Establishing of sustainable economic development of states national economies basing on a set of principles such as quality of life improvement, rationalization of output and consumption structure, ecosystems preservation and so on, is especially actual in the conditions of economy which is in growing extent based on knowledge, innovations and digital environment development. The concept of sustainability is most widely used in various fields of science and engineering where sustainability’s of elastic systems and fluid motion as well as sustainabilities of structures, automatic control systems, transport vehicles, electric power systems, thermodynamic stability, etc. are studied. Their definitions are based on the mathematical theory of stability where the latter is characterized as a term which does not have any clearly defined meaning and is applied to motion, to geometric or any other objects depending on parameters, e.g. statistics (statistical stability). However, it is emphasized that the above application fields of the term ‘sustainability’ do not fully cover its essence [1]. Among various concepts of motion stability the most popular are S. Poisson’s, G. Lagrange’s, A.A. Andronov’s and L.S. Pontryagin’s dsssefinitions as well as A.M. Lyapunov’s concept of stability. In analyzing sustainabilities of various systems concepts of local (in small) and global (in large) stabilities are of great importance [2]. A system is locally sustainable if the stability property is defined only for states sufficiently close in some appropriate sense to the initial state (or for trajectories lying "near" the initial trajectory). A system is globally sustainable if the stability property is applied to all states (trajectories) within the domain in which the system is studied. In our opinion, the obvious variety of definitions of stability includes some general feature that characterizes the essence of this category, namely, its idea of some object as an attribute of a given object to preserve (possibly with some deformations) its certain qualities, attributes or characteristics under uncertain conditions of its existence. According to this understanding equilibrium and homeostasis are a particular case of a stable state. In addition, viability can also be treated as a demonstration of the stability property of autonomous systems. Therefore the great importance is in economic tools of stability research including cost-benefit analysis, methods of assessment in conditions of uncertainty and risk as well as methods of assessment of economic policy tools impact on economic system sustainability [3].

In the economic literature the term "sustainability" is quite relative and allows various interpretations depending on the combination with various economic objects or categories such as market sustainability, economic sustainability in general, sustainability of development and economic growth, technological sustainability, financial sustainability, sustainability of money circulation, price, resource and ecological sustainability as well as some others. The basic and chronologically earliest concept which includes various aspects is the concept of market sustainability which permits the following interpretations: the property of a market system to finally attain price/equilibrium balance in the process of self-regulation (Walrasian stability); equilibrium of economic interests of interacting market subjects treated as: J. Nash equilibrium/ stability, that is such a market state (the situation in it) to change which is unprofitable to any participant of market relations; F. Edgeworth equilibrium/ stability, which is a non-blocked state of a market community of economic agents (players) when it is unprofitable for any coalition of this community to detach from other players and distribute the coalition gain between themselves. A lot of such states (non-dominated systems of contracts, allocations, etc.) form the "core of economy" or the C-core of the cooperative game (generally speaking, the latter can be not available, that is to be empty). The classic equilibrium interpretation of economic sustainability is most widely used in the social and general economic literature; it is formulated in the following way: ‘when social and economic parameters characterizing an economic entity keep an economic equilibrium state at any level under any internal and external environment perturbations [4]. However, we can and must speak about state sustainability as well as about development and functioning trajectory of an economic entity irrespectively of the fact if they are in equilibrium or not. Modern economic systems (firms, organizations, regional and national economy) are complex systems with a goal-oriented behavior, so not only organization but also self-organization are inherent to them in full extent. The matters of sustainability are especially important for them. The term sustainable development appeared in the last third of the 20th century. It means a process of economic and social change in which the use of natural resources, the direction of investment, technological progress and institutional changes are coordinated and strengthen national capacity to meet the needs of the present generation, without compromising the ability of future generations to meet their needs [5]. In other words, a joint development (coevolution) that does not contradict the further existence and development of mankind. A need to ensure harmony in economic, social, engineering and natural components is declared in sustainable development agenda of the Organization of United Nations [6]. There are plenty of works on sustainability. Not claiming it to be an all-inclusive survey we would like to mention for purposes of our research that some of them devoted to global and national sustainability [7-8]. While others with sustainability in industrial complexes and organizations [9-10]. As well as sustainability in innovative economyand environmental sustainability [11-13]. In modern conditions the situation is notably changing. Digital transformation means implementation of digital technologies and business into economy, creating ecosystems on the basis of digital platforms, alteration of people's quality of life etc. Three main problems are remaining: employment, trust in conditions of sharing processes and sustainability. The report of the World Economic Forum notes that ‘current business practices will contribute to a global gap of 8 billion tons between the supply of and demand for natural resources by 2030, translating to $4.5 trillion of lost economic growth [14]. In this regard it appears to be actual to research not only factors of systems economic sustainability, but also methods of its assessment in order to define in proper time deviations of target indicators from possible resource perturbations in ongoing transformation processes.

