Statistical Consultants Ltd

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.

Axioms

For 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

The index at the base period equals one.

identity1

Monotonicity in Current Period Prices

Monotonicity in Base Period Prices

Proportionality in Current Period Prices

Inverse Proportionality in Base Period Prices

Mean Value Test

The price index lies between the smallest and biggest price ratios.

mean value test

Symmetric Treatment of Establishment / Products

If there is a change in the ordering of establishments (or products within an establishment) the price quotations are obtained from for the two periods, the elementary index remains unchanged.

symmetric treatment

where

and

denote the same permutation of the components of

and

Price-Bouncing Test

A more generalised (and more controversial) version of the previous axiom, the price-bouncing test allows for different permutations between the base and current periods.

price bouncing test

where

and

denote possibly different permutations of the components of

and

Time Reversal

Circularity

Commensurability

The units of measurement for each product have no effect on the price index.

How the elementary indices compare

	Dutot	Jevons	Carli	Harmonic Mean of Price Relatives	Carruthers / Sellwood / Ward / Dalén	Ratio of Harmonic Means
*Continuity*	Y	Y	Y	Y	Y	Y
*Identity*	Y	Y	Y	Y	Y	Y
*Monotonicity in Current Period Prices*	Y	Y	Y	Y	Y	Y
*Monotonicity in Base Period Prices*	Y	Y	Y	Y	Y	Y
*Proportionality in Current Period Prices*	Y	Y	Y	Y	Y	Y
*Inverse Proportionality in Base Period Prices*	Y	Y	Y	Y	Y	Y
*Mean Value Test*	Y	Y	Y	Y	Y	Y
*Symmetric Treatment of Establishment / Products*	Y	Y	Y	Y	Y	Y
*Price-Bouncing Test*	Y	Y	N	N	N	Y
*Time Reversal*	Y	Y	N	N	Y	Y
*Circularity*	Y	Y	N	N	N	Y
*Commensurability*	N	Y	Y	Y	Y	N

Jevons index satisfies all of the axioms described here. In some countries (including USA and New Zealand), Jevons index has been used more in recent years, replacing the Dutot index at least for some categories.

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.

Statistical Consultants Ltd

Axiomatic Analysis of Elementary Indices

Axioms

Continuity

Identity

Monotonicity in Current Period Prices

Monotonicity in Base Period Prices

Proportionality in Current Period Prices

Inverse Proportionality in Base Period Prices

Mean Value Test

Symmetric Treatment of Establishment / Products

Price-Bouncing Test

Time Reversal

Circularity

Commensurability

How the elementary indices compare

See also: