Mathematical tools in economic research: conceptual transformations in the period of paradigmal changes

Сonsidered and characteristic the process of evolution and prospects of mathematical methods and models in economic research. The prospects of the synergetic direction as a basic direction in modern economic and mathematical modeling are revealed.

Рубрика Экономико-математическое моделирование
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Язык английский
Дата добавления 24.10.2022
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V.N. Karazin Kharkiv National University

Mathematical tools in economic research: conceptual transformations in the period of paradigmal changes

L.D. Filatova Ph.D., (Mathematics and Physics), Associate Professor, Associate Professor of the Department of Information Technology and Mathematical Modeling

O.V. Korzhova Senior Lecturer of the Department of Information Technology and Mathematical Modeling

Ukraine, Kharkiv

The article considered the process of evolution and prospects of mathematical methods and models in economic research. The advantages and disadvantages of mathematical formalization in the conditions of the new economic paradigm are analyzed. The necessity of searching for fundamentally new scientific approaches capable of ensuring the adequacy of economic and mathematical models in the study of modern socio-economic systems is substantiated. The prospects of the synergetic direction as a basic direction in modern economic and mathematical modeling are revealed. It is shown that its further development requires the use of the latest mathematical tools. The classification of mathematical methods and models that need to be included in the synergetic direction of economic research is given.

Key words: mathematical methods and models, mathematical formalization, economic research, economic paradigm, synergetic direction of research.

Л. Д. ФИЛАТОВА

кандидат физико-математических наук, доцент, доцент кафедры информационных технологий и математического моделирования Харьковского национального университета имени В. Н. Каразина, Украина, г. Харьков

О. В. КОРЖОВА

старший преподаватель кафедры информационных технологий и математического моделирования Харьковского национального университета имени В. Н. Каразина, Украина, г. Харьков

МАТЕМАТИЧЕСКИЙ ИНСТРУМЕНТАРИЙ В ЭКОНОМИЧЕСКИХ ИССЛЕДОВАНИЯХ: КОНЦЕПТУАЛЬНЫЕ ТРАНСФОРМАЦИИ В ПЕРИОД ПАРАДИГМАЛЬНЫХ ИЗМЕНЕНИЙ

В статье рассмотрен процесс эволюции и перспективы математических методов и моделей в экономических исследованиях. Проанализированы преимущества и выявлены недостатки математической формализации в условиях новой экономической парадигмы. Обоснована необходимость поиска принципиально новых научных подходов, способных обеспечить адекватность экономико-математических моделей при исследовании современных социально-экономических систем. Раскрыта перспективность синергетического направления как базового направления в современном экономико-математическом моделировании; показано, что его дальнейшее развитие требует применения нового математического инструментария. Приведена классификация математических методов и моделей, которые необходимо включить в синергетическое направление экономических исследований.

Ключевые слова: математические методы и модели, математическая формализация, экономические исследования, экономическая парадигма, синергетическое направление исследований.

Л. Д. ФІЛАТОВА

кандидатка фізико-математичних наук, доцентка, доцентка кафедри інформаційних технологій та математичного моделювання Харківського національного університету імені В. Н. Каразіна, Україна, м. Харків

О. В. КОРЖОВА

старша викладачка кафедри інформаційних технологій та математичного моделювання Харківського національного університету імені В. Н. Каразіна, Україна, м. Харків

МАТЕМАТИЧНИЙ ІНСТРУМЕНТАРІЙ В ЕКОНОМІЧНИХ ДОСЛІДЖЕННЯХ: КОНЦЕПТУАЛЬНІ ТРАНСФОРМАЦІЇ В ПЕРІОД ПАРАДИГМАЛЬНИХ ЗМІН

Постановка проблеми. На початку нового тисячоліття в умовах нестабільності соціально-економічного розвитку формується принципово нова економічна система, прискорюються парадигмальні зрушення в економічній науці. Своєю чергою, це вимагає нестандартних підходів до побудови економіко-математичних моделей та підтвердження їх адекватності, а перегляд теоретико-методологічних засад економіко-математичного моделювання в контексті нової економічної парадигми набуває особливої актуальності.

