Abstract
This note elaborates Suppes’ (1977, Erkenntnis Vol. 11, No. 1, pp 233-250) derivation of the logarithmic function as a consumer’s cardinal utility function on money income levels, in which the consumer’s preferences are specified by a level comparison relation and a difference comparison relation. Without assuming Suppes’ hypothesis (Bernoulli’s hypothesis or Weber-Fechner law), which asserts that the utility values are proportional to the logarithmic values of income levels, it is shown that the representability of the two relations by logarithmic utility function can be characterized only by the three (mutually independent) axioms on the relations.
Highlights
The logarithmic utility function on money income levels is widely adapted as a utility function representing a consumer’s preferences on income levels in the models of the welfare economics and the growth theory1
We introduce some fundamental concepts and definitions used in the cardinal utility theory based on the difference comparisons and show the characterization result for the logarithmic utility function4
A preference structure on X is a pair of a level comparison relation R and a difference comparison relation R∗ on X
Summary
The logarithmic utility function on money income levels is widely adapted as a utility function representing a consumer’s preferences on income levels in the models of the welfare economics and the growth theory. Suppes [7] Section 2 derives the logarithmic utility function based on a cardinal utility representation theorem ([7] Theorem 2) for a general class of preferences including non-monotonic preferences, in which individual preferences are specified by the difference comparison relation as well as the level comparison relation. A pair of the relations on the alternatives is called a preference structure, and it is shown that a preference structure is represented by the logarithmic function as a cardinal utility function if and only if the preference structure satisfies the (mutually independent) three axioms: monotonicity, consistency and homogeneity axioms. The homogeneity axiom is the new axiom introduced by this paper and the monotonicity and consistency axioms are standard axioms in the cardinal utility theory based on the difference comparisons
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