Key words: economic system, mathematical model, parametrical regulation, task of variational calculation, extremal, functional, bifurcation.
Abdykappar ASHIMOV, Kenzhegaly SAGADIYEV, Yuriy BOROVSKIY, Nurlan ISKAKOV, Askar ASHIMOV
ONTHE MARKET ECONOMY DEVELOPMENT PARAMETRICAL REGULATION THEORY
The paper offers the theory of a parametrical regulation of market economy development. This theory consists of the following sections: forming of a library of economic systems mathematical models; researches on rigidness (structural stability) of mathematical models; development of parametrical regulation laws; finding of bifurcation points of the extremals of one class of variation calculations tasks, etc. The report delivers some of the results obtained in the course of developing the theory under consideration. The work contains also example of a finding of bifurcation points for one model of economic systemand the results of the considered model’s rigidness study with and without parametrical regulation.
1. INTRODUCTION
Many dynamic systems (Gukenheimer, 2002) including economic systems of nations (Petrov, 1996), after certain transformations can be presented by the systems of non-linear ordinary differential equations of the following type:
,(1)
Here is a vector of a state of a system; - a vector of (regulating) parametrical influences; W, Хarecompact sets with non-empty interiors - and respectively; is an opened connected set; maps and , , are continuous in , - a fixed interval (time).
As is known (Pontryagin, 1970), solution (evolution) of the considered system of the ordinary differential equations depends both on the initial values’ vector, and on the values of vectors of controlled (u) and uncontrolled parameters. This is why the result of evolution (development) of a nonlinear dynamic system under the given vector of initial values is defined by the values of both controlled and uncontrolled parameters.
In the recent years the dynamic trends of such parameters are being actively researched as: various tax rates, state expenditures, accounting rate, reservation norm, currency rate and others. The influence of these parameters upon the evolution of economic processes is also in the focus of researches. Thus, in (Chernik, 2000) econometric methods are used for modeling the dynamic lines and statistical forecasting of tax revenues. In (Belenkaya, 2001) econometric methods are used for the analysis of dependences between the parameters of cash and loan policy (refinance rate, reservation norm) and economic development indicators (indexes of investment activity in the real sector etc.). Paper (Petrov, 1996) considers the impact of state expenditures’ share in the gross domestic product and of the state loans’ interest upon the average real income of the workers, upon average state expenses in stable prices and on the average gross domestic product. This is based on the mathematical model suggested by the authors (Petrov, 1996) and conducted after solving the task of parametrical identification.
Currently, due to the development of the theory of dynamic systems (Andriyevsky, 2004), the parametrical influence has become applied to the regulation of economic systems. The dynamics of these economic systems, according to the opinion of a number of experts, is described with the help of nonlinear models (Lorenz, 1997), whose behaviour can be chaotic. Thus (Yanovsky, 2002), the parametrical influence that is defined by the Otto-Gregory-Yorke method (Otto, 1990), was used in order to stabilize the unstable solutions in the models of a neoclassical theory of optimal growth.
In a number of papers (Popkov, 2005; Kulekeev, 2004; Ashimov, 2005a,b,c) the parametrical influences are being offered in order to effectively regulate the market economy development in the given range of the changes of main endogenic indicators of an economic system and to hinder its expansion outside the given range. The suggested parametrical influences are the extremals of the corresponding variation calculations tasks on the choice of optimal laws of parametrical regulations within the given finite set of algorithms. The functionals in the given tasks of variation calculations express certain (global, intermediate or tactical) goals of economic development. Phase restrictions and permitted restrictions are represented by mathematical models of economic systems from (Petrov, 1996). Mathematical models from (Petrov, 1996) contain a range of indexes, whose change within a certain interval leads to the deformation (perturbation) of the considered variational calculus tasks.
Nowadays, the parametrical deformation of the variational calculus tasks is being widely researched. Thus, the parametric perturbation in (Ioffe, 1974) is used in order to obtain the sufficient extremum conditions through construction of the correspondingS-functions and usage of the principle of restrictions removal.In (Ulam, 1964) the problem of the terms of stability of variations calculus tasks solutions is set (Ulam’s problem). The study of this problem is brought to finding the terms of regularity under which the functional of the perturbation task has the point of a minimum that is close to the point of a minimum of a non-perturbation task’s functional.In (Bobylev, 1998) the theorem about the terms of bifurcation point existence for one variations calculus task is proved. The functional of this task is considered in the Sobolev’s space and depends on the scalar parameter. Thus, it could be noted that among the existing sources there has been no evidence as to the terms of existence of the solutions to variational calculus tasks on the choice of optimal laws of parametrical regulation in an environment of the given finite set of algorithms. There are no researches on the influence of parametrical perturbations upon the solution of the given tasks.
Existence of the solutions to the variational calculus tasks on the choice of optimal laws of parametrical regulation in an environment of the given finite set of algorithms is under study in (Ashimov, 2006a,b,c). These research works also focus on the influence of parametrical perturbation (changes of uncontrolled parameters) upon the results of solving the tasks under consideration. In other words, the researches are conducted, in particular, as to the bifurcation of extremals of these tasks under parametrical perturbations.
