................................... |
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Sets of equilibria in the Wilson model of retail trade....... |
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Bifurcation analysis of the economic growth model .............................................. |
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The Theory of Singularities in Economic Analysis....................................................... ...... |
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Business cycles .................................................................. ................................................. ................. |
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Theories of economic cycles .............................................................. ................................ |
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Some mathematical results of the theory of limit cycles.................................... |
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The Poincaré-Bendixson theorem and its applications to economics.......... |
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Hopf's bifurcation theorem............................................................... ............... |
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5.8 Optimal economic growth associated with endogenous fluctuations 142
5.10 Competitive business cycles in an economy with overlapping
generations - discrete model .............................................................. ......................................... |
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Economic chaos in deterministic systems.................................................................. ................... |
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Chaos in deterministic systems............................................................... ...................... |
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Economic Chaos in a Discrete System....................................................... ............. |
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Aperiodic optimal economic growth .............................................................. |
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Dynamics of cities - Lorentz system .............................................. .................... |
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Chaos in the model of the international economy .............................................. .............. |
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Chaos and economic forecasting .............................................................. ................. |
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Remarks ................................................. ................................................. .............. |
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Appendix: Some criteria for the classification of attractors .................................................. |
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Lyapunov exponents of differential equations............................... |
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Lyapunov exponents for discrete mappings .......................................... |
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Signal, power spectrum, autocorrelation function and display |
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Poincare184 |
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Stochastic processes and economic evolution .............................................................. ................. |
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Random processes and economic evolution .............................................................. |
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Stochastic processes. Introduction ................................................ ............... |
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7.2.1. Some Concepts of Probability Theory .................................................................. |
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7.2.2. Stochastic processes .................................................................. ................... |
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Birth-Death Processes and the Master Equation .......................................................... |
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Schumpeter's Non-Equilibrium Clock Model............................................................... ........... |
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7.5. Effect of noise on the trajectories of nonlinear stochastic systems near
special points ................................................................ ................................................. ...................... |
7.6. The impact of random external factors on the second-order system in
neighborhoods of singular points ............................................................... ................................................. ....... |
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Conclusions................................................. ................................................. ............. |
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Urbanization - stability, structural changes and chaos .............................................. |
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8.1 Spatially continuous economics and process description
8.3 Economic cycles in the spatial model "multiplier-
accelerator" Puu .................................................... ................................................. ................... |
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Spatial diffusion as a stabilizer .............................................................. ...... |
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Separation and coexistence of heterogeneous groups of the population of the city .............................. |
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Urbanistic formations of the type of traveling waves .............................................. .. |
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Instability and city formation ............................................................... ...................... |
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Appendix: Structural changes in the two-component model............................................... |
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Model of morphogenesis .............................................................. ............................... |
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Brusselsator ............................................... ......................................... |
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Haken's principle of subordination and the time scale in economic analysis.................................................. |
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Haken's principle of subordination ............................................................... ................................. |
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The center manifold theorem............................................................... ................... |
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Singular perturbations .................................................................. ............................................... |
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Relationship between fast and slow variables in economic analysis.................................................. |
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Time scale in economic analysis .............................................................. ........... |
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Human dynamics. Attempt to comprehend .............................................................. ............ |
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Application: Subordination principle for stochastic differential equations |
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.................................................................................................................................................... |
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10 The synergy economy and its significance ............................................................... ............................................... |
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Synergetic economy and its connection with synergetics.................................................. |
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10.2 Connection of the synergetic economy with the traditional theory of economic dynamics | 297
10.3 Competitive and planned economy from the point of view of synergetic economy 303
10.4 developed and developing model economy from the point of view of synergetic economy 306
chance and necessity economic life................................. |
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The role of political decision in a chaotic world............................................................... |
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Correlation between micro- and macroeconomics .............................................. |
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11 Conclusions and prospects for further research ............................................................... ......................... |
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Foreword by the Translation Editors
All knowledge is only bringing the essence of life under the laws of reason.
