Model aspect of the study of the processes of sustainable development of socio-economic systems

The triune concept of sustainable development takes into account environmental, social and economic issues in general. At the same time, the solution of the problems of resource conservation and socio-economic development do not contradict each other, but contribute to mutual reinforcement. The purpose of this study is to build, describe, test and analyze economic and mathematical models of the processes of self-organization of the socio-economic system within the framework of the concept of sustainable development. The considered models take into account both quantitative and qualitative characteristics. The research methodology is based on the assumption that the system under consideration is dynamic. The study makes it possible to evaluate the process of sustainable development of the socio-economic system in terms of the formation and influence of its characteristics on the overall result. Various options for formalizing the level of motivation of economic activity are described and analyzed, one of which is based on the economic interpretation of the Weber-Fechner psychophysiological law. The assumption of the continuous development of innovative technologies made it possible to reveal the existence of a “fatigue point”, the onset of which indicates the possibility of increasing the growth rate of production activities due to its technological transformation.


Introduction
Human development in recent decades is often characterized by growth adverse impact on the external environment.Factors in this are also wars, political and socio-economic instability.People's actions have a negative impact on the environment.This threatens the survival of the Earth and future generations.This necessitates changes aimed at more rational and effective management of resources, which will reduce pressure and negative impact on the environment.
As a result, it became necessary to develop strategies for the development of mankind that would prevent the deterioration of the quality of the environment, eliminate the consequences of such deterioration, and contribute to the formation of conditions for providing resources for the current and future generations [1].
Such a viable behavior, to ensure the long-term exploitation of resources, without putting a threat to future generations, is considered within the framework of the concept of sustainable development [2].
The most progressive concept is the triune concept of sustainable development: environmental, social and economic issues are considered as a whole, and the goals of resource conservation and socio-economic development contribute to mutual reinforcement [3].
This understanding of sustainable development implies a balance between the three components of sustainability [2]: • environmental sustainability, which is aimed at maintaining the quality of the environment necessary for the implementation of economic activities and ensuring the quality of life of people; • social sustainability that seeks to ensure human rights and equality, the preservation of cultural identity, respect for cultural diversity, race and religion; • economic sustainability needed to maintain natural, social and human capital to ensure living standards.
Only such an interpretation of sustainable development makes it possible to form the conditions for achieving a high standard of living, a developed economy and conservation of resources.
This approach is consistent with the definition of Brundtland [4], who defined sustainable development as the process of meeting the needs of the subject in the current cycle without harming the ability to meet the needs in the next cycle.
This concept assumes that the effective solution of environmental problems should be accompanied by the development of environmental awareness in the performance of economic entities of their main functions, the use of appropriate forms of economic behavior [1].
Thus, it is relevant to study the forms of economic behavior in the context of sustainable development and develop methods for quantitative and qualitative assessment of the results of their use.
At the same time, the current stage of development of the concept of sustainable development is characterized by the transfer of the problem from the global and macroeconomic levels to the microeconomic level, the subjects of which are actively involved in causing harm to the environment and become subjects of its elimination [5].
With this approach, the sustainability of an individual microeconomic entity becomes not just a concept, but also an object of sustainable development, which forms the features of modeling, design, organization and management [6].
Thus, in the article [7], the problems of decision-making in the management systems for sustainable development of complex technological and socio-economic objects are studied, the limitations of traditional expert systems are shown.
The study [8] developed a methodological approach for conducting a comprehensive assessment of the socio-economic parameters of sustainable development.To obtain justified results, the work uses such modeling methods as a fuzzy logical model, the Saaty hierarchy method, the Mamdani algorithm, quantitative and qualitative methods of comparison.
In the article [9], the logistics system of enterprise management is presented in the form of an optimization model that combines economic and environmental factors.This model allows you to evaluate and take into account the impact on the environment.
Article [10] is devoted to the study of a dynamic model for the optimal use of water resources with taking into account the interests of regional counterparties of the two hierarchy levels.The problem is solved by simulation methods.
At the same time, the management of sustainable development processes often provides for such a type as organization by structures of the upper hierarchical level [8,[10][11][12].
The purpose of the study is to substantiate the strategies of economic behavior of socioeconomic entities within the framework of the concept of sustainable development.To achieve this goal, it is necessary to complete the following tasks: within the framework of existing methods of economic and mathematical modeling, describe an adaptive process model that takes into account the self-organizational and reflexive features of a socio-economic subject; 3 conduct model experiments to identify cause-and-effect relationships between the motives for the formation of economic behavior and results.

