nerally, the Hicks conditions on the left correspond to Hicksian adjustments when each currency (commodity) is treated in isolation; the adjacent conditions to their right are the Hicksian conditions when in the ith and ith goods move in the same proportion; and so on. The entire set of conditions are needed if the logic of Hicks' method is carried out to the bitter end.14
An important implication of the general conditions is that a system satisfying the Hicks conditions, but not the general conditions, will be stable or unstable depending on which currency is adopted as the key currency.15 Consider, for example, a world of three currencies, dollars (currency 0), pounds (currency 1), and francs (currency 2), and suppose that the balances of payments of the three countries are related to exchange rates according to the equations
where the exchange rates are defined in, and the Bi are expressed in, an abstract unit of account (IMF par value units). (Equilibrium exchange rates are unity or any multiple of unity, as the system is homogenous of degree 0.)
Let us consider a dynamic system in which the dollar is constant with respect to its par value so that the dollar becomes the effective numéraire. Let the par values of the pound and franc adjust in proportion to B1 and B2, respectively. We then have the following dynamic system:
In this system, the Hicks conditions of perfect stability, narrowly interpreted, are satisfied (since b11 = -2 < 0, b22 = -1 < 0), and
and the system is dynamically stable regardless of the (positive and finite) values of kl and k2 .
Consider, however, a system in which the par value of the franc is fixed so that it becomes the "key currency " instead of the dollar. The dynamic system then becomes
for which the Hicks conditions are not satisfied, It is dynamically stable or unstable according to whether k0 >< 4kl .
This result could be predicted at once by applying the general conditions as given in the stability triangle (32). The sum of the coefficients in (34a) are positive, so the general 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; t