On the attractivity of a class of homogeneous dynamic economic systems (Q1863625)

An \(n\)-firm industry is considered where each is labour-managed. It is assumed that each firm adjusts its output proportionally to its marginal profit. The assumption leads to an \(n\)-dimensional autonomous system of differential equations where the right-hand sides are homogeneous functions of degree 2. It is shown that the system has a half-line whose points are equilibria and that at each equilibrium point an eigenvalue is zero and the rest of the eigenvalues are negative. It is proved that the half-line of equilibria is an attractor and every solution from its basin tends to some point on this line. A great part of the paper consists of a summary of basic facts of the theory of dynamical systems. One may arrive to the results, perhaps, somewhat faster by applying \textit{B. Aulbach's} results [Continuous and discrete dynamics near manifolds of equilibria, Lecture Notes in Mathematics. 1058. Berlin etc.: Springer-Verlag (1984; Zbl 0535.34002)].