Index Numbers

INTRODUCTION

**Index numbers** are statistical tools designed to measure changes in the magnitude of a group of related variables over time. They summarize complex, diverse changes into single, easily comparable figures. This is essential because real-world situations involve multiple items changing at different rates—when prices of various commodities rise by different percentages, or when industrial outputs fluctuate unevenly across sectors, a single index number provides clarity.

Why Index Numbers Matter

**Salary vs. Cost of Living**: An industrial worker earning Rs 1,000 in 1982 and Rs 12,000 today has not necessarily become 12 times better off if prices have also risen. Index numbers reveal the real change in purchasing power.

**Stock Market Sentiment**: SENSEX movements indicate investor confidence and economic health.

**Inflation Measurement**: Governments track price changes across commodities to measure inflation without reporting every individual price change.

**Production Monitoring**: Index numbers track whether industrial output is rising or falling across different sectors simultaneously.

MEANING AND DEFINITION OF INDEX NUMBER

**An index number is a statistical measure of average change in a group of related variables over two different time periods or situations.**

Key Characteristics

**Base Period Definition**: The period chosen as the reference point (usually called year 0 or base year). The index value for the base period is always **100**.

**Current/Given Period**: The period being compared with the base period (usually called year 1 or current year).

**Expression in Percentage**: Index numbers are conventionally expressed as percentages relative to the base period value of 100.

**Interpretation**: An index of 250 means the variable is 2.5 times the base period value. An index of 80 means the variable is 80% of the base period value (a 20% decrease).

Types of Index Numbers

**Price Index Numbers**: Measure and compare prices of goods and services. Example: Consumer Price Index (CPI), Wholesale Price Index (WPI).

**Quantity Index Numbers**: Measure changes in physical volume of production, construction, or employment. Example: Index of Industrial Production (IIP).

**Value Index Numbers**: Measure combined changes in price and quantity (price × quantity).

CONSTRUCTION OF INDEX NUMBERS: THE AGGREGATIVE METHOD

Simple Aggregative Price Index

This is the simplest method, calculated using the formula:

**P₀₁ = (ΣP₁ / ΣP₀) × 100**

Where:

P₁ = Price in the current period

P₀ = Price in the base period

Σ = Summation symbol

Example Calculation

Consider four commodities:

| Commodity | Base Period Price (Rs) | Current Period Price (Rs) |

|-----------|------------------------|---------------------------|

| A | 2 | 4 |

| B | 5 | 6 |

| C | 4 | 5 |

| D | 2 | 3 |

**ΣP₀ = 2 + 5 + 4 + 2 = 13**

**ΣP₁ = 4 + 6 + 5 + 3 = 18**

**P₀₁ = (18/13) × 100 = 138.5**

This means prices have risen by **38.5%** from the base period to the current period.

Limitations of Simple Aggregative Index

**Unequal Units**: Prices of different commodities are measured in different units (kg, litre, piece), making direct summation problematic.

**Unweighted Nature**: Treats all commodities as equally important, ignoring that food items comprise a larger portion of household expenditure than clothing or entertainment.

**Misleading Results**: A large price increase in an unimportant item can distort the overall index.

Weighted Aggregative Price Index

This method assigns **weights** (importance values) to commodities, reflecting their relative importance in the basket of goods.

**P₀₁ = (ΣP₁q₀ / ΣP₀q₀) × 100** (Using base period quantities as weights)

Or

**P₀₁ = (ΣP₁q₁ / ΣP₀q₁) × 100** (Using current period quantities as weights)

Where:

q₀ = Quantity in base period

q₁ = Quantity in current period

Laspeyres' Price Index

Uses base period quantities as weights:

**P₀₁ = (ΣP₁q₀ / ΣP₀q₀) × 100**

**Interpretation**: If expenditure on the base period basket was Rs 100, this index shows how much must be spent in the current period to purchase the identical basket.

