1. INTRODUCTION
In the broad sense of the word, sustainable development is development based on the maintenance of a nondecreasing amount of fixed capital and ensuring nondecreasing utility. The end result of sustainable economic development is an increase in the quality and duration of life. The main indicator of sustainability, as recommended by the World Bank, is the indicator of the savings rate. Changes in the savings rate reflect the population's preferences between immediate and future consumption. The economic component of the sustainable development concept implies ensuring the wellbeing and resources of future generations while meeting the needs of current generations (Brundtland, 1987). Notice, that the economy with excessive savings it is dynamically inefficient, since the trajectory of per capita consumption lies below the possible alternative trajectories with higher consumption. The “golden rule of capital accumulation” in economic theory corresponds to the economic component of the sustainable development concept: the maximum “golden” volume of per capita consumption occurs when the members of the current and future generations are provided with the same consumption volume (Chatterjee et al., 2017;Dai, 2014;Villanueva, 2021). Please note that according to the transversality condition, an infinitely long accumulation of assets does not lead to their growth, since the total volume of assets has a different return rate, and the utility of assets will increase as a result of their consumption.
According to Pezzey (1989), various sustainability criteria can be derived from combining such general concepts as “maintaining fixed capital” and “ensuring nondecreasing utility”. Therefore, to characterize sustainable development, we focus on indicators of per capita consumption and savings rates in the region's economy. The authors of modern scientific publications explore numerous problems related to the savings rate in the economy: inflation (Jaya et al., 2021), population aging (Fukuda and Okumura, 2021), migration (Akay et al., 2021;Tan et al., 2021), land expropriation (Zhao and Hu, 2021). However, the available publications have shown a lack of research on the economic metrics of sustainable development. The study (Zhang et al., 2021) shows that low savings rates cause economic instability in developed countries, cause interest rate fluctuations in the global capital market, disrupt investment stability and increase economic instability in other countries, and inadequate financial development is the cause of economic instability in developing countries. The work (Ribaj and Mexhuani, 2021) shows a positive correlation between the savings rate and economic growth.
According to Barro and SalaiMartin (2004), increase in the savings rate, the property of economic growth convergence to a stationary state is preserved under general conditions of the Ramsey model. The study assumes that the preservation of the economic growth convergence property (in particular, consumption) will empirically confirm sustainable development: poor regions grow faster than rich ones, because they have cheaper resources and can copy the production methods, technologies and institutions of strong regions. In addition, resourcedependent economies are potentially stable if the rent for resources is invested in other productive assets (Hamilton, 2000).
The savings rate is physically limited to one. Therefore, for the longterm growth of per capita income and consumption within the framework of the neoclassical structure, it is advisable to use an unlimited resource  knowledge and technology. This means that growth and development in the long run is determined by technological progress, and not by the accumulation of physical capital. In this sense, digital technologies have valuable opportunities (Goldfarb and Tucker, 2019). The assessment and analysis of the digital technologies benefits for the global economy is tracked in the «SMARTer2030» report of the Strategic Partnership GeSi. For the consumer sector, digital technologies reduce a number of economic costs, expand the possibilities of individual consumer choice, accounting and analysis of consumer preferences (Schwab, 2017). Two or more manufacturers can use the same digital technology at the same time. Therefore, it can be theoretically assumed that the noncompetitiveness and nonexclusivity of digital technologies lead to the equalization of prices for factors of production: low wages and high rates of technological growth in developing economies with a negative impact on economies with high wages. An important consequence of this relationship is convergence in economic development and prosperity. Scientific reviews demonstrate that accessible digital technologies used in products, assets, infrastructure contribute to the development of sustainable development economic aspects (Ching et al., 2022;Kamble et al., 2022;Tsindeliani et al., 2022).
In addition, the sustainable development of the region is determined by the effective functioning of the basic industries, economic independence, the cost of technological innovation, and the income of the population. In a transitional economy, explicit processes of economic change bring more uptodate information about the laws of economic development. Therefore, we pay attention to the features in the development of Russian regions that are observed today.
