Advances in Systems Science and Applications (2014) Vol.14 No.3 199-229 The Theory of Parametric Control of Macroeconomic Systems and Its Applications(III) A. Ashimov, Zh. Adilov, R. Alshanov, Yu. Borovskiy and B. Sultanov Kazakh National Technical University named after K. Satpayev, 22 Satpaev Street, Almaty, 050013, Kazakhstan Abstract This work consists of three parts and presents the recent results of development of the theory of parametric control of macroeconomic systems and some its ap- plications for solving a number of concrete problems. Keywords Mathematical Model, Structural Stability, Parametrical Identification, Parametric Control Part 3. Applications of the Theory of Parametric Control of Macroeconomic Systems 3.1 Macroeconomic Analysis and Parametric Control of Macroeconomic System Based on an Econometric Model of Small Open Economy 3.1.1 Building an Econometric Model of a Small Open Economy General view of the model for small open economy of Kazakhstan, that describes equilibrium conditions in macroeconomic markets of goods, money, labor and of capital, taking into account its interaction with the Russian Federation and the rest of the world, is presented by the following relations [1-2]. Equilibrium in the goods market of the Republic of Kazakhstan is presented by the formula : Y D = Y S (1) where Y S - real supply of goods in the Republic of Kazakhstan, in billions of tenge (national currency of the Republic of Kazakhstan) ; Y D = C + I + G + NEFULL-real demand for goods in the Republic of Kazakhstan, in billions of tenge ; NEFULL = NE + NERU - real volume of goods net export from the Republic of Kazakhstan, in billions of tenge ; NERU = QRU ex − erQ RU im - real net export of goods from the Republic of Kazakhstan to the Russian Federation, in billions of tenge (indicator, that takes into account the terms of cooperation of the Republic of Kazakhstan within the regional Customs union) ; Qex - real volume of exports of goods from the Republic of Kazakhstan to the Russian Federation, in billions of tenge ; QRU im - real volume of imports of goods to the Republic of Kazakhstan from Russian Federation, in billions of tenge ; er = ePZ/P - real exchange rate in the Republic of Kazakhstan, tenge/US dollar ; e - exchange rate of national currency in the Republic of Kazakshtan, tenge/US dollar ; PZ - the general price level in the outside world ; P - the general price level in the Republic of Kazakstan ; NE = QW ex − erQ W im - real net export of goods from the Republic 200 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... of Kazakshtan to the rest of the world, in billions of tenge (indicator that takes into account the terms of cooperation between the Republic of Kazakhstan and the rest of the world) ; QW ex - real volume of exports of goods from the Republic of Kazakhstan to the rest of the world, in billions of tenge ; QW im - real volume of imports of goods to the Republic of Kazakhstan from the rest of the world, in billions of US dollars ; G - real volume of government expenditures in the Republic of Kazakhstan, in billions of tenge ; I - real volume of the Republic of Kazakshtan investments to the basic capital, in billions of tenge ; C - real volume of consumption by households in the Republic of Kazakhstan, in billions of tenge. All real data are presented for the year of 2000. Equilibrium in the money market of the Republic of Kazakhstan is presented by the following relation : M/P = L (2) where L is real cash balances in the Republic of Kazakhstan, in billions of tenge ; M - nominal money supply in the Republic of Kazakhstan (in billions of tenge). Equilibrium in the labor market of the Republic of Kazakhstan : PdY/dN = W (3) where W - nominal wage rate in the Republic of Kazakhstan, in thousands of tenge ; dY/dN - marginal productivity of labor in the Republic of Kazakhstan ; Y - real gross domestic product (hereinafter GDP) in the republic of Kazakhstan, in billions of tenge ; N - number of employed people in the Republic of Kazakhstan, in thousands of persons. Equilibrium in the capital market : PNEFULL = NKE (4) where NKE - net export nominal volume from the Republic of Kazakhstan, in billions of tenge. Let’s introduce additional notations for economic indicators used in develop- ment of a model : MRU - real money supply in Russian Federation (in billions of rubles) and GRU - real volume of government expenditures in Russian Federation (in billions of rubles), indicators allowing for conditions of operating of a country as part of regional customs union ; i - the average interest rate of banks for loans in the Republic of Kazakhstan ; iZ - interest rate of the outside world (Market yield on US, Treasury securities at 1-year constant maturity, quoted on invest- ment basis) ; Pavg = 0.6P + 0.4ePZ/e2000 - weighted average price level in the Republic of Kazakhstan ; - expected exchange rate in the Republic of Kazakhstan (tenge/US dollar) ; êe = (ee− e)/e - expected growth rate of the exchange rate in the Republic of Kazakhstan ; P oil - average oil price (in thousands of tenge for a Advances in Systems Science and Applications (2014) Vol.14 No.3 201 barrel) ; ∆ - operator of first difference for the series : ∆X = X −X−1;X−1- lag variable. Preliminary econometric analysis showed the possibility for evaluation of macroe- conomic indicators C, L, W, I, Y, QRU ex , QRU im , QW ex , QW im, NKE of equilibrium conditions in macroeconomic markets as a function of the regression on the basis of the following set of time series : Y, C, L, i, NEFULL, W, N, Pavg, I, QRU ex , P oil, MRU , GRU , QRU im , QW ex , er, QW im, NKE, iZ , êe for the years of 2000-2011 according to statistical data of national economies of the Republic of Kazakhstan and Russian Federation. In order to build non-spurious regression functions the considered sets were checked for stationary with the help of Augmented Dickey-Fuller method (ADF)and were decomposed [3]. According to the results of a check-up for stationarity of time series : - regression functions of the following type C = C(Y ), L = L(Y, i,NEFULL), W = W (N,Pavg), NKE = NKE(iZ , êe, i), I = I(i), Y = Y (N), QRU im = QRU im (Y,GRU ), QW im = QW im(Y, er) were built by the method of ordinary least squares on the basis of the following non-stationary series and the results of their analysis on the statistical significance are presented respectively in table 1 : Table 1 Regression functions for non-stationary series Consumption of domestic products in the : C= 558.3 + 0.38Y Republic of Kazakhstan(R2=0.99) (0.00) (0.00) Demand for real cash balances in the L=0.7Y -82.6 i -0.8 NEFULL Republic of Kazakhstan(R2=0.92) : (0.00) (0.02) (0.09) Price of labor supply in the Republic W =-0.15 N + 1174.8 Pm+0.4eZ/2000 of Kazakhstan, where Pm = 0.6P (R2 = 0.98) : (0.00) (0.00) Net capital export from the Republic of NKE = 291iz − 6090ẽe − 25.6i Kazakhstan (R2 = 0.65) : (0.03) (0.02) (0.35) Investments of the Republic of Kazakhstan I = 5885.5-291.5 i to the basic capital (R2 = 0.33) : (0.01) (0.05) Production function in the Republic of Y =-25255+4.27N Kazakhstan (R2=0,95) : (0.00) (0.00) Import of goods from the Russian QRU im = 0.07Y + 0.89∆GRU Federation (R2=0.89) : (0.00) (0.00) Import of goods from the rest of the world QW im = 0.22Y + 3.75er (R2 = 0.79) : (0.00) (0.01) - regression functions of the type QRU ex = QRU ex (er, P oil,MRU , GRU ), QW ex = QW ex(er, P oil) were built by the method of least squares on the basis of corre- sponding stationary series or stationary with respect to determined trends and the results of the analysis for statistical significance are presented respectively in table 2 : 202 