Example of Laspeyres' Index

| Commodity | P₀ | q₀ | P₁ | P₀q₀ | P₁q₀ |

|-----------|----|----|----|----- |------|

| A | 2 | 10 | 4 | 20 | 40 |

| B | 5 | 12 | 6 | 60 | 72 |

| C | 4 | 20 | 5 | 80 | 100 |

| D | 2 | 15 | 3 | 30 | 45 |

| **Total** | - | - | - | 190 | 257 |

**P₀₁ = (257/190) × 100 = 135.3**

Prices have risen by **35.3%** based on base period quantities.

Paasche's Price Index

Uses current period quantities as weights:

**P₀₁ = (ΣP₁q₁ / ΣP₀q₁) × 100**

**Interpretation**: If the current period basket were purchased in the base period at 100 units of spending, this index shows the required expenditure in the current period.

Example of Paasche's Index

| Commodity | P₀ | q₁ | P₁ | P₀q₁ | P₁q₁ |

|-----------|----|----|----|----- |------|

| A | 2 | 5 | 4 | 10 | 20 |

| B | 5 | 10 | 6 | 50 | 60 |

| C | 4 | 15 | 5 | 60 | 75 |

| D | 2 | 10 | 3 | 20 | 30 |

| **Total** | - | - | - | 140 | 185 |

**P₀₁ = (185/140) × 100 = 132.1**

Prices have risen by **32.1%** based on current period quantities.

Comparison: Laspeyres vs. Paasche

**Laspeyres' Index (135.3)** is higher because it uses fixed base quantities. When consumption patterns change to shift away from items with high price increases, Laspeyres' still uses the original quantities.

**Paasche's Index (132.1)** is lower because it uses current quantities. By the time prices have risen sharply, consumers may have reduced purchases of expensive items.

**In practice**: Laspeyres' is more commonly used because it's easier to maintain a fixed basket. Paasche's requires new quantity data every period.

METHOD OF AVERAGING RELATIVES

This method calculates the **price relative** (ratio of current to base price) for each commodity, then averages them.

Simple Price Relative Index

**P₀₁ = [Σ(P₁/P₀) × 100] / n**

Where n = number of commodities

Example Calculation

| Commodity | P₀ | P₁ | Price Relative (P₁/P₀) × 100 |

|-----------|----|----|------|

| A | 2 | 4 | 200 |

| B | 5 | 6 | 120 |

| C | 4 | 5 | 125 |

| D | 2 | 3 | 150 |

**P₀₁ = (200 + 120 + 125 + 150) / 4 = 595/4 = 148.75**

Prices have risen by **48.75%**.

Weighted Price Relative Index

Assigns weights to each commodity's price relative:

**P₀₁ = [Σ(W × (P₁/P₀) × 100)] / ΣW**

Where W = weight of the commodity

Example of Weighted Price Relatives

| Commodity | Weight (%) | P₀ (Rs) | P₁ (Rs) | Price Relative | W × PR |

|-----------|------------|---------|---------|-----------------|--------|

| Food | 40 | 100 | 120 | 120 | 4800 |

| Fuel | 15 | 50 | 55 | 110 | 1650 |

| Cloth | 25 | 80 | 90 | 112.5 | 2812.5 |

| Misc | 20 | 40 | 45 | 112.5 | 2250 |

| **Total** | 100 | - | - | - | 11512.5|

**P₀₁ = 11512.5 / 100 = 115.125**

Prices have risen by **15.125%**. Notice that this is lower than the simple average (148.75) because the higher-weighted commodities (Food at 40%) have lower price increases.

CONSUMER PRICE INDEX (CPI)

The **Consumer Price Index** measures changes in the average retail prices paid by consumers for a fixed basket of goods and services. It is also called the **cost of living index**.

Interpretation of CPI

If CPI (2001 = 100) = 277 in December 2014, it means:

A consumer who spent Rs 100 in 2001 would need Rs 277 in December 2014 to purchase the identical basket of goods and services.

This assumes a **2.77 times increase** in prices over this period.

**No actual purchase required**: The index measures purchasing power capability, not actual consumption patterns.

Wage Adjustment Formula

If CPI = 150 (increased by 50 from base of 100), wages must be increased by **50%** to maintain the same standard of living.

