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 