The approaches suggested in (Popkov, 2005; Kulekeev, 2004; Ashimov, 2005a,b,c) and the results obtained in (Ashimov, 2006a,b,c) could be treated as components of the parametrical regulation theory developed by the authors.Основными результатами доклада
2. COMPONENTS OF THE MARKET ECONOMY DEVELOPMENT PARAMETRICAL REGULATION THEORY
On the whole, the following components could characterize the initial version of the market economy development parametrical regulation theory:
1. Methods of formation of a macroeconomic mathematical models library. These methods are oriented towards a description of a variety of certain socio-economic situations with the account of environmental safety conditions.
2. Rigidness (structural stability) conditions assessment methods of mathematical models of an economic system out of the library without the parametrical regulation. At this, it is studied if the mathematical models under consideration belong either to the class of Morse-Smale system or to -rigid systems, or to the uniformly rigid systems, or to the class of У-systemsand others.
3. Methods of control or suppression of non-rigidness (structural instability) of mathematical models of an economic system. The choice (synthesis) of algorithms of control or structural instability suppression of a relevant mathematical model of a country’s economic system (Popkov, 2005).
4. Methods of choice and synthesis of the laws of parametrical regulation of market economy mechanisms based on the mathematical models of a country’s economic system (Kulekeev, 2004; Ashimov, 2005a,b,c).
5. Rigidness (structural stability) conditions assessment methods of mathematical models of an economic system out of the set/store with the parametrical regulation. At this, it is studied if the mathematical models under consideration belong either to the class of Morse-Smale system or to -rigid systems, or to the uniformly rigid systems, or to the class of У-systems.
6. Methods of clarification of the limits on the parametrical regulation of market economy mechanisms in case of a structural instability of mathematical models of a country’s economic system, with parametric regulation. Clarification of the limits on the parametrical regulation of market economy mechanisms.
7. Methods of study and research on bifurcation of the extremals of variational calculus tasks on the choice of optimal laws of parametrical regulation (Ashimov, 2006a,b,c).
8. Econometrical analysis, political and economic interpretation and coordination of the analytical researches findings and calculation experiments with the preferences of decision-makers.
9. Development of an informational system for the research and imitational modeling of market economy mechanisms with parametrical regulation.
10. Development of recommendations on elaboration and implementation of an effective state economic policy based on the theory of parametrical regulation of market economy mechanisms with the account of certain socio-economic situations.
Putting this newly-developed theory of parametrical regulation of market economy mechanisms into practice - elaboration and implementation of an effective state economic policy - could be as follows:
1. Choice of a state economic development direction (strategy), on the basis of a relevant assessment of a country’s economic status in the framework of economic cycle stages.
2. Selection of one or several mathematical models of economic systems that correspond to the tasks of economic development out of the library.
3. Assessment of suitability of the mathematical models to the given tasks: calibration the mathematical models (parametrical identification and retrospective prognosis according to the current evolution indicators of the economic system) and an additional verification of the selected mathematical models with the help of econometric analysis and political and economical interpretation of the sensibility matrixes.
4. Assessment of structural stability (rigidness) of mathematical models without the parametrical regulation according to the above given methods of assessment of rigidness conditions (see item 2). Structural stability of a mathematical model reflects the stability of the economic system itself. In this case a mathematical model can be used after the econometrical analysis, political and economic interpretation of the rigidness research findings, in order to solve the task of choice of optimal laws of regulating the economic parameters and forecasting the macroeconomic indicators.
5. If a mathematical model is structurally unstable, it is necessary to choose the algorithms and methods of stabilization of the economic system in accordance with the methods pointed out in section 3 of the theory under development. The result obtained in the course of the corresponding econometrical analysis, and political and economic interpretation could be approved and ready for practical implementation.
6. Choice of optimal laws of regulating the economic parameters.
7. Assessment of structural stability (rigidness) of mathematical models with the chosen parametrical regulation laws according to the above given methods of assessment of rigidness conditions (it.2). If the mathematical model is structurally stable under the selected parametrical regulation laws, then the results obtained in the course of the corresponding econometrical analysis, and political and economic interpretation, and after coordination with the preferences of decision-makers could be adopted for their practical implementation.
If the mathematical model under the selected parametrical regulation laws is structurally unstable, then the decision on the choice of parametrical regulation laws is clarified. The clarified decisions on the choice of parametrical regulation laws are also subject to be considered according to the above given scheme.
8. The study of dependence of the chosen optimal laws of parametrical regulation on the changes of uncontrolled parameters of an economic system. Here it is possible to replace some optimal laws by the others.
The given enlarged scheme on making decisions in the sphere of elaboration and implementation of an effective state policy through the choice of optimal values of economic parameters should be supported with the modern informational technologies of research and imitational modeling.
