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 attac