ant relative to that country. But the mathematical information is not given to us by the Hicks conditions. The reason is that the mathematical implications of the Hicks method have not been developed with respect to the currency adopted as numéraire.l0
When we do extend Hicks' method to make it " symmetrical " with respect to the numéraire (appreciating, say, the pound relative to, say, the franc, allowing various subsets of other currency markets to adjust), we get, of course, a set of conditions that specifies the signs of terms like those in (20) and (25). Along with the Hicks conditions, which can be written
[The next term requires that the ratio of the denominator of the second ratio and the sum of sixteen third minors be negative, and so on for successive ratios. The last term in the conditions of (27) specifies that the sum of the (n-l)th minors (n2 in number) be negative. But the (n-l)th minors are equivalent to the elements in the original determinant, so the last condition simply requires that the sum 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