Leo Tolstoy, War and Peace
characteristic feature modern stage development of economic science is its mathematization, which is manifested in the replacement of the studied economic process with an adequate mathematical model and the subsequent study of the properties of this model either by analytical methods or on the basis of computational experiments. Usage mathematical models in economics has more than a century of history. For example, one of the first models of market competition (O. Cournot) was published in 1838, and half a century later, L. Walras already applied mathematical models when reading a course in political economy at the University of Lausanne. To date, various models of interaction between labor markets, goods and money markets, models of single-product and multi-product firms, a model of consumer behavior, a model of firm competition in the goods market, and others, which, in essence, are equilibrium models, have firmly entrenched in economic theory.
However, the vast majority economic processes flows in time, as a result of which the corresponding mathematical models are, in principle, dynamic. One of the traditional approaches to forecasting the development of economic processes is the study of the shift in the equilibrium point of a dynamic system caused by a change in certain parameters of the model. This (quasi-stationary) approach is based on the key concept of classical political economy - the "invisible hand" of Adam Smith. As is known, this concept is based on the hypothesis of the existence of competitive markets automatic balance mechanism.
The use of a quasi-stationary approach to the analysis of the dynamic processes of the economy has led to the spread of the general
the idea that the development of any complex system can be viewed as a change from one stable state to another with a short period transition process between them. However, an analysis of real economic dynamics based on this approach may turn out to be erroneous, since the period of non-equilibrium development of many economic processes may turn out to be too long to be neglected. Perfectly understanding the importance of studying economic processes in dynamics, the classic of modern economic science A. Marshall justified the use of a quasi-stationary approach to assess changes in the market by the fact that "our analysis is still in its infancy."
Note that this approach is effective only for the time being, until, for some reason, the nature of the stationary state does not change radically. Such changes, called bifurcations, already belong to the field of application of methods of nonlinear dynamic analysis, the development of which leads to the increasing spread of this point of view: "The world is a constant development, eternal instability, and periods of stabilization are only brief stops along the way."
Dynamic mathematical models that have proven themselves in physics, and then in biology, have much in common, although they retain specific features each of these sciences. Now models of this class are increasingly used in sociology and economics. To date, the modern methodology for the analysis of nonlinear dynamic systems has taken shape in a new scientific direction called synergetics. This interdisciplinary science aims to identify general principles evolution and self-organization of complex systems in various fields of knowledge based on the construction and study of nonlinear dynamic mathematical models. Important concepts of synergetics are "catastrophe", "bifurcation", "limit cycle", "strange attractor", "dissipative structure", "traveling wave", etc. Arising from the use of relatively simple nonlinear models, these concepts allow us to penetrate deeper in the essence of many processes and phenomena. Physics, chemistry, and biology abound in examples of the successful application of this methodology. These include phase transitions between aggregate states of matter, turbulent fluid flows, structures in media in the presence of autocatalytic reactions, life waves and combustion waves, fluctuations in the number of natural populations, etc.
It is not surprising that this universal methodology, which has arisen relatively recently and has proven itself in the natural sciences, began to penetrate into the traditional humanities, and
primarily to the economy. Without fear of making a mistake, it can be argued that any branch of economic science can be attributed to the field of applications of synergetics, since when considering any dynamic economic process, some active, i.e., feedback element is always present as an acting factor. Therefore, if we want to look beyond the horizon of a narrow world in which everything seems to be stable and in which there is no place for catastrophes and restructuring, we cannot do without using a synergistic approach.
In the book offered to the attention of readers by V.-B. Zang "Synergetic Economy" an attempt is made to give a general idea of the possibilities of a synergistic approach in the economy. In this case, the main attention is paid to the consideration of relatively simple mathematical models of small dimensions, which, as a rule, can be investigated by analytical methods. The use of synergetic methods in the economy is not a tribute to fashion, but an urgent need to move forward beyond the limits outlined by the quasi-stationary approach, to look for new ways to use powerful modern computing tools to solve serious practical problems.
The mathematical toolkit of the book is a fairly compact set of methods that make it possible to carry out a very effective analysis of nonlinear models of real economic processes. The undoubted advantage of the approach used is that the analysis of the low-dimensional models discussed in the book is easy to comprehend, since the set of properties that are the most striking consequences of nonlinearity is rather limited. Therefore, the mathematical apparatus used in the book should become not only the alphabet for a new generation of economists, but at the same time a beacon to which the mathematical training programs of economic universities should be tuned. Apparently, it is in connection with this that V.-B. Zang recommends his book not only to specialists, but also to students of economic specialties.
