An Axiomatic Derivation of the Logarithmic Function as a Cardinal Utility Function on Money Income Levels

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.


Introduction
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 1 .Specifically, the logarithmic utility function is used as a typical utility function representing the law of diminishing marginal utilities, which is a key property of utility functions in the models leading some equity-regarding prescriptions such as the progressive income tax schedule.In the neoclassical (Paretian) utility theory, a consumer's preferences on income levels is specified by a level comparison relation and a level comparison relation is re-presented by the logarithmic utility function if and only if the relation satisfies the monotonicity axiom, i.e., the larger levels of income are more desirable.However, the relation is represented by the logarithmic function as an ordinal utility function 2 .Consequently, the equilibria or solutions of the specific economic models depend on the selections of the utility representations, if the definitions of the equilibria or solutions involve the cardinal properties of the utility functions.For example, the Nash bargaining solution is not well-defined, if the utility representations are determined unique up to monotone transformations.
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.Concretely, Suppes shows that the utility representation is determined unique up to positive affine transformations, 1 Watts [1] introduces a poverty index, which can be interpreted as the differences of the logarithmic utility values.See [2,3] for the Watts index.For the examples in the growth theory, see [4] Propositions 4 and 5, [5]  ; 3) the relation R is represented by any function in M and no function in M C represents R, where M is the set of all increasing functions on the income levels.
which means that the derived utility function is a cardinal utility function representing the two relations simultaneously, and then the equilibria or solutions involving the cardinal properties are well-defined under the utility representation.However, Suppes ([7] Section II Hypothesis U) assumes that the utility values are proportional to the logarithmic values of income levels to determine the logarithmic utility function among the general class of utility functions 3 .
In this paper, without assuming such a superficial hypothesis on the derived utility values, it is shown that the representability of the two relations by logarithmic function can be characterized only by the axioms on the relations.Specifically, 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.

The Model and Results
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 function 4 .

The Model
The set of all possible levels of money income is the set of positive real numbers denoted by X ≡ R ++ .A level comparison relation R is a complete and transitive binary relation on X.The expression xRy means that x is preferred to y.The symmetric and asymmetric parts of R are denoted by I and S, respectively.A difference comparison relation R * on X is a complete and transitive quaternary relation on X.The expression (x → y)R * (z → w) means that the transition (path) from x to y is preferred to the transition from z to w.We assume that all the possible transitions (x → y) are completed for a fixed length of time interval and the underlying price vector is fixed over the time interval; otherwise, the money income levels are assumed to be adjusted suitably (PPP adjustment).The symmetric and asymmetric parts of R * are denoted by I * and S * , respectively.A preference structure on X is a pair of a level comparison relation R and a difference comparison relation R * on X.The preference structure on X is denoted by (R, R * ).A preference structure (R, R * ) on X is represented by a real-valued function u on X if and only if the following two assertions hold: 1) X x y z w∈ .This definition implies that, for a given real-valued function u on X, there uniquely exists a preference structure (R, R * ) which is represented by the function u, i.e., if ( )

Results
We introduce the following axioms to characterize the representability of a preference structure by the logarithmic function: Monotonicity x y xSy > ⇒ for all , X x y ∈ .Consistency There exists some X z ∈ such that ( ) ( )

Homogeneity ( ) ( )
x y I tx ty * → → for all , X x y ∈ and all t > 0.
The monotonicity axiom means that the larger levels of income are more desirable, and the consistency axiom means that there exists some X z ∈ such that the difference comparison on ( ) coincides with the level comparison on x and y for all , X x y ∈ .The homogeneity axiom means that two transitions are indifferent if one transition is given by multiplying a positive number both for the initial and ending levels of another transition.Then we have the following proposition: Proposition Let (R, R * ) be a preference structure on X and let L be a set of real-valued functions f on X defined by L = {f: there exist a > 0 and b such that + for all X x ∈ }.Then the following three statements are mutually equivalent: 1) (R, R * ) satisfies the monotonicity, consistency and homogeneity axioms; 2) (R, R * ) coincides with ( ) 0 0 , R R * , i.e., (R, R * ) is represented by the logarithmic function; 3) (R, R * ) is represented by any function in L and no function in Proposition is proved in Appendix A. The three axioms in the assertion 1) of Proposition are mutually independent, which can be proved by constructing the counter examples: Define a preference structure ( ) for all , , , X x y z w∈ .The preference structure ( ) , R R * satisfies the homogeneity and consistency axioms, but it does not satisfy the monotonicity axiom, which implies that the monotonicity axiom is independent.Let 2 R * be a difference comparison relation defined by for all , , , X x y z w∈ , and let 3 R * be a difference comparison relation defined by for all , , , X x y z w∈ .Letting R 0 be the monotonic level comparison relation again, the preference , R R * satisfies the monotonicity and homogeneity axioms, but it does not satisfy the consistency axiom, which implies that the consistency axiom is independent.
In fact, it holds that 2S 0 1 and ( ) ( ) , R R * satisfies the monotonicity and consistency axioms, but it does not satisfy the homogeneity axiom, which implies that the homogeneity axiom is independent.

Conclusions
A consumer's preferences over money income levels are specified by a level comparison relation and a difference comparison relation.A pair of the relations is called a preference structure and it is shown that a preference structure is cardinally represented by the logarithmic function if and only if the preference structure satisfies the three mutually independent axioms on the preference structures.This result clarifies the mutually independent axioms characterizing the preference structure represented by the logarithmic function as a cardinal utility function.In particular, this result enables us to interpret the per capita national income measured by the logarithmic scale as the utility level of the income, and axiomatically cha-racterizes the simple rule evaluating the differences of per capita national income levels based on their growth rates under the monotonic level comparison relation.
We need a lemma proved in Appendix B: Lemma 2: 1) for all , , , f ⋅ is continuous and increasing on R.
Setting a * = f(1) − f(0) > 0 and b * = f(0), we will prove that f(t) = a * t + b * for all t ∈ R .Suppose that t is a rational number, i.e., there exists a pair of integers (p, q) such that , 0 and 0 t q p q p = ≥ ≠ .
Using the induction arguments with respect to 0,1, 2, q =  for a fixed p ≠ 0, it holds by Lemma 2 1) that ( ) ( ) ( ) ( ) for all integers 0 and 0 Case 1 (t ≥ 0) Setting q = p in (13), we have that Hence, we have by (13) and this that ( ) ( ) ( ) ( ) ( ) Case 2 (t < 0) It holds by (12) that p < 0. Setting q = − p > 0 in (13), we have that For , , , α β γ δ ∈ R , set x * = e α , y * = e β , z * = e γ and w * = e δ .Then we have by ( 14) that ( ) ( ) ( ) ( ) .2) Since g(x) is (strictly) increasing on X by (10), and since e t is (strictly) increasing on R, f(t) ≡ g(e t ) is (strictly) increasing on R. Hence it holds by [14] Chapter 5 Theorem 3 p.100 that there are at most countable number of points at which f is not continuous.Thus there is a point α in R at which f is continuous.Let β be a point in R, and let {β m } be a sequence in R converging to β.
Define a sequence {α m } in R by for limα m = α and f is continuous at α, we have that R * be the preference structure represented by the logarithmic function, i.e.,