By economic sustainability of social and economic systems we would mean their ability to ensure fulfillment of their target indicators in uncertain conditions of functioning or development (for example, fluctuations of marker conjuncture, unpredictability of partners behavior, industrial technologic malfunctions, unreliability of resource suppliers and other possible perturbations). Such aspects of sustainability as resource, technological, pricing, and financial and investment reflect manifestation of sustainability attribute towards either individual parameters or some fields of economy – output, consumption, logistics, and finances and so on. We do not equate stability with sustainability, as the former could be considered as a special case of the latter's manifestation. Economic sustainability can be classified into structural and functional-parametric ones. Structural sustainability is an ability of a system as a community of economic subjects to self-preservation and self-reproduction as well as to maintain its integrity, organizational unity given different interests of the subjects involved in the system. It suggests the preservation of structural integrity as a set of certain necessary subset of interrelations of independent component parts defining the given system. Structural sustainability is actually a necessary condition for ensuring economic sustainability of this system and embodiment of its structural aspect. In fact, if target indicators of the system are achieved, it may be said at least that it retains its integrity. However, if a system is disintegrating (decomposing), the matter of its target indicators achievement is discarded at all. Functional-parametric sustainability is subdivided into sustainability to small perturbations (sustainability in small) and sustainability to large deep perturbations which is formed due to managing ability and adaptability. Sustainability in small means that small changes in development conditions (functioning) of an economic object result in small deviations of actual values of its target indicators from the planned (program) ones. Managing ability is a property (capability) of a system to generate rational managerial decisions (within the managing subsystem) and to respond adequately to managing actions. Adaptability is a property to adapt (passively or actively) and to respond adequately to changes in external and internal environments. The process of this adaptation itself in general suggests both adaptation to changes in conditions and changes in themselves. Two aspects could be outlined here. The first aspect, namely the reliability aspect, is connected to the ability of a system to counteract decreasing the quality of its target indicators under negative (unfavorable) perturbations (economic reliability). We should note that the category of economic reliability is treated as reciprocal to the category of risk. So far economic reliability could be defined as risk sustainability. The second aspect of adaptability is related to the ability of the system to implement additional possibilities under positive (favorable) perturbations, for example, the improvement of market conjuncture, appearance of new sources of investment etc. Based on the above it is possible to call this aspect cumulative efficiency (implement ability). Both aspects are closely related and defined by the following factors: maneuverability, flexibility as well as functioning institutions and cybernetic principles of management [15]. Flexibility of an economic system is its ability to adapt without any structural changes, for example, by means of creating various kinds of redundancy (reserves of productive capacity, raw materials stocks, materials, fuel etc.) Maneuverability is an ability of a system to maneuvering, i.e. to making purposeful adjusting actions, to implementing active changes and to structural rebuilding in response to perturbations. Essentially these actions can be expressed as alterations in the composition of objects in the planned system, their rearrangement, as well as in changes in the directions of scientific and technological progress, organizational and economic characteristics of objects, the topology of relations between them, directions of perturbations distribution etc. Institutions are a set of formal and informal rules implemented by people as well as corresponding mechanisms of control for their observance and protection. Institutions create inertia by virtue of their propagation and rooting in time and space [16-17]. Cybernetic control principles – emergence, necessary variety, external addition, feedback, systematicity, hierarchy etc. – are well known and are used to plan and manage systems [18-19].

possible basing on development of theory of effective reliability and elasticity. Classical theory of reliability is developed for engineering (conditional in reliabilistic terminology) systems, their quantitative parameters are widely known (probability of failure, average non failure operating time, frequency of failures, availability factor etc.). In our days reliability of such systems is assessed through experiments, Kolmogorov’s differential quations, Bayesian networks, Markov process and GO methodology, hybrid methods of assessment and multilevel analytical processes [20-26]. However classical theory if reliability is not applicable, as a rule, to complex social and economic (or unconditional) systems. The case is that in conditional systems separate failures lead to two threshold extreme conditions: either the systems works correctly (in case of making reserves) or it doesn't work at all, while in unconditional systems separate failures cause any from continuous set of states, from full operability to full non-operability, i.e. from projected values of quality parameters to their going beyond acceptable limits [27-29]. The problem of unconditional systems research (effective reliability) is much more complex than that of conditional systems research. Criteria of effective reliability could be functional from difference or relation real and ideal characteristics of object performance quality 