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... Table 2 Regression functions for stationary series Import of goods from the Republic of : QRU ex = 4.4∆er + 31.4P oil Kazakhstaninto the Russian Federation −0.066MRU + 0.128GRU (R2=0.72) : (0.03) (0.04) (0.02) (0.00) Exportof goods from the Republic of Kazakhstan to the rest of the world QW ex = −11.2er + 278.0P oil + 1830.4 (R2=0.99) : (0.01) (0.00) (0.01) Regressions functions (where corresponding time series are stationary with respect to the determined trends) were checked on spuriousness by the T-test [4]. Spurious regressions functions (where time series were non-stationary) were checked on spuriousness by the Engle-Granger cointegration test [5]. The model of small open economy of the Republic of Kazakhstan based on the equilibrium conditions in macroeconomic markets of goods, money, labor and capital (1)-(4) and based on the built regression functions (tables 1, 2) has the following form :  Y D = 3201.8 + 0.84 M P − 17.63 ePZ P + 4.01 e−1P Z −1 P−1 + 281.8P oil − 0.06MRU − 0.694GRU + 0.811GRU −1 Y S = −25525 + 19967.4P + 13392.7PZ e e2000 Y ZB0 = 6311.72 + 1066.9P oil − 66.72 ePZ P + 15.17 e−1P Z −1 P−1 − 0.2276MRU + 3.07GRU −1 − 1003.45 iz P + 21000 ee e − 19369.8 1 P − 0.221 M P 2 − 1.04 ePZ P 2 + 0.205 e−1P Z −1 PP−1 + 19.77 P oil P − 0.034 MRU P − 0.04 GRU P + 0.0478 GRU −1 P Y S = Y D = Y ZB0 i = 18.5− 0.0025 M P − 0.0117 eP 2 P + 0.0023 e−1P Z −1 PP−1 + 0.19P oil − 0.00003MRU + 0.0005GRU −1 (5) Here Y ZB0 is the function of zero balance of payments of the Republic of Kaza- khstan. Advances in Systems Science and Applications (2014) Vol.14 No.3 203 3.1.2 Estimating Stability Indicators of the Model for a Small Open Economy The quality of the researched econometric model of the small open economy is es- timated by the stability indicator β (Section 1.3.2), which characterizes a change in the equilibrium model solutions by small deviations of the input parameters used. If the stability indicator of the model takes a small value for a small devi- ation of input variables used, it is considered that the model is qualitative in the sense of stability indicator. Estimation of stability indicators is made by the algorithm 5, where the vector X = {M,G,PZ , iZ , ee, P oil,MRU} has been considered as the vector of input pa- rameters, and the vector z = {Y, e, i} has been considered as the vector of output variables. Conducted computing experiments show that deviations of equilibrium solu- tions up to 1 % correspond to the deviations of input factors within 1%. This confirms the fact that the considered econometric model (5) is a qualitative model in sense of stability indicator β. 3.1.3 Parametric Control of the Country’s Export Depending on Uncontrollable Factors Based on the fact of dependence of the solution of algebraic equations on its coefficients, we propose an approach to parametric control of national economy evolution taking into account the requirements for equilibrium on macroeconomic markets, which comes down to making recommendations based on the optimal values of economic tools in the form of solutions of mathematical programming problems based on the econometric model of a small open economy. Let us consider the possibility of estimating the optimal values of M and G tools of economic policy for the given values of the uncontrollable input parameters P oil, iZ , PZ , ee, MRU and GRU that represent the values of these factors in the framework of the Customs Union by example of one country and the rest of the world in 2011 within the framework of the model IS-LM- ZB0 (built on statistics for 2000-2011) in sense of the maximum criterion : Qex = QW ex +QRU ex → max (6) Here Qex is the function of total exports of goods. The stated estimate can be obtained by solving the following problem of math- ematical programming. Problem 3.1. Based on the mathematical model (5) find values (M, G), that provide maximum to the criterion (6) under the constraints (7) Here M * and G* are accepted values of money supply and government expen- ditures respectively, for the years of 2008-2011 ; Y ∗, P ∗, e∗, i∗ - basic equilibrium solutions of the system (5) ; Y , P , e, i - optimal equilibrium solutions of the sys- tem (5). 204 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ...  |M −M∗| ≤ 0.1M∗, |G−G∗| ≤ 0.1G∗, |P − P ∗| ≤ 0.1P ∗, |e− e∗| ≤ 0.1e∗, |i− i∗| ≤ 0.1i∗, |Y − Y ∗| ≤ 0.1Y ∗, (7) The proposed approach to the parametric control of national economy evolu- tion consists in realization of the following algorithm : 1. Choice of mathematical model based on statistical analysis of the regression functions and estimation of stability indicator of the econometric model of eco- nomic general equilibrium for the open economy of the Republic of Kazakhstan ; 2. Statement of the mathematical programming problem ; 3. Prediction of uncontrollable factors PZ , iZ , P oil, MRU , GRU and ee for the period of choosing the recommendations on economic policy ; 4. Solution to the mathematical programming problem based on the selected mathematical model to forecast values of uncontrollable factors ; 5. Making recommendations on values of the M and G tools based on the anal- ysis of the results of the mathematical programming problem for predicted values of the uncontrollable factors and possible additional information on the economic conjuncture. Below we present an illustration of the proposed approach of parametric control of the national economy evolution for 2012. 1. Let the model of a small open economy be a mathematical model selected on the basis of estimation of stability indicators (less than 1%) in 2011. 2. As a statement of the optimization problem for the model of a small open economy in 2011 we take the statement of the problem 3.1. 3. Forecasted values of uncontrollable factors, obtained on the basis of the models built taking into consideration the results of time series decomposition into components, took the following values for 2012 : PZ = 1.32 ; P oil = 14.75 thousandtenge per barrel ; iZ = 0.17% ; ee = 148.68 tenge for one US dollar ; MRU = 27949.1 billion rubles and GRU =10898.3 billion rubles. 4. Solution to the mathematical programming problem on the basis of the model for a small open economy by example of the Republic of Kazakhstan and the predicted values of uncontrollable factors for 2012 are : M = 6733.0 billion tenge ; G = 1444.4 billion tenge ; the value of the criterion max(Qex W + Qex RU ) = 3465.1 + 335.6 = 3800.7 billion tenge ; 5. The following can be proposed as a recommendation : solutions obtained during the experiment M = 6733.0 billion tenge and G = 1444.4 billion tenge or Advances in Systems Science and Applications (2014) Vol.14 No.3 205 some correcting values, those can be obtained on the basis of the additional data analysis on economic conjuncture. 3.2 Macroeconomic Analysis and Parametric Control of Cyclical Dynamics The major section of modern macroeconomic theory is propositions on market cycles, in which the factors generating them are considered and different math- ematical models for their analysis are proposed [1,6-8]. Suppression of market cycles is the major field of stabilization policy [9-10]. 