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Axiomatic Analysis of
Elementary Indices
Economics, Economic / Financial Data Before reading this feature, it may help if you first read: Elementary Indices. Elementary index calculation methods can be tested against various axioms, to investigate whether or not they have certain desirable properties. Some axioms are more important than others. AxiomsFor each of the following axioms: n
is
the sample size of the good being surveyed
are the prices of the good at the current period are the prices of the good at the base period P is the elementary price index and is a function of the current and base period prices Continuity
is a
continuous function of n positive
base period prices and
n
positive current period prices.Identity
Monotonicity in Current Period
Prices
if Monotonicity in Base Period
Prices
if Proportionality in Current
Period Prices
if Inverse Proportionality in Base Period PricesifMean Value Test
Symmetric Treatment of
Establishment / Products
where and denote the same permutation of the components of and . Price-Bouncing Test
where and denote possibly different permutations of the components of and . Time Reversal
Circularity
Commensurability
How the elementary
indices compare
Jevons index satisfies all of the axioms described here. In some countries (including A problem with the Dutot index (and ratio of harmonic means) is that it doesn’t satisfy the commensurability axiom. This means that if the units of measurement change for a product, it would affect the price index. For some products, this won’t be a problem. The Carli index and the harmonic mean of price relatives don’t satisfy the price bouncing test, time reversal or circularity. The inability for time reversal is the biggest of these problems, but can be solved by taking the geometric mean of the two indices i.e. by using a Carruthers / Sellwood / Ward / Dalén index. The inability for time reversal has meant the Carli index is upwardly biased, and the harmonic mean of price relatives is downwardly biased. See
also:
Different ways of measuring the Consumer Price Index ( Index Numbers |
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