Аналіз останніх досліджень і публікацій. Тривалий час дослідження вітчизняних та зарубіжних науковців проводилися за двома напрямами: детальний аналіз проблеми застосування класичного математичного інструментарію в фундаментальних економічних дослідженнях; підтвердження адекватності такого інструментарію економічним реаліям сьогодення. Останнім часом усвідомлюється обмеженість таких досліджень і відбувається ілюстрація можливостей сучасного математичного апарату для вирішення економічних проблем. Одним із найбільш перспективних напрямів в економіко-математичному моделюванні, здатних урахувати нестабільність, нерівноважність та нелінійність сучасної економіки, є синергетичний напрям, але системного методологічного підґрунтя для його застосування на даний час не існує.

Формулювання цілей. Метою статті є виявлення суперечностей при застосуванні класичного математичного інструментарію в економічних дослідженнях та розкриття суті синергетичного підходу в економіко-математичному моделюванні як перспективного напряму дослідження сучасних економічних систем.

Виклад основного матеріалу. У сучасній економічній теорії домінує неокласична парадигма, основним інструментом аналізу якої є математичний. Однак математична формалізація економічної науки розвивається як суперечливий процес. Ця суперечливість проявляється передусім у неоднозначності впливу математичної формалізації на результати наукових досліджень.

До позитивних наслідків математизації економічної науки можемо віднести:

- універсальну мову спілкування, що забезпечує комунікацію поколінь наукової спільноти, а це, своєю чергою, полегшує накопичення і збільшення знань;

- можливість конструювання, оперування ідеалізованими моделями дійсності, що дозволяє виділити головне, більш точно описати закономірності, що існуюють, суворо визначити структуру тих чи інших явищ;

- можливість побачити спільні риси різнорідних явищ, тобто застосовувати одну й ту саму модель у різних задачах, змінюючи лише позначення символів.

Якщо систематизувати негативні сторони математичної формалізації в сучасній економічній науці, то вони зводяться до таких положень:

- комунікацію вчених за межами неокласичної парадигми утруднено;

- відбуваються відрив від реальних економічних проблем і втрата економічного змісту: основне завдання економічної науки - розуміння реальних економічних процесів і розроблення заходів економічної політики - залишається на задньому плані або зовсім відсутнє;

- з економічного аналізу виключаються явища, які не підлягають математичній формалізації;

- математичний формалізм не приносить нового знання, а лише дозволяє інтерпретувати найпростіші з наявних ідей, причому ця інтерпретація затребувана лише вузьким колом фахівців.

Такі суперечності сформувалися в результаті математичного підходу до економіки як рівноважної закритої системи, де всі її частини взаємопов'язано, причому рівновага ототожнюється з економічною статикою, динаміка ж розуміється як тимчасове порушення рівноваги. Але на зламі тисячоліть в економічній науці відбувається пошук нової парадигми - це поштовх для модифікації і вдосконалення її математичного інструментарію. Математична формалізація економічних задач в умовах парадигмальних зрушень повинна базуватися на таких постулатах: нестабільність, нерівноважність та нелінійність сучасної економіки.

