The Stability of Non-Walrasian Processes: Two Examples - Eckalbar, John C. (Econometrica, 1980)
As a non-Walrasian system tracks through the phase space, the differential equations which govern its motion will typically change as the system crosses certain borders. This increases the complexity of the stability problem considerably. In the present paper we find that some straightforward modifications to Lyapunov's method render the problem tractable. These methods are derived, and their use is illustrated in the case of two different systems which have trading out of equilibrium. 1 . THOUGH ECONOMISTS HAVE BEEN INTERESTED in the stability on non-Walrasian systems at least since Clower's paper' over a decade ago, we have yet to get very far with the inquiry. There may be a number of reasons for this, but the most important seems to be that we have not yet fully appreciated the differences between the methods of analysis suitable for studying the stability of Walrasian and non-Walrasian systems. It is widely known that the primary distinction between the two systems is that quantities actually traded enter as arguments in the non-Walrasian excess demand functions. These quantities will sometimes be demand quantities and sometimes supply quantities, depending upon the overall state of the markets-but then this implies that the excess demand functions themselves will be changing as the system moves through time and that the system itself is not everywhere differentiable. Take, for example, an output supply function which depends upon the actual quantity of labor hired. If the actual quantity hired is the lesser of the quantities supplied and demanded, then under the usual assumptions the partial derivative of output supply with respect to the price of labor will sometimes be positive, sometimes negative, and sometimes non-existent, depending upon whether the demand for labor is greater than, less than, or equal to the supply. What we have in effect is a dynamic system which has its endogenous variables sometimes governed by one set of equations and sometimes by another, with the overall system lacking differentiability at the points of changeover. This much is fairly clear, but the methods which can be used to study such systems have, with few exceptions,2 yet to be seriously explored. In the present paper we are interested in finding modifications to Lyapunov theorems which will render them suitable for the study of non-Walrasian systems. Two such modifications are found and their usefulness in studying non-Walrasian systems is illustrated by means of some relatively simple economic examples.