Introduction to Index Numbers

Introduction

We often encounter situations where a group of related variables changes over time, but the changes are not uniform. For example, the prices of some commodities in a market may rise while others fall. Describing the individual rate of change for every single item can be confusing and impractical, especially when the number of items is large. An index number is a statistical device that provides a single, summary measure for these changes.

Index numbers help us analyze questions such as:

• Has the standard of living improved if a worker's salary has increased 12 times over 30 years, considering the rise in prices?
• What does it mean when the Sensex (a stock market index) rises or falls?
• How does the government measure the rate of inflation?

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What is an Index Number?

An index number is a statistical device for measuring changes in the magnitude of a group of related variables over two different situations. It is a measure of the average change in variables like prices, production volume, or cost of living.

Key Concepts

• Base Period: The period with which a comparison is made. The value in the base period is conventionally given the index number 100.
• Current Period: The period for which the change is being measured.
• Expression in Percentage: Index numbers are usually expressed in terms of percentage. An index number of 250 for the current period means that the value has increased to two and a half times that of the base period (a 150% increase).

Types of Index Numbers

• Price Index Numbers: Measure and permit comparison of the prices of a specific list of goods.
• Quantity Index Numbers: Measure the changes in the physical volume of production, construction, or employment.

While price index numbers are more common, a production index is also a vital indicator of an economy's output level.

Construction of an Index Number

There are two main methods for constructing a price index number: the aggregative method and the method of averaging relatives.

1. The Aggregative Method

This method involves aggregating (summing up) the prices of commodities in the base and current periods.

Simple Aggregative Price Index

The formula for a simple aggregative price index is:

$P_{01} = \frac{\sum P_1}{\sum P_0} \times 100$

Where $\sum P_1$ is the sum of the prices of all commodities in the current period, and $\sum P_0$ is the sum of prices in the base period.

This index is of limited use because it treats all items as having equal importance or weight, which is often unrealistic. For instance, a 10% rise in the price of rice has a much greater impact on a family's budget than a 10% rise in the price of salt.

Weighted Aggregative Price Index

To address this limitation, a weighted index is used, where the relative importance of each item is taken into account. The weights are typically the quantities consumed.

• Laspeyre’s Price Index: This method uses the base period quantities ($q_0$) as weights. It answers the question: "If the base period basket of commodities cost ₹100, how much would the same basket cost in the current period?"

  $P_{01} = \frac{\sum P_1 q_0}{\sum P_0 q_0} \times 100$

• Paasche’s Price Index: This method uses the current period quantities ($q_1$) as weights. It answers the question: "If the current period basket of commodities were consumed in the base period for ₹100, how much would it cost in the current period?"

  $P_{01} = \frac{\sum P_1 q_1}{\sum P_0 q_1} \times 100$

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2. The Method of Averaging Relatives

This method involves calculating the price relative for each commodity and then taking their average. The price relative for a commodity is the ratio of its current period price to its base period price, expressed as a percentage.

Price Relative = $\frac{P_1}{P_0} \times 100$

Simple Average of Relatives Index

This is the simple arithmetic mean of the price relatives.

$P_{01} = \frac{1}{n} \sum \left( \frac{P_1}{P_0} \times 100 \right)$

Weighted Average of Relatives Index

This is the weighted arithmetic mean of price relatives. The weights (W) are usually determined by the proportion of expenditure on each commodity in the total expenditure.

$P_{01} = \frac{\sum W \left( \frac{P_1}{P_0} \times 100 \right)}{\sum W}$

Some Important Index Numbers in India

1. Consumer Price Index (CPI)

The Consumer Price Index (CPI), also known as the cost of living index, measures the average change in the retail prices of a specific basket of goods and services consumed by a particular group of consumers.

Interpretation: A statement like "The CPI for industrial workers (2012=100) is 131.4 in May 2017" means that a basket of commodities that cost ₹100 in 2012 would cost ₹131.40 in May 2017. CPI is crucial for adjusting wages and salaries to account for changes in the cost of living.