3.2.1 Macroeconomic Analysis and Parametric Control of Cyclical Dynamics Based on Kondratiev Cycle Model Model description This model combines descriptions of non-equilibrium economic growth and non- uniform scientific and technological advancement [11]. The model is described by the following system of equations, including two differential and one algebraic equation :  n(t) = Ay(t)a, dx/dt = x(t)(x(t)− 1)(y0n0 − y(t)n(t)), dy/dt = n(t)(1− n(t))y(t)2(x(t)− 2 + µ+ l0 n0y0 ), n0 = Ayα0 . (8) Here t is the time (in months) ; x is the efficiency of innovations ; y is the capital productivity ratio ; y0 is the capital productivity ratio corresponding to the equi- librium trajectory ; n is the rate of savings ; n0 is the rate of saving corresponding to the equilibrium trajectory ; µ is the coefficient of withdrawal of funds ; l0 is the job growth rate corresponding to the equilibrium trajectory ; A and a are some model constants. Estimation of the model parameters is carried out based on statistical informa- tion from the Republic of Kazakhstan for the years 2001-2005 [12]. The deviations in the observed statistical data and the calculated data do not exceed 1.9% within the considered period. As a result of solving the problem of parametric identification, the following val- ues of the exogenous parameters are obtained : α = −0.0046235, y0 = 0.081173, n0 = 0.29317, µ = 0.00070886, l0 = 0.00032161, x(0) = 1.911144. A retrospective prediction for 2006 and 2007 are characterized by errors equal to 6.1% and 12.1%, respectively, for the capital productivity ratio, and 2.3% and 11%, respectively, for the rate of savings. The respective cyclic phase trajectory of the Kondratiev cycle model is pre- sented in Fig.1. The period of cyclic trajectory corresponding to the statistical information of the Republic of Kazakhstan for the given years is estimated to be 206 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... 232 months. Fig.1 Cyclic phase trajectory of the Kondratiev cycle model Fig.2 Chain-recurrent set for the Kondratiev cycle model Estimating the robustness of the Kondratiev cycle model without parametric control The estimation of structural stability (robustness) of the mathematical model Advances in Systems Science and Applications (2014) Vol.14 No.3 207 is carried out according to the 4th component of the parametric control theory (Section 1.1) in the chosen compact set of the model phase space. Fig.2 presents an estimate of the chain-recurrent set R(f,N) obtained by the application of the chain-recurrent set estimation algorithm for the region N = [1.7; 2.3]×[0.066; 0.098] of the phase plane Oxy of system (8). Since the set R(f,N) is not empty, one can draw no conclusion about the weak structural stability of the Kondratiev cycle model in N on the basis of Robinson’s theorem. However, since there is a non-hyperbolic singular point in N , namely, the center (x0 = 2− µ+l0 n0y0 , y0), then system (8) is not weakly structurally stable in N. Parametric control of the evolution of economic system based on the Kondratiev cycle model Choosing the optimal parametric control laws is carried out in the environment of the following four relations : 1) n0(t) = n0 ∗ + k1 y(t)− y(0) y(0) ; 2) n0(t) = n0 ∗ − k2 y(t)− y(0) y(0) ; 3) n0(t) = n0 ∗ + k3 x(t)− x(0) x(0) ; 4) n0(t) = n0 ∗ − k4 x(t)− x(0) x(0) . (9) Here ki is the scenario coefficient ; n∗ 0 is the value of the exogenous parameter n0 obtained as a result of the estimation of parameters. The problem of choosing the optimal law of parametric control at the level of the econometric parameter n0 can be formulated as follows. On the basis of mathematical model (8), find the optimal parametric control law in the environment of the set of algorithms (9), ensuring reach of optimal values of the following criterion : K = 1 T T∑ t=1 (( x(t)− x0 x0 )2 + ( y(t)− y0 y0 )2 ) → min (10) (here T = 232 is the period of the cycle) under the constraints 0 ≤ y(t) ≤ 1, 0 ≤ n(t) ≤ 1, 0 ≤ x(t) (11) The base value of the criterion (without parametric control) is as follows : K = 0.0307. The value of criterion K = 0.007273 for the control law, that is optimal in 208 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... Fig.3 Capital productivity ratio without parametric control and with use of law 4, optimal in the sense of criterion K Fig.4 Efficiency of innovations without parametric control and with use of law 4, optimal in the sense of criterion K the sense of the criterion (10) of the 4th law, from the set (9) represented be- fore is obtained by solving the problem formulated above through application of the parametric control approach to the evolution of the economic system. Corre- sponding value of adjustable coefficient of this law is . The values of the model’s endogenous variables without applying parametric control and with use of the optimal parametric control law for criterion K are presented below in graphic form (Fig.3 and Fig.4). Estimating the structural stability of the Kondratiev cycle mathe- matical model with parametric control Advances in Systems Science and Applications (2014) Vol.14 No.3 209 To carry out this analysis, the expressions for optimal parametric control laws (11) with the obtained values of the adjustable coefficients are substituted into the right-hand side of the second and third equations of system (1) for the param- eter n0. Then, by using a numerical algorithm for estimating the weak structural stability of the discrete-time dynamical system for the chosen compact set N de- termined by the inequalities 1.7 ≤ x ≤ 2.3, in the state space of the variables (x, y), the estimation of the chain-recurrent set R(f,N) as the empty (or one- point) set is obtained. This means that the Kondratiev cycle mathematical model with optimal parametric control law is estimated as weakly structurally stable in the compact set N . Analysis of the dependence of the optimal value of criterion K on the parameter for the variational calculus problem based on the Kon- dratiev cycle mathematical model Let us analyze the dependence of the optimal value of criterion K on the exoge- nous parameters µ (share of withdrawal of capital production assets per month) and a for parametric control laws (11) with the obtained optimal values of the adjusted coefficients ki, where the values of the parameters (µ, a) belong to the rectangle A = [0.00063; 0.00147]× [−0.01; 0.71] in the plane. Plots of dependencies of the optimal value of criterion K (for parametric con- trol laws 0 and 2, yielding the maximum criterion values) on the uncontrollable parameters (see Fig.5) were obtained by computational experimentation. The pro- jection of the intersection line of the two surfaces in the plane (µ, α) consists of the bifurcation points of the extremals of the given variational calculus problem. 3.2.2 Macroeconomic Analysis and Parametric Control of Cyclical Dynamics Based on Dynamic Stochastic General Equilibrium Model for the Economy of Kazakhstan In nonlinear dynamical stochastic general equilibrium (DSGE) model is presented on the base of given composition and behavior of agents, their inter action in stochastic conditions and of taking the principle of rational expectations [13]. This nonlinear DSGE model of the economy consists of : - first-order aggregate conditions of optimization problems of agents (house hold sand intermediate product producers) [14] ; - description of government activity rules ; and - rules of shocks specifying in terms of either first-order auto regression s or Gaussian white noises. First-order aggregate conditions involve equilibrium conditions in market of la- bor, capital, intermediate and final goods. The nonlinear DSGE model that was built involves both the model of actual economy, and the model of potential econ- omy. The model of potential economy is similar to the model of actual economy by its composition, except that potential economy functions under flexible prices and wages (in the model of actual economy the se prices are not flexible), and also 210 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... Fig.5 Plots of the dependencies of the optimal value of