Chapter Introduction
Index numbers are specialised averages that measure the relative change in a variable (or group of variables) over time with respect to a base period. They are the most widely used statistical tool in economics — from measuring inflation (CPI, WPI) to tracking industrial production (IIP) to comparing stock market performance (Sensex). This chapter teaches you how index numbers are constructed, the different types and weighting methods, and how to interpret them. Index numbers form the last chapter of the Statistics for Economics book and tie together many concepts from earlier chapters.
Key Definitions
| Term | Definition |
|---|---|
| Index number | A statistical measure that shows the relative change in a variable (or group of variables) compared to a base period |
| Base year | The reference period against which comparisons are made; its index value is 100 |
| Current year | The period for which the index is being calculated |
| Price relative | The price of a commodity in the current year expressed as a percentage of its base year price: (p1/p0) x 100 |
| Price index | An index that measures changes in the price level of goods and services |
| Quantity index | An index that measures changes in the physical volume of production or sales |
| Value index | An index that measures changes in the total value (price x quantity) |
| Weighted index | An index that assigns different weights (importance) to different items |
| Laspeyres index | A weighted price index using base year quantities as weights |
| Paasche index | A weighted price index using current year quantities as weights |
| CPI | Consumer Price Index — measures changes in retail prices of consumer goods and services |
| WPI | Wholesale Price Index — measures changes in wholesale prices of goods |
| Sensex | Bombay Stock Exchange Sensitive Index — measures stock price movements |
| Deflating | The process of removing the effect of price changes from nominal values using a price index |
Key Formulas
Simple (Unweighted) Index Numbers
| Method | Formula |
|---|---|
| Simple aggregate index | P01 = (Sum of p1 / Sum of p0) x 100 |
| Simple average of price relatives | P01 = (Sum of price relatives) / n, where price relative = (p1/p0) x 100 |
Weighted Index Numbers
| Method | Formula |
|---|---|
| Laspeyres price index | Lasp = (Sum of p1q0 / Sum of p0q0) x 100 |
| Paasche price index | Paas = (Sum of p1q1 / Sum of p0q1) x 100 |
| Fisher's ideal index | Fisher = sqrt(Lasp x Paas) |
| Weighted average of price relatives | P01 = (Sum of WP / Sum of W), where P = price relative, W = weight |
Inflation Rate
Inflation rate = [(Current year index - Previous year index) / Previous year index] x 100
Real Income (Wage)
Real income = (Nominal income / CPI) x 100
Tests for Index Numbers
| Test | Condition | Satisfied by |
|---|---|---|
| Time reversal test | P01 x P10 = 1 | Fisher (geometric mean eliminates bias) |
| Factor reversal test | Price index x Quantity index = Value index | Fisher |
Detailed Concepts
Types of Index Numbers
| Type | What it measures | Example |
|---|---|---|
| Price index | Changes in price level | CPI, WPI |
| Quantity index | Changes in physical volume (production, sales) | Index of Industrial Production (IIP) |
| Value index | Changes in total value (price x quantity) | Index of foreign trade value |
Simple Index Number Methods
Simple aggregate method:
- Add all current year prices; divide by sum of base year prices; multiply by 100
- P01 = (Sum of p1 / Sum of p0) x 100
- Simple but misleading — items with higher absolute prices dominate the result regardless of their importance
- Example: If rice is Rs 40/kg and gold is Rs 50,000/10g, gold's price change will dominate the index
Simple average of price relatives:
- Calculate each price relative = (p1/p0) x 100
- Take the arithmetic mean of all price relatives
- P01 = (1/n) x Sum of price relatives
- Better than simple aggregate because each item gets equal weight
- But still unweighted — all items treated equally regardless of importance
- Example: A 100% rise in the price of salt gets the same weight as a 100% rise in the price of petrol
Weighted Index Numbers
Weighted indices assign different weights to different items based on their relative importance.
