Measuring utility from demand

This study presents a new method to calculate a preference relation from a demand function. Our method works well under the weak axiom and can calculate a smooth utility function if the demand function obeys the strong axiom. Further, if the demand function is derived from a customary utility function, our method restores the original preference. Our method provides a complete and rigorous proof of Samuelson’s conjecture. In addition to these results, we guarantee the recoverability: i.e., the uniqueness of the preference relation corresponding to a demand function.     

Recoverability revisited

This study considers the uniqueness problem of the preference relation corresponding to a demand function, which is called the “recoverability problem”. We show that if a demand function has sufficiently wide range and is income-Lipschitzian, then there exists a unique corresponding upper semi-continuous preference relation. Moreover, we explicitly construct a utility function that represents this preference relation. Compared with related research, a feature of our result is that it ensures not only the uniqueness, but also the existence of the corresponding upper semi-continuous preference relation. Further, we introduce two axioms related to demand functions, and show that these axioms are equivalent to the continuity of our preference relation in the interior of the consumption set. In addition to these results, we present three examples that explain why our requirements (including the upper semi-continuity of preference relations and the wide range requirement and income-Lipschitzian property of demand functions) are necessary, and a further two examples in which there is no continuous preference relation corresponding to the given demand function.