Budget Constraint Prices as Preference Changing Parameters of Generalized Fechner-Thurstone Direct Utility Functions

Let V(X) be any direct utility function characterizing consumer preferences that are reflexive, transitive, and complete, and whose parameters are fixed relative to the usual linear budget constraint p X = M. Here X designates the vector (X1,.. ., X,) and is confined to the positive n-orthant; Xi designates quantity of the ith commodity; p is the vector (P,, ...,pn) of positive commodity prices; and M designates total expenditure. Let X(p, M) designate the system of demand functions derived from V(X) subject to pX = M, which possess the properties of zero degree homogeneity, symmetry, and semidefinite negativity. Let W( X; 0) designate a direct utility function with (a) parameter vector 0 dependent on the price vector p that appears in the budget constraint and (b) that rationalizes the same system of demand functions as V(X), namely, X(p, M). Probably, many economists who have considered price-dependent preferences' would conjecture that every such price-dependent utility function W(X; p) satisfying (a) and (b) is of the rather restricted form