um of all the elements in the original determinant Delta be negative.]
This suggests an alternative-and simpler-way of developing the generalized conditions. Consider the augmented determinant
formed by bordering Delta with its column and row sums, with a change of sign, so that
Then the extended Hicks conditions can be stated simply as the requirement that principal minors of B arranged in successive order oscillate in sign, except for the (singular) determinant B itself.11
Because the elements in the augmented determinant B are interdependent the generalized conditions can be expressed entirely in terms of the elements of the "normalized" determinant Delta. The supplemental conditions are
with the last condition reducing to the basic determinant Delta itself. These forms are equivalent to (30) and imply the signs of the ratios in (27).
This representation has the intuitive appeal of starting with the matrix of all the currencies in the system.l2 Thus, instead of omitting the numéraire currency at the outset, we start with a nonnormalized system of n + 1 currencies, exchange rates being expressed in terms of an abstract unit of account (for example, IMF par values), and apply the Hicks conditions allowing each currency the role of numéraire in turn.
The above conditions are more general than the Hicks conditions. Yet they still do not exhaust the information inherent in the Hicks methodology. The Hicks method of holding various subsets of prices constant with respect to one another can be regarded as a device for constructing "composite commodities"; in the present context of currencies, we shall describe them as "currency areas." Now if we apply the Hicks method to a system based on arbitrary arrangements of countries in the currency areas, we get a further generalization of the results obtained by Hicks. When, for example, the mark is pegged to the dollar (the numéraire), the dollar and mark constitute a currency area. But there is no reason to restrict the formation of currency areas in a unidirectional attachment 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 ge