ABSTRACT
The expansion locus associated with any prices is the locus of consumption when income varies while the prices remain fixed. When demand is governed by utility, properties of the utility are reflected in the characteristics of the expansion loci. For instance, if the utility relation R ⊂ C ×C is homogeneous, or conical, having the property
xRy ⇒ xtRyt (t ≥ 0), bywhich its graph inC×C is a cone, then also the expansion loci in the commodity space C are cones, possibly just single rays, or lines through the origin. Correspondingly, the relation between incomes that have the same purchasing power at some different prices is homogeneous linear, and so is represented by a line through the origin, now in the income space, the space of two dimensions with the incomes as coordinates. We have also called this the Fleetwood space, because his method determines points in it, without the compulsion to take ratios of coordinates and otherwise abandon the points, as in effect is done with the CPI. The slope of the line, which in this case completely describes the relation of equivalent incomes, is the price index corresponding to the two different price situations. The idea of a price index for both theory and practice is encompassed in the idea that this relation between equivalent incomes should be homogeneous linear, and so capable of description by a single number-its slope-that number being the price index. At least, it seems, very suitable to think of it that way. In order for such a relation to be possible when expressed in terms of utility, the utility involved must be homogeneous or conical in the way just stated. Correspondingly, in the commodity space, expansion paths must be rays through the origin. Beyond that, no more can be said of the relation except that it should be monotonic. The price index thus represents a highly restricted idea about purchasing power, or cost of living, with implications about demand behaviour, and about utility when that is brought in.
