18.1 Mathematical methods and models in economics

Problem: The dynamic intersectoral model of the production system is described by a system of linear homogeneous differential equations: \[ \frac{d x(t)}{d x}=(E-A) B^{-1} \cdot x(t) \text {, где } \frac{d x(t)}{d t}=\left(\frac{d x_{1}}{d t} ; \frac{d x_{2}}{d t} ; \ldots ; \frac{d x_{n}}{d t}\right) \] is the column vector of absolute growth of the production volume; \( x(t)=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \) is the column vector of production volume: \( E \) is the identity matrix: \( A=\left(a_{i j}\right) \) is the matrix of coefficients of direct material costs; \( B=\left(b_{i j}\right) \) is the matrix of coefficients of the capital intensity of the production growth (the costs of production accumulation per unit of production of the corresponding form). Let a two-industry model of a production system be given with a matrix of direct costs \( A=\left(\begin{array}{ll}0,21 & 0,32 \\ 0,32 & 0,45\end{array}\right) \) and the matrix of coefficients of capital intensity of the production growth \( B=\left(\begin{array}{cc}0,52 & 0,52 \\ 1 & 0,83\end{array}\right) \). Find the production volumes of the industries, if at the initial moment \( x(0)=\left(\begin{array}{l}60 \\ 60\end{array}\right) \).