Application of the parametricalregulation theory for reduction the effect of a shadow sector of economy
AshimovA.A., SultanovB.T., AdilovZh.M., BorovskiyYu.V., MerekeshevT.B., AshimovAs.A
The work presents some results of the parametricalregulation theory which consider peculiarities of CGE models. The method of parametrical identification of macroeconomic models with large quantity of evaluation parameters is proposed and tested. The application efficiency of the parametrical control theory for reduction of the effect of a shadow sector of economy is shown by example of one CGE model. Optimum values have been obtained for regulation of economic system development based on examined mathematical model on the level of 19 parameters.
Key words: shadow economy, computable general equilibrium model, discrete dynamic system (semi-cascade), parametricalregulation, parametric identification.
1. Introduction
As it is known, national economies of many countries function with a shadow sector, which activity affects national economy development negatively [10-12]. A rational direction of evaluation and searching of the effective economic measures to reduce the effect of a shadow sector on development of a country’s economic system is application of the mathematical model of a national economy. [8] presents a computable general equilibrium model (CGE model) with a shadow sector with large quantity of adjustable parameters, which evaluation on the basis of computational algorithms requires application of effective methods of identification.
Thetaskofidentification (calibration) of exogenous parameters of the model comes to the searching of the global minimum of some objective function, which is set with the help of the CGE model itself. Meanwhile limitations on set of optimization are also set with the help of the model. The task of searchingthe global extremum in general case of high dimensionality is quite complex; random search methods, parallel algorithms of calculation and other methods are applied for its solving [14, 15]. The overview of numerous publications on searching the global extremum is shown in [13]. The paper contains unmentioned before in the literature the algorithm of parametric identification of a model, which considers peculiarities of macroeconomic models of high dimensionality and in some cases enables to find the global minimum of an objective function with large quantity of variables (more than one hundred). Two objective functions (two identification criterions – main and complementary) are used in the algorithm; this enables to obtain the exit of values from points neighborhood of local (and non-global) extremums, to continue searching the global minimum while keeping the conditions of corresponding movement to the global extremum.
[1, 3, 5, 4] present the elements of the theory of the effective parametricalregulation of market economy development described by the system of ordinary differential and algebraic equations. Application efficiency of parametrical regulation approach based on a number of models is showed in [2, 6, 7]. In the network ofthe proposed approach optimum (in sense of some criterion) values of parameters have been obtained with the help of a family of functions, defined with the help of endogenous factor of the mathematical model and adjustable coefficients. Development of the theory of parametricalregulation for the case when optimum (in sense of some criterion) values of controlled parameters are evaluated in some given set is of practical interest.
The present work contains results of development and application of the parametricalregulation theory for a given case based on the CGE model, in which shadow economy is considered to be of two type: “white-collar” and “grey” [10]. Such processes of white-collar shadow economy as transfer of asset share from budgets of production sectors and a consolidated budget to a budget of household were used in the work while modeling some scenarios of economic development of a state with negative influence of shadow economy and modeling neutralization of negative outcomes of such processes applying the parametrical regulation theory approach.
The CGE model with a shadow sector of economy, proposed in [8], is presented in the general view with the help of the following correlation system.
1) Subsystem of difference equations, binding the values of endogenous variables for the two consistent years.
(1)
Here t- number of a year, discrete time, ;- vector of endogenous variables of the system;
, , .(2)
Here ,variables include values of basic funds, agents’ remained cash in bank accounts and other; include agents’ demand and supply values in different markets and other, - different types of market prices and budget shares in markets with exogenous prices for different economic agents;u and - vectors of exogenous parameters, - vector of controlled (regulated) parameters; X1, X2, X3, W- compact sets with non-empty interiors- and respectively; vector of uncontrolled parameters, - open connected set; - continuous mapping.
2) Subsystemofalgebraicequations, describing agents’ behavior and interaction in different markets during the selected year, these equations allow expression ofvariables through exogenous parameters and rest endogenous variables
,(3)
Herecontinuous mapping.
3) Subsystem of recurrent correlations for iterative calculations of equilibrium values of market prices in different markets and values of a budget share in the market with state prices for different economic agents.