At the present time, the above given sections 1, 2, 3, 5, 6 of the theory of parametrical regulation are developed in the framework of the modern approaches of the theory of identification (Krasovsky, 1987; Samarsky,2002) and the theory of dynamic systems (Katok, 1999; Magnitsky, 2004).
This report presents some findings on the development and usage of the components of theoffered theory of parametrical regulation of market economy development as applied to an optimal growth model.
3. RESULTS OF THE PARAMETRICAL REGULATION THEORY COMPONENTSDEVELOPMENT
3.1 On mapping of the conjunction of some areas of a phased space for non-perturbed and perturbed dynamic systems
The following theorem has been formulated and proved in the framework of sections 2 and 5 of the parametrical regulation theory.
Theorem 1. Let and be fixed, and the inequality is fulfilled in area X of a phased space of system (1) for some i . Let be a solution to Cauchy’s task (1), determined for and located in Int(X). Let be a perturbed vector field under X that meets the requirements laid out in the Introduction; for and some . Let be the solution to the corresponding Cauchy’s task for the field .
Then (closed) areas and will be found, where and , for which there exists a homeomorphism , that at the same time realizes one-to-onecorrespondence between the phase curves in areas and keeping their direction (for the fields f and respectively).
3.2. Conditions of existence of a solution to the one variational calculus task
The variational calculus task on the choice of an optimal set of parametrical regulation laws at the multitude of combinations from p parameters on r in an environment of the given finite set of algorithms and the assertion on existence of a solution to the variational calculus taskin an environment of the given finite set of algorithms looks like this:
Let be a solution of the given task (1) in the interval under the constant values and . Let . Let us mark the solution (1) for the selected through . Further is fixed.
Let us mark through the closed set in the space of continuous vector-functions , which consists of all the continuous vector-functions that satisfy the following restrictions.
, , ,, .(2)
Let and be the finite set of continuous for real value functions. All the functions are also continuous in . An opportunity of choiceof an optimum set of parametrical regulation laws at the set of combinations from the p parameters on r and in the time interval is explored in the following algorithms (laws of control):
(3)
Here, are adjusted factors.
Handling of the set of r () laws from (3) under the fixed in the system (1) means the substitution set of functions in the right parts of the equations of the system for r different indexes , ). Herewith, the rest values of , where j does not enter into the specified set of indexes , are considered as constant and equals to the values .
For the solutions of system (1) under the usage of the set of r laws of control the following functional (criterion) is considered:
.(4)
Setting the task of a choice of a dynamic system parametric regulation set of laws in the environment of the finite set of algorithms looks like the following.
Under the fixed find the set of r laws from the set of algorithms (3), which provides the supremum of the values of criterion (4)
(5)
under the fulfilled conditions (1,2) for the given time interval.
Theorem 2.There is a solution to the task of finding the supremum of criterion K under the usage of any selected set of lawsfrom the set of algorithms (3) with the restrictions (1) and (2):
.(6)
Herewith, if the set of possible values of factors of laws of the considered task is limited, then the indicated supremum for the selected set of laws is reached. The task (1)-(5) has got the solution for the finite set of algorithms (3).
The proof of the theorem 2 is present in the(Ashimov, 2006a).
3.3.Sufficientconditions for the existence of a bifurcation point of extremals of one variational calculation task
Let us suggest the following definition, which characterizes the values of parameter , under which the replacement of one optimal law for another becomes possible.
Definition.Value would be called a bifurcation point of the task (1)-(5) extremal, if under there were, as minimum, two different optimal sets of the laws from (3), that would differ at least by one law , and if in each neighborhood of the point there is such value , for which the task (1)-(5) would have a single solution.
The following theorem provides sufficient conditions for the existence of a bifurcation point of the extremals for the considered variations calculus task at the choice of parametrical regulation set of laws in the given finite set of algorithms.
Theorem 3(about the existence of a bifurcation point). Let for the values of the parameter and , () task (1)-(5) has the relevant unique solutions for the two different sets of r laws from (3) that would differ at least by one law . Then there is at least one point of bifurcation .
The proof of the theorem 2 is present in the(Ashimov, 2006a).
4. DEVELOPMENT and USAGE of the THEORY on PARAMETRICAL REGULATION of an ECONOMIC SYSTEM EVOLUTION BASED on a MODEL of OPTIMAL GROWTH
Development and application of this theory to definite tasks of market economy development parametrical regulation involves choice of one or several mathematic models, conforming to the main objectives of economic system developmental directions. This requires such supplementary studies as assessment of rigidness (structural stability) of chosen mathematic model(s), choice of definite parametric regulation laws and analysis of their dependence upon the values of uncontrolled parameters .
4.1. Optimal growth model description
The mathematic model of economic growth (Yanovsky, 2002) represented by a following system of two ordinary differential equations, which contains time derivatives ():
(7)
Here indicates the ratio of capital () to labour (), i.e. amount of capital per an employee. This model does not differentiate between the population of a country and workforce (labour);
- average per capita consumption;
- level of growth (or reduction) of population, ;
- level of capital depreciation, ;