# Index number

| Key fact | Detail |
|---|---|
| Main formulas | Laspeyres (1871) prices the base-period basket, Paasche (1874) the current-period basket, and Fisher's 1922 ideal index is the geometric mean of the two<sup>[1](https://www.imf.org/-/media/files/data/cpi/companion-publication/chapter-1-basic-index-number-theory.pdf)</sup> |
| Substitution bias | A fixed-basket Laspeyres index tends to overstate inflation because it ignores substitution away from goods whose prices rise; the C-CPI-U vs CPI-U gap has averaged about 0.25 percentage points per year<sup>[2](https://www.bls.gov/osmr/research-papers/2025/pdf/ec250020.pdf)</sup> |
| Boskin estimate | The 1996 Boskin Commission put total CPI bias at about 1.1 percentage points per year (plausible range 0.8 to 1.6), with 0.6 from quality change and new goods<sup>[2](https://www.bls.gov/osmr/research-papers/2025/pdf/ec250020.pdf)</sup> |
| CPI vs PCE | The CPI uses a modified Laspeyres formula with household-survey weights; the PCE price index uses a chained Fisher ideal formula with business-survey weights; from 2002 to 2007 the CPI rose 0.4 pp/year faster<sup>[3](https://apps.bea.gov/papers/pdf/cpi_pce.pdf)</sup> |
| Modern data | BLS now uses web-scraped gasoline prices and a new-vehicle transaction database; the UK ONS was scheduled to adopt GEKS-Törnqvist grocery scanner data in March 2026<sup>[2](https://www.bls.gov/osmr/research-papers/2025/pdf/ec250020.pdf)</sup><sup> • </sup><sup>[4](https://www.ons.gov.uk/economy/inflationandpriceindices/methodologies/howmultilateralindexmethodshelpusunderstandgroceryscannerdata)</sup> |

## What an index number is

The fixed-basket construction, now called the Lowe index, dates back to the Middle Ages and was studied in detail by the English economist Joseph Lowe in the early 1800s: value the same basket of goods at two sets of prices and take the ratio.<sup>[5](https://www.imf.org/en/-/media/files/publications/manuals-and-guides/2025/english/cpimea2025.pdf)</sup> The invention of the device is credited to the Italian G. R. Carli, who in 1764 reduced the prices of grain, wine, and oil in 1750 to percentages of their 1500 prices and averaged them.<sup>[6](https://fraser.stlouisfed.org/files/docs/publications/bls/bls_0656_1938.pdf)</sup>

Two refinements define the modern theory. The Laspeyres index, proposed in 1871, prices the base-period basket at current prices; the Paasche index, from 1874, prices the current-period basket at base-period prices. The two measures of aggregate price change can differ, sometimes substantially.<sup>[1](https://www.imf.org/-/media/files/data/cpi/companion-publication/chapter-1-basic-index-number-theory.pdf)</sup> [Irving Fisher](https://www.edgepedia.org/irving-fisher)'s 1922 ideal index takes the geometric mean of the two, which Diewert-era theory treats as the best symmetric average.<sup>[1](https://www.imf.org/-/media/files/data/cpi/companion-publication/chapter-1-basic-index-number-theory.pdf)</sup> A further distinction matters for interpretation: the European HICP is designed as a cost-of-goods index, measuring the changing cost of a fixed basket, not a cost-of-living index, which asks how much income a household needs to reach a fixed level of well-being.<sup>[7](https://ec.europa.eu/eurostat/documents/3859598/18594110/KS-GQ-24-003-EN-N.pdf)</sup>

## How an index is built

Official consumer price indexes are assembled in two stages. In the US CPI, the first stage calculates basic indexes for each of 7,776 item-area combinations; the second aggregates them with expenditure weights.<sup>[8](https://www.bls.gov/opub/hom/cpi/calculation.htm)</sup> At the lower level, most item strata use a weighted geometric mean of price ratios, a formula adopted in 1999 specifically to address lower-level substitution bias; a limited number of strata use a modified Laspeyres formula.<sup>[8](https://www.bls.gov/opub/hom/cpi/calculation.htm)</sup><sup> • </sup><sup>[2](https://www.bls.gov/osmr/research-papers/2025/pdf/ec250020.pdf)</sup> The scale of collection is large: at the time of the Boskin report, BLS gathered about 71,000 price quotations at roughly 22,000 outlets, plus rent data, organized into 207 item strata in 44 areas.<sup>[9](https://stanford.edu/~boskin/Publications/CPI.pdf)</sup>