If CPI = 80 (decreased by 20), there is a **20% surplus** in purchasing power (wages could be decreased while maintaining living standards, though practically this rarely happens).

CPI in India: Current Framework

The **All-India Combined Consumer Price Index** with **base 2012 = 100** is now the primary measure. The basket composition (as per 68th Round NSS, 2011-12) includes:

| Item Category | Weight (%) |

|----------------------------|-----------|

| Food and Beverages | 45.86 |

| Pan, Tobacco, Intoxicants | 2.38 |

| Clothing & Footwear | 6.53 |

| Housing | 10.07 |

| Fuel & Light | 6.84 |

| Miscellaneous | 28.32 |

| **Total** | 100.00 |

**Key Point**: Food items (45.86%) have the highest weight, reflecting that food occupies the largest share of household expenditure for most Indian consumers.

Different CPI Series in India

**CPI for Industrial Workers** (base 2001 = 100): Value in May 2017 = 278

**CPI for Agricultural Labourers** (base 1986-87 = 100): Value in May 2017 = 872

**CPI for Rural Labourers** (base 1986-87 = 100): Value in May 2017 = 878

**All-India Rural CPI** (base 2012 = 100): Value in May 2017 = 133.3

**All-India Urban CPI** (base 2012 = 100): Value in May 2017 = 129.3

**Combined CPI** (base 2012 = 100): Value in May 2017 = 131.4

Consumer Food Price Index (CFPI)

Similar to CPI for the 'Food and Beverages' category, but **excludes alcoholic beverages** and 'Prepared meals, snacks, and sweets'. Useful for monitoring food inflation separately.

How to Construct CPI: Example Calculation

| Item | Weight (%) | Base Price (Rs) | Current Price (Rs) | Price Relative (R) | W × R |

|---------|------------|-----------------|--------------------|--------------------|---------|

| Food | 35 | 150 | 145 | 96.67 | 3883.45 |

| 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 | - | - | - | 9786.85 |

**CPI = 9786.85 / 100 = 97.86**

**Interpretation**: The cost of living has **declined by 2.14%** (100 - 97.86 = 2.14). A consumer who spent Rs 100 in the base period would need only Rs 97.86 in the current period.

RBI's Use of CPI

The Reserve Bank of India uses the **All-India Combined Consumer Price Index** as the main measure of inflation for monetary policy decisions (inflation targeting framework at 4% with ±2% band).

WHOLESALE PRICE INDEX (WPI)

The **Wholesale Price Index** measures changes in prices at the wholesale level, tracking overall inflation in the economy.

Key Differences from CPI

**No Reference Consumer**: Unlike CPI which assumes a typical consumer, WPI is economy-wide.

**Goods Only**: Does not include services (barber charges, repairs, etc.).

**Earlier Availability**: Data is available more quickly, making it useful for quick policy assessment.

Interpretation of WPI

"WPI with 2004-05 as base = 253 in October 2014" means:

General price level has risen by **153%** (253 - 100 = 153) from 2004-05 to October 2014.

Current WPI Framework

With **base 2011-12 = 100**, the structure is:

| Category | Weight (%) |

|----------------------|-----------|

| Primary Articles | 22.62 |

| Fuel and Power | 13.15 |

| Manufactured Products| 64.23 |

| **All Commodities** | 100.00 |

**Special WPI Measures**:

**Headline Inflation**: 'All Commodities Inflation Rate' = 100% weight (tracks overall WPI)

**Food Index**: 24.23% weight, comprising food from primary articles and food products from manufactured goods

**Core Inflation**: Approximately 55% weight (manufactured goods excluding food and fuel), focuses on underlying price pressures excluding volatile components

Example WPI Interpretation

If Headline Inflation (WPI) = 5.5% and Food Index = 7.2%, it indicates that food prices are rising faster than the overall average, creating concern about food inflation's impact on households.

