economists who signed the petition.
10 This is perhaps apparent immediately, since the Hicks conditions can be expressed in terms of the commodity chosen as numéraire, the excess demand for which is never allowed to go to zero. On this point (but without apparent recognition of its implications) see Lange (, p. 92).
11 More compactly, we can state the conditions as requiring that every first minor of B be "Hicksian." Not all the conditions are independent, however, since B00 = B11 = Bnn in view of the characteristics of B.
12 One might think that Hicks really intended his conditions to extend over the entire range of excess demand coefficients (including the numéraire), this would be incorrect since the last (augmented) determinant is singular. Alternatively, it might be thought that Hicks intended the conditions to apply no matter what commodity were taken as numéraire. The latter interpretation seems to be fortified by a footnote which states ( p. 75): " [this] can be seen at once if we adopt the device of treating X (momentarily) as the standard commodity, and therefore regarding the increased demand for X as an increased supply of the old standard commodity m ...." If this device were adopted to derive the Hicks conditions it would indeed result in conditions equivalent to the above stability conditions. Yet this interpretation would conflict, not only with all subsequent interpretations of the Hicks conditions by other writers on stability, but also with Hicks' own explicit statement in the Mathematical Appendix (p. 315), which specifies that the conditions must hold "for the market in every Xr (r = 1, 2, 3, . . ., n-1)"; that is, the market for the numéraire (the nth commodity) is omitted, In any case his discussion on pages 68-71 fails to make the point clear, while his third graph in Figure 16, which he asserts is stable, is actually totally stable only if the line along which there is zero excess demand for the standard commodity (not drawn in his figures) is inelastic referred to the abscissa. The discussion 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