The formula used is the weighted average of relatives, where the weights (W) are the percentage expenditure on each item group, and R is the price relative.

CPI = $\frac{\sum WR}{\sum W}$

In India, several CPIs are prepared, such as CPI for Industrial Workers (CPI-IW), CPI for Agricultural Labourers (CPI-AL), and the now prominent All-India Combined Consumer Price Index (CPI-C) with base year 2012=100.

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2. Wholesale Price Index (WPI)

The Wholesale Price Index (WPI) measures the change in the general price level at the wholesale level. Unlike the CPI, it does not have a reference consumer category and does not include services.

Interpretation: The statement "WPI with 2011-12 as base is 112.8 in May 2017" means that the general price level has risen by 12.8% during this period.

The WPI is widely used to measure the rate of inflation, often referred to as 'Headline Inflation'.

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3. Index of Industrial Production (IIP)

The Index of Industrial Production (IIP) is a quantity index that measures the changes in the physical volume of production in the industrial sector. With a base year of 2011-12=100, it tracks the output of various industries. The main branches are 'Mining', 'Manufacturing', and 'Electricity'. The Eight Core Industries (coal, crude oil, natural gas, etc.) have a combined weight of 40.27% in the IIP.

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4. Human Development Index (HDI)

The HDI is a composite index widely used to measure the overall development of a country, considering aspects like health, education, and standard of living.

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5. Sensex

The Sensex is the benchmark index of the Bombay Stock Exchange (BSE), with 1978-79 as its base year. It consists of 30 major stocks from leading sectors of the economy. A rising Sensex indicates a healthy market and growing investor confidence in the economy.

Issues in the Construction and Use of Index Numbers

Issues in Construction

Several important issues must be considered while constructing an index number to ensure it is meaningful and accurate.

• Purpose of the Index: The purpose must be clearly defined. For example, a value index cannot be used when a volume index is needed.
• Selection of Items: The items included in the index must be carefully selected to be as representative as possible for the group being studied.
• Choice of the Base Year: The base year should be a "normal" year, free from extreme fluctuations. It should also not be too far in the past, as consumption patterns and available goods change over time. Base years are routinely updated to maintain relevance.
• Choice of Formula: The choice between different formulas (e.g., Laspeyre's, Paasche's) depends on the specific question the index is intended to answer.
• Source of Data: The reliability of the data is crucial. Data from unreliable sources will produce misleading results.

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Index Numbers in Economics (Uses)

Index numbers are vital tools in economic policy-making and analysis.

• Policy Formulation: CPI is helpful in wage negotiations, income policy, price policy, and general economic policy formulation. WPI is used to measure the rate of inflation.
• Deflating Economic Aggregates: WPI is used to eliminate the effect of price changes on economic aggregates like National Income, converting them from nominal (current prices) to real (constant prices) terms.
• Calculating Purchasing Power and Real Wages: CPI is used to calculate the purchasing power of money and real wages.
  • Purchasing Power of Money = $\frac{1}{\text{Cost of Living Index}}$
  • Real Wage = $\frac{\text{Money Wage}}{\text{Cost of Living Index}} \times 100$
• Monitoring Production: The Index of Industrial Production (IIP) and Agricultural Production Index provide a quantitative measure of the performance of these sectors.
• Guiding Investment: The Sensex serves as a useful guide for investors in the stock market, reflecting market sentiment and economic health.

Conclusion

Index numbers are powerful statistical tools that enable us to calculate a single measure of change for a large number of related items. They can be constructed for various economic variables, including price, quantity, and volume. It is essential to construct and interpret them carefully, paying close attention to the items included, the choice of the base period, and the formula used.

Widely used index numbers like the WPI, CPI, IIP, and Sensex are indispensable in modern economic policy-making, helping to track inflation, adjust wages, monitor industrial performance, and guide investment decisions. They provide a concise and comprehensive way to understand complex economic changes over time.

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