Пошук новітнього математичного інструментарію, який враховував би такі особливості, відбувається за багатьма напрямами, але найперспективнішим, на нашу думку, є синергетичний, складниками якого є теорія катастроф, теорія хаосу, теорія клітинних автоматів, вейвлет-аналіз, фрактальна геометрія, нейронні мережі, нечітка логіка. Класифікація математичних методів синергетичного напряму виглядає так:

- принципи економіко-математичних методів: теорія економіко-математичного моделювання; теорія статистичного моделювання; теорія оптимізації економічних процесів;

- математична статистика: вибірковий метод; кореляційний аналіз; регресійний аналіз; дисперсійний аналіз; багатовимірний статистичний аналіз; факторний аналіз; теорія індексів;

- економетрика: теорія економічного зростання; теорія виробничих функцій; міжгалузеві баланси; національні рахунки; аналіз попиту і споживання; регіональний і просторовий аналіз; глобальне моделювання;

- методи прийняття оптимальних рішень: математичне програмування; мережеві методи управління; теорія і методи управління запасами; теорія масового обслуговування; теорія ігор; теорія рішень; теорія розкладів;

- моделі конкурентної економіки: моделі вільної конкуренції; моделі циклу обороту капіталу; моделі монополії, олігополії; моделі індикативного планування; моделі міжнародних відносин; моделі теорії фірм;

- економічна кібернетика: системний аналіз економіки; теорія економічної інформації; теорія керуючих систем; теорія інформаційних економічних систем; інформаційні технології в управлінні економікою; теорія імітаційного моделювання економіки; ділові ігри; експертні системи.

Висновки. Синергетичний напрям математичного дослідження - це потужний інструментарій, здатний урахувати основні особливості розвитку сучасної економіки. Характерною рисою такого інструментарію є те, що він займається аналізом переходу кількісного в якісне: будує й досліджує моделі, у яких описуються процеси переходу повільних, поступових, кількісних змін у докорінні, якісні. Це говорить про те, що сучасні математичні методи стають здатними досліджувати і якісні явища, завдяки чому межі їхнього застосування розширюються. Тому найсуттєвіша претензія до математичних методів - що вони не здатні охопити й описати якісні процеси в економіці, втрачає свою актуальність.

Але, незважаючи на те, що математика - точна наука, не варто чекати від її методів точних економічних прогнозів і рецептів ефективного управління економічними системами. Математичні методи можуть лише дати рекомендації для управління поведінкою системи і вказати на причини тих чи інших процесів і явищ.

У статті розглянуто процес еволюції та перспективи математичних методів і моделей в економічних дослідженнях. Проаналізовано переваги та виявлено недоліки математичної формалізації в умовах нової економічної парадигми. Обґрунтовано необхідність пошуку принципово нових наукових підходів, здатних забезпечити адекватність економіко-математичних моделей при дослідженні сучасних соціально-економічних систем. Розкрито перспективність синергетичного напряму як базового напряму в сучасному економіко-математичному моделюванні, показано, що його подальший розвиток вимагає застосування новітнього математичного інструментарію. Наведено класифікацію математичних методів та моделей, які необхідно включити до синергетичного напряму економічних досліджень.

Ключові слова: математичні методи та моделі, математична формалізація, економічні дослідження, економічна парадигма, синергетичний напрям досліджень.

Introduction

Problem setting. At the beginning of the new millennium, the world economics faced a number of systemic challenges. In the conditions of instability of social and economic development a fundamentally new economic system is formed, there is a rapid deepening of globalization processes. Against this background, «economic science is experiencing the longest and deepest crisis, which actualizes the formation of a new scientific paradigm...» (Bronytska, 2019, p. 21). Also paradigmatic shifts in economics itself lead to a rethinking of the possibilities and prospects for the use of mathematical tools in economic research.

Modern economics is a set of super-complex multicomponent systems exposed to constant structural and institutional transformation. The development of such systems can be both controlled and spontaneous, and the processes occurring in them are, for the most part, nonlinear. It requires non-standard approaches to the construction of economic and mathematical models and confirmation of their adequacy, and the revision of the theoretical and methodological principles of economic and mathematical modeling in the context of the new economic paradigm becomes especially relevant.

