| Non-Rationalised Economics NCERT Notes, Solutions and Extra Q & A (Class 9th to 12th) | |||||||||||||||||||
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Chapter 8 Index Numbers
1. Introduction
In the previous chapters, we explored how to summarize a single set of data using measures like mean, median, and mode. However, in economics, we often need to understand the collective change in a group of related variables. For instance, when we go to the market, we notice that the prices of many different commodities have changed—some have increased, some have decreased, and others have remained the same. Simply describing the price change for each individual item would be confusing and impractical.
Similarly, to understand the performance of the industrial sector, we need a way to summarize the output changes across numerous subsectors. A single, representative figure is needed to capture the overall trend. Index numbers provide this solution.
Consider these real-world questions:
- An industrial worker's salary increased from $\textsf{₹}$1,000 in 1982 to $\textsf{₹}$12,000 today. Has their standard of living truly increased 12 times? How do we account for the rise in prices (inflation)?
- Newspapers report that the SENSEX has crossed 8000 points. What does this number actually represent?
- How do governments measure the rate of inflation to make policy decisions?
The study of index numbers provides the tools to answer such questions by summarizing complex changes into a single, understandable figure.
2. What Is An Index Number
An index number is a specialized statistical device designed to measure the average change in a group of related variables over time or between different situations. It acts as a barometer, summarizing the general trend from a multitude of diverging individual changes.
Key characteristics of an index number include:
- It's a Measure of Relative Change: It compares the level of a variable in a current period with its level in a chosen base period.
- Expressed as a Percentage: For ease of comparison, the value of the index number for the base period is always taken as 100.
- Interpretation: The index number for the current period shows the percentage change relative to the base period. For example, if the price index for 2020 (with a 2010 base) is 150, it means that prices, on average, have increased by 50% since 2010.
Types of Index Numbers:
- Price Index Numbers: These measure the general change in the prices of a specific group of commodities. They are the most widely used type of index number.
- Quantity Index Numbers: These measure the change in the physical volume of items, such as the volume of industrial production, agricultural output, or employment.
3. Construction Of An Index Number
There are two primary methods for constructing a price index number:
- The Aggregative Method
- The Method of Averaging Relatives
The Aggregative Method
This method involves aggregating (summing up) the prices of commodities in the base and current periods.
Simple Aggregative Method:
This is the most basic method, where the sum of the prices of all commodities in the current period ($\sum P_1$) is divided by the sum of their prices in the base period ($\sum P_0$) and expressed as a percentage.
Formula: $P_{01} = \frac{\sum P_1}{\sum P_0} \times 100$
Limitations of the Simple Aggregative Method:
- It treats all commodities as equally important, which is unrealistic. For example, a 10% rise in the price of salt has a much smaller impact on a family's budget than a 10% rise in the price of wheat.
- It is influenced by the units in which prices are quoted. A commodity with a high price per unit (like ghee per kg) will have a much larger influence on the index than a commodity with a low price per unit (like matchboxes per pack), regardless of their actual importance.
Weighted Aggregative Method:
To overcome the limitations of the simple method, the weighted aggregative method is used. In this method, each commodity is assigned a weight according to its relative importance in the group. For price indices, these weights are usually the quantities consumed or produced.
There are two main formulas for this method, differing only in the period from which the weights (quantities) are taken:
- Laspeyre’s Price Index: This method uses base period quantities ($q_0$) as weights. It measures the change in the cost of purchasing the base period's "basket of goods" at current prices.
Formula: $P_{01} = \frac{\sum P_1 q_0}{\sum P_0 q_0} \times 100$
- Paasche’s Price Index: This method uses current period quantities ($q_1$) as weights. It answers the question of what the current period's basket of goods would have cost in the base period.
