An index number measures the relative change in a variable (or group of variables) between a base period (index = 100) and a current period. We covered simple, Laspeyres, Paasche, Marshall–Edgeworth, Fisher, time-reversal and factor-reversal tests, CPI and WPI.
This unit deals with operations on index-number series — how to change the base, build chain-base series, splice two series and deflate nominal monetary series.
Base shifting means converting an index-number series with one base period into a series with a different base period — usually a more recent one, so the index numbers stay close to 100.
If the new base is the average of several years, divide by the average of the old indices for those years.
Original CPI series with 2018 = 100:
| Year | Index (2018=100) |
|---|---|
| 2018 | 100 |
| 2019 | 105 |
| 2020 | 112 |
| 2021 | 118 |
| 2022 | 125 |
| 2023 | 132 |
Shift base to 2021 (= 100). Divide each by 118 and multiply by 100:
| Year | Index (2021=100) |
|---|---|
| 2018 | (100/118)×100 = 84.7 |
| 2019 | (105/118)×100 = 89.0 |
| 2020 | (112/118)×100 = 94.9 |
| 2021 | 100.0 |
| 2022 | (125/118)×100 = 105.9 |
| 2023 | (132/118)×100 = 111.9 |
If new base is the average of 2019, 2020, 2021: average old index = (105 + 112 + 118)/3 = 111.67. Divide every old index by 111.67 and multiply by 100.
All comparisons made with a single fixed base period. Formula:
Each year's index is computed with the previous year as base (year-on-year link relative); then linked into a chain.
For a simple price series (or simple index without weights), FBI and CBI coincide. For weighted indices they may differ slightly because weights change each year.
(when weights are unchanged).
Price (per kg) of a commodity:
| Year | Price | Link relative | CBI (2018=100) |
|---|---|---|---|
| 2018 | 100 | — | 100 |
| 2019 | 110 | 110 | 110 |
| 2020 | 121 | 110 | 110×110/100 = 121 |
| 2021 | 133 | 110 | 121×110/100 = 133.1 |
| 2022 | 140 | 140/133 = 105.3 | 133.1×105.3/100 = 140.2 |
FBI in 2022 = 140/100 × 100 = 140 — practically equal to CBI 140.2 (tiny rounding only).
A weighted CBI accommodates changing baskets each year. E.g., the modern Consumer Price Index in many countries follows a Laspeyres-style chained formulation.
Splicing means joining two index-number series that have different bases into one continuous series. It is needed when an official agency revises its base year and publishes two non-overlapping series.
Adjust the old series (with old base) to align with the new base.
The overlap year is when both series report a value; in the new series the overlap year is = 100 by construction.
Adjust the new series to be expressed on the old base.
Old series (2015 = 100):
| Year | Old (2015=100) | New (2020=100) |
|---|---|---|
| 2015 | 100 | — |
| 2016 | 108 | — |
| 2017 | 116 | — |
| 2018 | 122 | — |
| 2019 | 128 | — |
| 2020 | 134 | 100 |
| 2021 | — | 106 |
| 2022 | — | 112 |
Forward splice: divide old series by 134 (overlap year) and ×100:
Using the same data, backward splice (new on old base): multiply new index by 134/100.
Combined backward-spliced series (all on 2015 = 100): 100, 108, 116, 122, 128, 134, 142.04, 150.08.
Deflating converts a nominal (money-of-the-day) series into a real (constant purchasing power) series by dividing by a price index. This isolates the genuine quantity change from price-level inflation.
Common uses:
Money wage ₹50 000 in 2020 (CPI = 110) and ₹65 000 in 2024 (CPI = 145).
Real wage 2020 = 50000/110 × 100 = ₹45 455.
Real wage 2024 = 65000/145 × 100 = ₹44 828.
Nominally wages have risen 30 %, but in real terms purchasing power has slightly declined.
Nominal GDP in 2024 = ₹300 lakh crore; GDP deflator (2015 = 100) = 150. Real GDP = 300/1.5 = ₹200 lakh crore at 2015 prices.
The Index of Industrial Production (IIP) measures the volume of industrial output in an economy, relative to a base year. Unlike price indices, IIP focuses on physical production quantities.
An interim index is released with provisional data soon after the reference month. It uses estimated weights and incomplete reporting; subject to revision.
After the final data is collected (1–2 months later) and weights are confirmed, the revised IIP replaces the interim figure. The revised series is what economists use for long-term analysis.
3 industries with production relatives and weights:
| Industry | R | w |
|---|---|---|
| Cement | 118 | 15 |
| Steel | 132 | 25 |
| Textiles | 108 | 10 |
IIP = (15·118 + 25·132 + 10·108)/(15+25+10) = (1770 + 3300 + 1080)/50 = 6150/50 = 123.