Analysis of recent research and publications. Economic science, since the middle of the XVIII century, has never existed without a mathematical basis (Kantorovich, 2010). The result of the rapid mathematization of economic research in the 60s of last century was the question of creating a special field of knowledge called «economic and mathematical methods» (Nemchinov, 1965, pp. 11-12). For a long time, research by domestic and foreign scientists on this issue was conducted in two areas: a detailed analysis of the problem of using classical mathematical tools in basic economic research (Cherniak & Zakharchenko, 2018; Berezhnaja & Berezhnoj, 2007; Shyhun, 2007; Vlasiuk, 2011); confirmation of the adequacy of such tools to the economic realities of today (Samoilenko, 2015; Borovska, 2018; Taleb, 2007; Tarasevych, 2018; Umantsiv, 2019; Bazylevych & Yl'yn, 2010; Peters, 2004). But recently, the limitations of such research are realized and there is an illustration of the possibilities of the modern mathematical apparatus for solving economic problems (Vertelieva, 2019; Soloviov, 2015; Piddubna, 2017; Fadieieva, 2014; Vitlinskyi & Skitsko (2013); Polovtsev, 2010). One of the most promising areas in economic and mathematical modeling, able to take into account the instability, imbalance and nonlinearity of the modern economy, is the synergetic direction. Some developments on this issue are presented in a number of studies (Kniazevych, Kraichuk & Strilchuk, 2019; Derbentsev, 2010; Haken, 2005; Kapitsa, Kurdyumov & Malinetskiy, 2003), but a systematic methodological basis for the synergetic direction in economic and mathematical modeling currently does not exist.

An objective of the paper is to identify contradictions in the use of classical mathematical tools in economic research; revealing the essence of the synergetic approach in economic and mathematical modeling as a promising area of research of modern economic systems.

Presentation of basic material

The formation, development and current state of economic theory is a process and result of the struggle of multidirectional scientific schools ideas, where the object of research is the economics as the material basis of society.

Initially, economic theory, like physics, revealing the laws of nature, claimed the role of fundamental, universal knowledge, revealing the general laws of economic growth and development, and in crisis and transitional states of society sought to reach a new level of knowledge of the economic system, to find new methodological approaches to solving current and promising, theoretical and practical problems. But the object of study of physics and many other sciences is quite static in comparison with the economics, which is characterized by a high degree of variability, uncertainty, and complexity. Therefore, nowadays economic theory is a variety of directions and schools, each of which gives its own vision of reality and its own recipes for solving problems. Today's fragmented economic science has largely lost both interest and ability to analyze key economic problems and development trends of modern societies. It means that economic theory is in crisis (Hrytsenko, 2018).

The crisis of economic theory means that it cannot fully perform its functions or solve the tasks assigned to it using existing methods, that is, the crisis of a theory is a crisis of its method. In such a period, the problem of modification or complete change of the dominant paradigm is maturing in any science (Mazaraki, 2019; Hrytsenko, 2019). Modern economic theory is dominated by the neoclassical paradigm, the main tool for the analysis of which is the mathematical one. Speaking about the crisis of economic theory, as a rule, they assume the crisis of neoclassicism, which arose as a result of a one-sided application of this method (to the detriment of the substantive side of the method).

In this regard, critics believe that formalism has led to the emasculation of the essence of economic theory as a social science, replacing economic thought and the art of economic policy with a dead, purely academic form of social mathematics. Indeed, if we consider the real state of the economics from the point of view of the theory fulfilling its practical and predictive functions, which are carried out not without the help of the mathematical method, then the claims to this method are not groundless. Economics has not yet developed methods that would lead to sustainable crisis-free management and long-term forecasting of the economic system. The development of more sophisticated mathematical methods capable of adequately describing the ongoing processes is, in our opinion, only at the initial stage. And it is connected, first of all, with the formation of a new direction in economic science which is synergetics.

