Index numbers: Laspeyres, Paasche and Fisher

A rupee buys less than it did ten years ago, but how much less? You cannot answer by looking at one price. Rice, pulses, oil and vegetables rise at different rates, and a household buys them in different amounts. A price index answers by pricing a fixed shopping basket in two years and comparing the bills.

It is built to be relative: the base year is set to 100 and every other year is expressed against it (Hyderabad 2020). An index of 234.7 means the basket costs 2.347 times what it did in the base year, a rise of 134.7%, not 234.7% (UGC NET 2023). Mixing up the index level with the percentage change is the commonest slip in this topic.

Throughout the lesson we use a household’s monthly basket in Ashanagar. The goods and quantities are invented and the prices illustrative.

The price relative of one good is its price now over its price in the base year, times 100: rice at ₹44 against ₹40 has a relative of 110. There are two simple ways to combine goods:

• Simple aggregative index: add up the prices now, add up the base prices, and divide: Σp₁ ÷ Σp₀ × 100.
• Average of price relatives: take the mean (arithmetic, or geometric) of the individual relatives.

Suppose rice rises 10% to ₹44, pulses 30% to ₹130, oil 10% to ₹143 and vegetables 20% to ₹36. The simple aggregative index is 353 ÷ 300 = 117.7, and the average of the relatives is (110 + 130 + 110 + 120) ÷ 4 = 117.5.

Both have the same flaw: they treat every good as equally important. Pulses count as much as rice, though a household spends half as much on them. The simple aggregative also depends on the units. Quote rice per quintal instead of per kilo (₹4,000 rising to ₹4,400) and the same price changes give an index of 4,709 ÷ 4,260 = 110.5. Nothing about the economy changed; the units did. An index should not behave like that. The fix is to weight each price by how much of the good is bought. In fact every worthwhile index is weighted, and the simple ones are rough shortcuts.

A weighted index prices a basket of quantities in both years. The question is which basket. Let p₀, p₁ be prices and q₀, q₁ quantities in the base and current years. The two classic answers:

Laspeyres answers: what would the old basket cost at today’s prices? It is the base-weighted index (Hyderabad 2014), the one that uses the quantity of goods in the base year as weights (JAM 2026). Paasche answers: what does today’s basket cost now, compared with what it would have cost then? It is the current-weighted index.

Two more are built from these. Fisher’s ideal index is their geometric mean, not the arithmetic or the harmonic mean (JNU 2020, JNU 2021, Hyderabad 2015). The Marshall-Edgeworth index uses the average of the two years’ quantities as weights.

A Laspeyres index can also be written as a weighted average of price relatives, with the base-year values p₀q₀ as weights (Hyderabad 2017):

In the basket above, the relatives are 110, 130, 110 and 120 and the base spending is 1,200, 600, 520 and 600, so P_L = (110 × 1,200 + 130 × 600 + 110 × 520 + 120 × 600) ÷ 2,920 = 116.2. The basket that cost ₹2,920 now costs ₹3,392. Compare this with the simple aggregative index of 117.7. Pulses, which jumped 30%, are a small part of the bill, and the weighted index knows it.

When the price of a good rises, households buy less of it and more of cheaper substitutes. Laspeyres keeps pricing the old basket, as if nobody had changed what they buy, so it counts the rise on goods people have already cut back on. It overstates the increase in the cost of living. Paasche prices the new basket, which has already shifted toward the goods that got cheaper in relative terms, so it understates it. Fisher, in between, is a compromise.

Below, set the price changes and how strongly households switch away from goods that got dearer. With no switching the four weighted indices agree. With switching, Laspeyres rises above Paasche. The last column of the table is the time-reversal test of the next section.

The practical cost is that Laspeyres needs only base-year quantities, which a statistics office collects once every few years in a household survey, while Paasche needs fresh quantities every period. That is why consumer price indices are Laspeyres-type: the weights are the composition of household consumption in the base year, held fixed until the next revision.

Laspeyres and Paasche side by side

The comparisons examiners ask most often, in one table.

Economists ask an index number to pass a few common-sense tests. Two are examined again and again.

• Time reversal test. Swap the two years and the index should become the reciprocal: P₀₁ × P₁₀ = 1. If prices rose 25% from year 0 to year 1, they should have fallen 20% going the other way.
• Factor reversal test. Swap the roles of price and quantity, and the price index times the quantity index should equal the value ratio: P₀₁ × Q₀₁ = Σp₁q₁ ÷ Σp₀q₀.

