On the Stability of Generalized Metzlerian Systems

Abstract

The study of competitive equilibrium has always been concerned with the of that price structure which clears the commodity market. The earlier economists defined the concept of stability, and tended to assume it without giving a rigorous analysis of the stability hypothesis. Hicks [8] presented well-known conditions for stability, but in so doing failed to consider the dynamic process of price adjustment through time. Nonetheless, in a classic paper Metzler [12] pointed out that if all commodities are gross substitutes, the Hicksian conditions for perfect stability are identical to the conditions for stability in a dynamic sense. This discovery, brilliant as it was, turned out to be only a prelude to a more exciting theme. In later developments, it was established by Negishi 15], Hahn [7], Arrow and Hurwicz [2], and Arrow, Block and Hurwicz [1] that a competitive equilibrium is indeed dynamically stable for a gross substitute economy. Despite the central place it occupies, the assumption of gross substitutability represents a serious drawback to the stability literature. Although we can certainly dispose of complementarity of a technical nature often observed among several commodities by a simple process of aggregation, it is impossible to assume away weak gross complements without loss of generality. It should be noted, however, that the absence of gross complements is by no means a necessary condition for dynamic stability. Thus, the difficulty associated with universal gross substitutability should not totally undermine the stability hypothesis. This point, well recognized in the literature, is of course of little interest unless one is able to identify meaningful classes of competitive systems in which both complementarity and stability are present. This paper attempts to proceed a few steps forward in this direction via the natural route of extending the Metzlerian gross substitute system. We shall first propose three distinct, but closely related generalizations of the Metzlerian system. Subsequently, in Section III, these generalized systems will be shown to be locally stable even though they may include elements of gross complementarity. To indicate the extent to which the Metzler theorem carries through, we shall also demonstrate that the Hicksian conditions are necessarily satisfied for a particular pair of these generalized systems. From this, we can directly proceed to comparative statics. In Section IV we shall present a result of comparative statics closely analogous to the one obtainable for the original Metzlerian system. This result, far stronger than the one implied by the Hicksian conditions, will be seen to hold for a (different) pair of the proposed generalizations. To illustrate the nature of our concepts, a numerical example will then be considered briefly. Finally, in Section V, we shall discuss some related literature, especially an alternative line of generalization suggested by Morishima [13], [14].