Economics and Mathematical Methods

Marx proposed a bold conjecture in volume III of Das Kapital: in the process of total social production, the total value of all commodities equals the total price, and the average rate of profit of all capital equals the average rate of surplus value. For over a century, the academic community was debating the mutual compatibility of these "two equalities", and a rigorous mathematical proof remained elusive. By analyzing the inherent causes underlying the formation of constant and variable capital, distinguishing the dual attributes of capital in the forms of mechanical force and living labor, introducing the interest factor and labor productivity factor, and incorporating the dual relationship between productive capital and monetary capital, this paper constructs a complete mathematical framework for Marx's theory of production prices. Under this framework, the paper rigorously formulates and proves Marx's conjecture: equations in homogeneous sectors admit analytical solutions; equations in heterogeneous sectors can be solved via the "structural elimination-numerical approximation" method; and equations in different static and dynamic scenarios can be addressed using the corresponding fixed-point theorems. This lays a mathematical foundation for the theory of value transformation that integrates theoretical rigor with computational feasibility.

This article is a response by the authors of the shifting mode of reproduction (SMR) theory to recent publications by E.L. Goryunov (2023 and 2025), which attempts to use this theory to construct mathematical models describing the process of economic growth, taking into account the peculiarities of fixed capital renewal. Unfortunately, as the analysis showed, these attempts were unsuccessful, both due to an insufficient understanding of the key provisions of the theory and due to the inaccurate use of mathematical tools. As a result, the conclusions obtained differ radically from those in our publications and are incorrect. This situation necessitates, first, a clear definition of the key points of the SMR theory and the features of mathematical models that must be observed in order to comply with the provisions of the theory. Second, - to show, using the example of E.L. Goryunov's articles, that failure to comply with the principles of constructing SMR models leads to distorted conclusions and incorrect conclusions. Third, - to discuss the possibilities offered by SMR theory when used correctly and to provide a brief overview of the results obtained on its basis. An important result of SMR theory and models is the justification that, under certain conditions, money is not neutral in the long term, and therefore monetary stimulus policy in these conditions can stimulate economic growth in the long term. Economic history shows that monetary stimulus for growth is inherent in the real economy, and SMR models are capable of simulating it.

This paper examines the long-term effects of trade through global value chains (GVCs) on economic growth. The analysis covers 25 countries over 35-year period (1965–2000), including both advanced and developing economies with different trajectories of GVC integration. The period spans phases of trade stagnation and rapid globalization. The study combines theoretical and empirical perspectives. The literature review highlights two contrasting channels: productivity gains and technological upgrading versus dependence on foreign technologies and loss of domestic capabilities. The empirical strategy is based on recent approaches to international production networks, focusing on backward participation. Using input-output tables, a dynamic panel model with the System GMM is estimated. This approach addresses endogeneity concerns and yields consistent estimates. The results show a statistically significant positive effect of GVC participation on GDP per capita growth. Evidence of nonlinear effects is also found: stronger integration into production chains magnifies growth gains, consistent with economies of scale and knowledge accumulation. The findings indicate that GVC participation is an important source of long-run growth. At the same time, they suggest the need for policies that mitigate potential risks associated with technological dependence and external vulnerability.

The article proposes an approach to assessing the results of the economic comparison between Russia and Portugal and making forecast calculations until 2035. The peculiarity of the question is that we study mainly the comparative values of per capita GDP of Russia and Portugal in connection with the corresponding values of investment parameters - specific investments, incremental capital intensity. It was revealed that the noticeable lag in Russia's per capita GDP behind Portugal's that developed by the end of the 1990s was quickly overcome, but already in 2014 the reserves for preferential growth of the Russian indicator in question were exhausted. The investment impulse that generated a favorable trend for the Russian economy dried up: the ratio (RF/Portugal) of specific investment costs (per employed) consistently decreased from 1.04 in 2013 to 0.56 in 2023, and the proportions concerning incremental capital intensity also noticeably worsened. It is shown that the condition for achieving any significant and sustainable excess of Russian per capita GDP relative to the same indicator in Portugal by 2035 is the value of the accumulation rate in the Russian Federation of at least 25% by the end of the period, as well as the value of incremental capital intensity in 2024-2035 of no more than 10 USD/USD.