In response to criticism about the dominance of formalism in economic theory, Weintraub offers a better understanding of the historical evolution of mathematics and its application in economics (Weintraub, 2002; Rosser, 2003). Leont'ev (1990) wrote that among other social sciences, economics has rightly come to be regarded as primarily a quantitative science In this case, the methods of quantitative analysis are not just a methodological technique used by the researcher: they themselves are the subject of study.

In general, while agreeing with this approach, let us begin by considering the evolution of mathematical methods in the history of economic thought. We find the first ever application of the mathematical method in economic research in William Petty. In the 70s of the XVII century he wrote the book «Political Arithmetic», which can be considered the ancestor of statistics and econometrics. Petty (1940) explained his approach as follows: «... instead of using only words in the comparative and superlative degree and speculative arguments, I embarked on the path of expressing my opinions in the language of numbers, weights and measures . using only arguments coming from sensory experience, and considering only the reasons that have visible grounds in nature». He demanded precise observation and calculation of economic phenomena. In the «Treatise on Taxes and Duties» there is a characteristic phrase that can be considered the motto of Petty's entire theory: «The first thing to do is to count ...». He introduced the method of abstraction into political economics, when, abstracting from the external side of economic phenomena, from their empirical description, thought deepens into their internal causal relationships, that is, it learns the economic laws ruling the production of wealth. mathematical economic synergetic

Franзois Quesnay, a French economist and statistician of the 18th century, for the first time in his «Economic Table» attempted to calculate the «annual income and advances» of a country, similar to the calculations of GNP and NNP in modern macroeconomic analysis. The use of mathematics in the works of these economists can be attributed to the first stage of the mathematization of economics.

The second stage is associated with the emergence and development of the marginalist mathematical school. The predecessor of marginalism, the French economist Antoine Augustin Cournot, in his study of the mathematical principles of the theory of wealth, first began to use mathematical techniques to derive economic laws, and in particular, he formulated the law of aggregate demand, the theory of monopoly pricing, the competitive mechanism and costs in strict mathematical language. Cournot was the first to introduce the concept of «economic equilibrium» into scientific terminology. Outstanding representatives of the marginalist school are L. Walras, K. Wicksell, W. Jevons, V Pareto, F. Edjourt. They believed that only with the help of mathematics it is possible to explain economic phenomena, and economic science should take its rightful place along with the natural sciences.

As a result of the «marginalist revolution», an approach to the economics as an equilibrium closed system was formed, where all its parts are interconnected, there are no causes and effects, there is no primary and secondary (Lo, 2018). Moreover, equilibrium is identified with economic statics, while dynamics is understood as a temporary imbalance, during which the main postulates of the marginalist theory do not work. The concepts of indifference curves and the core of the economic system by F. Edjourt, the concept of multipurpose optimum by V Pareto, and the model of general economic equilibrium by L. Walras have entered into modern economic science and are widely used in studies. The main achievements of this school are: Slutsky's consumer budget stability equation, the mathematical calculus of useful labor costs taking into account Dmitriev's intersectoral relations and his mathematical analysis of the concept of marginal usefulness, the mathematical models of Stolyarov and Tugan-Baranovsky, synthesizing the marginalist and labor theories of value.

The third stage of mathematization began in the second half of the 20th century. If before the main mathematical apparatus were derivatives and equations, then at this stage set theory, vector algebra, and operations research began to dominate. The works of E. Domar, V. Leontiev, M. Morishma, H. Nikaido, J. von Neumann, P. Samuelson, R. Solow, J. Tinbergen, R. Frisch, R. Harrod, J. Hicks, K. Arrow and other foreign scientists gained fame and recognition.

This is the time for the recognition and wide productive application of the mathematical method in economic theory. It was during this period that various economic and mathematical models of game theory, econometric analysis, general market equilibrium, economic growth, input-output balance, linear and dynamic programming, optimal control, etc. were developed.

