Index numbers: meaning and construction

Detail

1. What an index number is

• An index number is a statistical tool. It measures the average change in a group of related variables between two situations.
• The two situations can be two years (time) or two places (space).

• It is usually written as a percentage, but the "%" sign is left out.

• Base period: the period we compare against. Its value is set at 100.

• An index of 250 means the value is 2.5 times the base value, a rise of 150%.

• An index of 90 means a 10% fall from the base.

• Current period: the period being compared with the base.

• Price index: measures changes in the prices of a chosen set of goods. This is the most widely used kind.

• Examples: CPI (Consumer Price Index, which tracks retail prices paid by households) and WPI (Wholesale Price Index, which tracks prices in bulk trade).

• Quantity index: measures changes in physical volume, such as production, construction or employment.

• Example: IIP (Index of Industrial Production). MoSPI describes it as a measure of "changes in the volume of production" of an item basket compared with its base year [6].

• Official terms in India's CPI 2024 series [3]:

• Index reference period is the period when the index equals 100. For CPI it is 2024 = 100.
• Weight reference period is the period used to work out the weights. For CPI it is HCES 2023-24 (Household Consumption Expenditure Survey).
• Price reference period is the period when base prices are collected. For CPI it is calendar year 2024.
• So the year the weights come from can differ from the year set at 100.

2. Why we need index numbers (Class 11)

• Money income is not real income.
• A worker earned ₹1,000 in 1982 and earns ₹12,000 today. That is 12 times more rupees.
• Suppose the consumer price index rose from 100 (1982) to 600 (today). This figure is only for illustration.
• Then the real wage = 12,000 ÷ (600/100) = ₹2,000 at 1982 prices.

• So the worker is 2 times better off, not 12 times. Dividing a money value by a price index in this way is called deflating.

• Measuring inflation. Thousands of prices move in different directions. We need one number that sums them all up.

• MoSPI's formula: Inflation rate (%) = (Iₜ − Iₜ₋₁₂) / Iₜ₋₁₂ × 100. Here Iₜ is the CPI for month t, and Iₜ₋₁₂ is the CPI for the same month last year [3].

• Example: CPI (2024 = 100) inflation was 2.75% in January 2026 (provisional). Rural was 2.73% and urban 2.77% [2].

• Other uses:

• Revising dearness allowance (DA, the extra pay given to cover price rises) through CPI-IW, the CPI for Industrial Workers.
• Guiding monetary policy, since the RBI targets CPI inflation.
• Converting nominal GDP into real GDP.
• Tracking industrial output through IIP.

3. Worked data used throughout (Class 11, Examples 1–3)

Commodity	p₀	q₀	p₁	q₁	Price relative (p₁/p₀×100)	Weight W
A	2	10	4	5	200	40
B	5	12	6	10	120	30
C	4	20	5	15	125	20
D	2	15	3	10	150	10

• p₀ and q₀ are the base-period price and quantity. p₁ and q₁ are the current-period price and quantity.
• W is the base-period expenditure share, as given by NCERT.

4. Aggregative methods (work with totals of prices)

(a) Simple aggregative price index

• Formula: ΣP₁ / ΣP₀ × 100
• Worked: (4+6+5+3) / (2+5+4+2) × 100 = 18/13 × 100 = 138.5. Prices rose 38.5%.
• Weaknesses:
• Prices are quoted in different units (per kg, per litre, per piece). Adding them together has little meaning, and changing a unit changes the answer.
• It is unweighted. Every item counts equally, even though food takes a far bigger share of spending than salt.

(b) Weighted aggregative price index

• A weighted index takes the relative importance of items into account. The weights can be quantities or expenditure shares.
• General formula: Σp₁q / Σp₀q × 100. The same fixed basket (q) is priced in both periods.
• Because the basket does not change, any change in its cost comes only from prices.

(c) Laspeyres price index (base-period quantities)

• Formula: Σp₁q₀ / Σp₀q₀ × 100
• Worked:
• Σp₁q₀ = 4×10 + 6×12 + 5×20 + 3×15 = 40+72+100+45 = 257
• Σp₀q₀ = 2×10 + 5×12 + 4×20 + 2×15 = 20+60+80+30 = 190

• Index = 257/190 × 100 = 135.3

• NCERT error: the denominator is printed as 100. The correct figure is 190.

• Meaning: if the base-year basket cost ₹100 then, the same basket costs ₹135.3 now.
• Why official statisticians like it: the base quantities are collected only once. Only prices need to be collected every month.
• In Indian practice:
• WPI and PPIs: compiled with the Laspeyres formula, including the new series with base 2022-23 [4][5].
• IIP: new series (base 2022-23) uses a "Laspeyres fixed base type" formula. Each item's weight is multiplied by its production relative (current production ÷ base production) [6].
• CPI 2024 (top level): the higher-level indices use the Young / Modified Laspeyres index [3]. A Young index takes its weights from one period (HCES 2023-24) and its base prices from another (2024).

