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 the augmented system.
17 I have applied this method to the problem of disentangling lags in expectational and cash balance adjustments .
18 More directly the matrix equationif k = B-1.
The usefulness of this result lies not so much in providing a policy maker with the rules of adjustment, since there is no problem if the basic matrix is known, and the rule gives insufficient information if the basic matrix is not known. The point is rather that pieces of information about the inverse may be sufficient to distract policy makers from mistakes, and that conditions like the Hicks conditions may be a sufficient guide to the relative importance to be attached to particular instruments.
19 I have discussed the implications of this condition in some detail .
20 Even the gap between the true dynamic stability and Hicksian stability may be bridged by integrating Hicksian stability conditions with the "Samuelson-Le Châtelier principle" (see, for example, Samuelson ) which can be expressed entirely in terms of the Hicksian determinants. Hicksian stability and dynamic stability mesh together under appropriate conditions of gross substitutes, gross complements, and symmetry, a common link being sign symmetry; and sign symmetry of the inverse of the basic Hicksian matrix is sufficient for at least one of the Le Châtelier conditions to hold (DeltaDeltaii,jj - DeltaiiDeltajj < 0 if Deltaij and Deltaji, have the same sign).
21 A further implication of the Hicks conditions was called to my attention by Daniel McFadden, who had proved, in a paper presented at the 1963 Econometrica Society meetings in Boston, Massachusetts, that if the Hicks conditions of perfect stability are satisfied, a stable dynamic system of the form can always be found for a diagonal K matrix with positive diagonal coefficients; the result also holds in the global form. Prior to McFadden's result, and unknown to him, a local version of the theorem had been published; see Fisher and Fuller .
In the context of the currency problem discussed above, the theorem mea