A. A. Ashimov*, Yu. V. Borovskiy*, As. A. Ashimov*
PARAMETRICAL REGULATION METHODS OF THE MARKET ECONOMY MECHANISMS
The tasks of parametrical regulation of mechanisms of market economy are considered on the basis of two mathematical models of economic systems. The models are described with the systems of the differential and algebraic equations, which can be reduced to systems of the differential equations. The solution and research of systems was carried out with using a method of Runge-Kutta. The approach to a choice of effective algorithms in environment of the given set of the laws of regulation as statement and decision of separate extreme tasks at a level one, two and three regulable economic parameters is offered.
1. INTRODUCTION
The development of economic processes within the framework of legislatively (normatively) established «rules of game», regulating relations (mechanisms) between the agents (enterprises, suppliers of resources and consumers) [1], is defined to a considerable extent by values of such economical parameters as different tax rates, the state expense, discount rate, norm of reservation, credit rates, rate of exchange and others. The specified parameters rather frequently vary depending on an economical conjuncture [2] and the methods of an estimation of their effective values are frequently unknown.
The active researches of dynamics of change of considered parameters and their influences on evolution of economical processes are carried out last years. So, in [3] econometrical methods are applied to modeling dynamic series and statistical forecasting of the tax incomes. In [4,5,6] econometrical methods also are used for the analysis of dependences between parameters of monetary-credit policy (refinancing rate, norm of reservation) and parameters of economical development (parameters of investment activity in real sector and etc.). In [7] on the basis of the mathematical model, which are offered by the authors, after the solution of a task of parametrical identification, the influence of the state expenses share on gross domestic product and percent on the state loans on the average real incomes of the employees, average state expenses in constant prices and on an average gross domestic product are investigated.
Nowadays, due to development of the theory of dynamical systems [8,9], parametrical influence begins to find a use in regulation of economic systems, dynamics of which, in opinion of many experts, is described by nonlinear models [10], which can have chaotic behaviour. So, in [11] parametrical influences determined by Otto-Gregory-York method, which is offered in [12] for stabilization of chaotic behaviour of the decisions in nonlinear dynamic system on an example of Anon’s discrete system, was used for stabilization of the unstable decisions in models of neoclassical theory of optimum growth.
The task of a choice of the effective laws of parametrical regulation (influence) in sense of some criterion of optimality, which can determine the definite (global, intermediate or tactical) purpose [2] of economical development, is considered in this paper on the basis of mathematical models [7].
* Institute of Informatics and Control Problems, Pushkin str., 125, Almaty, 480100, Republic of Kazakhstan
2. MATHEMATICAL MODEL OF ECONOMICAL GROWTH
The mathematical model, which is offered in [7] for researches of influence of the consumer expenses of the state on development of economy, after the appropriate transformation, has the following type:
(1)
(2)
, (3)
(4)
, (5)
(6)
, (7)
, (8)
, (9)
, (10)
, (11)
, (12)
, (13)
, (14)
, (15)
, (16)
. (17)
Here: М - total production capacity; Q – total reserve of goods in market; LG - total volume of the state’s debt; p – price-level; s - rate of wages; - volume of indebtedness of production; and - accordingly owner’s and bank’s dividends; Rd and Rs - accordingly demand and supply of labor; δ, v - parameters of function f(x), x - solution of equation; ФL and Ф0 - accordingly consumer expenses of employees and owners; ФI - investment flow; ФG - state's consumer expenses; ξ - rate of reservation; β - ratio of average rate of profit from commercial activity to rate of rentier profit; r2 – interest rate on deposits; r1 - interest rate for credit; rG - interest rate on government bonds; η0 - coefficient of owners disposition to consumption; π - state's consumer expenses share in gross domestic product; np, n0, nL - accordingly rates of taxes on cash flow, dividends and income of employees; b - rate of fund-capacitance of power unit; μ - factor of leaving of unit of capacity owing to degradation; μ* - rate of amortization; α - time constant; Δ - time constant, which is set of characteristic time scale of process relaxation of wage; - accordingly initial values of amount of employees and total amount of able-bodied people; λP>0 - appointed tempo of demographic growth; ω – per capita consumption in group of employees.