The scale of the task that the author set himself did not allow him to avoid some shortcomings. This concerns, first of all, the excessive conciseness of the presentation of fundamental hypotheses in the formulation of mathematical models, which, unfortunately, is inherent not only in this, but also in many other books on mathematical economics. You can also note that; what economic models often serve as illustrations of well-known mathematical results in the book. This puts the models under consideration in a subordinate position in relation to the mathematical apparatus, which, of course, cannot but cause some feeling of dissatisfaction among economist readers. However
in As a result of the author's approach to the presentation of the material, the reader discovers, for example, that economic cycles are as natural as population fluctuations, and "leaps" in society, that is, changes of a revolutionary type, are like phase transitions for matter. So this can be regarded as a deliberate methodological approach in presenting material, which forces readers to delve more carefully into those stingy lines that set out the main hypotheses and mathematical constructions of models, and to show maximum independence in understanding not only the results presented, but also mathematical formulation of the problem.
To some subjective assessments(and self-assessments) of the author, the reader should be treated quite critically. For example, speaking about Haken's subordination principle, it is impossible not to mention another formulation of this principle - Tikhonov's theorem for systems of equations with singular perturbations. And in general, speaking of synergetics, it should be remembered that many of its results are directly related to the development of mathematical modeling, at the origins of which in our country were A. A. Dorodnitsyn, N. N. Moiseev, A. A. Samarsky and others (for For the convenience of readers, we provide at the end of this preface a small list of literature in Russian on this topic).
At the same time, we would like to draw the attention of readers to the main advantage of the book: on the whole, the author managed to give a broad panorama of the state of affairs
in today's synergetics on the example of the analysis of relatively simple models of dynamic economic processes. Moreover, the book is aimed at developing a non-linear style of thinking among readers, which is important in any field of knowledge, including, of course, in modern economy.
When working on the translation manuscript, we corrected the observed inaccuracies of the original without any special reservations, and where necessary, made footnotes. It should be especially noted that the publication of the book in Russian was carried out thanks to the initiative of the translator of the book, N.V. Ostrovskaya, who supported her initiative. Russian fund fundamental research (head of the publishing department V. D. Novikov), employees of the Mir publishing house, as well as A. V. Fedotov, who took part in the translation of chapters 5 and 9.
We would also like to express our gratitude to the author of the book, Prof. V.-B. Zang for his attention to the Russian edition - he kindly sent, at our request, a list of typographical errors, which was taken into account in the Russian edition, and also answered a number of questions regarding the clarification of certain places in the text. In conclusion, we express the hope that the book will be useful to all readers interested in applications of nonlinear analysis methods in economics. Who knows, maybe among them will be those who, with its help, will find the very thread, unraveling which, it will be possible to get to a clear synergistic picture of the economic problems that we are all experiencing today and, having this picture in front of us, find real ways to decent economic development.
The use of natural science concepts, synergistic ideas and approaches allows in a new way look at such a complex area of scientific and practical activities man as an economy. Despite certain achievements of economic theories, economic forecasts very often do not correspond to the real development of the economy. To a large extent, this is due to the fact that the existing classical economic theory continues to remain in the block of the humanities. Built on disparate empirical facts, economic models of dynamic processes are based on linear concepts of reality. They satisfactorily reflect the dynamics of economic systems for narrow specific conditions and are not able to predict the ambiguous economic processes taking place in market economy. In particular, an economic development model developed and positively tested for the economy of one specific country leads to a completely different result after applying it to predict the development of another country, which actually develops according to a different scenario, different from that predicted by the “foreign” model.
In his book "Synergetic Economy" V.B. Zang notes that "in pre-synergetic theories, the most important results in economic analysis have been obtained on the basis of the concept of an equilibrium mechanism." This approach is applicable to describe the dynamics of a system that is in a weak non-equilibrium state, when a system previously taken out of equilibrium slowly returns to equilibrium. The development of such a system can be viewed as a succession of rapidly changing stable non-equilibrium states. Analysis based on this concept is effective as long as the system remains linear. With an increase in the degree of non-equilibrium, all systems begin to show their important property - nonlinearity. The behavior of a nonlinear system is complex and ambiguous. Systems of any nature, physical, chemical, biological, economic, social, etc., including the most gigantic and complex of the Universe known to us, are non-linear.