(d) Paasche price index (current-period quantities)

• Formula: Σp₁q₁ / Σp₀q₁ × 100
• Worked:
• Σp₁q₁ = 4×5 + 6×10 + 5×15 + 3×10 = 20+60+75+30 = 185
• Σp₀q₁ = 2×5 + 5×10 + 4×15 + 2×10 = 10+50+60+20 = 140

• Index = 185/140 × 100 = 132.1

• Meaning: how much more today's basket costs now than it would have cost in the base year.

• Drawback: quantity data are needed every period, which is costly. Also, each period's index uses a different basket.
• Link to the Deflator (Class 12, National Income):
• The GDP deflator = Nominal GDP ÷ Real GDP × 100.
• Nominal GDP values today's output at today's prices. Real GDP values the same output at base prices.
• So the deflator works like a Paasche-type index (current basket).
• CPI works like a Laspeyres-type index (fixed basket).
• The deflator covers all goods and services produced in the country. CPI covers only what households buy, and it includes imports.

(e) Substitution bias and Fisher's ideal index

• Substitution bias: when a good's price rises fast, consumers switch to cheaper goods.
• Laspeyres keeps the old basket, so it gives too much weight to goods that became dear. It tends to overstate the rise in the cost of living.
• Paasche uses the new basket, which is already tilted towards goods that stayed cheap. It tends to understate the rise.

• In our data: Laspeyres 135.3 > Paasche 132.1.

• Fisher's ideal index (beyond NCERT) = √(Laspeyres × Paasche) = √(135.3 × 132.1) ≈ 133.7.

• It lies between the two numbers and reduces the bias in both.

5. Method of averaging relatives (works with ratios, so units drop out)

(a) Price relative

• Formula: (p₁/p₀) × 100. It is the index for one good.
• Commodity A: 4/2 × 100 = 200. Its price has doubled.
• A relative has no unit, so it removes the "per kg vs per litre" problem of the simple aggregative method.

(b) Simple average of relatives (unweighted)

• Formula: (1/n) Σ(p₁/p₀) × 100
• Worked: ¼ (2 + 1.2 + 1.25 + 1.5) × 100 = ¼ (5.95) × 100 = 149 (148.75 rounded).
• Geometric-mean version (beyond NCERT):
• India's CPI 2024 uses the Jevons index for its elementary indices [3]. An elementary index is the lowest-level index, built from price quotes for a single item.
• The Jevons index is the geometric mean of price relatives, calculated where no weights are available.
• A geometric mean gives a lower figure than a simple average. This partly allows for shoppers switching to cheaper varieties.

(c) Weighted index of price relatives

• Formula: ΣWR / ΣW. R is the price relative and W is the weight, usually the base-period expenditure share.
• Worked: (40×200 + 30×120 + 20×125 + 10×150) / 100 = (8,000 + 3,600 + 2,500 + 1,500)/100 = 15,600/100 = 156, a 56% rise.
• Why it is higher than the unweighted 149: item A has the largest weight (40), and its price doubled.
• Link to Laspeyres: if W is the actual base expenditure share (p₀q₀), the weighted average of relatives gives exactly the Laspeyres index.
• This is why official indices can be built as "weights × relatives". IIP explicitly uses the weight-relative approach with production relatives [6].

6. Choosing weights and the base period

• Why base-period weights are preferred (NCERT):
• Recalculating weights every year is inconvenient.

• Indices with changing weights value different baskets, so they are not strictly comparable over time.

• Where weights come from in India:

• CPI 2024: weights come from HCES 2023-24 [3].

• IIP 2022-23: sector weights come from each sector's share in GVA (Gross Value Added, the value of output minus the cost of inputs) at current prices in FY 2022-23 [5]. Item weights inside a 4-digit industry are shared out by GVO (Gross Value of Output) as per ASI 2022-23 (Annual Survey of Industries) [6].

• Weights change as economies change. Share of Food & Beverages in CPI [3]:

• 45.86% in CPI 2012.
• 36.75% in CPI 2024, under the new classification.
• Using the old (2012) classification, the 2024 share would be 40.10%.

• Part of the fall comes from moving to COICOP 2018 (Classification of Individual Consumption According to Purpose), the UN Statistics Division's international grouping of household spending.

• Choosing a base year: it should be a normal, stable year, not a year of war, drought or pandemic. It should also match the base years of other indicators.

• MoSPI chose 2022-23 for IIP to line up with GDP and WPI [6].

• Why bases are revised: to reflect changes in the structure of the economy, new technology, and new products and industries [6].

