On the Stability of Dynamic Processes in Economic Theory

Abstract

The notion of stability in the sense of Lyapunov is applied to economic dynamic processes of the Champsaur-Dreze-Henry type. Our purpose in this note is to fill a small gap in the literature concerning dynamic processes in economic theory, of the type presented by Champsaur, Dreze, and Henry [3]. Indeed, these authors do not discuss stability in the sense of Lyapunov [7]. However, a recent result of Maschler and Peleg [9] on this kind of stability (presented in a discrete model) can easily be applied to both continuous and discrete processes used in economics. We shall present this result for a very general class of such processes and conclude with references to a few economic applications. For our purpose a (set valued) dynamic system is simply a pair 〈X,φ〉, where X is a compact subset of R and φ a correspondence from X to its nonempty subsets. Let T be a subset of [0,∞) containing 0 and x0 an element of X. Then a φ-process starting at x0 is a pair of functions: x(·) : T → X, ẋ(·) : T → R, such that: x(0) = x0 and, ∀ t ∈ T , ẋ(t) ∈ φ(x(t)). If T = {0, 1, 2, · · · , } and ẋ(t) = x(t + 1) then the process 〈x(·), ẋ)(·)〉 is called discrete. If T = [0,∞), if x(·) is absolutely continuous on any interval [0, τ ] in T , and ẋ(t) = dx(t)/dt for almost every t in T , then the process 〈x(·), ẋ(·)〉 is called continuous. In the first case the Econometrica, 47(3), 733-737, 1979. As pointed out by Negishi [10], this is the same as Samuelson’s stability of the second kind [11]. The term “stability in the sense of Lyapunov” is used by Arrow and Hahn [1]. Heal [5] and Hori [6] also use this concept of stability. See Champsaur, Dreze, and Henry [3, Section 5].