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Chapter 7 Index Numbers
In previous chapters, you learned to summarize data and describe changes in individual variables. This chapter introduces **index numbers**, which are statistical tools used to measure changes in a group of related variables collectively. Index numbers help summarize changes in price levels, production volumes, or cost of living over time, providing a single figure to represent the overall trend of change within a group of items.
Think about the bewildering experience of observing price changes for many items in a market, or the varied changes in output across different parts of an industry. Describing each individual change is impractical. Index numbers offer a way to summarize these changes into a single, comprehensible measure. They are useful for addressing everyday questions like the impact of inflation on living standards or understanding fluctuations in stock market indices like the Sensex.
Introduction
This chapter focuses on index numbers, statistical devices for summarizing changes in a group of related variables. Index numbers are useful for understanding overall trends when individual changes are numerous and varied. They are used in various contexts, such as measuring changes in prices, production, or the stock market.
What Is An Index Number
An **index number** is a statistical tool designed to measure relative changes in the magnitude of a group of related variables over different situations or periods. It captures the general trend of these changes and represents them as a single numerical measure. Comparisons are typically made between a base period (assigned an index value of 100) and subsequent periods (current periods). Index numbers are conventionally expressed as percentages.
If the base period has an index of 100, an index of 250 in a current period means the value has increased by 150% or is two and a half times the base period value. Index numbers can measure changes in various quantities, such as prices (Price Index Numbers), physical volume of production or employment (Quantity Index Numbers), cost of living, etc.
Construction Of An Index Number
The construction of an index number, illustrated through price index numbers, involves comparing values across a group of commodities between a base period and a current period. When percentage changes for individual commodities differ, a single index is needed to represent the overall change.
Example 1. Calculation of simple aggregative price index. Data for four commodities (A, B, C, D) with base period prices (P0) and current period prices (P1) are given.
Commodity A: P0=2, P1=4
Commodity B: P0=5, P1=6
Commodity C: P0=4, P1=5
Commodity D: P0=2, P1=3
Answer:
| Commodity | Base period price (Rs) (P0) | Current period price (Rs) (P1) | Percentage change |
|---|---|---|---|
| A | 2 | 4 | 100 |
| B | 5 | 6 | 20 |
| C | 4 | 5 | 25 |
| D | 2 | 3 | 50 |
The Aggregative Method
The aggregative method sums up the prices of commodities in the current period and the base period and compares the totals. A simple aggregative price index is calculated as $\frac{\sum P_1}{\sum P_0} \times 100$. Using Example 1: $\frac{4+6+5+3}{2+5+4+2} \times 100 = \frac{18}{13} \times 100 \approx 138.5$. This indicates a 38.5% price increase. However, this method has limitations as it doesn't account for units or the relative importance (weight) of items. In a weighted aggregative index, quantity weights ($q_0$ for base period, $q_1$ for current period) are incorporated.
Example 2. Calculation of weighted aggregative price index. Data for four commodities (A, B, C, D) with base period prices (P0) and quantities (q0), and current period prices (p1) and quantities (q1) are given.
Commodity A: P0=2, q0=10, p1=4, q1=5
Commodity B: P0=5, q0=12, p1=6, q1=10
Commodity C: P0=4, q0=20, p1=5, q1=15
Commodity D: P0=2, q0=15, p1=3, q1=10
Answer:
| Commodity | Base period price (P0) | Base period Quantity (q0) | Current period price (p1) | Current period Quantity (q1) |
|---|---|---|---|---|
| A | 2 | 10 | 4 | 5 |
| B | 5 | 12 | 6 | 10 |
| C | 4 | 20 | 5 | 15 |
| D | 2 | 15 | 3 | 10 |
Laspeyre’s price index (using base period quantities as weights): $P_{01} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$
$\sum p_1 q_0 = (4 \times 10) + (6 \times 12) + (5 \times 20) + (3 \times 15) = 40 + 72 + 100 + 45 = 257$
$\sum p_0 q_0 = (2 \times 10) + (5 \times 12) + (4 \times 20) + (2 \times 15) = 20 + 60 + 80 + 30 = 190$
$P_{01} = \frac{257}{190} \times 100 \approx 135.3$. Price rose by 35.3% using base period quantities.
Paasche’s price index (using current period quantities as weights): $P_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$
$\sum p_1 q_1 = (4 \times 5) + (6 \times 10) + (5 \times 15) + (3 \times 10) = 20 + 60 + 75 + 30 = 185$
$\sum p_0 q_1 = (2 \times 5) + (5 \times 10) + (4 \times 15) + (2 \times 10) = 10 + 50 + 60 + 20 = 140$
$P_{01} = \frac{185}{140} \times 100 \approx 132.1$. Price rose by 32.1% using current period quantities.
