ABSTRACT
M1 − m1 = P10(M0 − m0) (11) where the original incomes and the marginal price index are
m0 = µ(p0), m = µ(p1) and P10 = θ(p1)/θ(p0). (12) Let x0, x1 be taken now to be any pair of points on given expansion lines L0,
L1 . In the first caseL0, L1 are determined by joining x0, x1 to the origin. But now they are determined with a further pair of points which, subject to a simple requirement on (L0,p0), (L1,p1), given by the non-vanishing of a 2×2 determinant, can be chosen as the unique pair of critical points c0, c1 for which
p0c0 = p0c1, p1c0 = p1c1. (13) The original incomes then have particular determinations
m0 = p0c0, m1 = p1c1. (14) Then further, the marginal price index, limited by (10), has the particular determination
P10 = (p1(x1 − c1)p1(x0 − c0)/p0(x1 − c1)p0(x0 − c0)) 1 2 (15)
With these determinations, it is possible to identify (11)with the “NewFormula” of Wald (1939),2 obtained on the basis of a quadratic utility function, and thus to put that formula in the background of the general relation (11) whereP10 has any determination within the limits (10). These limits, with original incomes determined by (14), are particular generalizations of the Paasche and Laspeyres indices, these indices being obtained when c0 = c1 = 0, that is whenL0, L1 are rays through the origin. Correspondingly (15) is a marginal price generalization of Fisher’s “ideal index” where the data consists of a general pair of expansion lines, instead of a pair concurrent in the origin.