Let   to be project values of target parameters set on production output, volume of sales, market share, profit volume etc. Basic parameters to measure level of development (performance) reliability of a given project could be represented by variables  defined as:

Where  is mathematical expectation operator?  Is a normalized random value of underperformance (losses) of kth project order? Input reliability indicators characterize degree of confidence in the end targets of a system achievement, while, for instance,    could be used for risk evaluation. In the same way, reliability of a plan fulfillment could be determined through some cost characteristic of a system (investment or operational costs):

Is a random value of relative costs excess against ones set in the project 

The indicators shown are essentially momenta. If random function on the time interval [0, t] are considered instead of random values, then these indicators would change to interval ones. In addition let's note the following. Description of any economic process (including production one) in dynamics is possible in two forms: 1) volumes as a function of time  2) time as a function of volumes   

Correspondently description of discrepancies in performance development due to undefined factors is also possible in two forms. Basing on different interpretation of discrepancies different sets of reliability and risk indicators could be constructed as well as different groups of mathematical models suggesting their utilization. From this point onward only indicators and models of the first group allowing more natural reflection of different processes simultaneously proceeding in economic systems are to be considered. Let's return to the reliability indicator in the form (1). Its disadvantage is that it doesn't take into account dispersion of losses in achievement of target values. In principle, practically enough information on random numbers and random processes necessary for evaluation of economic indicators, development of different management systems etc. is contained in momenta characteristics of the first and the second order. From this perspective to characterize reliability of a system development (performance) the following expression could be suggested: 

Where   is a squared mathematical expectance?

  Is a variance operator?

  Is an operator of the second initial moment?

  degeneration into nonrandom variable equal to 1. The indicator (2) could be used if  When over performances are acceptable, i.e. when  another expression for   could be written: 

Where is some weight multiplier  Let 2 project variants of development are compared and   while  if their reliability is considered to be equal when 

Upper estimate of normalizing condition reachability  the random variable  discrete type looks like:

Increase of dispersion in (3) diminishes  , under any values of mathematical expectance (both positive and negative), which in general corresponds to qualitative understanding of this parameter. Elasticity is interpreted as ability of an economic system to neutralize perturbations suffering herewith some "deformations" (losses) in achievement of target indicators but avoiding however their complete non-achievement (it is clear that flexibility and maneuverability act like a sort of internal springs providing elasticity). Under otherwise equal conditions volume of losses characterizes elasticity degree as well, as it is the higher the lower losses are and controversially. The latter are a sort of "payment" for uncertainty used in information management. The attribute of elasticity is directly "adjacent" to risk-sustainability, as the latter is defined through elasticity and level of possible perturbations correspondent to evaluated variant of development / performance of social and economic system. Elasticity of some object with a given program of its development / performance can be described with some "payment" vector function or with its special type – elasticity function  approximating relation between input perturbations and deviations from planned (programmed) reference points. In this context elasticity determination and measuring appears as a mirrored transferring (expansion) of this term from the concept of economic system result formation (production function) to the concept of its adaptation. Some specific elasticity functions could be obtained with a correspondent reexpression from known typical production functions For example, in modern international research works production functions are used for evaluation of human capital influence on economic growth (Digital globalization, 2016). Indeed to be determined let   is a continuous scalar production function; a condition of initial balanced state of a program. Let’s assume we need to find elasticity function

  Expressing     and substituting this expression instead of  us have: 

From the perspective of abovementioned let's consider as examples some elasticity functions obtained by means of (4) from known typical production functions.

1. Linear production function:  Then

Elasticity function is not invariant to   

2. Production function with mutually complementary factors:

  (6)

It is visible from the construct   that in the presence of such elasticity function elimination (neutralization, extinguishing) is absent in relation to any of input perturbations. This example gives an idea to define in the correspondent way compensating ability of an economic object. Let us say that this object under a given development program (performance) possess compensating ability in kth target indicator in relation to realization 

3. Power production function (function with mutually complementary factors):

It is easy to figure out condition of neutralization of perturbations vector  

 Taking the logarithm of in equation 

As the right side of this formula is not less that any of multipliers in the brackets under the summation symbol, it would be true for   also present excluding the case with  when  trueness depends on interrelation between components of  Let us note considered above elasticity (6) and (8) functions are invariant relative to volume of program setting  

Let's illustrate application of elasticity functions to two econometric models in order to evaluate consequences of resource perturbations.

1. Econometric model of long-term type [30]. Is based on the classic Cobb-Douglas production function and is written as:  

Where Y is gross domestic product (GDP),

K–fixed capital,

L–employment,

G–products trade flow,

I–innovations,

M–migration flows,

D–data flows.