criterion K on exogenous parameters µ, a in absence of “extra charge” shocks (those are in the model of actual economy). The nonlinear DSGE model in question has the following vector form : EtF θ(Xt−1, Xt, Xt+1,H ΣH t ) = 0 (12) Here Et is sign of conditional mathematical expectation given information avail- able at the point of time t (t = 1, 2, ...) ; F θ is known vector function ; θ is pa- rameters set, consisting of structural parameters of the model and auto regression parameters of shocks ; Xt is vector, consisting of endogenous variables and shocks, determined by first-order auto regressions ; X0 is given ; HΣH t is vector, consisting of Gaussian white noises, ΣH is corresponding diagonal covariance matrix. According to the technique taken for DSGE model, linear approximation of nonlinear DSGE model (12) was built in neighborhood of its stationary point X [13]. Mentioned point is found by solving vector equation (15) obtained from (14) by dropping time subscripts and nulling white noises : F θ(X,X,X, 0) = 0 (13) Log-linearization of DSGE model of F . S mets and R. Wouters in neighborhood of its estimated stationary point gives linear DSGE model of the following form : AθX̂t−1 +BθX̂t + CθEtX̂t+1 +DθHΣH t = 0, t = 1, 2, 3... (14) Advances in Systems Science and Applications (2014) Vol.14 No.3 211 Here the sign ≪∧≫ corresponds to linearized variable, Aθ, Bθ, Cθ, Dθ are ma- trices of corresponding dimensions. In this example we consider the case of implementing state economic policy by the Taylor rule [15], describing behavior of National bank insetting interest rate, and rules for determining the size of government spending as well. In the framework of linear model (14) the Taylor rule for determining govern- mental bonds yield is presented in the following form : R̂t = ρR̂t−1 + (1− ρ) ( π̄t + rπ(π̂t−1 − π̄t−1) + rY (Ŷt − Ŷ p t ) + r∆π(π̂t−1 − π̂t−1) + r∆Y ( Ŷt − Ŷ P t − ( Ŷt−1 − Ŷ p t−1 )) + ηRt (15) and the rule of government spending in the following form : Ĝt = ρGĜt−1 + ηGt (16) Here R̂t, Ŷt, Ŷ p t , Ĝt, π̂t are variables, corresponding to : governmental bondsyield (1 + interest rate), output, potential output, government spending and inflation. ηRt is interest rate shock, given in terms of Gaussian white noise ; π̄t is inflation shock ; ηGt is government spending shock, ρ, rπ, r∆π, r∆Y , ρG are the parameters of the equations (15), (16). Estimating parameters of linear DSGE model on the basis of statis- tical data of the economy of the Republic of Kazakhstan Model (16) solution was obtained by the Blanchard-Kahn algorithm [16-17]. This solutionis presented in the form of first-order vector auto regression : X̂t R̂t Ĝt  = Qθ  X̂t−1 R̂t−1 Ĝt−1 + F θHΣH t , t = 1, 2, 3... (17) Hereinafter X̂t is vector-column consisting of all endogenous variables in the model (including shocks determined in terms of auto regression), excluding state policy tools of governmental bonds yield R̂t, and the size of government spending Ĝt. Vectors X̂0, R̂0, Ĝ0 are given ; Qθ, F θ are matrices of corresponding dimen- sions. Estimating parameters of the model in question (14) (using (17)) was made by the Bayesian estimation method (the Metropolis-Hastings algorithm with the number of simulations of 4 000 000) using the Kalman filter [18]. As observa- tions were used quarterly data for seven macroeconomic indicators of Kazakhstan (GDP, Investments, Consumption, Employment, Average wage, Refinancing rate, and Inflation) from 2002I till 2010III. We found the logarithm of mentioned sta- tistical indicators and linearly detrended them. For using the Kalman filter within 212 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... the Bayesian approach the model (17) was supplemented with vector equation of the dimension : Ŝt = M  X̂t R̂t Ĝt  (18) Here M is matrix, each row of which contains one unity, all of the rest its ele- ments are equal to 0. As the results of measuring of observed variables were taken log-deviations from its linear trends of macroeconomic indicators values (Con- sumption, Investments, GDP, Inflation, Average wage, Employment, Refinancing rate), corresponding to observed variables. Statistical data for the Republic of Kazakhstan from 2002I till 2011III was used in this study. For using the Bayesian approach there were given a priori density distribution p = p0(θ ′,ΣH) of parameters θ′,ΣH . The form and probabilistic characteristics of this distribution from were used in the research [13], with the exception of mathematical expectations of a priori distribution of parameters ΣH . Mentioned mathematical expectations were increased 2.5 times relative to corresponding val- ues from S mets F . and Wouters R. in connection with large sampled standard deviations of economic indicators of Kazakhstanin comparison with Eurozone. According to the Bayesian approach method [18], using like lihood function, obtained on the basis of the model(17), (18) using the Kalman filter, and a pri- ori distribution of parameters p0(θ′,ΣH) as well,was found posterior joint density distribution of initial estimates of parameters : p = p1(θ ′,ΣH). Then using the Metropolis-Hastings algorithm with density was p1(θ ′,ΣH) generated a sample, consisting of 4 000 000 sets of parameters θ′,ΣH . Finally, as required estimates of parameters were taken corresponding sampled averages. Quality of applied method for finding the parameters estimates was tested by retro prognosis. For this purpose there were made predictions for mentioned ob- served economic indicators for four periods from 2010 IV till 2011 III. Root mean square deviations of obtained expected predicted values of economic indicators from corresponding statistical data were about 3%. Analysis of shocks effectson GDP and inflation using estimating of impulse responses on disturbances within the framework of internal shocks of the economy of the Republic of Kazakhstan Fig.6 and Fig.7 present relatively impulse responses of real GDP and inflation on (unit positive) shocks. Each diagram presented in figures is obtained by cal- culation of linear model(17) for initial zero values of all endogenous variables of the model and the value of chosen shock equal to its standard deviation for zero period. Under this all of the values of this shock for non-zero time values, and also the values of all other shocks of the model for all of time values were taken as the null. Advances in Systems Science and Applications (2014) Vol.14 No.3 213 Fig.6 Impulse responses of real GDP on singular positive shocks Fig.7 Impulse responses of inflation on singular positive shocks Analysis of the diagrams of impulse response of real GDP presented in Fig.6 shows the following : • Given positive shocks of productivity (εAt ), labor supply (εLt ) and investments (εIt ) GDP increases. • Positive shocks of preferences (εBt ) and government spending (ηGt ) also in- 214 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... crease GDP (since these shocks increase,respectively, consumption and govern- ment spending) • Positive shock of extra charge for goods (ηPt ) decreases GDP, and shock of extra charge for wage (ηWt ) increases GDP. • Positive monetary shock (ηRt ) results in production decline (because of inter- est rate growth). Analysis of the diagrams of impulse response of inflation presented in the Fig.7 shows the following : • Given positive shocks of extra charge for goods and extra charge for wage inflation increases • Given positive shock of productivity inflation negligible decreases. The rest of shocks do not practically have an effect on inflation. These responses of the model to shocks correspond with theoretical proposi- tions. Decomposing