Asymptotic properties of optimal trajectories of discrete economic and demographic models (Q1824537)

The study of the asymptotics of the optimal trajectories of the von Neumann model, as a rule, is reduced either to analyzing the infinite sequence of difference equations of the form (1) \(x_{t+1}=Ax_ t\), \(t=0,1,...\), with a square matrix A or to the solution of an ordinary system of equations, with the help of the introduction of the algebraic field of formal power series. The analysis of a stable population model considering both sexes and the process of formation of conjugal pairs with the help of endogeneous domination also reduces to the study of an infinite sequence of difference equations of the indicated form. In this paper, necessary and sufficient conditions (Theorem 1) are found for the existence of a limit matrix \(\bar A\) of the sequences \((\alpha^{-1}A^ t)\), where \(\alpha\) is the largest eigenvalue in modulus of A. Theorem 1 is constructive, allowing one to compute any element of \(\bar A.\) Let us also note that from this theorem as a partial case is deduced the theorem of Kolmogorov on the limit of a sequence of powers of regular stochastic matrices. A further generalization of Theorem 1 is given in Theorem 2 and 3. One should take particular note of Lemma 3. It establishes a tie between the two (normal and Jordan) forms of matrices and gives a key to analyzing the asymptotics of the trajectories determined by equations of form (1). Lemma 3 makes possible the indication (see the corollary) of quite simple and easily verifiable necessary and sufficient conditions for diagonalizing a square matrix. The placing of a square matrix in the simplest diagonal form materially simplifies the analysis of the trajectories determined by equations of form (1).