INDEX OF INDUSTRIAL PRODUCTION (IIP)

The **Index of Industrial Production** measures changes in the volume of industrial output, tracking economic growth in the industrial sector.

Formula

**IIP₀₁ = [Σ(qᵢ₁ × Wᵢ) / ΣWᵢ] × 100**

Where:

qᵢ₁ = Quantity relative (ratio of current to base period quantity)

Wᵢ = Weight of the good (based on value added in base year)

Uses **Laspeyres' method** with quantity relatives

Base Year

With effect from **April 2017, base = 2011-12 = 100**. The base year is changed frequently because many items stop being manufactured while new items begin, making old comparisons less meaningful.

IIP Structure by Sectors

| Sector | Weight (%) |

|---------------|-----------|

| Mining | 14.4 |

| Manufacturing| 77.6 |

| Electricity | 8.0 |

| **Total** | 100.0 |

**Manufacturing is the dominant component**, reflecting that most industrial activity is in manufacturing rather than mining or electricity generation.

Core Industries

Eight core industries have a combined weight of **40.27%** in the IIP:

1. Coal

2. Crude Oil

3. Natural Gas

4. Refinery Products

5. Fertilizers

6. Steel

7. Cement

8. Electricity

These are monitored closely because they supply inputs to all other industries.

IIP by Use Classification

| Use Category | Weight (%) |

|---------------------------|-----------|

| Primary Goods | 34.1 |

| Capital Goods | 8.2 |

| Intermediate Goods | 17.2 |

| Infrastructure/Construction Goods| 12.3 |

| Consumer Durables | 12.8 |

| Consumer Non-durables | 15.3 |

| **Total** | 100.0 |

**Consumer Goods (28.1%)** comprise nearly 30% of the index, showing that consumer demand drives significant manufacturing activity.

Interpretation

**IIP rising**: Industrial sector expanding, more jobs, stronger economic growth likely.

**IIP falling**: Industrial slowdown, potential recession, employment concerns.

Different growth rates across sectors indicate structural shifts (e.g., if mining falls but manufacturing rises, the economy is diversifying).

SENSEX (BOMBAY STOCK EXCHANGE SENSITIVE INDEX)

**SENSEX** is the short form of Bombay Stock Exchange Sensitive Index, with **base 1978-79**.

Structure and Composition

**30 stocks**: Represents 13 major sectors of the Indian economy

**Companies**: Leading firms in their respective industries with high market capitalization

**Benchmark Status**: The primary indicator of Indian stock market performance

Market Signals

**Rising SENSEX**:

Market confidence increasing

Investors expect better corporate earnings

Economic growth prospects improving

Inflow of foreign investment

**Falling SENSEX**:

Market pessimism

Erosion of investor wealth

Economic slowdown expectations

Flight of capital

Example Application

When "SENSEX dipped 600 points, it eroded investors' wealth by Rs 1,53,690 crores," this demonstrates the immense value at stake in the stock market and why even small percentage movements in the index have massive financial implications.

Economic Significance

SENSEX movements precede real economic changes, making it a **leading economic indicator**. A sustained rise often signals upcoming economic expansion, while a sustained fall may warn of recession.

HUMAN DEVELOPMENT INDEX (HDI)

The **Human Development Index** measures overall development beyond just economic metrics, incorporating:

**Life Expectancy**: Health and longevity

**Education**: Mean years of schooling and expected years of schooling

**Income**: GNI per capita (adjusted for purchasing power)

HDI provides a comprehensive picture of human welfare that price or production indices cannot capture. Countries with high GDP but low HDI (and vice versa) highlight development imbalances.

ISSUES IN CONSTRUCTION OF INDEX NUMBERS

1. Purpose Clarity

**Issue**: An index must be constructed with a clear purpose.

Constructing a **quantity/volume index** when a **value index** is needed produces meaningless results

A **price index** cannot answer questions about production changes

**Solution**: Define the objective before selecting the methodology.

2. Selection of Items to Include

**Issue**: Different consumer groups have different consumption patterns.

A **rise in petrol prices** heavily impacts urban commuters but barely affects rural agricultural labourers

**Food price changes** affect the poor more than the wealthy (they spend larger proportion of income on food)

**Solution**: Create separate indices for different population groups (CPI for workers, for agricultural labourers, for rural vs. urban populations).