indicators evolution to shocks effect parts in retrospec- tive period Indicators decomposition in retrospective period, presented in Fig.8 and 9,il- lustrates the contribution (in percentage) of each shock effect of the estimated model (17) to deviations of actual values of GDP and inflation indicators from corresponding equilibrium values for the period 2002I-2012I. Presented diagrams show how deviations of GDP and inflation from their cor- responding trends in retrospective period (from 2002 I till 2012 III) emerged according to positive and negative effects of the shocks in question. For instance, deviation of GDP from trend in 2009IIIequal to -4.93% is the sum of positive summands : 1. Shock effects of labor supply (εLt ) equal to 3.02% deviation of GDP from trend, 2. Shock effects of interest rates (ηRt ) equal to 1.51% deviation of GDP from trend, 3. Shock effects of extra charge for goods (ηPt ) equal to 0.85% deviation of GDP from trend and negative summands : 4. Shock effects of preferences (εBt ) equal to -3.88% deviation of GDP from trend, 5. Shock effects of extra charge for wage (ηWt ) -3.63% deviation of GDP from trend, 6. Shock effects of extra charge for capital (ηQt ) -1.16% deviation of GDP from trend, 7. Shock effects of government spending (ηGt ) -1.12% deviation of GDP from trend, 8. Shock effects of productivity (εAt ) -0.69% deviation of GDP from trend (rest Advances in Systems Science and Applications (2014) Vol.14 No.3 215 shock effects of investments (εIt ) and shock effects of inflation (π̄t) are negligible). That is deviation of GDP from trend equal to -4.93% is the sum of all shock effects of the model for mentioned period. For instance, deviation of inflation from trend in 2007IV equal to 6.32% is the sum of only positive summands (all of the rest effects are negligible) : 1. Shock effects of extra charge for goods equal to 4.87% deviation of inflation from trend ; 2. Shock effects of extra charge for wage 1.43% deviation of inflation from trend. Analysis of the diagram in Fig.8 shows also that break-neck growth of GDP during the period 2004-2007 was mainly because of extra charge shocks (for wage, good, capital), lessening of GDP growth in 2008-2011 was mainly because of neg- ative shock effects of preferences, extra charge for capital and wage. Analysis of the diagram in Fig.9 shows that inflation deviation from the equi- librium level in the period 2002-2011 was almost fully because of extra charge shock for good and wage. In other words, all of the rest shocks of the model do not practically have an effect on inflation values in mentioned period. Fig.8 Shocks effects on GDP rate Fig.9 Shocks effects on the quarterly inflation 216 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... Prediction of shocks effects on economic indicators and suppression of their effects based on DSGE model of Smets-Wouters In the paper by estimated model (17) were obtained predicted values of macroe- conomic indicators (GDP and inflation) for 1, 4, 10, 20, 30 and 40 quarters (i.e. correspondingly for 2011IV, 2012III, 2014I, 2016III, 2019I, 2021III). For estimat- ing shock effects on error variances of economic indicators predictions were used the standard technique for defining decomposition of error variance of predictions for the models of vector auto regressions [19]. The results obtained by the soft- ware Dynare Matlab Toolbox [http ://www.dynare.org] are presented in Tables 1 and 2. There are no shocks, in these tables, which effects on variance less than by 0.01%. Analysis of the Tables 3 and 4 shows the following. Error variances of progno- sis of GDP generally are determined by preference shocks, government spending shock, and extra charge shocks on the cost of capital. Error variances of prognosis of inflation generally are determined by extra charge shocks on goods and extra charge shocks on wages. Table 3 Prognosis (billion tenge in average prices of 1994) and decomposition of variance of the quarterly GDP prognosis Time Prognosis Decomposition of variance (in %) hori Mathematical Standard ηAt ηBt ηGt ηLt ηIt ηRt ηQt ηPt ηWt-zon expectation deviation 1 271.413 8.17 2.25 22.90 17.57 1.47 0.09 1.07 45.32 6.13 3.19 4 292.166 10.960 4.04 21.73 15.30 3.25 0.15 1.56 41.99 8.27 3.71 10 327.034 13.229 4.04 21.73 15.30 3.25 0.15 1.56 41.99 8.27 3.71 20 385.810 16.067 4.09 20.69 14.20 3.70 0.14 1.51 39.84 7.90 7.92 30 456.450 19.136 4.03 20.39 13.98 3.67 0.14 1.49 39.47 8.01 8.82 40 542.395 22.751 4.02 20.37 13.97 3.67 0.14 1.49 39.45 8.03 8.86 Table 4 Prognosis (in %) and decomposition of variance of the quarterly inflation prognosis Time Prognosis Decomposition of variance (in %) horizon Mathematical Standard ηAt ηQt ηPt ηWtexpectation deviation 1 1.22% 0.75% 0.29 0.01 92.24 7.47 4 1.36% 0.78% 0.30 0.06 72.31 27.33 10 1.62% 0.80% 0.28 0.15 62.65 36.92 20 1.78% 0.81% 0.27 0.16 60.71 38.85 30 1.83% 0.81% 0.28 0.17 60.55 39.00 40 1.85% 0.81% 0.28 0.17 60.53 39.01 Advances in Systems Science and Applications (2014) Vol.14 No.3 217 In realization of the state policy for minimizing shock effects on economic in- dicators in the capacity of its tools we choose additive summands ηRt , η G t in the expressions (14), (15), desired values of which are searched in terms of determin- istic values instead of respective shocks. The parametric control approach for minimizing shock effects consists in the following. Let T be number of quarter, starting from which the state realizes parametric control policy for minimizing shock effects. At each time point t = T, T + 1, T + 2, ... on the basis of estimated model (17), written in the form X̂t+i R̂t+i Ĝt+i  = Qθ  X̂t+i−1 R̂t+i−1 Ĝt+i−1 + F θHΣH t+i , i = 1, 2, 3, ..., 40 (19) such deterministic values of tools ηRt+1, ..., ηRt+40, η G t+1,..., η G t+40 are defined, those give the minimum for criterion Lt = Et 40∑ i=1 βi(π̂2 t+i + λY Ŷ 2 t+i),min(ηRt+1, ..., η R t+40, η G t+1, ..., η G t+40)Lt (20) (characterizing expected discounted total deviation of GDP and inflation values from respective equilibrium values (trend)) under the following constraints on endogenous variables of the model. Mathematical expectations for inflationand bond yield in this time horizon should not deviate from respective equilibrium values more than by 0.5% : |Etπ̂t+i| ≤ 0.005, i = 1, 2, 3, ..., 40 (21) |EtR̂t+i| ≤ 0.005, i = 1, 2, 3, ..., 40 (22) and mathematical expectation for the size of government spending by 5.0% from their trend values : |EtĜt+i| ≤ 0.05, i = 1, 2, 3, ..., 40 (23) Moreover, at each specified time point t, the value of current state of economy for the time point t−(X̂t, R̂t, Ĝt) is known. After receiving a new information (the values of variables (X̂t+1, R̂t+1, Ĝt+1) at the next time point (t + 1)), the values of tools ηRt+2, ..., ηRt+41, ηGt+2,..., ηGt+41 are calculated again by solving the Problem (19)-(23) for respective period. Here discount factor, is some weight coefficient. Introduce a new minimization criterion : L̃t = 40∑ i=0 βi ( (Etπ̂t+i) 2 + λY (EtŶt+i) 2 ) ,min(ηRt+1, ..., η R t+40, η G t+1, ..., η G t+40)L̂t (24) 218 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... It is not difficult to check, that this criterion differs from by the value, inde- pendent from variables ηRt+1, ..., η R t+40, η G t+1, ..., η G t+40. From the relation (19), by taking mathematical expectations for both of its parts, we get Et  X̂t+i R̂t+i Ĝt+i  = QθEt  X̂t+i−1 R̂t+i−1 Ĝt+i−1 + F θ′′ [ ηRt+i ηGt+i ] , i = 1, 2, 3, ..., 40 (25) Here F θ ′′ is matrix, comprised by two corresponding columns of the matrix F θ Consequently, optimal values of variables of the problems (19)-(23)and (21)-(25) coincide between each other. Derived