A Foundation for Logarithmic Utility Function of Money

This paper presents a study on the optimization problem of a consumer’s choice constrained to a single time interval. In this problem, the choice is made over a set of perishable goods such that they do not retain value at the end of the period. Money has been introduced as the only means available to store that value for the future. Thus, consumer utility is measured on the possible combinations of goods consumed during the period and money held at the end of the period. Additionally, a set of simple conditions are assumed to the utility functions for goods and money given by: (1) Existence of a total utility that is additively separable with respect to the components of goods and money; (2) continuity of the derivatives of the utility functions of money and goods up to the second degree; and (3) non-uniqueness of the matrix obtained by differentiating the system of equations obtained by the condition of optimum. The article shows how the requirement of homogeneity conditions limits the possible expressions for the utility function of money. One of them is the frequently used logarithmic function.