e 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,