optimization problem (21)-(25) applies to classical (deterministic) type of variational calculus problems, which for each time point t is solved by the linear and quadratic programming method (with Matlab application). Below in the paper it is estimated the effectiveness of application of formulated above parametric control in assumption that the state will implement this policy during 30 years (120 periods). That is, assume that at each time point t (t = T, T +1, T +2, ..., T +119, T is number of quarter, corresponding to 2011III) the stated etermines the values of parameters ηRt+1, η G t+1 by solving above mentioned optimization problem(19)-(23) (or, that is the same, (21)-(25)). In computing experiment it is assumed that the economy is precisely described by estimated model :  X̂T+i R̂T+i ĜT+i  = Qθ  X̂T+i−1 R̂T+i−1 ĜT+i−1 + F θHΣH t+i , i = 1, 2, 3, ..., 160 where X̂T is known. In the paper, for estimating the effectiveness of application of formulated above parametric control approach the Monte-Carlo method was used with estimate of 100 development scenarios of economy. Let us present aggregative algorithm for estimating application of the parametric control approach. 1. Generation of the sample, consisting of 100 elements-sets of values of vec- tor Gaussian random values (white noises) {H ′ΣH T+1,H ′ΣH T+2, ..., H ′ΣH T+120}j where j = 1, ..., 100, HΣH T+i = [H ′ΣH T+i , η R T+i, η G T+i] T with known probabilistic characteristics of noises ΣH . 2.For each element of the sample {H ′ΣH T+1,H ′ΣH T+2, ...,H ′ΣH T+120}1, {H ′ΣH T+1,H ′ΣH T+2,..., H ′ΣH T+120}2,..., {H ′ΣH T+1,H ′ΣH T+2, ..., H ′ΣH T+120}120 : 2.1 The calculation of the model with parametric control : We solve optimization problem (21)-(25) for period t = T i.e. we find respective Advances in Systems Science and Applications (2014) Vol.14 No.3 219 ηRT+1, ..., η R T+40, η G T+1, ..., η G T+40 ; From obtained set of valueswe take ηRT+1, η G T+1 (we drop rest values ηRT+2, ..., η R T+40, η G T+2, ..., η G T+40. 2.2. The model is calculated for 1 step with shocks values HΣH T+1, which consist of tools values ηRT+1, η G T+1 determined by the state and shocks values H ′ΣH T+1,which were realized by economy independently (exogenously) from the state policy. X̂T+1 R̂T+1 ĜT+1  = Qθ  X̂T R̂T ĜT + F θHΣH T+1 2.3 The steps 2.1 and 2.2 are iteratedfor values t = T+1, T+2, T+3, ..., T+120. 3. On the bases of obtained 100 trajectories of GDP and inflation is built av- erage trajectory (expected prognosis value) and standard deviations of prognosis for the period t = T, T +1, T +2, T +3, ..., T +120. Obtained values are compared with basic prognosis. Realization results of formulated algorithm show that for used sample of shocks the parametric control of suppressing shocks effects provides diminishing predicted standard deviations of GDP by 58.3% at the average in prognosis horizon from 2011III till 2021III (see Fig.10). The parametric control of suppressing shocks effects provides diminishing pre- dicted standard deviations of inflation by 32.0% at the average in prognosis hori- zon from 2011III till 2021III and diminishing samples tan dard deviation of infla- tion by 47.8% in comparison with actual data during the period2002Itill 2011III (see Fig.11) 3.3 Macroeconomic Analysis and Parametric Control of the Economic Growth Based on Computable General Equilibrium Model for the Economic Sectors Presentation of computable general equilibrium model Non-autonomous computable general equilibrium model (CGE model) in gen- eral form is presented by the following system of relations [2], [20]. 1) Subsystem of differential equations, connecting endogenous variables values for two successive years : x1(t+ 1) = f1 (x1(t), x2(t), x3(t), µ(t), a(t)) (26) Here t = 0, 1, ..., n−1 is number of year, discrete time ; x(t) = (x1(t), x2(t), x3(t)) ∈ Rm is vector of endogenous variables of the system ; xi(t) ∈ Xi(t) ⊂ Rmi , i = 1, 2, 3 (27) Here the variables x1(t) involve the values of capital assets of the sectors-producers, budgets of 220 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... Fig.10 Prognostic values of real GDP for the basic scenario and the parametric control approach Fig.11 Prognostic values of inflation for the basic scenario and the parametric control approach (in %) economic agents and so on ; x2(t) involve the values of demand and supply of agents in different markets and so on ; x3(t) are different kinds of market prices and budget parts in markets with state-set prices for various economic agents ; m1 +m2 +m3 = m ; Advances in Systems Science and Applications (2014) Vol.14 No.3 221 u(t) ∈ U(t) ⊂ Rq is vector function of controllable (adjustable) parameters. Co- ordinate values of this vector correspond to various state economic policy tools, for instance, such as state budget parts and budget parts of economic agents, various tax rates, governmental bonds yield and so on ; a(t) ∈ A ⊂ Rs is vector function of uncontrollable parameters (factors). Coor- dinate values of this vector characterize various external and internal social and economic factors depending on time : export and import goods prices, population size of the country, production functions parameters and so on ; X1(t), X2(t), X3(t), U(t) are compact sets with non-empty interiors ; Xi =∪n t=1Xi(t), i = 1, 2, 3 ; X = ∪3 t=1Xi ; U = ∪n−1 t=0 U(t), is open connected set ; f1 : X × U ×A → Rm1 , is continuous mapping. 2) Subsystem of algebraic equations, describing behavior and interaction of agents in various markets within sampled year, these equations allow expressing the variables by exogenous parameters and rest endogenous parameters : x2(t+ 1) = f2 (x1(t), x3(t), x3(t), u(t), a(t)) (28) Here f2 : X1 ×X3 × U ×A → Rm2 is continuous mapping. 3) Subsystem of recurrence relations for iterative calculations of equilibrium values of market prices in various markets and budget parts in markets with state-set prices for various economic agents : x3[Q+ 1] = f3 (x2(t)[Q], x3(t)[Q], L, u(t), a(t)) (29) Here Q = 0, 1, ... is number of iteration ; L is the set of positive numbers (ad- justable constants of iterations, when their values decrease,economic system comes faster to its equilibrium condition, however,at the same time the risk of the case when prices go to negative range increases ; f3 : X2×X3× (0,+∞)ms ×U ×A → Rm2 is continuous mapping (that is compressing at fixed t ; x1(t) ∈ X1(t) ; u(t) ∈ U(t) ; a(t) ∈ A and some fixed L. In this case the mapping f3 has the only fixed point, to which converges the iterative process (28), (29). Computable model (26), (28), (29) under fixed values of functions u(t) and a(t) for each time point t determine the value of exogenous variables x(t), corre- sponding to price equilibrium of demand and supply in the markets of goods and services of agents in the framework of the following algorithm. 1) In the first step it is assumed that t=0 and it is determined the initial values of variables x1(0). 2) In the second step for current the initial values of variables x3(0)[0] are de- termined in various markets and for various agents ; the values x2(t)[0] = f2 (x1(t), x3(t)[0], x3(t), µ(t), a(t)) (the initial values of demand and supply of agents in the markets of goods and services) are calculated by (28). 3) In the third step for current t it is run iterative process (28), (29). In this, 222 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... for each value Q current values of demand and supply are found from (29) : x2(t) = f2 (x1(t), x3(t)[Q], x3(t), u(t), a(t)) by improvement of market prices and budget parts of economic agents. Condition for stopping iterative process is equality of demand and supply val- ues in various markets accurate within 0.01%. Consequently,there are determined equilibrium values of market prices in each market and budget parts in markets with state-set prices for various economic agents. We omit the index Q for such equilibrium values of endogenous variables. 