Results
It should be agreed that in socio-economic systems, the main system-forming element of models is meaningful or semantic approaches, and more complete models are built on target settings, which are based on value orientations, which are the main, fundamental basis for choosing system management models [13].Most of these systems are dynamic.
The two fundamental elements of the concept of sustainable development are development and sustainability.At the same time, classical economic theory considers development within the framework of economic growth [2].
In the study [14], an isomorphic connection between the concept of "dynamic system" and "autonomous system of ordinary differential equations" is established.The main mathematical apparatus in this case is the qualitative theory of differential equations [15], and the model is formalized by a system of autonomous differential equations where x i is some quantitative characteristic of the socio-economic agent, which takes part in the process, which corresponds to the coordinate of the vector in the n-dimensional economic space.The approach to building a possible full-factor model of the process of sustainable development consists in the balanced consideration of economic, social and environmental factors, among which the following are distinguished: individual; intrasystemic; external.At the same time, the external influence can be characterized by a known function or have a random nature.
In this case, system (1) takes the form and is classified as a system of ordinary differential equations.The two fundamental elements of the concept of sustainable development are development and sustainability.At the same time, classical economic theory considers development within the framework of economic growth [2].
If one characteristic is taken as an indicator of the level of development, then in equations ( 1), ( 2) n = 1.In this case, the development model (growth, increase, accumulation) can be represented in the form dx dt =kx, where k is a coefficient that characterizes the growth rate.Equation (3) corresponds to the Malthusian or exponential law of population growth [16].A particular solution to equation ( 3) is described by the function where x 0 is the value of the indicator under study at the initial time t = 0.
It is obvious that function (4) will correspond to development only when it increases.This is ensured by the condition k > 0, the fulfillment of which is accompanied by the theoretical possibility of an infinite increase in the indicator studied in the model lim t→∞,k>0 x 0 exp (kt) =∞. ( In this case, the management of the development process in the context of the concept of sustainable development can be accompanied by an organizational impact on the system, which is displayed in equation ( 3) by the transition k = k(t).That is It should be noted that some studies suggest modifications to model (3) [17,18], which allow correcting the dynamics of unlimited growth as a consequence of a lack of resources [19].At the same time, the limited resources are displayed in such models only indirectly, and the social and environmental components of the concept of sustainable development are absent.
Unlike classical and neoclassical economics, institutional economics provides for bounded rationality [20].For institutional economics, a characteristic type of management is selforganization [21] and a synergistic approach to management [22].Equilibrium in this case becomes the main goal of the functioning of the economic entity, which is consistent with the concept of sustainable development.
This principle corresponds to the form of the function k(t) in equation ( 6) where k max is a constant positive coefficient characterizing the maximum value of the growth rate; x Lim is the value of the indicator under study, limiting its increase.
In this case, equation ( 6) is transformed to the form which is a logistic (Verhulst-Pearl) model [23].
The essence of the presented equation is that changes in the system depend on factors that contribute to its development (growth) and factors that inhibit (limit) this development (growth).This means that within the system there is a certain regulatory mechanism that ensures the process of sustainable development at the level of self-organization.
The different influence of individual factors on the overall result is more clearly displayed if equation ( 8) is converted to the form The Verhulst-Pearl equation ( 8) is quite universal and can be used to describe various dynamic processes [10,[23][24][25][26].
Thus, in population models, as the x Lim value, which limits growth, an objective indicator of the capacity of the ecological environment is considered [23,24], and in models of production processes, in addition to environmental restrictions [10], the consumption potential in the market can be assessed [25].
At the same time, the concept of sustainable development of such socio-economic entities as households also implies a self-organizational type of management of their economic behavior.In this case, a unitary strategy of economic behavior may be characterized by the desire to achieve a level of opportunity x Lim (income) that allows them to satisfy economic needs that correspond to their social and cultural level [26].This approach to modeling rational economic behavior is consistent with the theory of Gossen [27], according to which rationality is determined by the desire to increase satisfaction (reduce dissatisfaction).
It should be noted that under the initial conditions x(t 0 ) = x 0 , a particular solution to equation (8) will be the logistic function for which lim Thus, the dynamics of the results of production activities for given model values of parameters and initial conditions is shown in figure 1.By controlling the parameters of the model, one can contribute to the formation of conditions for sustainable development.
A more general case is when the limiting parameter x Lim changes with time, that is, x Lim = x Lim (t).This situation may be a consequence of the development of technologies that allow gradually increasing the effectiveness of economic activity without increasing environmental damage and without increasing the ecological footprint.[28,29].We will use the assumption that there is a direct relationship between the innovative level of technology and the level of production acceptable from the point of view of environmental safety.
With the same increase in each period of time, a constant growth rate is observed, and the function itself x Lim = x Lim (t) has the form where x 0 Lim is the initial value of x Lim ; a 1 is the percentage of growth per unit of time.If development occurs at a constant rate, but relative to the level reached in the previous time interval, then the function has the form where a 2 is the percentage of growth per unit of time.
It should be noted that formulas (12), ( 13) imply discrete time.At the same time, function (13) more correctly reflects the process of sequential development.Passing to the continuous growth form, function ( 13) is transformed to the form Using function (14) we transform equation ( 8) The general solution of equation ( 15) has the form The joint dynamics of the results of production activity ( 16) and function (14) for given model values of parameters and initial conditions is shown in figure 2.  14), ( 16) at k max = 0, 2 (productivity growth rate); x 0 Lim = 800 pcs (initial value of maximum allowable capacity); x 0 = 5000 pcs (performance at the initial moment of time); a 2 = 0.01.Functions ( 14), ( 16) have the equality That is, an increase in the allowable production threshold is accompanied by a corresponding increase in production (figure 2).At the same time, at each moment of time there is some lag, which can be estimated by the function This function characterizes at each moment of time the value of the deviation of the real results of economic activity x(t) from the possible ones x Lim (t).It is not difficult to show that function (18) has an extremum, namely, a minimum.The visualization of this fact is shown in figure 3.
We propose to call the minimum point of function (18) the "fatigue point", which is characterized by the fact that at this point in time the growth of economic activity in a given mode ceases to have a critical impact on its potential development.This can be explained by the  Speaking about the self-organization of the processes of sustainable development of socioeconomic systems, it should be noted that consciously or subconsciously their motivators may depend on individual or unitary social, cognitive and emotional factors [30].
The level of motivation for the development of the system under the existing constraints x Lim in model ( 8) is represented by the function At the same time, psychological and social factors can influence the assessment of objective quantitative indicators and subjective qualitative indicators.This allows us to give an economic interpretation of the Weber-Fechner psychophysiological law [31] and quantify motivation by the function where r is the scale factor.Analysis of changes in functions (19), (20) (figure 4) displays their distinctive features.First, m(N ) ∈ [0; 1], and M (N ) ∈ (0; +∞).Secondly, with a sufficiently large deviation of the values of the indicator of the level of economic activity from the boundary value x Lim , the non-linear motivation function M (x) significantly exceeds the values of m(x), and with small deviations it can become smaller.This fact can be explained by the existence of features of a particular socio-economic system, which, in a critical situation, stimulate the search for additional opportunities to realize the existing potential.Such features are displayed by the scale factor r.
Taking into account the content and form of function (20), equation ( 8) can be modified The general solution of equation ( 21) has the form Let's compare functions (10) and ( 22) (figure 5).Obviously, depending on the individual characteristics of the socio-economic system, which are displayed in the model ( 21) by the scale factor r, the process of sustainable development can proceed less (figure 5a) or more (figure 5b) intensively.
It should also be noted that the use of the previously considered option x Lim = x Lim (t) (14) in model ( 21) also makes it possible to identify the "fatigue point".The results obtained can be explained by the presence of institutional features of socio-economic systems.

