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The paper discusses five early approaches to the price (and quantity) index number problem. The five approaches are: (1) the fixed basket approach; (ii) the statistical approach; (iii) the test or axiomatic approach; (iv) the Divisia approach and (v) the economic approach. The economic approach makes use of the assumption of optimizing behavior under constraint and the approach is discussed under four subtopics: (i) basic theoretical definitions; (ii) the theory of bounds; (iii) exact index numbers and (iv) econometric estimation of preferences. The paper also discusses several topics raised by Jack Triplett in a recent paper, including: (i) the merits of the test approach to index number theory, (ii) the chain principle and alternatives to it; (iii) the substitution bias and (iv) the new good bias. Although the paper is for the most part an extensive historical survey, there are a few new results in section 8 on multilateral alternatives to the chain principle. Also in section 6.3, it is shown that the Paasche, Laspeyres and all superlative indexes will satisfy the circularity test to the first order.

Promises, Promises: The States' Experience With Income Tax Indexing