Method Of Averaging Relatives
This method calculates the price relative ($\frac{P_1}{P_0} \times 100$) for each commodity and then takes the average of these relatives. For a simple average of relatives, the formula is $\frac{\sum (\frac{P_1}{P_0} \times 100)}{n}$. Example 1 data: $\frac{(\frac{4}{2} \times 100) + (\frac{6}{5} \times 100) + (\frac{5}{4} \times 100) + (\frac{3}{2} \times 100)}{4} = \frac{200 + 120 + 125 + 150}{4} = \frac{595}{4} = 148.75$. The text calculation is $\frac{1}{n}\sum \frac{p_1}{p_0} \times 100$, which for Example 1 would be $\frac{1}{4} \times (\frac{4}{2} + \frac{6}{5} + \frac{5}{4} + \frac{3}{2}) \times 100 = \frac{1}{4} \times (2 + 1.2 + 1.25 + 1.5) \times 100 = \frac{1}{4} \times 5.95 \times 100 = 1.4875 \times 100 = 148.75$. The text states 149 (rounding). A weighted price index using price relatives uses weights (W) and is calculated as $\frac{\sum W \times (\frac{P_1}{P_0} \times 100)}{\sum W}$. Weights are often expenditure shares.
Example 3. Calculation of weighted price relatives index. Data for four commodities (A, B, C, D) with weights (W in %), base year prices (P0), current year prices (P1) are given. Calculate the weighted price index.
Commodity A: W=40%, P0=2, P1=4
Commodity B: W=30%, P0=5, P1=6
Commodity C: W=20%, P0=4, P1=5
Commodity D: W=10%, P0=2, P1=3
Answer:
| Commodity | Weight (W) (%) | Base year price (P0) (in Rs) | Current year price (P1) (in Rs) | Price relative (R = P1/P0 × 100) | WR |
|---|---|---|---|---|---|
| A | 40 | 2 | 4 | 200 | 8000 |
| B | 30 | 5 | 6 | 120 | 3600 |
| C | 20 | 4 | 5 | 125 | 2500 |
| D | 10 | 2 | 3 | 150 | 1500 |
| Total | 100 ($\sum W$) | 15600 ($\sum WR$) |
$P_{01} = \frac{\sum WR}{\sum W} = \frac{15600}{100} = 156$.
The weighted price index is 156. Prices have risen by 56%.
Some Important Index Numbers
Several important index numbers are used in economics and policy-making.
Consumer Price Index
The **Consumer Price Index (CPI)**, or cost of living index, measures the average change in retail prices of a fixed basket of goods and services typically consumed by a specific category of consumers (e.g., industrial workers, agricultural laborers, rural/urban laborers). A CPI of 277 (base 2001=100) in Dec 2014 means that a basket costing Rs 100 in 2001 would cost Rs 277 in Dec 2014. It indicates the change in purchasing power required to maintain the same standard of living. An index above 100 indicates increased cost of living.
Example 4. Construction of consumer price index number. Data for items, weights (%), base period prices (P0), and current period prices (P1) are given. Calculate the CPI.
Food: W=35%, P0=150, P1=145
Fuel: W=10%, P0=25, P1=23
Cloth: W=20%, P0=75, P1=65
Rent: W=15%, P0=30, P1=30
Misc.: W=20%, P0=40, P1=45
Answer:
| Item | Weight (W) (%) | Base period price (P0) (Rs) | Current period price (P1) (Rs) | Price relative (R = P1/P0 × 100) (%) | WR |
|---|---|---|---|---|---|
| Food | 35 | 150 | 145 | 96.67 | 3383.45 |
| Fuel | 10 | 25 | 23 | 92.00 | 920.00 |
| Cloth | 20 | 75 | 65 | 86.67 | 1733.40 |
| Rent | 15 | 30 | 30 | 100.00 | 1500.00 |
| Misc. | 20 | 40 | 45 | 112.50 | 2250.00 |
| Total | 100 ($\sum W$) | 9786.85 ($\sum WR$) |
$CPI = \frac{\sum WR}{\sum W} = \frac{9786.85}{100} \approx 97.87$.
The cost of living index is approximately 97.87, indicating a decline of about 2.13 per cent. This means the cost of the basket is lower in the current period compared to the base period.
Consumer Price Index Number
Several CPI numbers are prepared in India for different consumer categories (e.g., industrial workers, agricultural laborers, rural laborers) with various base years. These are available at state and national levels. The All-India Combined CPI (base 2012=100) is widely used as the main measure of consumer price changes.