Laspeyres price index (base year weights):
- Lasp = (Sum of p1q0 / Sum of p0q0) x 100
- Uses base year quantities (q0) as weights
- Advantage: weights are fixed (base year), so comparison across years is consistent; base year data readily available
- Disadvantage: tends to overstate price increases because it does not account for substitution away from goods that became more expensive
- Most widely used method in practice (CPI uses a Laspeyres-type formula)
Paasche price index (current year weights):
- Paas = (Sum of p1q1 / Sum of p0q1) x 100
- Uses current year quantities (q1) as weights
- Advantage: reflects current consumption patterns; accounts for substitution
- Disadvantage: weights change every year, making comparison across years inconsistent; tends to understate price increases; requires current year quantity data, which may not always be available
Fisher's ideal index:
- Fisher = sqrt(Lasp x Paas)
- Geometric mean of Laspeyres and Paasche indices
- Called "ideal" because it satisfies two important tests:
- Time reversal test: P01 x P10 = 1 (reversing base and current years gives reciprocal)
- Factor reversal test: Price index x Quantity index = Value index
- Fisher avoids both the upward bias of Laspeyres and the downward bias of Paasche
- Disadvantage: requires both base year and current year quantity data; more complex to compute
CPI vs WPI
| Feature | CPI (Consumer Price Index) | WPI (Wholesale Price Index) |
|---|---|---|
| Level | Retail prices (what consumers pay) | Wholesale prices (bulk trading) |
| Coverage | Goods and services | Only goods (no services) |
| Base year | Currently 2012 (CPI for Industrial Workers) | Currently 2011-12 |
| Published by | Labour Bureau / NSO | Office of Economic Adviser, Ministry of Commerce |
| Use | Dearness allowance, consumer inflation | Headline inflation, policy decisions |
| Frequency | Monthly | Monthly |
| Categories | Food, clothing, housing, fuel, etc. | Primary articles, fuel & power, manufactured products |
| Population | Specific groups (industrial workers, agricultural labourers) | Economy-wide (wholesale level) |
| Services included | Yes (education, health, transport) | No |
Types of CPI in India
| CPI type | Target group | Published by |
|---|---|---|
| CPI-IW | Industrial Workers | Labour Bureau |
| CPI-AL | Agricultural Labourers | Labour Bureau |
| CPI-UNME | Urban Non-Manual Employees | NSO |
| CPI (Combined) | All India combined | NSO (used for inflation targeting by RBI) |
Uses of Index Numbers
1. Measuring inflation — CPI and WPI track price level changes over time 2. Policy formulation — RBI uses CPI for monetary policy; government uses WPI for trade policy 3. Dearness allowance — CPI-IW is used to calculate DA for government employees 4. Real income comparison — Convert nominal income to real income using CPI 5. Deflating nominal values — Remove the effect of price changes to get real values 6. Stock market analysis — Sensex and Nifty track market performance 7. Industrial production — IIP tracks growth in manufacturing, mining, electricity 8. International comparison — Compare economic performance across countries 9. Purchasing power of money — CPI indicates how much the rupee's value has changed
Limitations of Index Numbers
1. Based on sample items — Not all items are included; selection may not represent the whole economy 2. Base year choice — An abnormal base year distorts all comparisons 3. Quality changes — Index does not account for improvements in product quality 4. New products — Items not available in the base year cannot be properly included 5. Substitution bias — Laspeyres index overstates inflation because consumers substitute away from expensive goods 6. Weighting problems — Weights become outdated as consumption patterns change 7. Limited coverage — WPI excludes services, which are over 50% of India's GDP 8. International comparison difficulty — Different countries use different base years, items, and weights
Deflating and Real Values
Deflating means removing the effect of price changes from a nominal value:
Real value = (Nominal value / Price index) x 100
Example: If nominal income = Rs 1,00,000 and CPI = 150 (base year = 100), then: Real income = (1,00,000 / 150) x 100 = Rs 66,667
This means the income of Rs 1,00,000 in current prices buys the equivalent of Rs 66,667 in base year prices.
Worked Example: Laspeyres and Paasche Index
| Commodity | p0 (Base price) | q0 (Base qty) | p1 (Current price) | q1 (Current qty) | p1q0 | p0q0 | p1q1 | p0q1 |
|---|---|---|---|---|---|---|---|---|
| A | 4 | 50 | 6 | 40 | 300 | 200 | 240 | 160 |
| B | 3 | 60 | 5 | 50 | 300 | 180 | 250 | 150 |
| C | 2 | 40 | 3 | 30 | 120 | 80 | 90 | 60 |
| Total | 720 | 460 | 580 | 370 |
Laspeyres: Lasp = (720 / 460) x 100 = 156.5 Paasche: Paas = (580 / 370) x 100 = 156.8 Fisher: Fisher = sqrt(156.5 x 156.8) = sqrt(24538.2) = 156.6
Note: Laspeyres slightly lower than Paasche in this case. Usually the opposite.
Worked Example: Inflation Rate and Real Income
- WPI 2024 = 160, WPI 2025 = 168
- Inflation rate = [(168 - 160) / 160] x 100 = (8/160) x 100 = 5%
- If nominal income = Rs 5,00,000 and CPI = 150:
- Real income = (5,00,000 / 150) x 100 = Rs 3,33,333
Important Points for Exams
1. An index number measures relative change compared to a base period (index = 100). 2. Three types: price index, quantity index, value index. 3. Simple aggregate method: P01 = (Sum of p1 / Sum of p0) x 100. Simple but biased toward high-priced items. 4. Simple average of price relatives: equal weight to each item's relative price change. 5. Laspeyres index uses base year weights (q0); Paasche uses current year weights (q1). 6. Fisher's index = geometric mean of Laspeyres and Paasche; satisfies time reversal and factor reversal tests. 7. CPI measures retail prices; WPI measures wholesale prices. 8. Inflation rate = [(current index - previous index) / previous index] x 100. 9. Real income = (nominal income / CPI) x 100. 10. Limitations include: sample bias, quality changes, new products, substitution bias, outdated weights. 11. Fisher's index is called "ideal" because it satisfies both the time reversal and factor reversal tests.
