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# Index Theory and Price Statistics

## Peter von der Lippe

This textbook integrates mathematical index theory and its application in official price statistics. It tries to bridge theory and practice, due to the apparent divergence between mathematicians with ever more sophisticated and complex models and practitioners with problems that are more and more difficult to understand without broad knowledge and some experience. The text offers an introduction into axiomatic, microeconomic and stochastic reasoning as regards index numbers, with moderately difficult mathematics. It also summarizes many ongoing discussions concerning methodological merits and demerits of specific indices, such as consumer price-, producer price-, unit value- and chain indices, in official price statistics. The book is comprehensive and presents a readable overview of a great number of topics in modern price index theory and their application in inflation measurement, deflation of aggregates in National Accounts, sampling and quality adjustment in price collection and other important though controversial issues.

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# 3 Axioms and more index formulas 165

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165 Chapter 3 Axioms and more Index formulas In the framework of the so called axiomatic approach Index functionsl are explored with reference to a set of axioms, some of them introduced already in preceding chapters. The primary concern of this chapter is to present some more details con- cerning the Interpretation of axioms, relationships among various axioms, and selec- tions among them made by index theoreticians in order to form their "systems of axi- oms" as weil as to present some additional index formulas and "approaches" in Index theory.2 3.1 The axiomatic approach, some theorems and fundamental axioms a) Uses of "axioms" and axiomatic systems An axiom is a functional equation, which is an equation describing a relation which holds for a function, Ilke the index function. An example of a functional equation would be (3.1.1) 9(Y,x) = UP(x,Y)I1 which states that upon interchanging the arguments x and y in the function 9 (a func- tion of x and y, 9(x,y)) the result is the inverse value of 9(x,y) since 9(x,y) = [9(y,x)]-1. Eq. 3.1.1 applies for example to w(pt,p0) = Zpit/Epio which is Dutot's price index p12,„ as cp(po,pt) = EpiciElDit = [c(Pt,Po)] -1 = ppo , however, lt would not apply for example to the exponential mean pr (see below eq. 3.2.15), again an unweighted index function which, however, does satisfy the time reversal test prj # 1 / pgtx (as opposed to Dutot's...

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