(4)
Here - number of iteration. L –set of positive numbers (adjustable constants of iterations). When their values decrease, economic system comes to equilibrium faster, however the risk that prices go to the negative range increases at that time. - continuous mapping (contractive when , and some L are fixed). In this case Zmapping has a single fixed point, where an iteration process converges (4, 3).
CGE- model (1, 3, 4) when exogenous parameters are fixed in every point of ttime defines values of endogenous variables, corresponding to demand and supply prices equilibrium in markets of agents’ goods and services in the bounds of the following algorithm.
1) On the first step it is assumed that t=0 and initial values of x0 variables are specified.
2) On the second step initial values of variables are specified for current tin different markets and for different agents; values of are calculated with the help of (3) (initial values of demand and supply of agents in markets of goods and services).
3) On the third step the iteration process (4) is run for current t. Meanwhile for each Q current values of demand and supply are calculated from (3): through specifying market prices and budget shares of economic agents.
The condition for the iteration process termination is equality of demand and supply values in different markets. As a result, equilibrium values of market prices in every market and budget shares in markets with state prices for different economic agents are defined. Qindex is omitted for such equilibrium values of endogenous variables.
4) On the next step values of variables for next moment of time are found regarding equilibrium solution for t moment with the help of difference equations (1). The value of tincreases by unit. Transition to the step 2.
Number of reiterations of steps 1, 3, 4 is defined according to the problems of calibration, forecast and regulation for time intervals selected in advance.
2. Development of parametrical regulation theory for the class of CGE models of a general view (1, 3, 4)
Examined CGE-model of the (1, 3, 4) type can be expressed as continuous mapping, specifying conversion of values of the system’s endogenous variables for the null year to the corresponding value of the following year according to the set above algorithm. Here compact Xin the phase space of endogenous variables is determined by a set of possible x variables’ values (X1compact with non-empty interior) and corresponding equilibrium values of yand z variables calculable with the help of (3) and (4) correlations.
Let us assume that for point inclusion is right, when and are fixed for (N – fixed natural number). This fmapping determines the discrete dynamic system (semi-cascade) in the X set.
(5)
Such description of a state’s economic system (1, 3, 4, 5) differs from description of economic system with the help of continuous dynamic system in [1] and validate necessity of the parametricalregulation theory development for a discrete case of a semi-cascade.
Let us denote points of corresponding trajectory of semi-cascade by for selected .
Let us denote the closed set in space((N+1) sets of the variables for ), determined by limitations
, ,(6)
through .Last inequality in (6) is used for some values, when are positive;.
Fortheassessmentof efficiency of economic system evolution during the period of time, (N- fixed), let us use the criterion of type , where K -the continuous function in XN+1.
The definition of the problem of finding optimum values of a controlled vector of parameters for the semi-cascade (5) is of the following type. When is fixed, to find set out of N values of controlled parameters , which provide the lower boundary of (6) criterion’s values –
(7)
under constraints (6). The analogous task is set for the case of the criterionK maximization.
The following theorem is true.
Theorem. For the defined semi-cascade (5) under constraints (6) the problem (5-7) solution of finding the lowerboundary of K criterion exists.
The proof.Matchingof set of values (where) of regulated parameters to the corresponding output values of the discrete dynamic system (5) (under the regulation by means of this set of parameters) sets the continuous mapping H of some subset in the space .
The complete preimage of set with H mapping is compacted according to the theorem about compactness of the complete preimage of the compact set under continuous mapping. The set is nonempty, as it contains at which constraints (6) are satisfied.
The function, which sets the value of K criterion for each point of the set by the corresponding trajectory of the system (5) is continuous in the compact, and therefore at some point of this set it takes its minimum value. The theorem is proved.
3. Example
Efficiency of the obtained results is illustrated below by the example of the CGE model with a shadow sector [8]. This model describes behavior and interaction of the following agents in 13 markets (final goods, investment and capital goods, and the market of labor force):
Economic agent № 1- a state sector of an economy. This sector includes entities government share in which is more than 50%.
Economic agent № 2- a market sector, which consists of legally existing entities and organizations with private and mixed ownership.