However, the mathematical formalization of economics is developing as a contradictory process. This inconsistency is manifested primarily in the ambiguity of the impact of formalization on the results of scientific research. The positive consequences of the mathematization of economic science can be attributed to:

- precise and clear language of scientific communication, ensuring communication between generations of the scientific community, which facilitates the accumulation and augmentation of knowledge;

- the possibility of constructing, operating with idealized models of reality, which allows us to highlight the main thing, more accurately describe the existing patterns, strictly determine the structure of certain phenomena;

- the ability to see the features of commonality in dissimilar phenomena, that is, to apply the same model to analyze various phenomena, changing only the notation of symbols.

If we systematize the negative aspects of mathematical formalization in economic science, then they can be described boil in the following provisions:

- communication of scientists outside the neoclassical paradigm is difficult;

- there is an exaggeration of the meaning of the form to the detriment of the content of economic arguments;

- there is a separation from reality, real economic problems and the loss of economic content; the main task of economic science (which is understanding real economic processes and developing economic policy measures) remains in the background or is completely absent;

- the observed phenomena that are not subject to formalization are excluded from the economic analysis;

- formalism does not bring new knowledge, but only allows one to interpret the simplest of existing ideas, and this interpretation is in demand only by a narrow circle of specialists.

These problems, especially communication between economists, generated by mathematization can be solved provided a common view of the nature of complex economic processes and phenomena is developed. And the study of such phenomena needs the skills of dealing with modern mathematical methods, or at least the knowledge of a concept about them. It is important for not only the researcher benefits from knowledge and the ability to apply mathematical methods, but for the entire economic theory as a whole. The contradictory results of formalism in the performance by the economic theory of its explanatory, practical and predictive functions are caused by the fact that economic processes and phenomena, as a rule, are modeled and investigated using linear mathematical models.

They are a special case of non-linear models and consider and explain the economic process only for a short period of time. In reality, however, most economic phenomena, processes and systems are complex, and their study using nonlinear mathematical theory is more adequate. Building and researching nonlinear models that fit the real economics is a very laborious process. In it, the analysis of the simplest linear models that do not have the ability to deeply investigate complex phenomena can help to determine the direction of the search.

Today's neoclassical economics is a world of models. The American economist Krugman (2015), the Nobel Prize laureate in economics 2008, considers the future economic science as a set of formal models that make it possible to reproduce research results in a transparent form. As a rule, the verbal analysis is considered an alternative to modeling as a research method. Krugman admits the verbal expression of economic ideas, but only for the transmission of scientific results to the public, and within science he does not recognize any other way of expression than formal modeling. Apparently, the inconsistency of the process of mathematical formalization of economic science is a particular manifestation of an objective fundamental contradiction between the method and the subject of research as a whole, which has a global character and is inherent in any science.

On the one hand, this is due to the fact that social systems, including economic ones, are complex developing systems in which the pace of change in modern conditions has significantly accelerated. Therefore, the transformations in the object of research, the complication of the processes occurring in it, require constant improvement of existing and the creation of new methods of cognition.

Otherwise, the adequacy of understanding and theoretical substantiation of reality may not be achieved. Also, cardinal qualitative changes in social systems suppose revolutionary changes in the methods of cognition.

On the other hand, cognition in the social sphere has a reflexive character, that is, the one who comprehends constantly influences the object of research, in accordance with his understanding and vision of the surrounding reality. A person is included in the system that he is studying; he actively interacts with other elements of this system. Therefore, the social system is strongly dependent on the activities of people, on their level of consciousness and thinking, cultural and ethical values, mentality. Reflexivity means that cognition supposes the existence of various visions of the social environment, ways of ordering it, developing, improving, etc. This is reflected in the pluralism of the worldview, scientific approaches, and real social systems. Therefore, the adequacy of methods of scientific knowledge of the social world is determined by that specific concrete reality that is being investigated. Methods that are effective in some cases may be completely inapplicable in others. This also applies to mathematical methods in economics.