The article examines the role of regional and municipal bonds in the system of debt financing of the subjects of the Russian Federation, as well as their investment attractiveness for private investors. Special attention is paid to the analysis of key factors affecting the prospects for issuing and investing in these instruments, including macroeconomic stability, regional budget discipline and financial market conditions. The purpose of the study is to develop two econometric models: a model for generating regional budget revenues and a model for determining bond yield spreads. These models allow to assess the potential of debt instruments both for issuers seeking sustainable budget planning and for investors who consider the risk-return ratio. The results of the study demonstrate that regional and municipal bonds are an effective mechanism for attracting additional resources for the implementation of infrastructure and social projects, but their issuance is fraught with risks of increasing debt burden. For investors, the key factors in determining the risk premium are the credit quality of the issuer, the liquidity of securities and their sensitivity to changes in market conditions over time. The practical significance of the work lies in the development of recommendations for regional authorities on debt policy management, as well as for private investors on the formation of portfolio strategies. The study fills a gap in the Russian literature on the specifics of sub-federal bond loans.

The aim of the study is to analyze the level of predictive power of network indicators for the oil market. The article analyzes trade links in the international oil market for the period 2000-2020 using network analysis tools and considers the possibility of using similarity measures as one of the factors in forecasting trade links. The following indicators of changes in links in the graph are analyzed: link density, centrality, transitivity, and reciprocity. Key changes in the structure of links in the considered period are identified. Disadvantages of similarity indicators and the possibility of their interpretation for market analysis are discussed. An algorithm for selecting a similarity measure with the greatest predictive power, taking into account a given network topology, is presented, based on the AUC (Area under the ROC curve) indicator. It is shown that the "average commute time" and "common neighbors" measures have the greatest predictive power with AUC values from 0.55 and to 0.66. Cosine similarity and Jaccard indices, based on an assessment of the similarity of the circle of partners, on the contrary, demonstrate low AUC values and are not suitable for analysis. The applied method has significant limitations for forecasting because of low AUC values and needs to be improved. However, it is shown that the presented method allows taking into account additional important factors when assessing trade relations. The paper discusses potential ways to increase the predictive power through additional modification of similarity indicators, complication of the graph structure, and the possibility of using similarity measures in econometric analysis.

The problems of developing a system of insurance against environmental pollution risks are considered. The dependences of the tariff rates on the form of organization of environmental insurance (voluntary, compulsory and imputed) are shown, as well as the following parameters: the probability of occurrence of an insured event, the load in the gross rate, the number of insurance contracts, and the safety factor. The incentive function of environmental insurance is revealed using calculation examples, and the advantages of mandatory and imputed forms of organization compared to voluntary ones are shown. The aim of the study is to reveal, on the basis of comparative actuarial calculations, the possibilities of insuring the risk of environmental pollution for the effective application and development of this economic regulator. The main scientific tools of the work were economic and ecological-economic types of analysis, including content analysis, matrix-morphological, actuarial and information modeling. It is shown that a competently organized system of environmental risk insurance with a scientifically substantiated, widely differentiated tariff structure allows performing not only compensatory, but also stimulating and preventive. This will give impetus to the development of the economic regulator under consideration, and regardless of the form of its organization, will attract interested parties: economic entities — sources of increased danger and insurance companies. Wide application of environmental insurance is beneficial to the population — recipients experiencing negative consequences of increased anthropogenic load of economic activity. As well as to budgets of all levels by attracting private funds to compensation of environmental damage and protection of nature and saving budget resources, which are in short supply, especially in the present period.

Accounting and planning of labor migration of the population between regions is an essential factor in territorial development. A model of interregional (internal) labor migration is presented depending on the structure of job distribution and the qualifications of workers. The model allows us to identify the volume of migration between regions, monitor population migration on a GIS map, to look at the dynamics of changes in the distribution of vacant jobs, and observe changes in the age and sex diagram of the population of the regions. For each region, the actual number of residents is set according to Rosstat data. Each resident is modeled separately. The initial qualification of residents is distributed according to the binomial distribution law by talent and age. Qualification increases over time. Each region has its own distribution of jobs by qualification level. Residents try to take jobs according to their qualification, but the corresponding jobs may be occupied. The discrepancy between the job and the qualification level leads to a gradual increase in the resident's impatience. Exceeding the individual threshold of patience forces the resident to look for a new job. The salary supplement in the region increases for unoccupied jobs. The choice of region for migration depends on the availability of a vacant job corresponding to the level of qualification, the distance to the new place of work, the level of the salary, and the individual attachment to the resident's home. External migration (from other countries) is not taken into account, since it is displayed as a separate column in the Rosstat data. The model was developed in Anylogic, the code is on GitHub.