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