خود سازماندهی در مدل نظام اقتصادی با تعاملات ثابت مقیاس

The method of constructing the local scale invariant stochastic models is proposed. The possible extension of minimal scale-invariant interaction principle for stochastic systems is formulated. A simple scale invariant model that possesses an economical interpretation is considered. Essential characteristics of its self-organization mechanisms are discussed.

The gauge invariance is considered as one of the fundamental interaction principle in the theory of elementary particles [1]. There are reasons to suppose that it can be used for the construction of the models of basic self-organization mechanisms of complex adaptive natural system [2], [3], [4] and [5] and particularly for modeling of economical behavior [6]. We propose a simple way to construct models of such a kind using the principle of scale invariance in the stochastic dynamics. We suggest the use of ideas of minimal gauge interaction developed in the quantum field theory. It makes it possible to essentially reduce the number of arbitrary parameters so that the dynamical variables and parameters of the models allow simple and clear interpretation. The most complicated problem in the studies of complex system dynamics arises for unordered non-stable interaction structure. It is mostly the case in real economics. The activities of elements of economical system cannot be presented as being placed on the sites of a D-dimensional lattice and the basis for the solid state physics near neighbors interaction principle seems to be non-adequate in this situation. The interaction structure in economics is also usually time dependent on complex laws of dynamics. The simple and often seemingly fruitful investigation approach for the system of such a kind is based on the idea to replace the interaction between elements with an interaction of them with an external stochastic communication field. The main problem is to define correct statistical properties of communication field, but in many cases the fundamental physical principles help to solve it. In our consideration, we use the following conception. The essential feature of the economical interactions is that they are realized on the basis of comparison of results of activities of the system elements. The convenient method of comparison is to use the scale. Thus, the scale becomes the transfer-agent for element interactions. The most universal economical scale is money, and from the physical point of view it is the “field” realizing interactions in economics. We obtain the following picture. The interaction of elements in economical system is described with “scale field”. It is natural to suppose that the local scale invariance of the system, means that the system behavior must be independent of the way in which it is described by a chosen scale field. It could be considered as a particular formulation for economical system modeling of general Einstein relativity principle. Our aim is to investigate the main features of the models with local scale invariance and to construct the most general local scale invariant stochastic model. It will be done in the next section. We show that the local scale invariance makes a rather strong restriction on the possible interactions structure and on the form of an observable in stochastic dynamical systems. In Section 3, we consider the simplest local scale invariant model and after that we discuss the possible ways of investigations in the framework of proposed approach.

The stochastic dynamical model which can be constructed in the framework of the proposed approach seems to have a perspective for theoretical studies of universal self-organization dynamical mechanisms in complex economical systems. The models with local scale invariance can be easily interpreted in terms of economic problems. For example, the local scale invariant field is the simplest “economical observable” View the MathML source having as mean, the efficiency of activity of the ith element. The fields vi(t),hi(t) present individual stimulus of activity, and the scale invariant field u(t) in (12) describes the fonds. The matrix Λ(t) in general formulation of the model could be used for presentation of mechanisms of direct exchange in economics. Model (12) could be considered as a simple model of monetary economics. The total efficiency of a model of the exchange economics appears to be essentially defined by a consistent individual stimulus, presented by interaction matrix Λij. In the model of monetary economics (12), the efficiency can be essentially defined with finance flows in the system, described by the stochastic field u(t) and coefficients View the MathML source (presenting the indexes of the efficiency of ith activity). Analysis of correlation function (13), (14), (15), (16), (17) and (18) shows that the dynamics of the system in a general situation is non-equilibrium. One can retain the problem in order to formulate laws of dynamics for the system parameters modeling the mechanisms of self-organization in real economics. For system (12), one can use the ideas of P. Bak and K. Sneppen for modeling the evolution of parameters View the MathML source on the basis of natural selection principles. This paper will be the model of self-organized criticality which could help to understand mechanisms of price formation and appearance of the power laws in the behavior of economic systems. In statistical studies of empirical data of real economics, the most universal and stable results are obtained for the scale invariant characteristics of economical processes. It seems to be in agreement with that obtained in our consideration rule that all the observable quantities are scale invariant. One can hope that the local scale invariance as interaction principle captures the essential features of the nature of economics and that the proposed approach may be useful for a better understanding.