Index Numbers – Chapter Notes

1. Meaning and Interpretation of an Index Number

An index number is a specialised average that shows the relative change in one variable or a group of related variables between two periods or places. The period being compared is called the current period, while the reference period is called the base period. The base-period index is normally fixed at 100 so that the current index can be interpreted directly as a percentage of the base level.

Suppose the price of industrial oil rises from ₹200 per litre to ₹250. The price relative is 250 ÷ 200 × 100 = 125. The current price is therefore 125% of the base price. Since the first 100 represents the original level, the actual increase is only 25%.

A simple index measures one variable, such as the price of one grade of steel. A composite index combines several related variables, such as a consumer price index based on food, housing, clothing, fuel and transport. Most economic indices are composite because a single commodity cannot represent the movement of an entire market or the cost of living.

2. Decisions Before Constructing an Index

An index cannot be designed correctly until its purpose is clear. A cost-of-living index should contain items that affect the expenditure of the relevant class of consumers. Capital machinery would not normally belong in that basket. A production index, on the other hand, must represent the major outputs of the industry whose physical growth is being studied.

Selection of data and commodities

The commodities, markets, price quotations and units must be representative and comparable. If retail prices are collected in the base year but wholesale prices are used in the current year, the two observations do not measure the same thing. Index numbers are often constructed from samples, so the sample must properly represent the population.

Choice of base period

The base period should be normal, reasonably recent and unaffected by extraordinary conditions such as war, famine, a major strike or an abnormal boom. The base is only a reference point. As products and consumption patterns change, an old base becomes less representative and may need to be shifted.

Selection of weights

Weights show the economic importance of different items. In a household budget, cereals should normally influence the index more than an item purchased rarely. Equal treatment would distort the result even when every calculation is arithmetically correct.

Choice of average and formula

The geometric mean is theoretically more appropriate for averaging relatives, while the arithmetic mean is frequently used because it is simple. Different formulae can give different answers from the same data. The choice must therefore follow the purpose of the index and the information available.

3. Relatives and Simple Index Numbers

A relative compares a current observation with its base observation. Because it is a ratio, it is free from the original unit of measurement. This permits relatives based on kilograms, litres and pieces to be compared or averaged.

Simple aggregative price index

The current prices are added and compared with the total of the base prices. The method is easy but defective because an item having a larger numerical price receives more influence. If eggs are changed from a per-piece quotation to a per-dozen quotation, the index changes even though the actual price movement remains identical. The formula therefore fails the unit test.

Simple average of price relatives

This method first converts every price into a relative and then takes their arithmetic mean. It removes the unit problem, but it gives every commodity equal importance. A rarely purchased item can therefore influence the result as much as a major item.

The geometric-mean method is important for objective questions because the simple geometric mean of price relatives satisfies the circular test.

4. Weighted Price Index Numbers

Weighted indices correct the false equality of simple methods by giving every commodity influence according to its economic importance. In the notation used below, P means price, Q means quantity, 0 means the base year and n means the current year.

Laspeyres' Price Index

Laspeyres uses base-year quantities as weights. It asks what the old basket would cost at current prices compared with what that same basket cost in the base year. A useful memory link is Laspeyres = Last-year quantities. It may overstate a price rise because it ignores consumers substituting away from commodities that have become expensive.

Paasche's Price Index

Paasche uses current-year quantities. It values the present basket at both current and base prices. Remember Paasche = Present quantities. It can understate a price rise because the current basket may already reflect substitution towards cheaper goods.

Marshall–Edgeworth Price Index

This formula combines the base-year and current-year quantity patterns. Although it is described as using their average, the factor one-half is unnecessary because it would appear in both numerator and denominator and cancel.

Fisher's Ideal Price Index

Fisher is the geometric mean, not the arithmetic mean, of Laspeyres and Paasche. It is called an ideal index because it satisfies both the time reversal and factor reversal tests.

Weighted average of price relatives

When the weights are base-year values, W = P₀Q₀. The weighted arithmetic mean of price relatives then gives the same result as the Laspeyres price index because P₀ cancels when R is multiplied by P₀Q₀.

5. Quantity Indices and the Value Index

A price index measures price movement, whereas a quantity index measures changes in physical volume or output. The form of the formula remains the same, but prices and quantities exchange their roles. Prices become the weights used to combine quantities.

Value index

Value equals price multiplied by quantity. A value index therefore records their combined effect. If sales value rises, the value index alone cannot tell whether the rise came from higher prices, greater quantities or both.

6. Fixed Base, Link Relatives and Chain Indices

In a fixed-base series, every year is compared with one common base. In a chain-base series, every year is first compared with the immediately preceding year. This year-to-year comparison is called a link relative.

If the link relatives for 2025 and 2026 are 110 and 120, with 2024 = 100, the chain index for 2025 is 110. The 2026 index is 120 × 110 ÷ 100 = 132. The two changes must be multiplied rather than added: a 10% rise followed by a 20% rise produces a total increase of 32%.

