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′r xr 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, λr Drs +λs Dst +·· ·+λq Dqr > 0 (II) for all distinct r,s, t, . . . ,q taken from 1, . . . ,k.