Common Mistakes
| Mistake | Why it is wrong | Correct approach |
|---|---|---|
| Confusing Laspeyres and Paasche weights | Laspeyres uses q0 (base year); Paasche uses q1 (current year) | Remember: Lasp = p1q0, Paas = p1q1 in the numerator |
| Not multiplying by 100 in index number formulas | Index numbers are expressed as percentages with base = 100 | Always multiply by 100 at the end |
| Using Laspeyres formula for Paasche or vice versa | The weight (q0 vs q1) determines which formula is used | Check which year's quantities are given before choosing the formula |
| Computing Fisher's index as arithmetic mean of Lasp and Paas | Fisher uses geometric mean (square root of product), not arithmetic mean | Fisher = sqrt(Lasp x Paas) |
| Confusing CPI and WPI | CPI is retail, includes services; WPI is wholesale, goods only | CPI for consumer inflation and DA; WPI for headline inflation |
| Calculating inflation rate using absolute change instead of percentage | Inflation rate is a percentage change, not absolute change | Use: [(current - previous) / previous] x 100 |
| Forgetting to deflate nominal values | Nominal values are distorted by inflation | Always use Real value = (Nominal value / CPI) x 100 |
NCERT Exercise Focus
Focus on these NCERT exercise questions while preparing:
1. "Calculate Laspeyres and Paasche price index from the given data" — Most commonly tested 5-mark numerical. 2. "Calculate Fisher's ideal index" — Must compute both Lasp and Paas first, then geometric mean. 3. "Distinguish between CPI and WPI" — Prepare a comparison table. Frequently asked 3-mark question. 4. "Calculate inflation rate from index numbers" — Simple but commonly tested. 5. "What are the uses and limitations of index numbers?" — Give at least 5 each for a balanced answer.
Exam Tips
1. Numerical on Laspeyres and Paasche — Practice computing both from a table of prices and quantities. Very common 5-mark question. 2. Fisher's index — Calculate both Lasp and Paas first, then take geometric mean (square root of product). 3. CPI vs WPI distinction — Prepare a comparison table. This is a frequently asked 3-mark question. 4. Uses and limitations of index numbers — List at least 5 uses and 5 limitations. Balanced answers score higher. 5. Inflation rate calculation — Simple but commonly tested. Formula: [(current - previous) / previous] x 100. 6. Real income/wage — You may be asked to calculate real income given nominal income and CPI. 7. Time reversal test and factor reversal test — Know what they mean and that Fisher satisfies both. 8. Simple aggregate vs simple average of relatives — Know the difference and which is better.
Practice Questions for Self-Test
1. Calculate Laspeyres and Paasche price index numbers from the given data. (5 marks) 2. Distinguish between CPI and WPI. (3 marks) 3. What is Fisher's ideal index? Why is it called ideal? (3 marks) 4. Calculate inflation rate if WPI was 150 last year and 165 this year. (2 marks) 5. Calculate real income if nominal income is Rs 6,00,000 and CPI is 180. (2 marks) 6. What are the uses and limitations of index numbers? (5 marks) 7. Distinguish between simple aggregate index and simple average of price relatives. (3 marks) 8. What is the time reversal test? Which index satisfies it? (2 marks) 9. Explain the concept of deflating with an example. (3 marks) 10. Differentiate between price index, quantity index, and value index. (3 marks)
Quick Revision Notes
- Index number: measures relative change; base year = 100.
- Types: price index (CPI, WPI), quantity index (IIP), value index.
- Simple aggregate: P01 = (Sum of p1 / Sum of p0) x 100. Biased toward high-priced items.
- Simple average of relatives: P01 = Sum of price relatives / n. Equal weight per item.
- Laspeyres: base year weights (q0). Lasp = Sum of p1q0 / Sum of p0q0 x 100. Overstates inflation.
- Paasche: current year weights (q1). Paas = Sum of p1q1 / Sum of p0q1 x 100. Understates inflation.
- Fisher: sqrt(Lasp x Paas). Ideal — satisfies time reversal and factor reversal tests.
- CPI: retail prices, includes services. Used for DA and consumer inflation.
- WPI: wholesale prices, goods only. Used for headline inflation and policy.
- Inflation rate = [(current index - previous index) / previous index] x 100.
- Real income = (Nominal income / CPI) x 100. Deflating removes price change effect.
- Limitations: sample bias, quality changes, new products, substitution bias, outdated weights.