we have
More generally, if the basic matrix [bij] is Hicksian, the matrix of the speeds required for hyperstability must satisfy the conditions that every principal minor be positive, that is,
Analogous conditions hold for the extended Hicks conditions if the system is to be hyperstable regardless of the currency used as the key currency. In this sense the Hicks conditions alone are sufficient to establish in a weak sense the correctness of exchange rate policies directed at correcting "own" balances of payments.
These developments conform to the conclusions of economic intuition indeed, they may be interpreted as bringing the mathematical treatment of the subject closer to the level of common sense. They nevertheless suggest that the Hicksian stability analysis is not lacking in significance for the integration of dynamical and statical theory.20, 2l
1 Adapted from: a forthcoming article in Essays in Honor of Sir John R. Hicks (N. Wolfe, ed.).
2 I am happy to acknowledge many helpful conversations on the subject 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 constan