Mathematical controllability theory of capital growth of nations

摘要

In this report, we derive a nonlinear functional differential equation of neutral type as the dynamics of the growth of capital stock of n firms linked up in solidarity in a region. These equations are generalizations of ordinary differential systems studied by Takayama [21, p. 685–706], Arrow [2, p. 184], Knowles [17, p. 3], and Isaacs [13], and delay systems studied by Kalecki [14], Allen [1, p. 254], and Cooke and Yorke [9]. A neutral version (in which there is delay in the derivative) is implicit in the suggestion by Arrow [2, p. 184]. Given an initial capital endowment φ and an initial investment r, we investigate what conditions on the systems parameters and on the external interventions will guarantee that φ can be steered to some fixed target in some finite time. The controls, u, (m-vectors) are investments(if 0⩽ uj(t)⩽ 1) and consumption (if −1 ⩽ uj(t) ⩽ 0). From the controllability theorems proved for the hereditary systems, we derive universal laws for the control of the growth of capital stock. These laws provide broad, startling, but obvious policy prescription for the growth of companies and for national economies. For example, it describes how big external or governmental interventions on the firms (in the form of taxation or subsidy) must be, relative to the firms' investment and consumption to ensure the growth of capital stock to the given target. It also states when this growth is impossible. We use fixed point theorems and arguments from the theory of games of pursuit [10] to establish constrained controllability of our dynamics. Problems of optimality and interpretations are pursued in a subsequent communication.