hird, because slight errors in manipulating rates of changes in instrumental variables (interest rates, exchange rates, and so on) may turn a weakly stable system into an unstable system; and, finally, because the time involved in approaching equilibrium may be less under strongly stable systems and rapid adjustment may be preferred to slow adjustment.
We can consider, therefore, the problem of choosing a dynamic control mechanism with "hyperstable" properties, in the sense that variables rise or fall whenever they are out of equilibrium, and show how the Hicks conditions can be of some help in constructing such a system when the precise location of an equilibrium is not known.
We take again as our example an international currency system. In a general hyperstable system it will be necessary to vary exchange rates, taking into account the balances of payments of each country. The problem is to find the weights each central bank should give their own balance of payments disequilibrium and that of the other countries.
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.
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 