Stability in the Barro-Grossman Model

Abstract

In a recent paper in this journal L6fgren (1979) has shown, within the BarroGrossman framework, that a Walrasian equilibrium which is locally asymptotically stable under notional excess demand adjustment specifications may be unstable (in the Liapunov sense) under effective excess demand specifications. This result is interesting in its own right, and as Ldfgren suggests,' it shows that both Leijonhufvud (1973) and Grossman (1974) have been hasty in presuming that the stability of a Walrasian equilibrium is preserved when the notional excess demand hypothesis is replaced by the effective excess demand hypothesis. In the dynamical systems considered by Ldfgren, instability arises in the following form: a small displacement from the Walrasian equilibrium to a state of effective excess demand for output and labor initiates a trajectory which is directed away from the Walrasian equilibrium and which in an appropriate sense exhibits no immediate tendency to approach the (upward sloping) border between the region of effective excess demand for output and labor (as in Lofgren (1979), the H-region) and the region of effective excess supply of output and labor (the K-region). The question which Ldfgren's analysis leaves open is whether such an H-region trajectory reaches the border between the H-region and the K-region eventually (that is, in finite time) thereby becoming a K-region trajectory directed toward the Walrasian equilibrium. This question is important because, for many systems of interest, an affirmative answer implies that the Walrasian equilibrium is asymptotically stable (in the sense that all trajectories converge to it) even though it may not be Liapunov stable.2 For such systems, replacement of the notional excess demand hypothesis with the effective excess demand hypothesis preserves asymptotic stability even if it does not preserve Liapunov stability. The present note provides two results on the behavior of H-region trajectories. First, a set of plausible sufficient conditions is given guaranteeing that all trajectories which begin in the H-region (and which do not converge within the H-region to the Walrasian equilibrium and do not otherwise leave the Hregion) enter the K-region in finite time. The most significant assumptions