Editing Price index (section)

=== Unweighted indices ===
These compare prices of single goods between periods without quantity or expenditure weights, often as building blocks for indices like Laspeyres or Paasche within broader measures like CPI or PPI. For example, a Carli index of bread prices might feed into a Laspeyres index for a food category. They are called "elementary" because they’re applied at lower aggregation levels (e.g., a specific brand of peas), assuming prices alone capture consistent quality and economic importance—a simplification that fails if quality changes (e.g., better peas) or substitutes shift demand, unlike weighted indices (e.g., Fisher) that adjust via quantity or expenditure data.<ref>PPI manual, 598.</ref>

==== Carli index ====
From Gian Rinaldo Carli (1764), an arithmetic mean of price relatives over a set of goods <math>C</math>:<ref>PPI manual, 597.</ref>

: <math>P_{C} = \frac{1}{n} \cdot \sum_{c \in C} \frac{p_{c,t}}{p_{c,0}}</math>

Simple and intuitive, it overweights large price increases, causing upward bias. Used in part in the British [[retail price index]], it can record inflation even when prices are stable overall because it averages price ratios directly—e.g., a 100% increase (2) and a 50% decrease (0.5) yield 1.25, not 1.<ref>[https://www.bbc.co.uk/programmes/p02rzwrl More or Less], 17 August 2012, 17:58</ref>

==== Dutot index ====
By Nicolas Dutot (1738), a ratio of average prices:<ref>{{Cite web |title=The Life and Times of Nicolas Dutot |url=https://www.chicagofed.org/~/media/publications/working-papers/2009/wp2009-10-pdf}}</ref>

: <math>P_{D} = \frac{\sum p_{t}}{\sum p_{0}}</math>

Easy to compute, it’s sensitive to price scale (e.g., high-priced items dominate) and assumes equal item importance.<ref>PPI manual, 596.</ref>

==== Jevons index ====
By W.S. Jevons (1863), a geometric mean:<ref>PPI manual, 602.</ref>

: <math>P_{J} = \left(\prod \frac{p_{t}}{p_{0}}\right)^{1/n}</math>

It’s the unweighted geometric mean of price relatives. It was used in an early [[Financial Times]] index (the predecessor of the [[FTSE 100 Index]]), but it was inadequate for that purpose because if any price falls to zero, the index drops to zero (e.g., one free item nullifies the cost). That is an extreme case; in general, the formula will understate the total cost of a basket of goods (or any subset) unless their prices all change at the same rate. Also, as the index is unweighted, large price changes in selected constituents can transmit to the index to an extent not representing their importance in the average portfolio.<ref>PPI manual, 596.</ref>

==== Harmonic mean indices ====
Related unweighted indices include the harmonic mean of price relatives (Jevons 1865, Coggeshall 1887):<ref name="PPI manual, 6002">PPI manual, 600.</ref><ref>Export and Import manual, Chapter 20, p. 8</ref>

: <math>P_{HR} = \frac{1}{\frac{1}{n} \cdot \sum \frac{p_{0}}{p_{t}}}</math>

and the ratio of harmonic means:<ref name="PPI manual, 6002" />

: <math>P_{RH} = \frac{\sum \frac{n}{p_{0}}}{\sum \frac{n}{p_{t}}}</math>

These dampen large price drops, offering stability but less economic grounding than weighted indices.<ref>PPI manual, 600.</ref>

==== CSWD index ====
Named for Carruthers, Sellwood, Ward, and Dalén, a geometric mean of Carli and harmonic indices:<ref>PPI manual, 597.</ref><ref>Export and Import manual, Chapter 20, p. 8</ref>

: <math>P_{CSWD} = \sqrt{P_{C} \cdot P_{HR}}</math>

In 1922 Fisher wrote that this and the Jevons were the two best unweighted indexes based on Fisher’s test approach to index number theory, balancing Carli’s bias with harmonic stability, though it lacks economic weighting.<ref>PPI manual, 597.</ref>