Weights come from household spending surveys. The CPI's expenditure reference period as of 2023 was 2021, and BLS rotates it annually with the January release.<sup>[8](https://www.bls.gov/opub/hom/cpi/calculation.htm)</sup> In the European Union, HICP weights are mandated by Implementing Regulation No 2020/1148 to come primarily from national accounts data, supplemented by household budget surveys.<sup>[7](https://ec.europa.eu/eurostat/documents/3859598/18594110/KS-GQ-24-003-EN-N.pdf)</sup>

## The formula problem: Laspeyres, Paasche, Fisher, and beyond

**Why formulas disagree.** When households substitute away from goods whose relative prices rise, price and quantity changes are negatively correlated, and under these normal conditions the ranking of fixed-basket formulas is Lowe > Laspeyres > Fisher > Paasche; the gap grows the further back the weight period sits.<sup>[10](https://webapps.ilo.org/CPI/doc/ch1.pdf)</sup> A Laspeyres index therefore tends to overstate inflation, because it keeps weighting goods consumers are buying less of, while a Paasche index understates it.<sup>[11](https://www.newyorkfed.org/medialibrary/media/research/current_issues/ci3-6.pdf)</sup> The difference between the chained CPI-U (C-CPI-U, a Törnqvist formula) and the official CPI-U is a direct measure of this upper-level substitution bias and has historically averaged around 0.25 percentage points per year.<sup>[2](https://www.bls.gov/osmr/research-papers/2025/pdf/ec250020.pdf)</sup>

**Tests and superlative indices.** Laspeyres and Paasche both fail the time reversal test, which requires that swapping the two periods yield the reciprocal of the original index; the [Fisher index](https://www.edgepedia.org/fisher-index) is the only price index satisfying four minimal tests: positivity, time reversal, quantity reversal, and factor reversal.<sup>[1](https://www.imf.org/-/media/files/data/cpi/companion-publication/chapter-1-basic-index-number-theory.pdf)</sup><sup> • </sup><sup>[10](https://webapps.ilo.org/CPI/doc/ch1.pdf)</sup> W. Erwin Diewert's 1976 paper defined an aggregator as flexible if it gives a second-order approximation to an arbitrary twice-differentiable linearly homogeneous function, and an index as superlative if it is exact for a flexible aggregator; for the order-2 case the quadratic mean of order r reduces to Fisher's ideal index.<sup>[12](https://bpb-us-e1.wpmucdn.com/sites.psu.edu/dist/c/13885/files/2014/07/Diewert1976_Exact-and-Superlative-Index-Numbers.pdf)</sup> The Fisher, Törnqvist–Theil, and Walsh bilateral indices are all superlative and approximate each other to the second order, so the choice among them is of secondary importance.<sup>[5](https://www.imf.org/en/-/media/files/publications/manuals-and-guides/2025/english/cpimea2025.pdf)</sup><sup> • </sup><sup>[1](https://www.imf.org/-/media/files/data/cpi/companion-publication/chapter-1-basic-index-number-theory.pdf)</sup>

**Bounding the truth.** Under the economic approach, the observable Laspeyres index bounds the true cost-of-living index from above and the observable Paasche index bounds it from below, which is why a superlative index between them is attractive.<sup>[5](https://www.imf.org/en/-/media/files/publications/manuals-and-guides/2025/english/cpimea2025.pdf)</sup> The catch is timeliness: superlative indexes approximate true inflation very well but cannot be computed until the period's expenditures are observed, which is why initial PCE estimates are in fact computed as Laspeyres-type indexes.<sup>[11](https://www.newyorkfed.org/medialibrary/media/research/current_issues/ci3-6.pdf)</sup>