The equations and the ratios from mathematical model (1-17) represent the appropriate expressions from [7] or these expressions after simple transformations.
So, the differential equation (1) is received from (3.2.18), (3.2.6); (2) - from (3.2.19) and (3.2.8); (3) - from (3.2.26) by substitution of expressions for () from (3.2.25); (4) represents (3.2.9); (5) represents (3.2.30); (6) represents expression from page 150 [7]; (7) and (8) -expressions from (3.2.39); (9) represents the decisions of the equation from (3.2.10): , where the function (11) is defined on page 157 [7]; (10) represents one of expressions (3.2.10); (12) - from (3.2.15) and (3.2.8); (13) - from (3.2.16) and (3.2.8); (14) - from (3.2.22); (15) represents the expression (3.2.36); (16) is (3.2.11); (17) - from (3.2.12), (3.2.13), (3.2.14).
3. IDENTIFICATION AND RESEARCH OF PARAMETRICAL SENSITIVITY OF THE MODEL
The numerical integration of system of the differential equations (1-5), after substitution of algebraic ratios (6-17) in it, was carried out with Runge-Kutta method. The parameters of model were estimated by the task solution of parametrical identification on the basis of the data of economy of the Republic Kazakhstan for 1996-2000 [13].
With research of values of the following parameters were accepted from [13] accordingly equal:
r1=0.13; r2=0.12; rG=0.12; β =2; np=0.08; nL=0.12; s=0.1; n0=0.5; μ=μ*=0.012; Δ=1.
For an estimation of values of other parameters of mathematical model: ξ; π; δ; v; η0; b; α; Q(0) was solved the task of parametrical identification by a search method in sense of a minimum of the sum of discrepancy squares
. (18)
Here Mj*, Mj**, pj*, pj** - accordingly observed and model (calculated) values of total productive capacity and price-level; N - number of observations (36 months).
As a result of the task solution of parametrical identification on the basis of the statistical data [13] the following estimations of values of unknown parameters were received:
ξ=0.1136; π=0.1348; δ=0.3; v=34; η0=0.05; b=3.8; α=0.008; Q(0)=-125000.
The relative value root-mean-square deviation of calculated values variables from appropriate observed values has made less than 5%, that is illustrated on a part of the observations, covered with parametrical identification, in table 1.
Table 1
Years / M* / M** / p* / p**1998 / 144438 / 158576 / 1.071 / 1.09
1999 / 168037 / 183162 / 1.16 / 1.20
2000 / 216658 / 212190 / 1.31 / 1.29
In table 1: М*, М**, p*, p** - accordingly values of observed and model (calculated) total productive capacity and price-level.
The results of researches of parametrical sensitivity for small (of 10%) deviations around found estimations of values of parameters: ξ, π, rG of mathematical model (1-17), are presented in the following table 2, where increments of parameters of a state of economic system are calculated on the data of the year 2000.
Table 2. A matrix of sensitivity
Parameters / Output dataМ / LG / p
ξ / / /
rG / / /
π / / /
The analysis of table 2 shows, that:
1) with rise of reservation rate total productive capacity and price-level are decreased, and the state debt is increased;
2) with rise of the interest rate under the government bonds the state debt is increased and the price-level is decreased;
3) with rise of a share of the consumer expenses of the state from gross domestic product total productive capacity, state debt and price-level are increased.
It is necessary to note, that the basic conclusions from table 2 correspond with known economical statements. So, for example, estimation of function of sensitivity on the rate of reservation corresponds with statements of realization of «cheap» («expensive») money policy [1].
4. TASC OF A CHOICE OF THE EFFECTIVE LAWS OF PARAMETRICAL REGULATION
As possible intermediate criterion of an optimality at the choice of parametrical regulation laws it is offered to accept an average price-level on an interval three years (Т), taking into account that in the period, covered for research 1996-2000, the economy of Kazakhstan was on rise [13] and the average level of the prices can serve some measure of a production efficiency of the goods and services, and also can characterize presence of process of inflation [1].