4) In the following step on the basis of obtained equilibrium solution for the time point using differential equations (26) we define the values of variables x1(t+ 1). The value increases by unity. Transition to the step 2. Quantity of iterations of steps 2, 3, and 4 are determined in accordance with the parametric identification problems, prognosis and control in chosen in advance periods. Considered CGE model can be presented in the form of continuous mapping f : X × U × A → Rm, determining transformation of the values of endogenous variables of the system for zero year to respective values of the next year according to presented above algorithm. Here the compacts X(t) = X1(t)×X2(t)×X3(t), determining the compact X in the space of endogenous variables are defined by the set of possible values of variables x1 and respective equilibrium values of vari- ables x2 and x3 calculated by relations (30)-(31). We will assume that for chosen point x1(0) ∈ Int(X1) and corresponding, cal- culated by (28), (29) points x(0) = (x1(0), x2(0), x3(0)), the inclusion x(t) = f t(x(0)) ∈ Int(X(t)) is true under some fixed u(t) ∈ Int(U(t)), a(t) ∈ A for t = 0, ..., n. (n-fixed positive integer). This mapping f defines discrete dynami- cal system in the set X, on the trajectory of which imposed appropriate initial condition : {f t, t = 0, 1, ...}, x|t=0 = x0 (30) Based on this conception specific CGE model of economic sectors is considered below. Parametric identification of CGE model of economic sectors The model in question by statistical data of the republic of Kazakhstan is pre- sented by the following 19 economic agents. Economic agent No.1. Agriculture, hunt and forestry ; Economic agent No.2. Fishery, fish breeding ; Economic agent No.3. Mining ; Economic agent No.4. Manufacturing ; Economic agent No.5. Production and distribution of electricity, gas and water ; Economic agent No.6. Construction ; Economic agent No.7. Trade ; automobile and house articles maintenance ; Advances in Systems Science and Applications (2014) Vol.14 No.3 223 Economic agent No.8. Hotels and restaurants ; Economic agent No.9. Transportation and communication ; Economic agent No.10. Financial activities ; Economic agent No.11. Transactions with real estates, lease and services to enterprises ; Economic agent No.12. Public administration ; Economic agent No.13. Education ; Economic agent No.14. Public health and social services ; Economic agent No.15. Other municipal, social and personal services ; Economic agent No.16. Housekeeping services ; Economic agent No.17. Aggregate consumer, combining households ; Economic agent No.18. Government, presented by the sum of central, regional and local governments, and non-budget funds as well. Government determines tax rates and amount of subsidies for agents-producers and the size of social trans- fers for households. Moreover, this sector includes non-commercial organizations, serving households (political parties, labor unions, social associations, etc.) ; Economic agent No.19. Bank sector, involving National bank and commercial banks. Economic agent No.20. Outside world. Here,economic sectors No.1-16 are agents-producers. The considered model is presented as general expressions of relations of (26), (28), (29) respectively m1 = 67, m2 = 597, m3 = 34 by expressions, which help to calculate values of its 698 endogenous variables. This model contains also 2045 estimated exogenous parameters. In the result of combined solution of the problems A and B as per the con- structed algorithm of parametric identification (Section 1.2) using statistical data on evolution of the economy of the Republic of Kazakhstan. The relative value of deviations of estimated values of variables used mostly as criteria of corresponding observed values was equal to less than 0.63%. Further the calculation of the model outside the period of parametric identifi- cation (forecast calculation) using extrapolated for the forecast period values of functions u(t), a(t) will be called as basic calculation. The results of calculation and retrospective basic calculation of the model for 2008 partially presented in the Table 5 show estimated values, observed values and deviations of estimated values of main output variables of the model from corresponding observed values. Here, the period 2000-2007corresponds to the pe- riod of parametric identification of the model ; 2008 is the retroprognosis period ; Y is gross output (in prices of 2000) ; Yg is GDP (in prices of 2000) ; P is consumer price index in percentages relative to the previous year ; sign ≪ ∗ ≫ corresponds to observed values, sign ≪ a ≫ corresponds to deviations (in percentages)of esti- 224 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... Table 5 Observed, calculated values of output variables of the model and corre- sponding deviations Indicator Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 Y ∗(t) 5.44 6.32 6.47 6.86 7.72 8.52 9.25 9.69 9.84 Y (t) 5.38 6.32 6.47 6.86 7.72 8.52 9.27 9.64 9.82 ∆Y (t) -1.22 -0.02 0.00 0.00 0.05 0.08 0.21 -0.51 -0.26 Y ∗ g (t) 2.45 2.78 3.05 3.36 3.72 4.09 4.55 5.01 5.18 Yg(t) 2.47 2.78 3.05 3.35 3.72 4.09 4.55 5.01 5.20 ∆Yg(t) 0.88 0.07 -0.04 -0.02 -0.02 -0.02 -0.04 -0.15 0.38 P ∗(t) 106.4 106.6 106.8 106.7 107.5 108.4 118.8 109.5 P (t) 107.6 106.8 106.9 106.7 107.3 108.2 118.6 109.4 ∆P (t) 1.13 0.18 0.08 -0.05 -0.23 -0.22 -0.24 -0.05 mated values from respective observed values. Analysis of sources for economic growth based on computable gen- eral equilibrium model of economic sectors Now, let us analyze sources for economic growth of economic sectors based on CGE model of economic sectors and based on retrospective data for 2000-2009. For this, using expressions of production functions of economic agents of the model, we estimate the effect of change of arguments of these functions on the rates of growth of GAV of sectors V Y _i[t + 1] in assumption about constancy of coefficients CA_z_ji[t] under consumed by sector intermediate products V D_pj_iz[t], co- efficients CA_k_i[t] under capital (V K_i[t] + V K_i[t + 1])/2 and coefficients CA_l_i[t] under labor V D_pi_il[t]. Here i, j = 1, ..., 16 are numbers of eco- nomic agents ; t is number of year ; Power (X,Y ) corresponds to XY , Exp(X) corresponds to eX ; CA_r_i is coefficient, characterizing technical progress in i-th sector. V Y _i[t+ 1] = CA_r_i× Exp(V D_p1_iz[t]× CA_z_1i)× Exp(V D_p2_iz[t] × CA_z_2i])× Exp(V D_p3_iz[t]× CA_z_3i)× Exp(V D_p4_iz[t]× CA_z_4i) × Exp(V D_p5_iz[t]× CA_z_5i)× Exp(V D_p6_iz[t]× CA_z_6i) × Exp(V D_p7_iz[t]× CA_z_7i)× Exp(V D_p8_iz[t]× CA_z_8i) × Exp(V D_p9_iz[t]× CA_z_9i)× Exp(V D_p10_iz[t]× CA_z_10i) × Exp(V D_p11_iz[t]× CA_z_11i)× Exp(V D_p12_iz[t]× CA_z_12i) × Exp(V D_p13_iz[t]× CA_z_13i)× Exp(V D_p14_iz[t]× CA_z_14i) × Exp(V D_p15_iz[t]× CA_z_15i)× Exp(V D_p16_iz[t]× CA_z_16i) × Power(((V K_i[t] + V K_i[t+ 1])/2), CA_k_i)× × Power(V D_pi_il[t], CA_l_i) (31) Advances in Systems Science and Applications (2014) Vol.14 No.3 225 After finding the logarithms of both parts (31) and total increment of the function ln (V Y _i) and dropping members of the highest order infinitesimal, we get the following estimate of the growth rate yi of real GAV of i-th sector depending on growth of arguments of production function : CA_r_i, V D_pj_iz, Kim = (V K_i[t] + V K_i[t+ 1])/2, and V D_pi_il[t]. yi = ∆V Y _i V Y _i = ∆CA_r_i