Modern methods for the analysis of nonlinear dynamic systems have taken shape in a special scientific direction - synergetics. As already noted, synergetics studies the principles of evolution and self-organization of complex systems of various nature based on the construction of nonlinear models of the behavior of these systems. Dynamic models developed in natural sciences (physics, biology, etc.) for describing complex processes are now increasingly used in economics. Complex economic processes generated by the nonlinearity and instability of systems cannot be understood and predicted at the phenomenological level of classical economics. Such processes include, for example, economic cycles, economic crises, fluctuations in competition, pricing, urban development dynamics, international economy, and many other processes. The regularities of a number of such real phenomena of the modern economy can be established at the quantitative physical and mathematical level within the framework of synergetics. Consequently, the synergetic methodology for studying economic phenomena and processes allows you to gradually "translate" economics from the block of humanities to the block of natural sciences. Such a task is solved by sections of synergetics - economic synergetics and synergetic economics.
At one time, the concepts of equilibrium and disequilibrium came to classical economics from natural science. In recent years, a stream of new terms and concepts has come into the economy from modern natural science, such as self-organization, nonlinearity, order parameters, chaos, entropy, bifurcation, catastrophe, limit cycle, phase space, dissipative structure, attractor, and many others. The influence of natural science, and in particular physics, and the methods of physical and mathematical sciences on the development of the economy is also evidenced by the fact that out of 40 Nobel laureates in economics, almost all have a physical and mathematical education. Developed by the remarkable Soviet physicist Academician L.I. Mandelstam, “nonlinear physical thinking” begins to penetrate into economic science, and thus physics influences the formation of “nonlinear thinking” and the culture of mathematical thinking in modern economics, the development of “nonlinear intuition” among economists.
The main tool of a “nonlinearly thinking” specialist (physicist, chemist, economist, etc.) is the corresponding physical and mathematical models. Such models of systems describe whole classes of phenomena, united according to some attribute. Even the most successful model is not a copy of the real phenomenon, but only an expedient approximation. Mathematical models of economic processes are a system of non-linear equations of various types. Modern synergetic models are constructed by combining numerical and analytical methods. A synergistic approach to non-linear mathematical and physical problems can be defined as the modern use of analysis and numerical machine mathematics to obtain solutions to reasonably posed questions regarding the mathematical and physical content of equations. The use of synergetic methods will allow the economy to go beyond the quasi-statistical approach and introduce the physical and mathematical language to solve real scientific and practical problems of economic development.
Year of issue: 1999Genre: Economy
Publisher:"World"
Format: PDF
Quality: OCR
Number of pages: 335
Description: In the book offered to the attention of readers by V.-B. Zang "Synergetic Economy" an attempt is made to give a general idea of the possibilities of a synergistic approach in the economy. In this case, the main attention is paid to the consideration of relatively simple mathematical models of small dimensions, which, as a rule, can be investigated by analytical methods. The use of synergetic methods in the economy is not a tribute to fashion, but an urgent need to move forward beyond the limits outlined by the quasi-stationary approach, to look for new ways to use powerful modern computing tools to solve serious practical problems.
The mathematical toolkit of the book is a fairly compact set of methods that make it possible to carry out a very effective analysis of nonlinear models of real economic processes. The undoubted advantage of the approach used is that the analysis of the low-dimensional models discussed in the book is easy to comprehend, since the set of properties that are the most striking consequences of nonlinearity is rather limited. Therefore, the mathematical apparatus used in the book should become not only the alphabet for a new generation of economists, but at the same time a beacon to which the mathematical training programs of economic universities should be tuned. Apparently, it is in connection with this that V.-B. Zang recommends his book not only to specialists, but also to students of economic specialties. The scale of the task that the author set himself did not allow him to avoid some shortcomings. This concerns, first of all, the excessive conciseness of the presentation of fundamental hypotheses in the formulation of mathematical models, which, unfortunately, is inherent not only in this, but also in many other books on mathematical economics. You can also note that; that the economic models in the book often serve as illustrations of well-known mathematical results. This puts the models under consideration in a subordinate position in relation to the mathematical apparatus, which, of course, cannot but cause some feeling of dissatisfaction among economist readers. However, as a result of this approach of the author to the presentation of the material, the reader discovers, for example, that economic cycles are as natural as population fluctuations, and "leaps" in society, that is, changes of a revolutionary type, are like phase transitions for matter. So this can be treated as a deliberate methodological approach in presenting material, which forces readers to delve more carefully into those stingy lines that set out the main hypotheses and mathematical constructions of the models, and to show maximum independence in understanding not only the results presented, but also the mathematical task setting.
