Axiomatic Approach

This chapter investigates the axiomatic approach in index number theory. This approach, also known as the test approach, aims to determine the most appropriate index number formula by listing and satisfying various axioms or tests. The chapter begins by addressing the multitude of existing index number formulas and emphasizes the importance of selecting one based on specific objectives. Edgeworth’s view, which advocated for choosing an index number formula according to its purpose, contrasts with the opinions of contemporaries like Walsh and Fisher. The axiomatic approach is central to modern index number theory, narrowing down desirable indices through axioms such as transitivity, monotonicity, linear homogeneity, identity, and independence from the measurement unit. The chapter introduces some basic axioms, elaborating on their significance and the only index number formula (the Cobb–Douglas-type index) that satisfies all these axioms. It also discusses the implications of abandoning certain axioms, such as transitivity, leading to a variety of index formulas that meet other criteria. Further, the chapter explores indices like the Walsh, Fisher, Törnqvist, and Sato–Vartia indices, highlighting their axiomatic properties and practical applications. It explains the concept of characterization, where an index number is defined through its foundational axioms, identifying specific index number formulas that uniquely fulfill these conditions. The chapter concludes by presenting the characterization of well-known index number formulas within the axiomatic framework, discussing various debates related to them, and emphasizing the ongoing importance and centrality of the axiomatic approach in current index number theory.