Editing Price index (section)

== Price index formulas ==
Over 100 formulas exist for calculating price indices, aggregating price (<math>p_0, p_t</math>) and quantity (<math>q_0, q_t</math>) data differently. They typically use expenditures (price × quantity) or weighted averages of price relatives (<math>p_t / p_0</math>) to track relative price changes. Categories include unilateral (single-period weights), bilateral (two-period weights), and unweighted indices, with modern applications favoring Laspeyres for simplicity and superlative indices like Fisher for accuracy in GDP and inflation metrics.

=== Unilateral indices ===
These indices use quantities from a single period—either the base (<math>t_0</math>) or current (<math>t_n</math>)—as fixed weights, meaning they do not adjust for changes in consumption patterns over time.

==== Laspeyres index ====
Developed in 1871 by [[Étienne Laspeyres]],<ref>{{Cite web |title=Q&A about the Consumer Price Index |url=http://www.stat.go.jp/english/data/cpi/1587.htm}}</ref> it uses base-period quantities:

: <math>P_L = \frac{\sum (p_{c,t_n} \cdot q_{c,t_0})}{\sum (p_{c,t_0} \cdot q_{c,t_0})}</math>

It measures the cost of a fixed <math>t_0</math> basket at new prices. This often overstates inflation because it does not account for consumers reacting to price changes by altering quantities purchased (e.g., substituting cheaper goods when prices rise). For example, when applied to an individual consumer’s bundle, a Laspeyres index of 1 means the consumer can afford to buy the same bundle in the current period as consumed in the base period, assuming income hasn’t changed.<ref>Statistics New Zealand; ''Glossary of Common Terms'', [http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/paasche+index "Paasche Index"] {{Webarchive|url=https://web.archive.org/web/20170518063121/http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/paasche+index|date=2017-05-18}}</ref>

==== Paasche index ====
Introduced in 1874 by [[Hermann Paasche]],<ref>{{Cite web |title=Q&A about the Consumer Price Index |url=http://www.stat.go.jp/english/data/cpi/1587.htm}}</ref> it uses current-period quantities:

: <math>P_P = \frac{\sum (p_{c,t_n} \cdot q_{c,t_n})}{\sum (p_{c,t_0} \cdot q_{c,t_n})}</math>

It understates inflation by assuming consumers instantly adjust to new quantities, ignoring that higher prices might reduce demand over time. For example, a Paasche index of 1 indicates the consumer could have consumed the same bundle in the base period as in the current period, given unchanged income.

==== Lowe index ====
Named after [[Joseph Lowe (economist)|Joseph Lowe]], this uses fixed quantity weights from an expenditure base period (<math>b</math>), typically earlier than both the base (<math>t_0</math>) and current (<math>t_n</math>) periods, where the principal modification is to draw quantity weights less frequently than every period:<ref name="hill23">Peter Hill. 2010. "Lowe Indices", chapter 9, pp. 197–216 in W.E. Diewert et al., ''[http://www.indexmeasures.ca/Vol6_10,09,26.pdf Price and Productivity Measurement: Volume 6]''. Trafford Press</ref>

: <math>P_{Lo} = \frac{\sum p_{c,t_n} q_{c,b}}{\sum p_{c,t_0} q_{c,b}}</math>

Unlike Laspeyres or Paasche, which draw weights from indexed periods, Lowe indices inherit weights from surveys (e.g., household budgets), often conducted every few years, while prices are tracked each period.<ref>https://www.bls.gov/pir/journal/gj14.pdf, citing International Labour Office (2004) paragraphs 1.17-1.23</ref> For a consumer price index, these weights on various expenditures are typically derived from household budget surveys, which occur less often than price data collection.<ref name="hill23" /> Used in most CPIs (e.g., [[Statistics Canada]], [[U.S. Bureau of Labor Statistics]]), it’s a "modified Laspeyres" where Laspeyres and Paasche are special cases if weights update every period.<ref>{{Cite web |date=19 December 2014 |title=Consumer Price Index |url=http://www.statcan.gc.ca/pub/62-553-x/2014001/chap/chap-6-eng.htm}}</ref><ref>{{Cite web |title=Different ways of measuring the Consumer Price Index (CPI) |url=http://www.statisticalconsultants.co.nz/blog/different-ways-of-measuring-the-cpi.html}}</ref><ref>[http://www.imf.org/external/pubs/ft/wp/2012/wp12105.pdf Post-Laspeyres], IMF WP/12/105</ref><ref>Bert M. Balk, [https://www.jstor.org/stable/23813500 Lowe and Cobb-Douglas], Jahrbücher für Nationalökonomie und Statistik, 230:6, 726–740</ref> The [[Geary–Khamis dollar|Geary-Khamis method]], used in the [[World Bank]]’s [[International Comparison Program]], fixes prices (e.g., group averages) while updating quantities.<ref name="hill23" />

=== Bilateral indices ===
These indices compare two periods or locations using prices and quantities from both, aiming to reduce bias from the single-period weighting of unilateral indices. They incorporate substitution effects by blending data symmetrically or averaging across periods, unlike unilateral indices that fix quantities and miss consumer adjustments.

