Qualitative Economics and the Stability of Equilibrium.

In this paper, we investigate the stability problem in a purely qualitative environment, i.e., we assume that no quantitative information is available to the economist. (This is perhaps not quite correct, since we interpret the term purely qualitative to refer to a knowledge of the signs (-, +, 0) of the entries in certain matrices and vectors, so that 0 is treated as a sign, not as a quantitative magnitude.) In Samuelson's use of the correspondence principle, stability is hypothesized and the consequences for comparative statics are then investigated. In this use of stability analysis, the assumption of stability might well—in fact, usually does—involve introducing additional quantitative information concerning the characteristics of the system under study, in addition to any qualitative information the economist is assumed to possess. On the other hand, the analysis of the present paper is concerned with a much less complex problem, since we are going to treat our assumption of a purely qualitative environment very strictly—stability, if and when it occurs, must be qualitative stability, i.e., stability determined solely by the sign patterns (and not the quantitative magnitudes) of the elements of the matrices with which we deal. As a consequence of this, the comparative statics theorems that will be presented are derived under conditions such that stability is guaranteed, not assumed. As the language of the above paragraphs has perhaps indicated, we will be concerned in what follows with investigating the local stability properties of an economic system (Samuelson's "stability of the first kind in the small"), for a particular dynamic adjustment mechanism, i.e., a tâtonnement mechanism. The far more complicated problems associated with global stability are not considered. Section 2 of the paper states the characteristics of the model we will employ; section 3 states the basic theorem of the paper relating to qualitative stability giving necessary and sufficient conditions for a matrix of signs to be stable; section 4 relates qualitative stability to the presence of comparative statics information, and section 5 contains some comments on applications of the concept of qualitative stability. [ABSTRACT FROM AUTHOR]