The reader should be quite critical of some subjective assessments (and self-assessments) of the author. For example, speaking about Haken's subordination principle, it is impossible not to mention another formulation of this principle - Tikhonov's theorem for systems of equations with singular perturbations. And in general, speaking of synergetics, it should be remembered that many of its results are directly related to the development of mathematical modeling, at the origins of which in our country were A. A. Dorodnitsyn, N. N. Moiseev, A. A. Samarsky and others (for For the convenience of readers, we provide at the end of this preface a small list of literature in Russian on this topic).
At the same time, we would like to draw the reader's attention to the main advantage of the book: on the whole, the author managed to give a broad panorama of the state of affairs in today's synergetics using the analysis of relatively simple models of dynamic economic processes as an example. Moreover, the book is aimed at developing a non-linear style of thinking among readers, which is important in any field of knowledge, including, of course, in the modern economy.
When working on the translation manuscript, we corrected the observed inaccuracies of the original without any special reservations, and where necessary, made footnotes. It should be especially noted that the publication of the book in Russian was carried out thanks to the initiative of the translator of the book N.V. Ostrovskaya, who supported her initiative to the Russian Foundation for Basic Research (head of the publishing department V.D. Novikov), employees of the Mir publishing house, as well as A.V. Fedotov, who took part in the translation of chapters 5 and 9.
We would also like to express our gratitude to the author of the book, Prof. V.-B. Zang for his attention to the Russian edition - he kindly sent, at our request, a list of typographical errors, which was taken into account in the Russian edition, and also answered a number of questions regarding the clarification of certain places in the text. In conclusion, we express the hope that the book will be useful to all readers interested in applications of nonlinear analysis methods in economics. Who knows, maybe among them will be those who, with its help, will find the very thread, unraveling which, it will be possible to get to a clear synergistic picture of the economic problems that we are all experiencing today and, having this picture in front of us, find real ways to decent economic development. Contents of a book
«Synergetic economy
»
Time and change in economic theory
ECONOMY
SYNERGETICS AND ECONOMY: PRINCIPLES OF INTERACTION
O.Yu. Chernyshov
The article describes a new approach to the study of economic processes and phenomena - synergetic. The author gives the principles of interaction of fundamental mechanisms, laws, theories, hypotheses and methods of the traditional economy with the mechanisms, laws, theories, hypotheses and methods of the synergetic economy as a universal approach to the study of economic processes and phenomena. As a result, the author derives the advantages of the new approach, describes its principles and universality.
Key concepts: synergetic economy, traditional economy, nonlinearity, dynamic chaos, open economic system.
In this article, we will try to justify a synergetic approach to a complex open system - the economy. The very fact that the economy of a country, region, city or any other territorial entity in the age of globalization cannot exist as a closed system, but develops as an open system, a priori suggests the possibility of applying a synergistic approach to the study of economic processes and phenomena. However, first you need to understand what “synergetics” is.
Synergetics as a science of the development and self-organization of complex systems adapts the interdisciplinary approaches of its predecessors: A.I. Bogdanov,
systems theory L. von Bertalanffy, cybernetics N. Wiener.
Synergetics (from the Greek "synergen" - joint action) is an interdisciplinary area of scientific research, the task of which is to study natural phenomena and processes based on the principles of self-organization of systems (consisting of subsystems). This is a science that studies the processes of self-organization and the emergence, maintenance, stability and decay of structures of a very different nature.
As G. Haken, professor at the Institute of Synergetics and Theoretical Physics in Stuttgart, believed, synergetics is the study of systems consisting of a large (one might say even huge) number of parts,
components or subsystems, parts that interact in a complex way with each other. Synergetics assumes the coordination of the functioning of these elements, which, ultimately, is reflected in the functioning of the system itself as a whole. The basic concept of synergetics is the definition of structure as a state resulting from the behavior of a multi-element or multi-factor environment that does not demonstrate a tendency to average.
