l conditions are not satisfied.
The general conditions are necessary conditions for a system to be stable regardless of the currency (commodity) chosen as key currency (standard commodity) and regardless of how quickly the various exchange rates adapt to disequilibrium. This proposition is perfectly general in the sense that it is valid in the n-currency case.16
I shall conclude this book by showing how the Hicks conditions, extended as above, can be useful in devising dynamic mechanisms that are "strongly stable." The problem could be approached from the direction of the correspondence principle, which, in a narrow version of it, suggests that we apply to comparative statics the conditions that the characteristic equation of the systems have negative real parts. The justification for this narrow version lies in the observation that the systems we know are not characterized by instability. But there is no reason, in principle, why stronger conditions could not be applied. We could, following Hicks (and Samuelson), require that the system be stable no matter which subsets of market variables are held constant. Alternatively, we could use conditions that the roots be real or complex according to whether we observe cycles in the system under investigation.17 If, for example, we observe an absence of cycles in the real world, we know at once that the Hicks conditions are sufficient conditions for dynamic stability.
In the theory of policy (under incomplete information) the problem is often to choose among different dynamic systems, or different degrees of centralization of a given dynamic system; this is often expressed in terms of allocating, dynamically, instruments to targets (the problem of effective market classification), and we may want to construct strongly stable systems: first, because systems near the borderline of stability may become unstable if disturbed by outside shocks; second, because the cost of adjustment may be higher if the system, even though stable, oscillates in its approach to equilibrium; third, 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 dy