Lets consider now an opportunity of effective state policy realization through a choice of the optimum laws of regulation on an example of the following economical parameters: a share of the consumer expenses of the state from gross domestic product (), interest rate under the government bonds () and rate of reservation (). These parameters are accepted for research with the account [7] and analysis of a matrix of sensitivity of parameters: total productive capacity (М), volume of the state debt (LG) and price-level (p).
There should be estimation of a choice opportunity of the optimum laws of parametrical regulation in the following sequence:
· choice of the optimum regulation law at a level of one of economic parameters (ξ, π, rG);
· choice of optimum pair laws of parametrical regulation on set of combinations from three economic parameters to two;
· choice optimum three of the laws of parametrical regulation for three economic parameters.
In work the choice of the optimum laws of parametrical regulation is carried out in environment of a set of the following dependences:
(19)
Here: Uij – i’s regulation law of j’s parameter (); the case j = 1 corresponds to parameter ξ; j = 2 – to parameter π; j =3 – to parameter rG; kij - set up coefficient of i’s law of regulation j’s parameter, ; constj - constant, equal to an estimation of value j’s parameter by results of parametrical identification;
t0 - time of a beginning of regulation - corresponds 1.01.1998; ; Mij(t), pij(t) - accordingly are values of total productive capacity and price-level with Uij’s regulation law; - interval of regulation - 36 months.
The task of a choice of the optimum parametrical regulation law at a level of one of economical parameters (ξ, π, rG) can be formulated in the following type. To find on the basis of mathematical model (1-17) optimum law of parametrical regulation at a level of one of three economic parameters (ξ, π, rG) in environment of a set of algorithms (19), that is, to find the optimum law from set {Uij}, which would ensure a minimum of criterion
, (20)
with restrictions
(21)
Here - top value of j’s parameter, - model (calculated) values of total productive capacity without parametrical regulation.
The formulated task is resolved in two stages:
· at the first stage the optimum values of coefficients kij for each law Uij by a way selection of coefficients values in intervals of a type quanted with step 0.01, ensuring a minimum K are determined with restrictions (21). Here - the first value of coefficient, with which is broken (21);
· at the second stage the law of optimum regulation of concrete parameter (from three) on base the results of first stage on minimum value of criterion K is chosen.
The results of the calculated decision of the first stage of the task for {Uij} are presented in table 3.
Table 3
Designations of laws of parametrical regulation / Optimum values of coefficients of laws / Values of K criterionU11 / 0.22 / 1.098
U21 / 0 / 1.1734
U31 / 1.56 / 1.037
U41 / 0 / 1.1734
U51 / 0.16 / 1.09
U61 / 0 / 1.1734
U12 / 0 / 1.1734
U22 / 0.11 / 1.09
U32 / 0 / 1.1734
U42 / 0.84 / 1.023
U52 / 0 / 1.1734
U62 / 0.08 / 1.084
U13 / 0 / 1.1734
U23 / 0.29 / 1.17
U33 / 0 / 1.1734
U43 / 0.39 / 1.1701
U53 / 0 / 1.1734
U63 / 0.23 / 1.1702
The analysis of table 1, according to the requirements of the second stage of the solution of the formulated task, allows to offering at a level of one-parametrical regulation of the market economy mechanism the law for parameter π of the following type:
,
which provides at the least value K=1.023 among all laws Uij.
The task of a choice of optimum pair laws for simultaneous regulation of two parameters can be formulated in the following type. To find an optimum pair of the laws of parametrical regulation (Uij,Uνμ) on set of combinations from three economical parameters to two on the basis of algorithms set (19), which would ensure(supply) a minimum of criterion
(22)
with restrictions
, p>0. (23)
Here , - value of total productive capacity for pair of regulation laws (Uij, Unm).