ject matter of this chapter with J. C. Weldon of McGill University.
3 Strictly, the problem is not trivial even in the case of exchange of two commodities; first, because of complications associated with the possibility of time derivatives of various orders of price changes entering the excess demand (X) functions, in a system such as second, because of nonlinearities in the excess demand function; and, third, because market exchange of two commodities among many people may involve adjustments of quantities toward budget constraints, giving rise to the more complicated dynamics such as Marshall postulated in his foreign trade analysis.
4 Let Xi = Xi(p, . . ., pn) be the excess demand functions for commodities i = 1, . . ., n. By differentiation . Solving for the dpi we get , where Delta is the determinant of the system and Deltaji, its first cofactors. When all other prices adapt so that every dXj = 0 except dXi, we have the solutions dpi = (Deltaii /Delta) dXi; if various prices k, . . ., n, are held constant with respect to the numéraire we get instead
Imperfect stability requires that Deltaii /Delta < 0 for i = 1, . . ., n, whereas perfect stability requires that Deltaii,kk, ... ,nn / Delta kk ... ,nn < 0 for any subset of commodities k, ...,n, excluding commodity i. The Hicksian perfect stability conditions are thus
the first condition being the condition of imperfect stability.
5 It is slightly ironic that Marshall in his 1879 manuscript on foreign trade utilized the link between dynamics processes and stability, but not explicitly the link between stability and comparative statics, whereas Hicks utilized the link between stability and comparative statics but not the link between dynamics and stability. Samuelson used both links in his integration of dynamics and statics.
6 Actually, Metzler's example (p. 285) that the Hicks conditions are not sufficient for stability to be independent of the speeds of adjustment contains a numerical error that mars his demonstration [the second cubic of his footnote 12 shou