Formula: $P_{01} = \frac{\sum P_1 q_1}{\sum P_0 q_1} \times 100$
Example 2. Calculation of weighted aggregative price index.
| Commodity | Base Period | Current Period | ||
|---|---|---|---|---|
| Price ($P_0$) | Quantity ($q_0$) | Price ($P_1$) | Quantity ($q_1$) | |
| A | 2 | 10 | 4 | 5 |
| B | 5 | 12 | 6 | 10 |
| C | 4 | 20 | 5 | 15 |
| D | 2 | 15 | 3 | 10 |
Answer:
Calculation for Laspeyre's Index:
We need $\sum P_1 q_0$ and $\sum P_0 q_0$.
$\sum P_1 q_0 = (4 \times 10) + (6 \times 12) + (5 \times 20) + (3 \times 15) = 40 + 72 + 100 + 45 = 257$
$\sum P_0 q_0 = (2 \times 10) + (5 \times 12) + (4 \times 20) + (2 \times 15) = 20 + 60 + 80 + 30 = 190$
Laspeyre's Index = $\frac{257}{190} \times 100 \approx 135.3$
This means prices have risen by 35.3% based on the base year's consumption pattern.
Calculation for Paasche's Index:
We need $\sum P_1 q_1$ and $\sum P_0 q_1$.
$\sum P_1 q_1 = (4 \times 5) + (6 \times 10) + (5 \times 15) + (3 \times 10) = 20 + 60 + 75 + 30 = 185$
$\sum P_0 q_1 = (2 \times 5) + (5 \times 10) + (4 \times 15) + (2 \times 10) = 10 + 50 + 60 + 20 = 140$
Paasche's Index = $\frac{185}{140} \times 100 \approx 132.1$
This means prices have risen by 32.1% based on the current year's consumption pattern.
Method of Averaging Relatives
This method calculates the price change for each individual commodity first, and then averages these changes.
A price relative for a single commodity is calculated as: $\frac{P_1}{P_0} \times 100$
Weighted Average of Price Relatives:
This is the most common method in this category. The price relative for each commodity is multiplied by its weight (W), and the sum of these products is divided by the sum of the weights.
Formula: $P_{01} = \frac{\sum (R \times W)}{\sum W}$ where $R$ is the price relative.
The weights (W) are typically the value shares of each commodity in the total expenditure during the base period ($W = P_0 q_0$).
Example 3. Calculation of weighted price relatives index.
| Commodity | Weight (W) | Base Price ($P_0$) | Current Price ($P_1$) |
|---|---|---|---|
| A | 40 | 2 | 4 |
| B | 30 | 5 | 6 |
| C | 20 | 4 | 5 |
| D | 10 | 2 | 3 |
Answer:
Step 1: Calculate the Price Relative (R) for each commodity.
A: $(\frac{4}{2}) \times 100 = 200$
B: $(\frac{6}{5}) \times 100 = 120$
C: $(\frac{5}{4}) \times 100 = 125$
D: $(\frac{3}{2}) \times 100 = 150$
Step 2: Calculate WR for each commodity and find $\sum WR$ and $\sum W$.
$\sum WR = (200 \times 40) + (120 \times 30) + (125 \times 20) + (150 \times 10) = 8000 + 3600 + 2500 + 1500 = 15600$
$\sum W = 40 + 30 + 20 + 10 = 100$
Step 3: Apply the formula.
Weighted Index = $\frac{\sum WR}{\sum W} = \frac{15600}{100} = 156$
The price index has risen by 56%.
4. Some Important Index Numbers
Consumer Price Index (CPI)
The Consumer Price Index (CPI), also known as the cost of living index, is designed to measure the average change in the retail prices of a specific basket of goods and services consumed by a particular group of people (e.g., industrial workers, agricultural laborers).
The government of India compiles several CPIs, including:
- CPI for Industrial Workers (CPI-IW)
- CPI for Agricultural Labourers (CPI-AL)
- CPI for Rural Labourers (CPI-RL)
Since 2012, a new series of CPI (Rural, Urban, and Combined) has been introduced, with the base year 2012=100. The CPI-Combined is now used by the Reserve Bank of India as the main measure of inflation affecting consumers.