7. Base revision and splicing (India's index series)

• Linking (splicing) factor: when a new base is adopted, old figures are converted to the new base so the time series stays continuous.
• Formula: LF = (average of the new series in the overlap period) ÷ (average of the old series in the same period).
• The average used is the geometric mean [3][6].

• Old index × LF = the same value expressed on the new base.

• CPI (MoSPI, Rural/Urban/Combined):

• First released in January 2011 with base 2010 = 100 [3].
• Base moved to 2012 = 100.
• Base revised to 2024 = 100, released 12 February 2026 [2].
• Weighted items rose from 299 to 358: goods from 259 to 308, services from 40 to 50 [2][3].
• Price collection covers 1,465 rural markets and 1,395 urban markets in 434 towns, plus 12 online markets [2][3].
• Structure: 12 Divisions, 43 Groups, 92 Classes and 162 Sub-classes [3].
• Prices are collected on tablets through CAPI (Computer Assisted Personal Interview) [3].
• The overlap year for linking is 2025. The general-level linking factors for the Rural, Urban and Combined series are about 0.52–0.53 (0.5222, 0.5320, 0.5267) [3].

• (NCERT: base 2012 = 100.)

• CPI-IW (Labour Bureau):

• Base moved from 2001 = 100 to 2016 = 100 in 2020 [7].
• Linking factor = 2.88 [7].

• Coverage: 88 centres (up from 78), 317 markets (up from 289) and 463 items (up from 392) [7].

• WPI (Office of the Economic Adviser, DPIIT):

• Base revised from 2011-12 to 2022-23. Approved 25 May 2026 and released 15 June 2026, together with new Producer Price Indices (PPIs) [4][5].
• Items rose from 697 to 957. Price quotations rose from 8,331 to 15,254 [4][5].

• (NCERT: base 2011-12.)

• IIP (NSO, MoSPI):

• Base revised from 2011-12 to 2022-23. Release was planned for 1 June 2026 [6].
• This is the 10th revision. The first IIP had base 1937 [6].
• Item groups rose from 407 to 463, including 120 new item groups [6].
• New coverage: minor and rare-earth minerals, gas supply, and water supply, sewerage and waste management [6].
• Renewable electricity now has its own separate index [6].
• Released monthly, 28 days after the reference month [6].
• (NCERT: base 2011-12.)

Prelims Hooks

• An index of 250 means the value is 2.5 times the base value. The base period always equals 100.
• Laspeyres = Σp₁q₀/Σp₀q₀ uses base quantities and tends to overstate inflation. Paasche = Σp₁q₁/Σp₀q₁ uses current quantities and tends to understate it. Fisher = √(L × P).
• A weighted average of price relatives with base expenditure-share weights gives the Laspeyres index.
• GDP deflator is a Paasche-type index. It covers all domestic output and excludes imports. CPI is a Laspeyres-type, fixed-basket index that includes imported consumer goods.
• IIP is a quantity (volume) index, not a price index. It is compiled by NSO, MoSPI. The new base is 2022-23 with a Laspeyres fixed-base formula [6].
• WPI is compiled by the Office of the Economic Adviser, DPIIT, not MoSPI. The new base is 2022-23, with 957 items [4][5].
• CPI (2024 = 100): weights from HCES 2023-24, 358 items, COICOP 2018. Elementary indices use Jevons (geometric mean). Higher levels use Young / Modified Laspeyres [2][3].
• CPI-IW (Labour Bureau) moved to base 2016. Linking factor is 2.88. Main use is setting DA [7].
• Trap: the "weight reference period" (HCES 2023-24) and the "index reference period" (2024) of the new CPI are not the same [3].

Mains Points

• Choice of formula affects policy.
• Fixed-basket (Laspeyres-type) indices can overstate the cost of living because of substitution bias.
• This matters when CPI is used for the RBI's inflation target, for DA through CPI-IW, and for indexing wages and pensions.

• Regular base revision and geometric-mean elementary indices (Jevons in CPI 2024) reduce this bias [3].

• Old weights distort the picture.

• CPI 2012 carried a food weight of 45.86%. The 2024 series carries 36.75% [3].
• A food-heavy index lets vegetable-price shocks drive headline inflation and, through it, RBI decisions.

• Updating weights from HCES 2023-24 makes the index match what households actually buy today.

• Aligned base years improve the data.

• GDP, WPI and IIP now share the base 2022-23 [4][6].
• Common bases make deflation (nominal to real) and comparisons across indicators consistent.

• The WPI-to-PPI shift moves India towards international practice (IMF manuals, COICOP) [3][4].

• Limits of index numbers:

• Quality changes are hard to measure. A price rise may partly reflect a better product.
• New goods enter only when the base is revised.
• Free public services are left out. The CPI excludes free social transfers, following the IMF CPI manual [3].
• Rural–urban and regional differences exist.
• So one headline index should be read together with sub-indices and other data.