A Note on Some Perturbation Theorems for Price Dynamics in an Exchange Economy

Abstract

Mukherji [1974] has shown that, in the Edgeworth process, price path converges to the equilibrium price vector.2 His result may be best interpreted in the context of stability theory under perturbation. It is well known that a tatonnement price adjustment is globally stable if the initial allocation is Pareto-efficient. An allocation different from the Pareto efficient limit allocation introduces a perturbation to the price adjustment function. When this perturbation converges to zero over time, Mukherji's thoerem shows, the global stability property of the original dynamical system obtains. A close reading of Mukherji's proof suggests, however, a perturbation theorem of wide applicability which contains slightly more information. This may be loosely stated as follows: If C' Lyapunov function is defined for an equilibrium of a differential equation system on a compact phase space, then, for perturbations to the differential equations, the positive limit set of any point in the compact set is the equilibrium poin1t. In our framework, the above statement will be made precise and proved (Lemma 2). One of the objectives of this paper is to apply this lemma to a result of M. Sattinger to obtain the global asymptotic stability property of the equilibrium price when perturbation (introduced by non-Pareto-efficient property of allocation) is small but unchanging. Sattinger [1975] proved that when an initial allocation is nearly Pareto-efficient, the corresponding equilibrium price vector is locally stable in a tatonnement process.3 Since the above-mentioned lemma shows that the positive limit set of an initial price vector in an appropriate compact set lies near the equilibrium price vector in a tatonnement process when initial allocation is nearly Pareto-efficient, it may be asked whether the lemma may be combined with the Sattinger result to produce the global stability property of price dynamics in the same environment. It turns out that the combination is possible if all equilibrium price vectors corresponding to initial allocations in a neighborhood of a given Pareto efficient allocation contain s-balls (with centers at respective equilibrium prices), for some fixed s >0, in their regions of attraction (Theorem 2). That this is actually the