ssion is not completely clear, however, and I am therefore inclined to interpret Hicks as actually intending to give full coverage to the numéraire but not completing the mathematical implications of doing so.
13 I have considered the problem of " optimum" currency areas in terms of its attributes for stabilization policy, optimum exploitation of the functions of money, and so on in Chapter 12; the present analysis suggests an additional criterion in terms of stability conditions.
14 Stability conditions should, in principle, be "invariants" in the sense that they are independent of the choice of units in which commodities are measured; this corresponds to Lange's "principle of invariance" (, p. 103). The above stability conditions are invariants only if, as is assumed in the case of the foreign exchange market analysis, the bij's are all measured in equivalent currency units.
In the general case, however, where prices respond to excess demands denominated in physical quantity units, the units in which bij( j = 1, . . ., n) is measured differ from the units in which bkj(j = 1, . . ., n) is measured. This means that invariant stability conditions require that each row of every stability determinant involving the sum of elements be multiplied by an arbitrary number reflecting units of measurement. Thus "unit-invariant" stability in the general case involves the sign of terms such as
For arbitrary values of ki and kj extended over the entire range of the (n+ 1) x (n+ 1)-B determinant, and given the homogeneity postulate, the general conditions then imply that all goods are gross substitutes.
15 From a historical point of view, it is somewhat amusing that although the Hicks conditions are not symmetrical with respect to the commodity chosen as numéraire, neither are the (normalized) dynamic systems used to refute the validity of the Hicks conditions as true dynamic stability conditions.
16 The proof in the general case follows by analogy to the proof of Metzler (, pp. 280-285) when his method is extended to include t