So, the interest in predicting possible consumption inequality in regions under the assumption of a fair distribution of income determined the performance of the study in the context of βconvergence. We formulated the research questions: Is there a βconvergence of the average consumption growth rates in the long term, confirming sustainable development? Does the savings rate affect the convergence of the average consumption growth rates in the regions? The mutual influence of regions on each other can be a significant external determinant of consumption and accumulation of fixed capital, the life cycle of innovations, increasing the convergence of their growth rates, the mobility of factors and production results. This means that the inclusion of a spatial lag in econometric models will make it possible to obtain unbiased estimates of explanatory factors. The main purpose of the study is to measure the convergence of consumption growth rates in Russian regions on the basis of spatialeconometric models of conditional βconvergence (Barro and SalaiMartin, 1992;Rey and Montouri, 1999).
2. DATA AND METHODOLOGY
The data sample was obtained from the “Regions of Russia. Socioeconomic indicators” edition on the website of the Federal State Statistics Service across 81 regions (except for the Kaliningrad region, Nenets Autonomous Okrug, Crimea, Sevastopol) from 2014 to 2019. To estimate the models, we used the logarithm of consumer spending growth rate per capita to measure the conditional βconvergence of consumer spending from 2014 to 2018. The independent variables are: cash income per capita, rubles; share of households using the Internet, %; number of active users of broadband Internet connection per 100 of the population, people; ruble deposits of physical persons, million rubles; savings rate, % ((cash income per capita  consumer spending per capita) /cash income per capita).
To reflect the modern features of the Russian regions, the calculation of the variation coefficient of GRP per capita, average per capita income, savings rate, a graphical method was used, cartograms were built in the R software environment of expenses for technological innovations per capita of the employed population, of the share of Russian population using the Internet, of the consumer spending.
We used the boundary weight matrix and the Moran (Anselin, 1995) and Geary (LeSage and Pace, 2009) indices to account and detect spatial dependencies. We used the local Moran’s indices (LISA) to identify the spatial clustering of regions (Anselin, 1995):
If a given region differs significantly from its neighbors (outlier), then the negative value of the local Moran's index belongs to it. A positive correlation indicates that the region is similar to neighboring territories (cluster). The larger the LISA value in modulus, the stronger the similarity / difference between the region and its neighbors.
To identify spatial dependence, global Moran's indices are defined (Anselin, 1995):
where N – the number of regions, X  average value of an indicator X, w_{ij}  elements of the boundary matrix of weights, W indicates the amount for all w_{ij}.
The Giri spatial correlation indices are also calculated (LeSage and Pace, 2009):
where W denotes the sum over all w_{ij}, the other notation corresponds to that of the Moran’s index.
Moran’s index takes values in the interval [−1; 1]. The Giri index takes values in the range [0; 2], where values from 0 to 1 indicate positive spatial correlation, and values from 1 to 2 indicate negative spatial correlation. A positive spatial correlation coefficient means that a growing region contributes to the growth of its neighbors; a negative value means that a growing region “takes” the resources of its neighbors. The insignificance of the coefficient indicates the lack of interrelation of processes in different regions.
In the R software, models of conditional β  convergence (package splm) are constructed on panel data:

 SARmodel with spatial autoregressive lag (Anselin, 1988):
where i=1,…81 – region number, [t_{0} + T] – year number for convergence period from year 2014 to 2019, y_{i,t0} – consumer spending per capita in region i at the initial moment of time (2014), k – explanatory variable number, K – number of explanatory variables, α_{i} – vector of regional fixed effects, which allow to control for unobserved spatial heterogeneity, ${\delta}_{\text{t}0+\text{T}}$ – time fixed effects, set by a number of dummy variables for years, is a time effect in order to control for common country factors affecting dynamics of considering factors, β – parameter to be estimated for the consumer spending per capita at the initial moment of time (2009), γ_{k}  parameters to be estimated for explanatory variables; W_{ij}  boundary weighting matrix for i,j = 1, ..., 81, ρ – spatial autoregressive coefficient, ${\epsilon}_{\text{i},\text{t}0+\text{T}}$ – random error, which are normally distributed. β represents the convergence. If β <0 , then there is conditional beta convergence. This means that poorer regions have higher growth rates than richer regions — which is why they are able to ‘catch up’.

