Maximizing gross product for the macroeconomic system with consumption proportional to labor resources

A new problem of managing a macroeconomic system with a linearly homogeneous production function is considered, taking into account the balance equation. A gross income of a year is divided into an investment and a consumption, while the volume of the total consumption is proportional to labor resources. A criterion for optimal control is the total value of gross income for a given time interval. As a research apparatus, the maximum principle is applied, due to which the optimal control problem is reduced to a variational problem with a nonholonomic constraint. Its solution is expressed in terms of the quadrature of the Cauchy problem for one equation with separable variables. The values of coefficients of proportionality of consumption, tax and depreciation deductions, ensuring the non-decline of fixed assets, are found. A system with a Cobb—Douglas production function is considered as an example.