namics:
We shall also find it convenient to consider a reduced system in which we choose the alphai, the rate at which each pi is restored to equilibrium (with a negative sign), to be equal to the corresponding Hicksian conditions of imperfect stability as given in (9), that is, we equate
Two observations can immediately be made about (44). First, if the elements of the inverse matrix all have the same sign, hyperstability implies that positive weights be assigned to each balance of payments. Thus, "Britain" should depreciate (appreciate) more rapidly the greater the deficits (surpluses) in the balance of payments of other countries, for any given deficit in her own balance (this implies corresponding changes in the U.S. balance). But from Mosak's theorem the elements in the inverse will all have the same sign if the original currency matrix [bij] is a gross substitute matrix provided [bij] is Hicksian; and it will be Hicksian provided the dollar is also (reciprocally) a substitute for all other currencies.
The second point to notice about (44) is that " Britain" should attach less weight to the balance of payments of other countries' currencies than to her own balance if currencies are all substitutes for one another; this follows because every ratio Deltaji / Deltaii < 1 for j not= 1.19
Leaving now the special case of the gross substitutes to return to the more general case represented by equations (42), we can find immediate implications of the Hicks conditions. First, if the Hicks conditions are satisfied, " normal" adjustments are implied in the sense that, ceteris paribus, a deficit in a country's balance of payments suggests depreciation and a surplus appreciation; this follows because every kii = Deltaii /Delta > 0 given the Hicks conditions of imperfect stability and alpha i< 0.
But more than this can be said. The following identities have to hold, from our definitions and Jacobi's ratio theorem:
the inequality following at once from the Hicks conditions of perfect stability. Proceeding to the last term 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