7. Deflating, Shifting the Base and Splicing

Deflating a monetary value

A current money value contains the effects of both real change and price change. Deflation removes the price effect and expresses the current value at base-period prices.

If nominal salary is ₹30,000 and the consumer price index is 150, real salary at base prices is ₹30,000 × 100 ÷ 150 = ₹20,000. Purchasing power moves inversely with prices.

At an index of 125, the purchasing power of one current rupee is 100 ÷ 125 = ₹0.80 in base-year terms.

Shifting the base

Dividing every index by the index of the selected new base year makes that year exactly 100 and rescales all other years consistently.

Splicing two index series

Splicing joins two series that have different bases, normally because weights, commodities or the method of construction have changed. The common overlapping year is used as the linking point. One series is proportionately converted to the base of the other before the two are joined.

8. Four Tests of Adequacy

The tests of adequacy examine whether an index-number formula behaves consistently when the unit, time direction, contributing factors or base period changes.

Time reversal test

The forward index and the backward index should be reciprocals. If percentage-form indices are used rather than ratio-form indices, their product will be 10,000 instead of 1. Fisher satisfies this test because reversing time reverses both component ratios inside its geometric mean.

Factor reversal test

A price index isolates price change and a quantity index isolates quantity change. Their product should reconstruct the total change in value. Fisher satisfies this condition, which is one of the reasons it is described as ideal.

Circular test

This is an extension of time reversal across several periods. It permits consistent shifting of the base from one period to another. Laspeyres, Paasche and Fisher do not satisfy the circular test.

9. CPI, WPI and Stock-Market Indices

Consumer Price Index

The Consumer Price Index, also called the cost-of-living or retail price index, measures how a change in the prices of a representative basket affects the purchasing power of a specified class of consumers. Under the family-budget method, group indices are combined using expenditure weights.

Wholesale Price Index

The Wholesale Price Index measures relative changes in the prices of commodities traded at the wholesale level. The distinction between CPI and WPI is therefore not merely mathematical; their baskets, market levels and purposes are different.

Stock-market indices

A stock-market index summarises the movement of a selected basket of shares and acts as a benchmark. In a market-capitalisation-weighted index, companies having a larger market value receive a larger weight. The constituent companies need not remain unchanged permanently, so suitable adjustments preserve continuity when the basket changes.

10. ICAI MCQ Decoder

The attached question booklet repeatedly tests interpretation and formula selection, not merely arithmetic. The following explanations address the logic behind the most common distractors.

An index of 250 is a 150% increase

The base index already contains the original 100%. Therefore, index 250 means that the current level is 2.5 times the base and the increase is 250 − 100 = 150%.

“Current price is 1.25 times” and “increased by 1.25 times”

If current price is 1.25 times the base price, the index is 125. If it has increased by 1.25 times the base, current price becomes 1 + 1.25 = 2.25 times the base and the index is 225. Some objective questions depend upon this exact wording distinction.

Equal percentage rise and fall do not cancel

A 20% increase followed by a 20% decrease gives 1.20 × 0.80 = 0.96. The final level is therefore 4% below the original. The decrease is calculated on the increased base, not on the original base.

Real wages and compensation

Real wage equals money wage × 100 ÷ CPI. If a worker must retain the original purchasing power, required current wage equals base wage × CPI ÷ 100. Additional dearness allowance is the difference between the required wage and the actual current wage.

Fisher is a geometric mean

Fisher's index must normally lie between Laspeyres and Paasche and is calculated as √LP. An option based on (L + P) ÷ 2 is an arithmetic-mean distractor.

Recognising the formula before calculating

First identify whether the question asks for a price, quantity or value index. Then examine the weight. Q₀ signals Laspeyres price index, Qₙ signals Paasche price index, P₀ signals Laspeyres quantity index, and Pₙ signals Paasche quantity index.

11. Worked Examples

Example 1: Arithmetic mean of price relatives

The base and current prices of four commodities are (5, 7), (8, 10), (25, 32) and (6, 12). Their price relatives are 140, 125, 128 and 200 respectively.

The phrase “method of relatives using arithmetic mean” tells us to average the individual relatives. Adding the raw current prices and dividing by the raw base-price total would be a different method.

Example 2: Laspeyres, Paasche and Fisher

Laspeyres = 63 ÷ 52 × 100 = 121.15. Paasche = 59 ÷ 52 × 100 = 113.46. Fisher = √(121.15 × 113.46) = approximately 117.24.

Example 3: Purchasing power and real wage

If CPI is 313 and the money wage is ₹160, real wage = 160 × 100 ÷ 313 = ₹51.12. The reciprocal relationship explains why real purchasing power falls when the price index rises.

Example 4: Shifting the base

Suppose the old-base indices for 2024, 2025 and 2026 are 120, 150 and 180. If 2025 becomes the new base, the converted series is 120 ÷ 150 × 100 = 80, 150 ÷ 150 × 100 = 100, and 180 ÷ 150 × 100 = 120.