Economic agent № 3 – a shadow sector. This sector describes the type of economic activities that are not taken into account in official statistics, i.e. which are hidden from statistical reports. The model considers two types of shadow economy:“white-collar” and “grey” economy.
Economic agent № 4- aggregate consumer, which joins households.
Economic agent № 5- government. Additionally this sector includes non-commercial organizations, serving household (political parties, labor unions, public associations etc.).
Economic agent № 6- banking sector.
Examined model contains 144 exogenous parameters (which values require to be evaluated by solving a task of parametric identification) and 123 endogenous variables. Theconsidered CGE model with shadow sector is presented in the frameworks: relations (1) – by means of11 expressions (); relations (3)-by means of98 expressions (); relations (4) - by means of14 expressions ().
3.1. Parametric identification and retrospective forecast based on the CGE model with a shadow sector
The task of parametric identification of the examined macroeconomic mathematical model consists of finding estimates of unknown values of its parameters, that enables to reach minimum value of an objective function, which defines deviations of output variables’ values from a corresponding observed value (known statistical data). This task comes to finding a minimum value of a function of several variables (parameters) in some closed domain of Euclidean space under constraints of type (2), imposed on values of endogenous variables. In case of high dimensionality of possible values’ range of desired parameters, standard methods of finding functions’ extremums are often inefficient due to existence of several local minimums of an objective function. The algorithm which considers special features of macroeconomic models’ parametric identification task and which enables to bypass the mentioned problem of “local extrema”.
For the assessment of possible values of exogenous parameters, the range of type has been considered.Here- interval of parameter’s possible values.Meanwhile, assessment of parameters, for which observed values existed, were searched in intervals with centers in corresponding observed values (in case if there is one such value) or in some intervals covering observed values (in case if there are several such values).Other intervals for parameters searchinghave been chosen with the help of indirect assessments of their possible values. In computing experiments the Nelder-Mead [16]algorithm of the directed search has been applied for finding the minimum values of a continuous function with severalvariables with additional constraints on endogenous variables of type (2). Applicationof this algorithm for the starting point can be interpreted asa sequencewhich converges to the point of local minimum of function Fwhere , While describing the following algorithm let us assume that point can be found sufficiently accurately.
FortheassessmentofthequalityofretrospectiveforecastingbasedonthedataoftheRepublicofKazakhstanfortheperiodof 2000-2004 forsomestartingpointthetask (taskA) ofthemodel’sparametersassessmentandassessmentofinitialconditionsfordifferenceequationshas beensolvedwiththehelpoffindingcriterion’s minimum:
.(8)
Heret – numberofayear; basic macroeconomic measures:
- estimatedGDP in milliards of tenge, prices of 2000;
- estimatedlevelofconsumer prices.
Here and below the sign «*» corresponds to observed values of corresponding variables. The task of parametric identification for the model (1), (3), (4) is assumed to be solved, if point is found where for sufficiently small.
Along with the task A for the point the analogous task (task B) has been solved with application of extended criterion instead of criterion.
(9)
Here:
L1t – numberofemployeesina statesector;
L2- number of employees in a market sector;
K1t -basic funds of a state sector;
K2t–basic funds of a market sector;
Y1t- gross value added of a market sector;
Y2t –gross value added of a state sector;
Y3t –gross value added of a shadow sector.
Values reducing weights in criterion (9) are defined during the process of identification of parameters for the concrete dynamic system.
While solving the task of parametric identification for each of these criterions separately, due to the existence of several local minimuma of functions and , it is quite difficult to achieve values of these criterions that are sufficiently close to zero.Therefore the ultimate algorithm of solving the task of parametric identification of the model has been chosen with the help of the following steps.
1. The A and B tasks for some vector of initial values of parameters are solved simultaneously, as a result points and are found.
2. If or, then the task of parametric identification of the model (1, 3, 4) is solved.
3. Otherwise, the task A is solved takingpoint as initial point and the task B is solved takingpoint as initial . Transitiontothestep 2.
Quite large number of iterations of 1, 2, 3 steps in some cases provides an opportunity for desired values to come out from the point neighborhood of non-global minimum of one criterion with the help of another criterion, thereby the task of parametric identification can be solved.