 SEMmodel with spatial interaction in errors with fixed effects (Elhorst, 2014):

 SEMmodel with spatial interaction in errors with random effects (Elhorst, 2014):
where λ – spatial autocorrelation coefficient for shock, ${v}_{i,{t}_{0}+T}=\mu +{u}_{i,{t}_{0}+T}$ (Kapoor et al., 2007).

 SACmodel with spatial autoregressive lag and spatial interaction in errors with fixed effects (Elhorst, 2014):
$$\begin{array}{l}\frac{1}{T}\mathrm{ln}\frac{{y}_{i,t+T}}{{y}_{i,{t}_{0}}}={\alpha}_{i}^{}+{\delta}_{{t}_{0+T}}+\beta \mathrm{ln}({y}_{i{t}_{0}})\\ \text{}+\text{}{\displaystyle \sum _{k=1}^{K}{\gamma}_{k}{X}_{kit}}+\rho {W}_{ij}\mathrm{ln}\frac{{y}_{i{t}_{0}+T}}{{y}_{i{t}_{0}}}\\ \text{}+\lambda W{\epsilon}_{i,{t}_{0}+T}+{u}_{i,{t}_{0}+T}\end{array}$$(4)
The SACmodel allows identifying the impact of both average consumption growth rates per capita and shocks from other regions on the studied indicator of this region.
The dependent variable autoregression coefficient ρ for the spatial lag allows one to identify the influence of the consumer spending per capita in other regions on the studied region. The statistical insignificance of the spatial autoregression coefficient means that the processes of increasing consumer spending per capita in different regions are not related to each other, a positive value indicates regional cooperation, and a negative value indicates regional competition. The spatial autocorrelation coefficient for shock λ reveals the influence of the spatial structure of errors. The statistical insignificance of λ means that the shocks of neighboring regions that affect the consumer spending growth rates in a given region are not related to each other.
3. RESULTS AND DISCUSSION
Firstly, Figure 1 shows the increase in variation in the level of per capita income (Figure 1). Growing income inequality in the future may weaken the convergence of consumption in the regions (Wan, 2005). Uneven consumption can lead to a deterioration in wellbeing and social stability in the regions, to the need for targeted protection of the population by the authorities.
Secondly, Table 1 shows the increase in savings rate variation across Russian regions.
Third, Figure 2 demonstrates that an increase in the savings rate leads to a decrease in consumption.
Finally, Figure 3 demonstrates the technological inequality of the regions. We revealed the clusters of regions with a higher level of expenses for technological innovations in the raw materials sector, negative spatial correlation, a predominance of peripheral regions with low expenses for technological innovations (Bagautdinova and Kadochnikova, 2020).
In the consumer sector there is the insufficient use of digital technologies and digital inequality in the regions (Figure 4). In 2019, 52% of Internet users in Russia carried out online banking operations, 37%  searched for healthrelated information, 33 %  purchased goods and services online, 8%  searched for the job, 7%  used online goods and services, 5%  rent housing, 3%  were learning online (Digital economy indicators: 2020: Statistical collection, 2020).
Figure 5 demonstrates the strong differentiation of consumer spending per capita in the regions of Russia.
Local spatial clusters of regions similar to neighboring territories were found in Moscow and the Moscow Region, the Ural Federal District, the Siberian Federal Okrug and the North Caucasus Federal District in terms of consumer spending per capita. The global Moran and Geary indices indicated an increase in positive spatial autocorrelation, when strong regions contribute to the growth of consumption in their neighboring regions (Table 2).