CA_r_i + 16∑ j=1 (CA_z_ji× V D_pj_iz) ∆V D_pj_iz V D_pj_iz + CA_k_i ∆Kim Kim + CA_l_i ∆V D_pi_il V D_pi_il (32) Denote by ai = ∆CA_r_i CA_r_i the rate of technical progress in i-th sector ; zij = ∆V D_pj_iz V D_pj_iz is the rate of intermediate products consumed by i-th sector, and produced by j-th sector ; ki = ∆Kim Kim is the rate of capital accumulation in i-th sector ; li = ∆V D_pi_il V D_pi_il is the growth rate of labor inputs in i−th sector, where the sign ”∆” means change of variable ; time values in (32) were dropped for short. Coefficients on the right-hand-side of formula (32) at stated above rates and characterize degree of effect of the factors in question on economic growth and allow comparing their effect with effect of technical progress, coefficient at which is equal to 1. By denoting these coefficients by aij = CA_z_ji × V D_pj_iz, βi = CA_k_i, γi = CA_l_i, from (32) we get its reduced writing : yi = ai + 16∑ j=1 αijzij + βiki + γili (33) Let us present the values of coefficients, defining contributions of sources of eco- nomic growth of sectors based on the model in question for 2008. (see Table 6). Coefficients in the Table show how much the growth rate of GAV would increase given the one percent increase of growth factors (fixed assets, labor or demand for intermediate goods of economic agents). Analysis of coefficients table βi, γi, αs i = ∑16 j=1 αij of the Table 4 shows that if we exclude the rate of technical progress, effect of which on the rate of growth of all sectors in given model is the same, then from rest three factor rates of economic growth, the largest effect on the rate of real output of sectors 1, 5, 7, 12, 13, 16 of the economy has the rate of labor inputs ; for sectors 4, 6, 8, 10, 15-the rate of capital accumulation ; and for other sectors 2, 3, 9, 11, 14 -the rate of consumed by the sector intermediate products, produced by all of the sectors. Also note that for sectors 5, 7 12, 16 the rates of capital accumulation practi- cally do not have effect on corresponding rate of output growth ; the rates of labor inputs haven on-zero effect on the rates of output growth of all sectors ; for sectors 226 A. Ashimov : The Theory of Parametric Control of Macroeconomic Systems and ... 6, 16 the rates of consumed intermediate good shave no effect on corresponding rate of output growth. The results of analysis allow choosing the following budget parts of 16 economic sectors in the capacity of tools for solving the problems of economic growth. Oij-budget part of i-th sector, which is for payment for goods and services, bought from j-th sector ; Ol i-budget part of i-th sector, which is for payment for labor ; On i -budget part of i-th sector, which is for payment for investment goods. Table 6 Coefficients, characterizing an impact of economic growth factors Number of βi γi αi1 αi2 αi3 αi4 αi5 αi6 αi7sector i 1 0.3089 0.9051 1.345· 9.480· 2.897· 2.171· 1.602· 2.028· 1.345· 10−18 10−02 10−14 10−13 10−14 10−14 10−12 2 0.2426 2.4964 1.590· 7.308· 7.087· 9.884· 1.390· 1.120· 1.590· 10−16 10−01 10−16 10−15 10−15 10−15 10−16 3 0.9650 0.6886 2.886· 0.000 2.970· 8.269· 9.894· 1.478 2.886· 10−15 10−12 10−13 10−14 10−15 4 1.2900 0.0805 2.227· 1.343· 8.634· 9.989· 9.029· 2.641· 2.227· 10−13 10−1 10−13 10−13 10−03 10−16 10−13 5 1.0· 2.4083 3.199· 5.537· 8.913· 6.720· 5.406· 5.300· 3.199· 10−10 10−17 10−17 10−14 10−4 10−3 10−4 10−17 6 0.9343 0.7721 2.078· 9.186· 3.313· 3.940· 2.467· 1.324· 2.078· 10−16 10−18 10−14 10−13 10−15 10−14 10−16 7 1.0· 1.8792 8.133· 1.061· 3.115· 2.660· 4.124· 2.508· 8.133· 10−10 10−16 10−15 10−14 10−13 10−14 10−13 10−16 8 1.0691 0.4706 2.003· 1.498· 4.343· 6.171· 3.375· 7.288· 2.003· 10−02 10−15 10−17 10−14 10−15 10−15 10−02 9 0.8660 0.2153 4.289· 2.265· 1.666· 8.753· 3.057· 4.005· 4.289· 10−16 10−13 10−15 10−14 10−14 10−16 10 0.6702 0.5492 0.000 0.000 0.000 4.397· 5.602· 8.811· 0.000 10−15 10−15 10−15 11 1.2022 0.1006 2.929· 2.267 5.064· 1.030· 1.664· 5.308· 2.929· 10−05 10−14 10−12 10−04 10−13 10−05 12 1.0· 2.5822 5.415· 0.000 0.000 4.255· 2.573· 0.000 5.415· 10−10 10−14 10−13 10−14 10−14 13 0.2635 1.7177 2.370· 0.000 0.000 8.847· 1.142· 5.739· 2.370· 10−14 10−13 10−13 10−15 10−14 14 0.0227 1.7814 6.736· 2.018· 6.280· 1.288· 1.919· 4.809· 6.736· 10−14 10−1 10−15 10−12 10−13 10−13 10−14 15 0.9304 0.2173 1.041· 1.408· 7.681· 3.712· 3.926· 1.579· 1.041· 10−15 10−3 10−15 10−13 10−14 10−13 10−15 16 0 1.9372 0 0 0 0 0 0 0 Continuation of Table 6. Advances in Systems Science and Applications (2014) Vol.14 No.3 227 Number of αi8 αi9 αi10 αi11 αi12 αi13 αi14 αi15 αi16sector i 1 4.325· 4.595· 1.807· 1.177· 0 1.681· 1.178· 9.304· 0 10−15 10−2 10−14 10−14 10−16 10−15 10−18 2 8.054· 1.087· 2.616· 2.280 0 0· 0 0 0 10−17 10−14 10−17 3 8.054 2.221· 1.436· 4.626· 0 1.581· 4.991· 1.571· 0 10−14 10−2 10−13 10−12 10−15 10−4 10−14 4 8.038· 1.444· 4.353· 6.338· 0 1.247· 3.379· 5.445· 0 10−15 10−13 10−14 10−16 10−16 10−03 10−16 5 1.642· 2.760· 6.132· 2.512· 0 3.273· 2.026· 5.494· 0 10−15 10−1 10−3 10−14 10−5 10−17 10−4 6 7.977· 1.117· 6.525· 2.014· 0 8.436· 1.554· 1.089· 0 10−15 10−2 10−16 10−14 10−17 10−17 10−16 7 6.067· 4.548· 3.407· 1.505· 0 5.394· 8.576· 4.404· 0 10−14 10−1 10−13 10−12 10−16 10−5 10−16 8 1.528· 8.267· 3.053· 1.507· 0 2.339· 1.338· 2.244· 0 10−15 10−15 10−14 10−14 10−17 10−17 10−16 9 3.346· 1.002· 3.203· 4.339· 0 3.657· 4.803· 1.091· 0 10−14 10−12 10−13 10−13 10−15 10−16 10−15 10 1.991· 3.270· 2.862· 5.408· 0 1.347· 0 8.805· 0 10−14 10−01 10−13 10−15 10−14 10−18 11 6.664· 6.596· 2.965· 2.646· 0 9.260· 1.661· 9.599· 0 10−14 10−13 10−13 10−12 10−14 10−03 10−06 12 4.185· 1.548· 9.506· 7.964· 0 2.126· 1.408· 2.886· 0 10−1 10−1 10−13 10−13 10−14 10−15 10−15 0 13 4.820· 5.807· 1.794· 1.331· 0 3.939· 7.738· 1.836· 0 10−14 10−1 10−14 10−14 10−15 10−17 10−14 14 1.891 2.118· 4.749· 2.743· 0 2.219· 1.782· 6.646· 0 10−13 10−17 10−13 10−16 10−14 10−14 15 6.285· 2.139· 9.739· 8.436· 0 9.538· 1.416· 1.222· 0 10−14 10−1 10−14 10−14 10−17 10−17 10−13 16 0 0 0 0 0 0 0 0 0 Finding the optimal parametric control laws on the basis of CGE model of economic sectors In computational experiments with CGE model of economic sectors,the crite- rion below was used as maximization criterion K = 1 6 2015∑ t=2010 V Y [t] (34) Here K is average value of country’s gross output for 2010-2015 in prices of 2000. In experiments with optimization criterion (34), the constraints on growth of consumer price index of the following form were used : V Pr[t] ≤ 1.0 228 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ... Here is calculated consumer price index of the model without parametric control, is consumer price index with parametric control. In computational experiments control was performed for 1536 exogenous pa- rameters - j -th agent-producer’s budget parts for purchasing goods and ser- vices, produced by i -th agent-producer for 2000-2015 : Oj i [t] ; t=2010,...,2015 ; i,j=1,...,16. Here ∑16 i=1O j i (t) ≤ 1 for mentioned values of t. Basic values of men- tioned parts, obtained by solving the parametric identification problem of the model on data of 2000-2008, we will denote by Ōj i ; i,j=1,...,16. The following problem of finding optimal values of adjustable parameters vec- tors was considered. 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[18] Fernández-Villaverde J. (2010), “The econometrics of DSGE models, SE- RIEs”, Vol.48, No.5, pp.1305-1312. [19] Hamilton J.D. (1994), “Time series analysis”, New Jersey: Princeton. [20] Makarov V.L., Bahtizin A.R., and Sulakshin S.S. (2007), “Application of computable models in state administration”, Moscow: Nauchniy expert. Corresponding author A. Ashimov can be contacted at: ashimov37@mail.ru