 the dollar and the other currencies pegged to it as a composite currency. By analogy the price of the pound should rise when demand shifts from a currency other than the dollar, say, the franc, while other currencies (for example, the mark) are pegged to the franc.
Thus consider a shift of demand, at constant exchange rates, among three currencies i, j and k, such that
and every other deltaBr / delta alpha = 0. Then, from (13), we have
Applying the restrictions that and setting
we can deduce the change in price of the pound relative to the mark and franc.
where A is the sum of the cofactors of the elements in the following matrix:
The cofactor of, say, the element Deltajk can be related to the second cofactors of Delta by Jacobi's ratio theorem,
so that A /Delta can be written entirely as the sum of second cofactors, and (24) can be rewritten
The inequality sign should hold if an increase in demand for one country's currency occurs at the expense of any other country, one other currency price remaining constant 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, 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