hment to the dollar. A group of currencies could be "attached" to the pound or the franc or any other currency, in principle. More importantly, we can then allow entire currency areas to appreciate and require that the balance of payments of the areas worsen, while various subsets of other currencies in the currency areas remain unchanged.13
The remarkable fact is that the conditions resulting from making arbitrary currency alignments among the (nondollar) countries and applying the Hicks method to the resulting matrix incorporate the conditions just developed as a special case. Thus consider the denominator of the first term in (27),
This term is the result of combining the ith and jth currencies together to form a currency area of those two countries. With no loss of generality we can write i = 1 and j = 2. Then if the first and second rows and columns are replaced by their combined rows and columns, we have
as can be proved by straightforward expansion. Similarly, it can be shown that the denominator of the second term in (27) is the determinant formed by replacing the ith, jth, and kth rows and columns of Delta by the amalgamated row and column.
When we now carry out Hicks' method for arbitrary arrangements of currency areas, extended over the whole range of currencies, we get a new set of conditions on the original (n x n) price matrix. These conditions can be expressed in a triangular arrangement of principal minors as follows:
The conditions on the left side of the stability triangle are the conditions of perfect stability Hicks developed; they do not provide the information implicit in extending the analysis to the numéraire commodity. The conditions on the base of the triangle result from extending the Hicksian method to the numéraire; they ignore the experiments resulting from allowing currency areas to appreciate. Finally, the conditions on the right side of the triangle are the conditions applicable when various sets of prices are raised in the same proportion, other prices remaining constant. More generally, 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 genera