Editing Price index (section)

== Calculation methods ==

=== Normalization ===
Indices are often normalized so the base period equals 100, with later values as percentages. For example, if 2010 = 100 and prices rise 5% by 2011, the index is 105, showing a 5% increase.<ref>Diewert (1993), 75–76.</ref>

=== Chained vs. unchained calculations ===
Unchained indices compare all periods to a fixed base (e.g., 2010), amplifying bias over time as quantities diverge. Chained indices update the base each period, calculating period-to-period changes (e.g., 2010 to 2011, 2011 to 2012) and multiplying them:<ref>ILO CPI manual</ref>

: <math>P_{t_n} = \prod_{i=1}^{n} \frac{\sum (p_{c,t_i} \cdot q_{c,t_{i-1}})}{\sum (p_{c,t_{i-1}} \cdot q_{c,t_{i-1}})}</math>

Chaining reduces bias (e.g., reflecting recent substitution) but requires more data and can drift if errors accumulate.<ref>ILO CPI manual</ref>

=== Relative ease of calculating the Laspeyres index ===
The Laspeyres index is simpler to compute than Paasche or bilateral indices because it fixes quantities at <math>t_0</math>, requiring only price updates (e.g., monthly price surveys) without new quantity or expenditure data each period. Paasche needs current quantities, and superlatives like Fisher demand both, increasing data and computational demands.<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>

=== Calculating indices from expenditure data ===
Sometimes, especially for aggregate data, expenditure data (<math>E_{c,t} = p_{c,t} \cdot q_{c,t}</math>) are more readily available than quantity data. If expenditure and base prices are known, indices can be computed without direct quantities. For example, Laspeyres becomes:<ref>Statistics New Zealand; ''Glossary of Common Terms'', [http://www2.stats.govt.nz/domino/external/omni/omni.nsf/0/7d11773239ed84eecc256f350016b841?OpenDocument "Laspeyres Index"] {{Webarchive|url=https://web.archive.org/web/20120206210828/http://www2.stats.govt.nz/domino/external/omni/omni.nsf/0/7d11773239ed84eecc256f350016b841?OpenDocument|date=2012-02-06}}</ref>

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

This uses expenditure shares and price relatives, a practical approach when quantities are hard to measure, though it retains Laspeyres’ fixed-weight assumptions.<ref>Statistics New Zealand; ''Glossary of Common Terms'', [http://www2.stats.govt.nz/domino/external/omni/omni.nsf/0/7d11773239ed84eecc256f350016b841?OpenDocument "Laspeyres Index"]</ref>