position holds, if the Hicksian perfect stability conditions are satisfied, when the shift of demand is interpreted as being from the dollar and all currencies whose exchange rates are kept fixed to the dollar. Note, however, that the Hicks conditions do not give us the sign of
so that we cannot specify, on the grounds of the Hicks conditions alone, whether a shift of demand from dollars to pounds raises or lowers the price of (say) the franc relative to the dollar.
But now we are in a position to see the narrow form of the mathematical implications of the Hicks conditions. Consider a shift of demand from the franc (currency j) to the pound (currency i). Then deltaBs / delta alpha = 0 for s not= i, j, while deltaBi / delta alpha = - deltaBj / delta alpha >0 in view of (16); with no loss of generality we can make deltaBi / delta alpha = - deltaBj / delta alpha = 1. Substitution in (13) then gives the change in the dollar price of the pound and the franc:
The Hicks conditions do not provide us with the sign of either (18) or (19), nor, by analogy to (17), should we expect them to. But, by analogy with (14), we should expect the difference
to be unambiguous in sign for any system in which units are chosen so that each ps = 1, initially. When demand shifts from the franc to the pound, we should not expect to be able to predict the sign of the change in the dollar price of the pound or franc, but we should be able to determine, on the basis of the Hicks conditions, the sign of the change in the franc price of the pound, the expression given in (20). But the Hicks conditions are no help here, and this means that Hicks has not developed the mathematical implications of extending his method to the standard commodity.
The same information problem applies, a fortiori, when various subsets of prices are held constant. An implication of the Hicks condition of perfect stability is that a shift of demand onto pounds raises the price of the pound even when various currencies remain pegged to the dollar; this amounts to treating 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 s