МОДЕЛЮВАННЯ ОПТИМАЛЬНИХ ЗВ’ЯЗКІВ МІЖ ТІНЬОВОЮ ТА ЛЕГАЛЬНОЮ ЕКОНОМІКАМИ

Abstract

This article is devoted to modelling the optimal links between the shadow and legal economies in order to understand their interaction and impact on the main patterns of changes in the savings of employees and business owners, and in the price of aggregate goods. The study focuses on finding optimal management strategies that will reduce the size of the shadow economy and improve the economic situation. The article uses the theory of optimal control to build a mathematical model of the interaction between the shadow and legal economies. A number of assumptions are taken into account for construction the mathematical model, in particular, firstly, the rate of change of savings of real sector employees is proportional to the difference between their total income, consisting of legal and shadow salaries, reduced by income taxes, and total expenditures (expenditures on consumption of legal and shadow products); secondly, the dynamics of owners' savings depends on the difference between their total income from the sale of legal and shadow products, reduced by the amount of relevant income taxes, and total expenditures on employee salaries, production needs and taxation in the legal and shadow sectors and thirdly, the rate of change in the price of legal and shadow products is proportional to the difference between total demand and total supply of these products on the market. The assumptions are added with initial conditions for the dynamics of cash savings and the price of the aggregate product, and restrictions are imposed on the consumer shares of savings used for consumption of legal and shadow products for employees and for business owners. The goal criterion is to maximise the average (integral) total welfare of employees and entrepreneurs over a certain period of time. Sufficient conditions of optimality were used for the study. The findings may be useful for economists, policy makers and researchers interested in the shadow economy and looking for effective ways to influence this sector. The results of the study can serve as a basis for the development of policies and recommendations aimed at reducing the shadow economy and stimulating legal economic activity.