Since the basis of the state structure of a country is its national economy, then economic threats to the state structure are formed, when the national economy of a country begins to function in a «circle of poverty». To identify economic threats, which may arise to the state structure and developing preventative measures, first of all, it is necessary to determine the emergence of the «circle of poverty» in the activities of the country’s national economy. This work presents the methodology of the study of the emergence of the «circle of poverty» in the activity of the national economy of the country at the stage of declining development rates and stagnation. The methodology (algorithm) of assessing the criteria for the functioning of the national economy of the country in the «circle of poverty» is proposed. A research methodology and evaluation methods were developed using a systematic approach, comparative analysis and methods of economic and mathematical analysis.