## Chain-weighting, chain drift, and fixed baskets

Chaining updates the basket every period, which reduces fixed-basket substitution bias but creates a new problem. A chain Laspeyres index drifts further and further above 100 when prices fluctuate and periodically return to their original levels, so chaining is not advised with fluctuating or seasonal prices.<sup>[10](https://webapps.ilo.org/CPI/doc/ch1.pdf)</sup> Chain drift is the gap between the direct and chained approaches: prices returning to base-period levels do not return the index to unity.<sup>[4](https://www.ons.gov.uk/economy/inflationandpriceindices/methodologies/howmultilateralindexmethodshelpusunderstandgroceryscannerdata)</sup> The 2004 CPI Manual endorsed superlative formulas and chained indexes, but with high-frequency retail scanner data the chaining approach fails Walsh's Multiperiod Identity Test, sometimes quite spectacularly.<sup>[5](https://www.imf.org/en/-/media/files/publications/manuals-and-guides/2025/english/cpimea2025.pdf)</sup>

The fix is multilateral. Methods such as GEKS-Fisher, CCDI, and GEKS-Walsh are free from chain drift by construction, unlike chained bilateral superlative indexes, and product churn is a key driver of drift bias.<sup>[13](https://www.tandfonline.com/doi/full/10.1080/07350015.2025.2537392)</sup> The HICP takes an intermediate route: it is a Laspeyres-type index obtained by annually chain-linking price indices, with weights updated every year and the price reference period December of the previous year.<sup>[7](https://ec.europa.eu/eurostat/documents/3859598/18594110/KS-GQ-24-003-EN-N.pdf)</sup>

## Quality change, new goods, and hedonics

The hardest measurement problem is that products change. The Boskin Commission estimated that of the +1.1 percentage point annual CPI bias, +0.6 came from quality adjustment and new goods and +0.10 from outlet substitution; Lebow and Rudd (2003) later estimated the overall bias at +0.87 and the new goods and quality component at +0.37, a substantial downward revision.<sup>[2](https://www.bls.gov/osmr/research-papers/2025/pdf/ec250020.pdf)</sup> Scanner data aggregated to the UPC level can form unit values used to generate superlative or hedonic price indexes, and a 2002 National Research Council panel argued scanner technology had the potential to improve the entire CPI collection process.<sup>[14](https://www.nationalacademies.org/read/26485/chapter/4)</sup>

## By the numbers

The fiscal stakes of index error are large because so much of the budget is indexed. The Boskin Commission concluded the CPI overstates the cost of living by about 1.1 percentage points per year, with a plausible range of 0.8 to 1.6, and calculated that over a dozen years the cumulative additional national debt from overindexing the budget would exceed $1 trillion.<sup>[9](https://stanford.edu/~boskin/Publications/CPI.pdf)</sup> The commission recommended a superlative formula for a retrospective annual index and a geometric mean formula for the monthly CPI, and estimated upper-level substitution bias at 0.15 percentage points per year.<sup>[15](https://www.nber.org/system/files/chapters/c5068/c5068.pdf)</sup> The gap between the chained and official CPI has been measured two ways: the BLS puts the historical average at about 0.25 percentage points per year,<sup>[2](https://www.bls.gov/osmr/research-papers/2025/pdf/ec250020.pdf)</sup> while an NBER analysis reports an initial-year difference of 0.8 percent coming down to about 0.3 percent per year.<sup>[15](https://www.nber.org/system/files/chapters/c5068/c5068.pdf)</sup> Between the two headline US measures, the CPI-U rose 0.4 percentage points per year faster than the PCE price index from 2002:Q1 to 2007:Q2, with almost half the gap explained by index-number formula differences alone.<sup>[3](https://apps.bea.gov/papers/pdf/cpi_pce.pdf)</sup>

## How it compares: CPI, PCE, and GDP deflator

The CPI and the PCE price index measure overlapping concepts with different machinery. The CPI is based on a modified Laspeyres formula with weights from household surveys; the PCE price index is based on a chained Fisher ideal formula with weights from business surveys.<sup>[3](https://apps.bea.gov/papers/pdf/cpi_pce.pdf)</sup> Scope differs sharply: in December 2004 weights, housing was 42.0 percent of the CPI but 23.2 percent of the PCE, while medical care was 6.1 percent of the CPI versus 20.3 percent of the PCE.<sup>[16](https://bea.gov/sites/default/files/papers/P2006-2.pdf)</sup> The PCE also includes expenditures made on behalf of households, by employers and government programs such as Medicare and Medicaid, where the CPI covers only out-of-pocket spending; about 74 percent of the PCE is CPI-type data.<sup>[16](https://bea.gov/sites/default/files/papers/P2006-2.pdf)</sup>