==== Marshall-Edgeworth index ====
Credited to [[Alfred Marshall]] (1887) and [[Francis Ysidro Edgeworth]] (1925),<ref>PPI manual, Chapter 15, p. 378.</ref> it averages quantities:

: <math>P_{ME} = \frac{\sum [p_{c,t_n} \cdot (q_{c,t_0} + q_{c,t_n})]}{\sum [p_{c,t_0} \cdot (q_{c,t_0} + q_{c,t_n})]}</math>

It uses a simple arithmetic mean of base and current quantities, making it symmetric and intuitive. However, its use can be problematic when comparing entities of vastly different scales (e.g., a large country’s quantities overshadowing a small one’s in international comparisons).<ref>PPI manual, 620.</ref><ref>PPI manual, Chapter 15, p. 378</ref>

==== Superlative indices ====
Introduced by [[Walter Erwin Diewert|W. Erwin Diewert]] in 1976,<ref name="Diewert197642">{{cite journal |last=Diewert |first=W. Erwin |year=1976 |title=Exact and Superlative Index Numbers |journal=Journal of Econometrics |volume=4 |issue=2 |pages=115–145 |doi=10.1016/0304-4076(76)90009-9}}</ref> superlative indices are a subset of bilateral indices defined by their ability to exactly match flexible economic functions (e.g., cost-of-living or production indices) with second-order accuracy, unlike the Marshall-Edgeworth index, which uses a basic arithmetic average lacking such precision. They adjust for substitution symmetrically, making them preferred for inflation and GDP measurement over simpler bilateral or unilateral indices.<ref name="Hill200442">{{cite journal |last=Hill |first=Robert J. |year=2004 |title=Superlative Index Numbers: Not All of Them Are Super |url=https://doi.org/10.1016/j.jeconom.2004.08.018 |journal=Journal of Econometrics |volume=130 |issue=1 |pages=25–43 |doi=10.1016/j.jeconom.2004.08.018|url-access=subscription }}</ref><ref>Export and Import manual, Chapter 18, p. 23.</ref>

===== Fisher index =====
Named for [[Irving Fisher]], it’s the geometric mean of Laspeyres and Paasche:<ref>{{cite book |last1=Lapedes |first1=Daniel N. |url=https://archive.org/details/mcgrawhilldictio00iona/page/367 |title=Dictionary of Physics and Mathematics |publisher=McGrow–Hill |year=1978 |isbn=0-07-045480-9 |page=[https://archive.org/details/mcgrawhilldictio00iona/page/367 367]}}</ref>

: <math>P_F = \sqrt{P_L \cdot P_P}</math>

It balances Laspeyres’ base-period bias (overstating inflation) and Paasche’s current-period bias (understating it), offering greater accuracy than Marshall-Edgeworth’s arithmetic approach. It requires data from both periods, unlike unilateral indices, and in chaining, it multiplies geometric means of consecutive period-to-period indices.<ref>PPI manual, p. 610</ref>

===== Törnqvist index =====
{{Main|Törnqvist index}}
A [[geometric mean]] weighted by average value shares:<ref>PPI manual, p. 610</ref><ref>[http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/tornqvist+index+and+other+log-change+index+numbers "Tornqvist Index"] {{Webarchive|url=https://web.archive.org/web/20131224111339/http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/tornqvist+index+and+other+log-change+index+numbers|date=24 December 2013}}</ref>

: <math>P_{T} = \prod_{i=1}^{n} \left(\frac{p_{i,t}}{p_{i,0}}\right)^{\frac{1}{2} \left[\frac{p_{i,0} \cdot q_{i,0}}{\sum (p_{0} \cdot q_{0})} + \frac{p_{i,t} \cdot q_{i,t}}{\sum (p_{t} \cdot q_{t})}\right]}</math>

It weights price relatives by economic importance (average expenditure shares), providing precision over Marshall-Edgeworth’s simpler averaging, but it’s data-intensive, needing detailed expenditure data.<ref>PPI manual, p. 610</ref>

===== Walsh index =====
Uses geometric quantity averages:<ref>PPI manual, p. 610</ref>

: <math>P_{W} = \frac{\sum (p_{t} \cdot \sqrt{q_{0} \cdot q_{t}})}{\sum (p_{0} \cdot \sqrt{q_{0} \cdot q_{t}})}</math>

It reduces bias from period-specific weighting with geometric averaging, outperforming Marshall-Edgeworth’s arithmetic mean in theoretical alignment, though it shares superlative data demands.<ref>PPI manual, p. 610</ref>

=== 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>

=== Geometric mean index ===
Weighted by base-period expenditure shares:<ref>PPI manual</ref>

: <math>P_{GM} = \prod_{i=1}^{n} \left(\frac{p_{i,t}}{p_{i,0}}\right)^{\frac{p_{i,0} \cdot q_{i,0}}{\sum (p_{0} \cdot q_{0})}}</math>

A [[geometric mean]] of price relatives, it weights by economic importance, offering stability over arithmetic means like Laspeyres, but it’s fixed to base-period behavior.<ref>PPI manual</ref>