Let's analyze the Prospects for the development of mathematical methods in economic science. The need of economic systems for long-term planning and strategic management forces scientists-economists to turn to interdisciplinary research carried out at the intersection of economics and other sciences. One of these areas, which involves further strengthening the integration of economic science and mathematics, is synergetic economics (Zang, 1999). As a special case of the general theory of synergetics by Haken, it studies evolution and changes in nonlinear unstable economic systems (Kolomiiets, 2020). A synergistic approach to the study of the functioning, development, management, forecasting of economic systems is based on the following provisions:

- the economics is a self-organizing system, a change in the state of which is caused not only by the influence of the external environment, but also by the interaction of the elements of the system with each other;

- the elements of the system are interdependent, therefore, management decisions made in some areas of the system affect decisions made in other areas;

- the system is in equilibrium only for a short period of time, in a nonequilibrium state it obeys laws of a nonlinear nature;

- in nonlinear systems, there is the so-called phenomenon of resonant excitation, which, even being weak, leads to a better result than a strong, but not consistent with the system impact;

- the task of state management of the economics in a situation of uncertainty is to try to maintain the stability of the system, while simultaneously searching for alternative ways to solve the problem.

In conditions of instability, the task of economic forecasting should take into account the following points:

- it makes no sense to demand an accurate forecast for a long period of time. We can only concider forecasts for a short period, about the presence or absence of stable states of the economic system. The forecasting strategy involves identifying the main parameters of the system and their combinations necessary to get to the desired limiting state;

- forecasting is not predicting future events, but identifying emerging problems and finding possible ways to solve them.

In fact, we are talking about the formation of the forecasting technology as an iterative process, at each step of which the prediction technique is adjusted. This technology supposes the availability of access to reliable information, the construction of adequate models, the training of specialists in modeling and forecasting (Kuznecov, 2020) .

The modified paradigm of postneoclassical economic theory should make its main object of study the economic system as a complex, open, nonlinear, selforganizing, nonequilibrium, unstable system (Yakimtsov, 2018). It is supposed to study such systems using the latest mathematical tools of economic synergetics, developed in such areas as catastrophe theory, chaos theory, cellular automata theory, wavelet analysis, fractal geometry, neural networks, fuzzy logic. The existing classification of the methods looks like this:

- principles of economic and mathematical methods: theory of economic and mathematical modeling; theory of statistical modeling; theory of optimization of economic processes;

- mathematical statistics: sampling method; correlation analysis; regression analysis; analysis of variance; multivariate statistical analysis; factor analysis; index theory;

- econometrics (mathematical economics): the theory of economic growth; theory of production functions; intersectoral balances; national accounts; analysis of demand and consumption; regional and spatial analysis; global modeling;

- methods of making optimal decisions: mathematical programming; branch and bound methods; network management methods; theory and methods of inventory management; queuing theory; game theory; decision theory; scheduling theory;

- models of a competitive economy: models of free competition; capital cycle models; models of monopoly, duopoly, oligopoly; indicative planning models; models of international relations; models of the theory of firms;

- economic cybernetics: systems analysis of the economics; theory of economic information; theory of control systems; theory of information economic systems; information technology in economic management; theory of simulation modeling of the economics; business games; expert systems.

Conclusions

The modern mathematical theory of economic synergetics deals with the transition from quantitative to qualitative. It builds and explores models that describe the processes of transition of slow, gradual, quantitative changes into fundamental, qualitative ones. This suggests that quantitative and qualitative methods act in dialectical interconnection and interaction. Therefore, claims to mathematical methods that they are not able to cover and describe qualitative processes in the economy are losing their relevance. Mathematical methods become capable of investigating qualitative phenomena, due to which the boundaries of their application expand.

But despite the fact that mathematics is an exact science, we should not expect accurate economic forecasts and recipes for effective management of economic systems from its methods. Mathematical methods can only give recommendations for controlling the behavior of the system and indicate the causes of certain processes and phenomena.

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