Wholesale Price Index (WPI)
The Wholesale Price Index (WPI) measures the change in the general price level of goods at the wholesale level. Unlike the CPI, it does not include services and does not have a specific reference consumer group. The WPI is often used by the government to monitor inflation and is referred to as 'headline inflation'.
Index Of Industrial Production (IIP)
The Index of Industrial Production (IIP) is a quantity index. It measures the changes in the physical volume of output in the industrial sector. It is a key indicator of industrial growth and performance. It is compiled for major sectors like 'Mining', 'Manufacturing', and 'Electricity', and for specific "core" industries like coal, steel, and cement.
Human Development Index (HDI)
The HDI is a composite index used to measure the overall development of a country. It combines indicators of life expectancy, education, and per capita income.
Sensex
The SENSEX (short for Bombay Stock Exchange Sensitive Index) is a benchmark stock market index for India. It tracks the performance of 30 large, well-established, and financially sound companies listed on the BSE. A rising Sensex generally indicates investor optimism and a positive outlook for the economy, while a falling Sensex suggests the opposite.
5. Issues In The Construction Of An Index Number
Constructing a reliable and meaningful index number involves several critical considerations:
- Purpose of the Index: The objective must be clear. A price index cannot be used to measure changes in volume, and vice-versa.
- Selection of Items: The items included in the index must be representative of the group they are intended to cover. For a CPI, this means selecting goods and services that are actually consumed by the target population.
- Choice of Base Year: The base year should be a "normal" year, free from extreme events like wars, famines, or economic crises. It should also not be too far in the past, as consumption patterns and products change over time. Base years are routinely updated to maintain relevance.
- Choice of Formula: The appropriate formula (e.g., Laspeyre's, Paasche's) depends on the specific question being addressed and the availability of data.
- Source of Data: The data used must be reliable. Using data from untrustworthy sources will lead to misleading results.
6. Index Number In Economics
Index numbers are vital tools in economics and policy-making with numerous applications:
- Wage and Salary Adjustments: The CPI is widely used for wage negotiations. Governments often link dearness allowance (DA) for employees to changes in the CPI to protect their real income from inflation.
- Measuring Inflation: The WPI and CPI are the primary indicators used to measure the rate of inflation in an economy.
- Deflating Economic Data: Index numbers are used to adjust nominal values (which are at current prices) to real values (which are at constant prices). This process, known as deflating, removes the effect of price changes, allowing for a true comparison of physical output over time. For example, Real National Income = (Nominal National Income / Price Index) $\times$ 100.
- Calculating Purchasing Power and Real Wages:
- Purchasing Power of Money = 1 / Cost of Living Index
- Real Wage = (Money Wage / Cost of Living Index) $\times$ 100
- Gauging Economic Performance: The IIP provides a measure of industrial sector performance, while the SENSEX acts as a barometer of investor confidence and overall economic health.
7. Conclusion
Index numbers are indispensable statistical tools that allow us to distill complex changes in a large number of related items into a single, manageable measure. Whether it's price, quantity, or volume, an index number provides a clear picture of the trend over time.
The construction of an index number requires careful consideration of its purpose, the items to be included, and the choice of the base period and formula. When interpreted correctly, index numbers play a crucial role in economic analysis and are essential for sound policy-making.
Recap
- An index number is a statistical tool for measuring the relative change in a group of items.
- There are various formulas for constructing index numbers (e.g., Laspeyre's, Paasche's), and each requires careful interpretation.
- The choice of formula and methodology depends on the specific purpose of the index.
- Widely used index numbers in India include the Consumer Price Index (CPI), Wholesale Price Index (WPI), Index of Industrial Production (IIP), and SENSEX.
- Index numbers are indispensable for economic policy-making, including inflation management, wage adjustments, and assessing economic performance.
Exercises
This section contains questions for practice and self-assessment, designed to test the learner's understanding of the concepts discussed in the chapter, such as the properties of index numbers, the differences between various indices, and practical calculations involving price and quantity data.
Suggested Additional Activities
This section is not included in the provided text.