Elasticity function (type 8)   possible perturbation (deviations) in arguments-factors would cause correspondent deviations of target indicator GDP (Y). Therefore when  When  When      

Evaluating calculations for the resources volumes decreased by 20% are shown in the Table 1.

Table 1: GDP change as a response of resulting indicator to resource perturbation involving decrease of resources utilization in a non-linear production function.

2. Econometric model of innovative products output in Russia [31]. Basin on Russian statistics agency date 12 factors determining innovative products output in Russia were detached:

x1 – share of personnel engaged in scientific research and development, per 1000 inhabitants, %;

?2 – share of researcher having scientific degree, per 1000 inhabitants, %;

?3 – internal costs on scientific research and development, mln rubles;

?4 – patents granted, pcs;

?5 – expenditures on technological innovations, mln rubles;

?6 – advanced production technologies used, pcs;

?7 – number of organizations engaged on scientific research and development, pcs;

?8 – share of post-graduate students per 1000 inhabitants, %;

?9 – innovative activity of organizations, %;

?10 – turnover of medium organizations, bln rubles;

?11 – average income per capita, rubles;

?12 – investments in basic capital, mln rubles.

In order to make different-scale indicators comparable the input data were normalized according to formula:

Where σ_x – mean square deviation.

To describe formation of innovative products volume let's turn to regression analysis methods. In general multiple regression equation looks like: 

Where Y is a vector of response values; X is a matrix of independent variables values for N experiments; Θ is a vector of unknown parameters of a model; E is a vector of random errors.

Using the least square method allowed to obtain estimation of regression equation for the model (10): 

Determination coefficient and Fishers ratio are significant for this model, i.e. the model is suitable for description of studied phenomena. In order to range the factors by their influence on volumes of innovative products in Russia the method ‘Least Absolute Shrinkage and Selection Operator’ (LASSO) suggested by Tibshirani was used. It turned out the strongest influence on response is shown by the 5^th, 6^th and 11^th factors. In order to build a respective elasticity function the formula (5) could be used [32].

Let's suggest that x₅⁰=100, x₆⁰=100 and x₁₁⁰=200; then Y₀ = 144. When ?₅=0.10 (10%) (almost 5%). When ?₆=10% =1.11%. When ?₁₁=10% = 4%.

It follows thence that in the economy of Russia the most desirable is to increase in the first turn expenditures on technologic innovations of organizations, as when they grow by 10% the volume of innovative products in economy would increase by almost 5%.

In real practice for many random perturbations the most typical distributions are those which decrease quite steeply when deviation values (both positive and negative) increase. It looks quite natural as planned variants which are combined with high probabilities to large expected perturbations are most often thrown away a priori. Such a character of perturbations distributions may also serve as an indirect proof of existence of mechanical compensation of these perturbations in economy. Also let us note that a formal relationship between a production function and an elasticity function should not conceal their principal difference, implying that they represent different aspects of the program of development or functioning of an economic system: the former simulates the mechanism of planned (program) tasks forming while the latter simulates the mechanism of their stabilization or, more precisely and in a wider interpretation, the mechanism of adaptation built-in in the program. The latter manifests only when the assumed conditions of program implementation are violated and thus generally speaking, when the conditions of generating the production function of the chosen program version are violated. Therefore, the transformation (4) fixes sort of general part or an "intersection" of above aspects in ƒ. So far mutual substitutability of production factors is one of the characteristics of production process, process of results obtaining and at the same time a prerequisite of maneuvering. However, reserves in their principal meaning are an attribute of only an adaptive aspect of a program because the production function always describes production possibility limits. Application of the elasticity model gives an estimation of the target indicator deviations in various changes of external environment comparing to the predicted values. Hereafter reliability indicators can be calculated for various levels of perturbations. Calculation results are used for scientific substantiation of improvement in the project management mechanism aimed at increasing its economic reliability and efficiency and hence adaptability and sustainability of the respective economic system.

The essence of economic sustainability is defined through such mutually related characteristics as managing ability and adaptability, and the latter splits into economic reliability (risk sustainability) and cumulative efficiency (performance). At the same time the above characteristics and economic sustainability itself are formed due to a whole set of factors all before due to the flexibility and maneuverability, functioning of certain institutions, application of cybernetic principles of efficient management. An approach to evaluation of social and economic system sustainability basing on the theory of reliability and elasticity is proposed. It could be applied both at the stage of elaboration of programs and projects of economic systems development and at the stage of implementation in the process of revealing possible resource limitations and correction of targets and ways of their achievement.

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