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Paper abstract "Cobweb Model for Economic Dynamics of the Prices" Plan:Introduction
Cobweb theory is rapidly developing over the past two decades. The problem of the market equilibrium stability is of great economic importance. When market reaches equilibrium under the influence only of its internal forces (i.e., through self-regulation), that means that additional external market regulation is not required: the market is able to maintain its balance itself. If the equilibrium is unstable, then the market regulation becomes imperative. Taking into account the crisis in our country, the Ukrainian market requires competent mechanism of its regulation and restoration.
Figure 1 - The process of searching an equilibrium in the cobweb model (gif-animation, 11 shots, 6 cycles, size 76 Кб, software MP GIF Animator) Topicality The idea of the general economic balance goes back to the works of the classical economists beginning with the works of Smith. The fundamental possibility to achieve a general balance in conditions of perfect competition in a mathematical form was expressed first by L. Walras. Hereafter the topic was considered by Marshall ("fundamental symmetry"), and later - by his oppositionist Italian economist Piero Sraffa. A modern analysis of market equilibrium came away from its initial review, but all that was laid earlier, is an essential foundation for further research in this area. We analyze the nonlinear cases, with emerging market instability and chaos. The most prominent scientists, cobweb model researches are: C. Chiarella, C. H. Hommes, B. Finkenstadt, R. V. Jensen, R. Urban, A. Matsumoto and others. Among native researchers can be distinguished N.V. Chumachenko, A.I. Lysenko, V.V. Shevchenko, T.M. Borovskaya and others. Active investigations take place at the Center for Nonlinear Dynamics in Economics and Finance in Amsterdam (CENDEF). The scientific value of study Currently, the building of nonlinear cobweb models for the native market is not popular either among the Ukrainian economists, nor among Ukrainian mathematicians (as opposed to foreign scholars). We would like to prescribe a new research direction in our country by researches, that conducted in this paper. This direction requires more serious further development. To study this problems we would like to use already known mathematical packages, and also new software, which has not been applied in Ukraine until now. A review of existing researches and studies on the topic As already mentioned, among the native investigations on this topic, was not found any interest work. Nevertheless, I can provide research of National Technical University of Vinnitsa, namely of T.M. Borovska, which is summarized in her study functions of supply and demand in cobweb theorem. The author introduces a demand function as parameterized logistic dependence, which has the following form: Lv(p,n,ω,a,s)=monotone_decreasing _function(price, concave, frequency, amplitude, initial_price). And gives a generalized function of production, which is as follows: Lv(x,n,ω,a)=monotone_increasing _function(price, concavity, frequency, amplitude). The main constraint in the model is supposed the size of the market. On this list of Ukrainian accomplishments ends (at least in my search ability). A completely different situation is observed in the scientific thought of foreign scientists. Whole world occupies the cobweb theorem, its various modifications and complications. Special studies conducted in North-East Asia (China, Japan, Korea, etc.), in Western Europe (Netherlands, etc.) and in the south-west Pacific (New Zealand, Australia, etc.). A significant step was made by professor in the economic dynamics Dr Cars H. Hommes. This man founded in October 1998 at the University of Amsterdam Center for Nonlinear Dynamics in Economics and Finance, which conducts extensive research on nonlinear economic models. Cars Hommes wrote a considerable number of works on studying and analyzing the cobweb model. He has developed various types of web models:
Researches of the cobweb model with heterogeneous producers also were made by scientists of the Australian School of Finance and Economics and University of Technology in Sydney — Carl Chiarella, Xue-Zhong He and Peiyuan Zhu. Paul Bedford and Chris Bloor built cobweb pricing model for financial stability in New Zealand.This model was built for different markets:
Another guideline in the analysis of cobweb models is the study of chaotic behavior; states in which appear the bifurcations of price fluctuations. Cobweb theorem can be viewed as a discrete nonlinear system. And as we know, such systems are prone to chaotic behavior u with certain parameters. Such systems are very sensitive to initial conditions. Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Sensitivity to initial conditions is popularly known as the "butterfly effect", so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C. entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior. Chaotic motion is described by a strange attractor, which are very complicated and have many options. Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Henon map). Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points — Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and a fractal dimension can be calculated for them. Poincare-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system (specified by differential equations) if it has three or more dimensions. Finite dimensional linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be either nonlinear, or infinite-dimensional. Discrete two-and even-dimensional systems can have strange attractors. Therefore, one of the challenges of today's research of cobweb model is an analysis of emerging attractors (regular and chaotic) to study the sensitivity of prices over time, and the emphasizing of key factors and parameters that must be carefully controlled. The next problem is the continuation of attempts to stabilize the chaotic behavior of the identified systems, turning it into a regular. This area is developing rapidly, and to date has already written a sufficient number of works related precisely with the stabilization of chaotic behavior in the cobweb model. Among the main methods used to solve this problem mostly apply the method of delayed feedback, or another name - the Pyragas method. It is surprising that the method of feedback was proposed in 1990 by J. York with co-authors, which asserted the possibility of significant changes in the properties of chaotic systems using a very small change in its parameters. The article concludes that even a small management in the form of feedback applied to the nonlinear (chaotic oscillating) system, can radically change the dynamics and properties (for example, to convert the chaotic motion of a periodical). The work has generated an avalanche of publications. Some of them were made in experimental way, but more often through computer simulations. They which demonstrate how a control (with feedback or not) can affect to the behavior of a variety of real and model physical systems. Proposed control method became known as OGY method to the initial letters of authors' names. Number of references to work by 2002 exceeded 1300. Later were proposed and other methods to transform the chaotic motions into periodic. One of the most effective and proved method of delayed feedback (or Pyragas method). In the Pyragas method, an appropriate continuous controlling signal is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit. There are used and some other existing methods of nonlinear and adaptive control. Findings СToday the study of cobweb models gone far ahead of its classical presentation. This model allows to get internal pricing mechanism seriously, to determine the main parameters affecting the system and analyze its behavior. It is not just an economic model — it is a complicated mathematical analysis, allows to explain changes of price and withdraw the scheme of properly control of the market prices. During writing this abstract the master work was not finished yet. Final completion: December 2010. Full text can be obtained from the author or her teacher аfter that date. Literature
© A. Mospan 2010 |
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