However, the number of regions in the LL quadrant increases – regions with a low level of consumer spending in the environment of the same weak regions (Figure 6).
In Tables 3 and 4, all types of conditional β convergence models predict the convergence of consumption in the regions in the long term, which corresponds to the assumption of their sustainable development. This result is consistent with the works (Kopoteva, 2020;Michail, 2020;Ozturk et al., 2021). We measure the convergence of consumption as a whole, without singling out individual product categories. This approach does not make it possible to measure the equilibrium state in consumption by different groups of the population by income level, in different price segments of goods. Interest in such measurements is determined by works in which the authors use the budget distribution gap as a measure of consumption convergence (Ozturk et al., 2021), measure consumption convergence separately for consumer goods and more expensive goods (Wan, 2005), the consequences of food consumption convergence (Fukase and Martin, 2020).
We also found the statistically significant positive effect of average per capita monetary income and the share of the population using the Internet on the average growth rate of consumer spending. As expected, the negative impact of individual deposits and the savings rate was confirmed. We do not use consumer behavior variables in the study. This limits our study. The work (De Mooij, 2003) shows that in conditions of income convergence, consumer behavior is determined by the culture of consumption, the authors come to the same conclusion in the work (Aizenman and Brooks, 2008). They find that the integration of territories enhances the convergence of cultural values in consumption. Therefore, to account for cultural diversity in different regions, it is advisable to use consumption culture variables.
Positive significant spatial coefficients (p) and (λ) confirm the assumption of regional cooperation and the influence of shocks from neighboring regions on the growth rate of consumption in this region. What are the consequences of the convergence of consumption expected by the Russian regions? Ozturk et al. (2021) show the impact of technological advances and make an important conclusion that the higher the level of consumption convergence, the higher the market concentration is. This conclusion is important for policy makers, since increased market concentration can disrupt the balance and proportionality of regional development.
In Table 5, the direct effects (effects within the region) confirmed the positive impact of per capita monetary income and deposits on the growth rate of consumer spending in this region, with a savings rate negative impact. Indirect effects (the effects of the consumption determinants influence of among neighbors) show a negative impact of the average per capita monetary income and deposits of neighboring regions on the growth rate of consumer spending in this region, and the savings rate positive impact.
4. CONCLUSION
The conducted research allowed formulating the following conclusions and recommendations. All types of conditional βconvergence models predict the convergence of consumption in the regions in the long term. We found the statistically significant positive effect of average per capita monetary income and the share of the population using the Internet on the average growth rate of consumer spending. As expected, the negative impact of individual deposits and the savings rate was confirmed. We found the regional cooperation and the influence of shocks from neighboring regions on the growth rate of consumption in this region.
The peculiarities of regional development, when strong regions “tighten” the costs of technological innovations, can predict an increase in consumption variation and a transition to divergence. As expected, the assumption about the insufficient use of consumer sector digitalization to achieve sustainable development was confirmed. The negative impact of the size of deposits and the savings rate on the average growth rate of consumer spending per capita may indicate the dynamic inefficiency of the economy in the regions and inequality in the distribution of income. Increased market concentration, as a result of the equilibrium state of consumption, the involve ment of monetary liquidity in economic turnover, the use of the mechanism of regional cooperation, the development of online services in the systematic practices of households may require state policy measures to achieve sustainable development in the regions.
As the results of previous studies have shown, in future studies, the analysis of predictors of consumption convergence should be supplemented with variables of price segments of goods, population groups by income level, variables of the culture of consumer behavior.
Consumption and the savings rate, as characteristics of the sustainable development of regions, require control and regulation by the state. Uneven consumption increases economic instability, labor migration, a gap in the quality of life and requires state financial support for lagging regions. Therefore, the results obtained in the study should be used in the preparation of measures aimed at increasing consumption in the regions.