Conclusions
1.The proposed methodology for modeling the processes of sustainable development of socioeconomic systems is based on a combination of neoclassical and institutional approaches and is characterized by limited rationality, which implies the achievement (maintenance) of an equilibrium state of the socio-economic system.2. The motivational and stimulating factor in the formation of sustainable development processes is the desire to achieve (maintain) a possible (satisfactory) level of development.
Under such conditions, the quantitative assessment of motivation is defined as the magnitude of the deviation between the possible (satisfactory) level and the real one.3. To take into account socio-economic features, an economic interpretation of the Weber-Fechner psychophysiological law was proposed, which made it possible to formalize motivation in a logarithmic form, which, unlike a linear one, allows taking into account individual (group) features of the socio-economic system under study.4.An analysis of the constructed models made it possible to identify the presence of a "fatigue point", that is, the moment in time when the growth of economic activity in a given mode ceases to have a critical impact on its potential development.This moment testifies to the possibility (necessity) of increasing the growth rate of production activity due to its technological transformation.5.The constructed models can be attributed to simulation.They make it possible to study transitional processes within the framework of the concept of sustainable development.6.According to the authors, the development of a methodology for determining the scale factor will allow taking into account the individual features of the economic behavior of the modeling object.

Figure 1 .
Figure 1.Graph of function (10) at k max = 0.2 (productivity growth coefficient); x Lim = 8000 pcs (maximum permissible productivity in terms of environmental safety); x 0 = 5000 pcs (productivity at the initial moment of time).