ntries.
Let Bi represent, as before, the balance of payments of the ith country, dependent upon the n exchange rates, according to the equation
that will make the system hyperstable. (The "speed" kij can be interpreted as the weight that country i has to give to the condition of the balance of payments of the jth country in adjusting its own exchange rate.)
It is readily shown that the system (36) is hyperstable if the k's are chosen so that
where alphai is a negative real constant and the Deltaji's are, as before, the cofactors of Delta. The condition (37) means that the hyperstable speeds are weighted elements of the inverse of the price matrix.
To prove this proposition we need to prove that the dynamic system
But, from the properties of any determinant, the typical term
has a value of unity for k = i and a value of zero for k not= i. The system (39) therefore reduces to
It is instructive to write out the hyperstable system (38) in detail to see clearly the implications of Hicksian perfect stability for dynamics:
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.
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