Synergetics refers to new sciences. However, unlike other new sciences, it did not arise at the junction of two old ones. Synergetics in each of the sciences sees its own internal points, from which it repels. A physicist, chemist, biologist, mathematician, economist or any other scientist sees his material in synergetics, uses it, developing the methodological base of synergetics and his science.
G. Haken, who came up with the name of this new science, noted that “there are truly amazing analogues between the behavior of completely different systems studied by different sciences ... Of course, synergetics does not exist on its own, but is associated with other sciences, at least doubly. First, the systems studied by synergetics are within the competence of various sciences. Secondly, other sciences bring their ideas to synergetics. A scientist trying to break into a new field naturally
regards it as a continuation of his own field of sciences.
Synergetic methods are based on nonlinear mathematics and the results of natural sciences that study the development of complex systems. In recent years, synergetic methods have been actively transferred to the study of the humanities, such as history, sociology, psychology, pedagogy, economics, etc. These sciences are also studying complex systems in which the principle of self-organization plays an important role.
The history of the development of synergetic methods is associated with the names of many prominent scientists of the 20th century. So, the great French mathematician, physicist and philosopher A. Poincaré at the end of the 19th century. laid the foundations for the methods of nonlinear dynamics. He introduced the concepts of "attractors", "bifurcation points", "unstable trajectories" and "dynamic chaos".
It is important to note that a significant contribution to the development of synergetics at the beginning of the 20th century. introduced by Soviet scientists: A.M. Lyapunov,
N.N. Bogolyubov, L.I. Mandelstam, A.A. Andronov, A.N. Kolmogorov. Among Western scientists, first of all, one should name
A.M. Turing and E. Fermi. These scientists developed the so-called. “synergetics before synergetics”, since the term itself has not yet been used.
In the second half of the XX century. there is a significant leap in the study of the processes of self-organization of various systems. Thus, in 1963, dynamic chaos was discovered in problems of weather forecasting. Then the study of strange attractors begins, the “butterfly effect” is discovered as the instability of the solution according to the initial data of strange attractors, a universal theory of catastrophes is created by R. Thom and V.I. Arnold and for the first time a synergistic approach is used to explain the processes and phenomena taking place in the field of study of the humanities. In 1970, G. Haken will call the range of these methods "synergetics" or "the theory of collective, complex behavior of systems." In the future, synergetics develops in such directions as a more complete study of dynamic chaos in fractal geometry, the phenomenon of self-organized criticality is discovered. Today, synergetics is becoming more and more deeply integrated into the field of the humanities: new areas of evolutionary economics, socio-
synergetics, it is used by psychologists and educators, applications are being developed in linguistics, history and art history.
The field of research in synergetics has not yet been fully defined, since the subject of its interests lies among various sciences, and the main methods of synergetics are taken from non-linear non-equilibrium thermodynamics. There are several schools within which a synergistic approach is being developed:
1) The Brussels school of I. Prigogine, in line with which the theory of dissipative systems was developed, the historical background and ideological foundations of the theory of self-organization were revealed;
2) the school of G. Haken, professor at the Institute of Synergetics and Theoretical Physics in Stuttgart;
3) the mathematical apparatus of the theory of catastrophes for describing synergetic processes was developed by a Russian mathematician
B.I. Arnold and the French mathematician R. Thomas;
4) within the school of Academician A.A. Samara and Corresponding Member of the Russian Academy of Sciences
C.P. Kurdyumov developed a theory of self-organization based on mathematical models and a computational experiment (including the theory of development in blow-up mode).
5) contribution to the development of synergetics was made by Academician N.N. Moiseev - ideas of universal evolutionism and co-evolution of man and nature;
6) a synergistic approach in biophysics is developed in the works of Corresponding Members of the RAS M.V. Volkenstein and D.S. Cher-Navsky;
7) a synergetic approach in theoretical history is developed in the works of D.S. Cher-navsky, G.G. Malinetsky, L.I. Borodkina, S.P. Kapitsa, S.Yu. Malkova, A.V. Korotaeva, P.V. Turchin, V.G. Budanova, A.P. Nazarene and others.
