Economics and Mathematical Methods

The search for a standard measure of value remains a significant theoretical challenge. Classical economists such as Smith, Ricardo, and Marx believed in a uniform measure of value. However, neoclassical economists argued that the social value of wealth cannot be measured in an objective and uniform way. In recent years, a new theory of value emerged. This theory, building on the foundations of classical economics and drawing inspiration from Newtonian mechanics, constructed an axiomatic system of value mechanics. It also deduced a system of value dimensions that can be applied in the field of economics. This new approach preliminarily solved the problem of calculating value using a uniform measure. Accordingly, we can now use the Euler–Lagrange equation to explore the process of optimal capital motion. This equation allows us to analyze theoretical problems of capital, including the value calculation of absolute surplus value, relative surplus value, and irrational economic behavior. This paper aims to establish a system for the measure of value, with two-fold character — natural and social, similar to the measurement system in natural sciences, offering standardized material properties for measurement units corresponding to value dimensions.

One of the main factors affecting the export of pharmaceutical products is the exchange rate and its changes. Most studies confirm that there is a significant correlation between export volume and exchange rate. Changes in the exchange rate have a direct and indirect impact on country’s exports. This effect is manifested through the state of the balance of payments, the formation of supply and demand in the money market at the macro level, inflation, the state budget and other similar macroeconomic processes and indicators. Therefore, this article uses the case of Uzbekistan to carry econometric analysis of the relationship between the export of pharmaceutical products and exchange rate dynamics. First, the dynamics of the two variables were tested for nonstationarity using the Augmented Dickey–Fuller test, or ADF-test. Based on the results obtained, the hypothesis about the existence of cointegration in time series was tested using the Engel–Granger method. In the study, the direction of correlation was determined using the Granger causality test. All analyzes were performed using EViews software. The results confirmed the presence of a positive relationship between the export of pharmaceutical products and the dynamics of the exchange rate.

Coherent risk measures are widely used in practice to calculate risk. Multidimensional coherent risk measures are important to use for companies and banks operating in different international markets. Here we introduce an example which clearly shows that multidimensional coherent risk measures reduce capital requirements on multi-asset or multi-currency markets. In this paper we consider two different approaches to define multidimensional coherent risk measures. The first approach is based on a cone-based exchange rate set, whereas the second employs random sets to account for liquidity constraints and other market frictions. Also constructive approach for multidimensional risk measures using one-dimensional law invariant ones is considered. We introduce the example of coherent risk measure which cannot be represented by the constructive approach. Two important properties of multidimensional risk measures such as law invariance and space consistency are investigated and their equivalent properties in terms of determining set and acceptance set are given. Then we consider a multidimensional generalization of Tail VaR and show that it satisfies space consistency and law invariance properties and provide an example which shows that multidimensional portfolio risk estimation using this risk measure gives an adequate result. JEL Classification: C39, D81. UDC: 519.25. For reference: Kulikov A. V., Volkov N. V. (2025). Multidimensional coherent risk measures and their properties. Economics and Mathematical Methods, 61, 3, 116–125.