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There are four main approaches to bilateral index number theory: the fixed basket, stochastic, test, and economic approaches. The paper reviews the contributions of Irving Fisher to these approaches to index number theory, which are still in use today. The paper also reviews Fisher's contributions to multilateral index number theory. The main themes of the paper are developed in the context of a review of the early history of index number theory: a history that conveys a wealth of information and insight into the making and use of index numbers today.

Abstract  Traditional index number theory decomposes a value ratio into the product of a price index times a quantity index. The price (quantity) index is interpreted as an aggregate price (quantity) ratio. The present paper takes an alternative approach to index number theory, started by Bennet and Montgomery in the 1920s, which decomposes a value difference into the sum of a price difference plus a quantity difference. Axiomatic and economic approaches to this alternative branch of index theory are considered in the present paper. The analysis presented has some relevance to accounting theory in which revenue, cost, or profit changes need to be decomposed into price quantity components or where standard or budgeted performance is compared with actual performance (variance analysis). The methodology presented in the paper is also relevant for consumer surplus analysis.

Abstract  A comment on the article by W. Erwin Diewert in this volume.

"Dieser Beitrag stellt neuere Entwicklungen in der Theorie der Produktions- und Kostenfunktionen sowie der Theorie der Indexziffern zusammenfassend dar. Die Indexzifferntheorie bietet das nötige Instrumentarium, um Probleme zu lösen, denen sich Wirtschaftshistoriker schon lange gegenübersahen. Hier wurden diese Methoden angewendet, um die Kostenunterschiede bei der Stahlherstellung in Deutschland, Großbritannien und in den Vereinigten Staaten in den Jahren 1907 und 1909 zu analysieren. Dabei ergab sich, daß sowohl die amerikanische als auch die deutsche Stahlindustrie um 15 Prozent effizienter produzierten als die britische. Darüber hinaus vermochte Deutschland seine Position auf dem Weltmarkt noch durch besonders niedrige Inputpreise zu verbessern, während Amerika seinen Produktivitätsvorteil durch das dort vorherrschende hohe Lohnniveau ziemlich wieder einbüßte." (Autorenreferat)

Index number theory informs us that if data on matched prices and quantities are available, a superlative index number formula is best to aggregate heterogeneous items, and a unit value index to aggregate homogeneous ones. The formulas can give very different results. Neglected is the practical case of broadly comparable items. This paper provides a formal analysis as to why such formulas differ and proposes a solution to this index number problem