Let us note the following principles of the synergetic approach in modern scientific knowledge:
Science deals with systems of different levels of organization, the connection between them is carried out through chaos;
The union of elements, parts, subsystems, systems does not mean that the whole (new system) obtained as a result of the union is equal to the sum constituent parts;
All systems are subject to spontaneous formation, changes, which causes the emergence of new qualities. These processes occur due to self-organization. The behavior of systems in the transition from a disordered state to a state of order is the same for all systems;
Disequilibrium in the system is the source of the emergence of a new organization (order);
Systems are always open and exchange energy with the external environment;
The processes of local ordering are carried out due to the influx of energy from outside;
Under highly non-equilibrium conditions, systems begin to perceive those factors that they would not perceive in a more equilibrium state;
In non-equilibrium conditions, the independence of elements gives way to corporate behavior;
Far from equilibrium, the consistency of the behavior of elements increases;
In conditions that are far from equilibrium, bifurcation mechanisms operate in systems - the presence of bifurcation points, the continuation of development. Options for the development of the system are practically unpredictable.
The economy belongs to such a class of systems, which are characterized by distinctive features - non-linearity and probability of development, fluctuation of indicators in real-time dynamics, inability to assess the behavior of parameters economic categories for the long term. The development of such systems is possible only in the field of evolution - some systems die off, others gain strength according to the collector principle of distribution.
In order to consider the significance of the synergetic economy as a new approach to studying various economic problems, processes and phenomena, one should consider how the transition from the traditional economy to the new synergetic economy took place, consider how the traditional and synergetic economies correlate.
Over time, our ideas about the formation and development of economic processes and phenomena change and improve. Thus, the traditional model of economic theory is based on the summation of equilibrium elements, while the synergetic model is based on
turnover is based on disequilibrium, dynamic chaos and self-organization. However, it should be noted that the synergetic economy does not reject the methods of the traditional economy. Synergetic economics deals with the evolution of economic systems, i.e. with economic dynamics. Many theories related to the traditional economy are studying the dynamics of the development of economic systems. These include: the theory of business cycles, the theory of economic growth, the theory life cycles, a variety of analytical methods such as the correspondence principle. As we can see, all these theories and methods belong to the traditional economy, which, unfortunately, cannot be considered as universal. To make the theoretical and methodological set of the traditional economy universal and applicable to the explanation of many economic phenomena and processes is the main task of synergetics. As noted above, synergetics enriches the methodological set of any other science, including economics.
Synergetics expands the apparatus of economic dynamics, as it tries to explain many other economic processes and phenomena that traditional theory misses. A remark should be made right away: synergetic economics cannot answer all the questions that arise in the study of economic systems, but it gives a clearer idea of a larger number of studied economic systems. economic issues. "Synergetic Economics Offers Encouraging New Direction for Explaining Complex Economic Phenomena".
Based on the traditional economy, fundamental mechanisms of processes and phenomena occurring in economic systems arose: competition, cooperation, equilibrium, rational behavior of objects of economic processes.
Synergetics offered a new direction in the economy. The concepts of rational behavior, stability and balance, competition play an important role in a synergetic economy, but by no means the most important one. Here such mechanisms as instability, non-equilibrium, dynamic chaos, self-organization are put at the head.
The synergetic economy sees the evolution of economic systems from a somewhat different angle, not from the standpoint of stability and linearity, but vice versa - instability and non-linearity.
Here, synergetics relies on a mathematical apparatus that studies non-linear functions. Before the advent of modern non-linear dynamical theory, chaos was practically inaccessible to study. The synergetic approach in economics tries to prove that chaos underlies any evolutionary economic system.
Synergetics arose in response to the crisis of stereotyped, linear thinking that has exhausted itself, the main features of which are: the idea of chaos as an exclusively destructive beginning of the world; considering randomness as a secondary, side factor; a look at disequilibrium and instability as annoying troubles that must be overcome, since they play a negative, destructive role; the processes taking place in the world are reversible in time, predictable and retropredictable for infinitely long periods of time; development is linear, progressive, non-alternative (and if there are alternatives, then they can only be random deviations from the mainstream, subordinate to it and ultimately absorbed by it); what has been passed is of exclusively historical interest; the world is bound by rigid causal relationships; causal chains are linear, and the effect, if not identical to the cause, is proportional to it, that is, the more energy is invested, the greater the result.
