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 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 should read lambda3 + 4lambda2 + 3.4lambda + 13.2 = 0 instead of lambda3 + 4lambda2 + 2.6lambda + 13.2 = 0 and in the first (correct) cubic the Routh conditions are satisfied]. But a slight adjustment to his counterexample can nevertheless demonstrate his point. If, for example, speeds are chosen so that, in his terminology, kl = k3 = 1, giving the cubic lambda3 + (2 + k2)lambda2 + (1+ 1.2k2) lambda + 6.6k2 = 0, the Routh conditions will not be satisfied for values of k2 somewhat less than 1.
7 In this connection Samuelson writes (, p. 554): "In principle the Hicks procedure is clearly wrong, although in some empirical cases it may be useful to make the hypothesis that the equilibrium is stable even without the 'equilibrating action' of some variable which may be arbitrarily held constant." Samuelson provides an example in connection with the Keynesian system later in his paper.
8 See, for example, Mundell  and the references there to the principle of effective market classification.
9 See page 111 for a list of 