Laspeyres and Paasche pass neither. Their reversal product is P_L ÷ P_P and P_P ÷ P_L, which is 1 only when the two agree. Fisher’s index passes both (JNU 2020, Hyderabad 2016), which is why it is called ideal: the lab’s table shows its reversal product equal to 1.000 however you move the sliders. The simple aggregative and Marshall-Edgeworth indices pass the time reversal test too, though the factor reversal test is Fisher’s alone among these.

A third requirement, the circularity test, asks that P₀₁ × P₁₂ = P₀₂ so that a chain of year-on-year indices agrees with a direct comparison. It is what makes chain-base indices meaningful: each year is compared with the one before and the links are multiplied.

Inflation, and real versus nominal

The inflation rate is the percentage change in the price index between two dates. A real quantity is the nominal one divided by the price index (as a ratio to the base): the same rupees, valued at base-year prices.

That is the whole answer to the opening question. The CPI rises from 300 to 330: prices are up 10%. The money wage rises from ₹40 to ₹47: up 17.5%. The wage has grown faster than prices, so the real wage has risen, by 1.175 ÷ 1.10 = 1.068, nearly 7%. Nothing else follows: the data say nothing about employment, poverty or productivity (Hyderabad 2011). Another example: money wages up from ₹40 to ₹50 (+25%) while the CPI goes from 125 to 150 (+20%) gives a real wage index of (50 ÷ 150) ÷ (40 ÷ 125) = 104.2, a 4.2% gain.

The same deflating idea gives the GDP deflator, nominal GDP over real GDP, a Paasche-type index of the prices of everything produced. In an economy with apples and bananas where prices double between 2011 and 2012 and quantities stay put, nominal GDP doubles and real GDP, measured at base-year prices, does not, so the deflator goes from 100 to 200 and inflation is 100% (JAM 2023).

Which index measures what

• The consumer price index (CPI) is the most commonly used measure of the change in the cost of living for households (Hyderabad 2015). It is an aggregate index, a weighted combination of prices, and not a simple one (Hyderabad 2016).
• The wholesale price index (WPI) does not reflect the cost of living well because it includes producer goods, such as raw materials and intermediate goods, as well as consumer goods (Hyderabad 2017).
• The GDP deflator covers everything produced in the economy, while the CPI covers what households buy.

Contributions to the index

With fixed weights, a group’s contribution to the overall change is its weight times its own change. If fuel and power have a weight of 15% and their price index rises 20%, they add 0.15 × 20 = 3 percentage points to inflation, exactly (JNU 2013).

Relative prices and the terms of trade

The ratio of two indices is a relative price. If the agricultural price index goes from 200 to 215 and the manufactured goods index from 150 to 165, both rose by 15 points. But agriculture rose 7.5% and manufactures 10%. The terms of trade of agriculture, the ratio 200 ÷ 150 = 1.33 falling to 215 ÷ 165 = 1.30, moved in favour of industry (JNU 2013).

Base shifting and splicing

An index can be moved to a new base year. It is possible, contrary to what one distractor claims (Hyderabad 2020): divide the whole series by the value at the new base and multiply by 100. If 2010 = 100 and the index in 2015 is 144 and in 2020 is 180, then with 2015 as the base the 2020 index is 180 ÷ 144 × 100 = 125. When an agency revises its basket and starts a new series, splicing joins the old and new series using the year they overlap, in the same way.

An index is a time series: one number per period. Economists find it useful to split any time series into four pieces, and the first lesson on time series in an exam is usually to name them (CUET 2025):

• Trend, the steady long-run movement. It is also called the secular trend.
• Seasonal variation, the movement that depends on the time of year: umbrellas in the monsoon, gold in the wedding season.
• Cyclical variation, the periodic swings of the business cycle, longer than a year and not tied to the calendar.
• Irregular, or random, fluctuation, the movements due to unpredictable events.

A moving average smooths out the short-run noise by replacing each value with the average of its neighbours, which brings out the trend. Exponential smoothing is a forecasting method: tomorrow’s forecast is a weighted average of today’s observation and today’s forecast, giving more weight to recent data (Hyderabad 2014). Anything more, unit roots and models of the series, is for the econometrics module.