CPI Combined (Base 2012=100) Weightage Pattern: Food & beverages (45.86), Pan, tobacco, intoxicants (2.38), Clothing & footwear (6.53), Housing (10.07), Fuel & light (6.84), Misc. (28.32). General (100.00).
The Consumer Food Price Index (CFPI) is a sub-index of CPI, measuring changes in food prices excluding alcoholic beverages and certain prepared items.
Wholesale Price Index
The **Wholesale Price Index (WPI)** indicates changes in the general price level at the wholesale stage. It doesn't refer to a specific consumer category and includes only prices of goods, not services. WPI is commonly used to measure the rate of **inflation** (general and continuous increase in prices). The WPI (base 2011-12=100) is updated regularly.
WPI (Base 2011-12=100) Weightage Pattern: Primary Articles (22.62), Fuel and Power (13.15), Manufactured Products (64.23). All Commodities (Headline Inflation) (100.00). WPI Food Index (24.23%). Core Inflation (excluding food and fuel).
A WPI value of 253 (base 2004-05=100) in Oct 2014 means the general price level rose by 153% during that period.
Index Of Industrial Production
The **Index of Industrial Production (IIP)** measures changes in the physical volume of industrial production. With base 2011-12=100 (since April 2017), it is a weighted average of quantity relatives, with weights based on value added in the base year. IIP is available for industrial sectors (Mining, Manufacturing, Electricity) and use-based groups (Primary Goods, Consumer Durables, etc.).
IIP (Base 2011-12=100) Weightage Pattern: Mining (14.4), Manufacturing (77.6), Electricity (8.0). General Index (100.0). Use-based Groups: Primary (34.1), Capital Goods (8.2), Intermediate (17.2), Infrastructure/Construction (12.3), Consumer Durables (12.8), Consumer Non-durables (15.3). General Index (100.0).
The Eight Core Industries (coal, crude oil, natural gas, refinery products, fertilizers, steel, cement, electricity) have a combined weight of 40.27% in the IIP.
Human Development Index
The **Human Development Index (HDI)** is a composite index measuring a country's overall development based on indicators like life expectancy, education, and income. Discussed in Class X, it is a crucial indicator of well-being.
Sensex
Sensex is a benchmark index for the Indian stock market, representing 30 leading company stocks on the Bombay Stock Exchange (BSE), with base 1978-79=100. A rising Sensex indicates a positive market trend, investor optimism, and confidence in the economy.
Issues In The Construction Of An Index Number
Constructing an index number involves several considerations that can affect its reliability and interpretation.
- Purpose: The purpose of the index must be clear (e.g., measuring price changes vs. quantity changes).
- Item Selection: Items included must be representative of the basket being measured.
- Base Year: The base year should be a normal year without extreme fluctuations and not too distant in the past. Base years are regularly updated.
- Formula Choice: The choice of formula depends on the nature of the study. Different formulas (like Laspeyre's and Paasche's) use different weights (base period vs. current period quantities), leading to different index values.
- Data Reliability: The accuracy of the index depends on the reliability of the data sources. Careful collection or selection of reliable secondary data is crucial.
Index Number In Economics
Index numbers are essential tools in economic analysis and policy formulation. WPI, CPI, and IIP are widely used.
- CPI: Used in wage negotiations, income and price policies, rent control, taxation, and overall economic policy.
- WPI: Used to remove the effect of price changes from economic aggregates like national income and capital formation. Widely used to measure inflation rate. Inflation is calculated as $\frac{\text{WPI}_t - \text{WPI}_{t-1}}{\text{WPI}_{t-1}} \times 100$. Inflation lowers the value of money.
- CPI: Used to calculate the purchasing power of money (1/Cost of living index) and real wage (Money wage / Cost of living index × 100).
- IIP and Agricultural Production Index: Measure changes in the volume of production in industrial and agricultural sectors.
- Sensex: Guides investors in the stock market, reflecting economic health and investor confidence.
Economic Survey provides data for many widely used index numbers.
Conclusion
Index numbers are valuable tools for summarizing relative changes in a group of related items. They can measure changes in prices, quantities, volumes, etc. Correct interpretation of index numbers is vital, considering the items included, the base period chosen, and the formula used. Index numbers play an indispensable role in economic policy making due to their various applications in measuring inflation, cost of living, production changes, and guiding economic decisions.
Recap:
- An index number measures relative change in a large number of items.
- Different formulae exist for index numbers, requiring careful interpretation.
- Formula choice depends on the question of interest.
- WPI, CPI, IIP, agricultural production index, and Sensex are widely used index numbers.
- Index numbers are essential tools in economic policy making.