The prerequisites for detecting instability in economic systems can be found in the works of economists of the traditional era, namely, in the works of K. Marx, J. Keynes, I. Schumpeter and others, although all their research did not contribute to the formation of new hypotheses or theories.
The first significant attempt to study the dynamics of economic development was undertaken by K. Marx in Capital. The author tried to open economic law society movements. For many decades this attempt remained the only one. However, studies of this theory of development
The expected economic processes and phenomena were linear in nature, that is, they were based on the idea of the unidirectionality and predetermination of history. At the same time, the following processes and phenomena of no small importance for the study were missed: scientific and technological progress, unemployment, foreign trade, equity capital, crises, changes in state regulation, monopolies. In this theory National economy was presented as a closed, equilibrium economic system, which to a greater extent contributed to the formulation of a number of laws (the impoverishment of the proletariat, the downward trend in the rate of profit), the effect of which is observed only if the system of assumptions is observed, which in reality is practically unattainable.
I. Schumpeter's theory of the dynamic development of the economy was based on the same. The starting point of the study was the "economic cycle" of a closed economy, which is an equilibrium state. As well as in the theory of K. Marx, we find a number of restrictions: profit and interest are absent, producers reimburse only costs, economic relations in the national economy are unchanged, complete freedom of competition reigns, the volume of production is determined from previous sales experience, supply and demand are equal, the economy is stationary . However, it should be noted that Schumpeter, without realizing it, identified bifurcation points in the economy through innovative shocks (shocks), which lead to qualitative changes in the economic system.
J. Keynes considered the national economy as a system that requires outside intervention in order to acquire new qualities that are needed at a given time. Thus, he assumed the active participation of the state in the economic life of the country, which before him was considered impossible and extremely harmful. However, Keynes also believed that intervention is necessary in order for the national economy to reach an equilibrium state.
In the future, many prominent scientists dealt with the problems of the instability of economic systems, especially in the study of the theory of business cycles. This is N.D. Kondratiev, S. Kuznets, K. Zhuglyar, J. Kitchin,
Toffler and others. However, their attempts were reduced mainly to describing the processes of returning economic systems to an equilibrium, stable state.
It should also be noted that the aforementioned economists did not actively use mathematical tools that help to accurately express instability, cyclical development, and dynamic chaos. Also, the emphasis in the works of these scientists was placed on the development of closed, linear systems. Nevertheless, the identified errors during the testing of these theories pointed to a new approach in describing the development of economic systems - synergetic.
Synergy economics emphasizes the interaction of different variables and different levels of the system. Although the significance of such interactions is also recognized by "systems analysis", but, assuming stability, it remains within the framework of traditional economics, which makes it difficult to understand the processes of economic evolution.
So, from all of the above, we can draw the following conclusions about the interaction of synergetics and economics:
1) a synergistic approach is a completely new approach to the study of economic processes and phenomena occurring during the dynamic development of economic systems;
2) this approach does not cancel the fundamental mechanisms, laws, theories, hypotheses developed within the framework of the traditional economy, but organically uses and improves them, which makes it possible to apply these mechanisms as universal in the study of economic processes and phenomena;
3) a synergistic economy, unlike the traditional economy, is based on the openness of economic systems
topics, their nonlinearity, instability and self-organization;
4) a synergetic economy assumes that the economic system is complex, consists of many interrelated elements, and the system as a whole is not equal to the equilibrium sum of its elements;
5) in a synergetic economy, dynamic chaos is put at the head, and not the desire for an equilibrium state, as is assumed in the traditional economy;
6) in the synergetic economy, the non-linear mathematical apparatus is very widely used, which was practically not used in the traditional economy. This makes it possible to describe dynamic chaos in economic systems.
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3. Erokhina EL. Theory of economic development: system-synergetic approach.
Tomsk, 1999.
Received August 4, 2008
Chernyshova O.Y. Synergy and economy: interaction principles. The article describes a new approach to study of economic processes and occurrences (a synergetic approach). The author gives the principles of interaction of fundamental mechanisms, laws, theories, hypothesis and methods of traditional economy with mechanisms, laws, theories, hypothesis and methods of synergetic economy as a universal approach to study of economic processes and occurrences. The author outlines advantages of the new approach of study, describes its principles and universality.
Key words: synergetic economy, traditional economy, nonlinearity, dynamic chaos, open economic system.
