Stability of Equilibrium and the Value of Positive Excess Demand
I SHALL PROVE two theorems using a new method in the problem of stability of equilibrium based upon the second method of Liapounov [4, p. 256ff.]. The novelty of method lies in the selection of the function V(p) whose decrease with time leads to the equilibrium position.2 This is the price weighted sum of the positive excess demands. I shall first prove the existence and stability3 in the large of the set of equilibrium points in the case of cross-elasticities which are nonnegative. The set of equilibrium points is compact and convex, and if the gross substitution matrix is indecomposable at equilibrium, the equilibrium is unique. When a numeraire is not present, it is possible to proceed beyond the limitation of nonnegative cross-elasticities to consider cases where certain weighted sums of the partial derivatives of excess demands with respect to prices are positive. This appears to be a natural generalization. Although the second theorem is primarily of local interest, one hardly need apologize for that. Global stability is not to be expected in general. This type of study was initiated by Walras [8, p. 170) and given its present formulation by Samuelson [5, p. 269]. I shall not elaborate on its limitations. Suffice it to say that, strictly interpreted, the groping for equilibrium which
- DOI
- 10.2307/1910134
- Volume
- 28 (3)
- Pages
- 606
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