The GDP price index, though also chain-weighted, may be a worse cost-of-living measure than the CPI because it includes acquisition prices of long-lived and one-of-a-kind goods, and hard-to-measure government prices.<sup>[11](https://www.newyorkfed.org/medialibrary/media/research/current_issues/ci3-6.pdf)</sup> Despite the alternatives, a [Federal Reserve Bank of New York](https://www.edgepedia.org/federal-reserve-bank-of-new-york) review concluded there is no thoroughly viable alternative to the CPI as an easy-to-use, fairly reliable guide to inflation movements.<sup>[11](https://www.newyorkfed.org/medialibrary/media/research/current_issues/ci3-6.pdf)</sup>

## References

1. [Basic Index Number Theory (Diewert), CPI Manual companion chapter, IMF](https://www.imf.org/-/media/files/data/cpi/companion-publication/chapter-1-basic-index-number-theory.pdf)
2. [Measurement Issues in Consumer Price Indexes, BLS OSMR working paper (2025)](https://www.bls.gov/osmr/research-papers/2025/pdf/ec250020.pdf)
3. [A Reconciliation between the Consumer Price Index and the PCE Price Index, Bureau of Economic Analysis](https://apps.bea.gov/papers/pdf/cpi_pce.pdf)
4. [How multilateral index methods help us understand grocery scanner data, Office for National Statistics](https://www.ons.gov.uk/economy/inflationandpriceindices/methodologies/howmultilateralindexmethodshelpusunderstandgroceryscannerdata)
5. [Consumer Price Index Manual: Theory, 2025, IMF](https://www.imf.org/en/-/media/files/publications/manuals-and-guides/2025/english/cpimea2025.pdf)
6. [The Making and Using of Index Numbers, BLS Bulletin No. 656 (1938)](https://fraser.stlouisfed.org/files/docs/publications/bls/bls_0656_1938.pdf)
7. [Harmonised Index of Consumer Prices Methodological Manual 2024, Eurostat](https://ec.europa.eu/eurostat/documents/3859598/18594110/KS-GQ-24-003-EN-N.pdf)
8. [Calculation, CPI Handbook of Methods, U.S. Bureau of Labor Statistics](https://www.bls.gov/opub/hom/cpi/calculation.htm)
9. [Consumer Prices, the Consumer Price Index, and the Cost of Living (Boskin Commission report)](https://stanford.edu/~boskin/Publications/CPI.pdf)
10. [An Introduction to Consumer Price Index Methodology, ILO CPI Manual, Chapter 1](https://webapps.ilo.org/CPI/doc/ch1.pdf)
11. [Are There Good Alternatives to the CPI? Federal Reserve Bank of New York](https://www.newyorkfed.org/medialibrary/media/research/current_issues/ci3-6.pdf)
12. [Diewert (1976), Exact and Superlative Index Numbers, Journal of Econometrics 4](https://bpb-us-e1.wpmucdn.com/sites.psu.edu/dist/c/13885/files/2014/07/Diewert1976_Exact-and-Superlative-Index-Numbers.pdf)
13. [Inflation Measurement with High-Frequency Data, Journal of Business & Economic Statistics (2025)](https://www.tandfonline.com/doi/full/10.1080/07350015.2025.2537392)
14. [Modernizing the Consumer Price Index for the 21st Century, National Academies (2022)](https://www.nationalacademies.org/read/26485/chapter/4)
15. [Price Index Concepts and Measurement, NBER](https://www.nber.org/system/files/chapters/c5068/c5068.pdf)
16. [Comparing Price Measures: The CPI and PCE Price Index, BEA (Moyer)](https://bea.gov/sites/default/files/papers/P2006-2.pdf)

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