ABSTRACT
In the investigation of preference orders which are explanations of expenditure data which associates a quantity vector xr with a price vector pr (r = 1, . . . , k), in respect to some n goods, there is considered the class of functions φ, with gradient g, which are increasing and convex in some convex region containing the points xr, such that gr = g(xr) has the direction of pr.† Let ur = pr/er, where er = p′rxr so that u′rxr = 1. Then
gr = urλr, for some multipliers λr > 0, since φ is increasing. Also, if φr = φ(xr) then
(xr − xs)′gs > φr − φs, since φ is convex. Accordingly
λr > 0, λsDsr > φr − φs (r = s), (I) where Dsr = u′sxr − 1. With the number Drs given, there has to be considered all solutions = {λr}, = {φr} of the system of inequalities (I). With this system, there is involved a consideration of systems of the form ars > Xr − Xs (where ars = λsDsr ,Xr = φr), the theory of which is going to be developed here. It is remarked, incidentally, that the existence of , satisfying (I) is equivalent to the existence of satisfying
λr > 0, λrDrs + λsDst + · · · + λqDqr > 0 (II) for all distinct r, s, t, . . . , q taken from 1, . . . , k.