3. Choice of Base Year

**Important Considerations**:

**Should be Normal**: Avoid years with extreme values (recession, harvest failure, war, pandemic)

**Should be Recent**: Comparisons between 1993 and 2005 are more meaningful than 1960 and 2005

**Outdated Items**: In 1960, a typical consumption basket included items that no longer exist or are irrelevant (typewriters, transistor radios). A 1960 basket cannot represent 2005 consumption patterns.

**Routine Updates**: Base years are regularly updated (from 2001=100 to 2012=100 for many Indian indices) to remain relevant

**Problem of Chain Base**: Changing base years makes long-term comparisons difficult (must convert all figures to a common base year using chain indices).

4. Choice of Formula

**Laspeyre's vs. Paasche's**:

**Laspeyre's Index** (P₀₁ = ΣP₁q₀/ΣP₀q₀ × 100):

Uses fixed base year quantities

Easier to maintain (no new quantity data needed yearly)

Tends to **overstate inflation** because it assumes consumers don't change consumption patterns even as prices change

**Preferred in practice** for cost of living indices

**Paasche's Index** (P₀₁ = ΣP₁q₁/ΣP₀q₁ × 100):

Uses current year quantities

Reflects actual consumption patterns

More difficult to compute (requires new quantity data every period)

Tends to **understate inflation** because it assumes consumers shift to cheaper goods

Requires less frequent base year updates

**Middle Ground**: In practice, a compromise using **average weights** from two periods or **chained indices** (yearly updates with base shifting forward) balances both approaches.

5. Treatment of New and Obsolete Items

**Issue**: New products (smartphones, electric vehicles) and obsolete items (typewriters, VCRs) constantly enter and exit the market.

**Solutions**:

**Frequent Base Updates**: Allows new items to enter and old items to exit with updated weights

**Splicing/Chaining**: Connect different time periods with different base years using ratios

**Hedonic Pricing**: Adjust prices for quality changes (a smartphone with better features should not be treated as a pure price increase)

PRACTICAL APPLICATION AND EXAM-RELEVANT POINTS

Key Formulas to Memorize

1. **Simple Aggregative Index**: P₀₁ = (ΣP₁/ΣP₀) × 100

2. **Laspeyres' Index**: P₀₁ = (ΣP₁q₀/ΣP₀q₀) × 100

3. **Paasche's Index**: P₀₁ = (ΣP₁q₁/ΣP₀q₁) × 100

4. **Simple Price Relative Index**: P₀₁ = [Σ(P₁/P₀) × 100] / n

5. **Weighted Price Relative Index**: P₀₁ = [Σ(W × (P₁/P₀) × 100)] / ΣW

Common Exam Questions

**Calculate an index number** using given data (requires formula application)

**Interpret index values** (What does an index of 150 mean? What percentage change?)

**Compare different indices** (Why is Laspeyre's different from Paasche's?)

**Discuss limitations** (Why is the simple aggregative index problematic?)

**Apply CPI concepts** (What wage increase is needed if CPI rises from 100 to 125?)

**Analyze inflation trends** (What does rising Food Index within WPI indicate?)

**Real-life application** (How do SENSEX changes affect investor wealth? How does IIP indicate economic health?)

Data Interpretation Tips

When calculating **percentage change from index values**: If Index₀ = 100 and Index₁ = 150, the percentage change = (150-100)/100 × 100 = **50% increase**

**Higher weights** for items with larger price increases will produce higher overall indices

**Changing base year**: If old base was 2000 = 100 and you want to rebase to 2005 = 100, divide all old index values by the 2005 index value (in terms of old base) and multiply by 100

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**END OF CHAPTER NOTES**

These comprehensive notes cover every concept, definition, formula, calculation method, Indian application, and exam-important aspect of Index Numbers required for CBSE Class 11 board examination success.