As a result of simultaneous solving of the A and B tasks in accordance with the stated algorithm the values andhave been obtained. Thisimpliesthatdeviation’s relative size of variables’ calculatedvalues used in criterion (8) from corresponding observed values is less than 0.25%.
Results of the retrospective forecast of the model for the period of 2005-2008 presented in the table 1 show calculated, observed values deviations of calculated values of main output variables of the model from corresponding actual values.
Table 1. Results of the retrospective forecast of the model.
Year / 2005 / 2006 / 2007 / 2008/ 4258.03 / 4715.65 / 5136.54 / 5303.27
/ 4221.69 / 4586.33 / 5004.12 / 5478.31
Error(%) / -0.861 / -2.820 / -2.646 / 3.195
/ 107.6 / 108.4 / 118.8 / 109.5
/ 108.4 / 109.5 / 112.6 / 112.0
Error(%) / 0.706 / 1.017 / -5.528 / 2.240
For conducting the following experiments the task of parametric identification of a model for 2000- 2008 time interval was solved repeatedly using solving the tasks of type A and B. As a result of the solution of the task of parametric identification of a model for the stated time interval, values of criterions of type and turned out to be 0.015 и 0.15 respectively.
3.2. Scenario approach and finding of optimum values of parameters based on the CGE model with shadow sector
Result of the shadow economy (in opinion of financial flows) is redirection of funds share, assigned to the budgets of production sectors of legal economy and consolidated state budget to the budget of households(players of shadow economy) [8]. This transfer of funds can be made through economic activity in a network of a shadow sector of economy, as well as directly with the help of some illegal acts (thefts, bribes, kickbacks, etc.)
In the network of examination of analysis of the connection between some effects of shadow economy and basic macroeconomic measures of a state (GDP and index of consumer prices), a number of computable experiments has been conducted (scenarios estimations, which consider some possible negative phenomenon in economy of a state), analogous to experiments in [8].
The work examines the following 6 scenarios.
1) Imitation of a process of cash withdrawal (10%, 20%, 30%) from a consolidated budget of a state and transfer of it to households from 2003 (scenario 1, 2, 3). The process of theft has been imitated as direct or as quite legal process of budgetary funds spending (process of kickback).
2) Imitation of cash withdrawal (10%, 20%, 30%) from producer and its redirection to households starting from 2003 (scenario 4, 5, 6). In this case the process of giving (by producer) and taking a bribe (finally by households) has been imitated.
Results of the listed 6 scenarios of economic development of a state with negative effect of shadow economy in comparison with initial variant of evolution are presented in tables 2 and 3.
Table 2. GDP values (in milliards of tenge using prices of 2000) in initial variant and applying scenarios 1-6.
GDPYear / 2005 / 2006 / 2007 / 2008
Initial variant / 4300103 / 4618653 / 4963707 / 5337048
Scenario 1 / 4301026 / 4623221 / 4975060 / 5357813
Scenario 2 / 4301887 / 4627487 / 4985442 / 5376495
Scenario 3 / 4302752 / 4631527 / 4994972 / 5393343
Scenario 4 / 4298244 / 4612732 / 4953878 / 5324870
Scenario 5 / 4296520 / 4607483 / 4945717 / 5315665
Scenario6 / 4294935 / 4602927 / 4939176 / 5309146
Table 3. Values of consumer price indexes (in % to preceding year) in initial variant and applying scenarios 1-6.
Price indexYear / 2005 / 2006 / 2007 / 2008
Initial variant / 107.624 / 108.602 / 109.334 / 108.816
Scenario 1 / 115.575 / 109.706 / 109.986 / 108.989
Scenario 2 / 123.530 / 109.761 / 110.470 / 109.044
Scenario 3 / 131.481 / 108.962 / 111.001 / 109.006
Scenario 4 / 138.576 / 118.506 / 113.760 / 111.462
Scenario 5 / 171.450 / 123.439 / 115.029 / 111.904
Scenario6 / 206.522 / 125.441 / 114.879 / 111.508
Analysis of table 2 and 3 shows that examined scenarios affect GDP insignificantly, at the same time the indexes of consumer prices increase significantly during the first year of 1-6 scenarios application, during the following years their influence on price indices decreases.