, 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 th