What is Fisher's Ideal Index?

Fisher's Ideal Index is the geometric mean of Laspeyres and Paasche indices: P₀₁F = √(Lasp × Paas). It satisfies both the time reversal and factor reversal tests, making it the most balanced index number formula.

What is CPI and how is it different from WPI?

CPI (Consumer Price Index) measures the average change in retail prices of goods and services consumed by a specific group. WPI (Wholesale Price Index) measures changes in wholesale prices. CPI is used for calculating dearness allowance; WPI is used as an indicator of inflation.

Index Numbers — NCERT Class 11 Economics

Exercise Solutions Available

Step-by-step answers for all textbook questions in this chapter

View Solutions →

How to Use This Page

This page provides complete NCERT exercise solutions for CBSE Class 11 Economics Chapter 8 — Index Numbers. Review the Key Concepts and formulas first, then attempt each numerical. Step-by-step working is shown for all index number calculations. Verify your answers against these solutions.

Key Concepts (Quick Recap)

Index Number: a statistical measure showing relative change in a variable over time; expressed as a percentage
Base Year: the reference year with index = 100; should be a normal year
Price Relative: P₀₁ = (p₁/p₀) × 100 (price in current year relative to base year)
Quantity Relative: Q₀₁ = (q₁/q₀) × 100
Value Relative: V₀₁ = (p₁q₁)/(p₀q₀) × 100
Simple Index Number: unweighted average of price relatives
Weighted Index Number: uses weights (quantities or values)
Laspeyres Index (L): base year quantities as weights
P₀₁L = Σp₁q₀/Σp₀q₀ × 100
Paasche Index (P): current year quantities as weights
P₀₁P = Σp₁q₁/Σp₀q₁ × 100
Fisher's Ideal Index (F): geometric mean of L and P
P₀₁F = √(Lasp × Paas)
Consumer Price Index (CPI): measures retail price changes; used for DA calculation
CPI = ΣWI/ΣW where W = weight, I = price relative
Wholesale Price Index (WPI): measures wholesale price changes; inflation indicator
Tests of Index Numbers: Time Reversal Test (P₀₁ × P₁₀ = 1), Factor Reversal Test (P₀₁ × Q₀₁ = V₀₁)
Uses of Index Numbers: measure inflation, adjust wages (DA), deflation of national income, compare cost of living

Intext Questions

Section 8.1 — Meaning and Types

Q1. What is an index number?

AnswerAn index number is a statistical measure designed to show changes in a variable or a group of related variables over time, with reference to a base period. It is expressed as a percentage of the base period value (which is taken as 100).

Q2. Why is the base year important?

AnswerThe base year serves as the reference point against which all comparisons are made. It should be a normal year — free from economic abnormalities like war, famine, or boom — so that the index number provides meaningful comparisons.

Section 8.2 — Construction of Index Numbers

Q3. What is the difference between simple and weighted index numbers?

AnswerSimple index numbers give equal importance to all items. Weighted index numbers assign different weights to different items based on their relative importance. Weighted indices are more realistic because all items do not have equal significance in consumption or production.

Chapter-End Exercise

Question 1

Calculate the price index number for 2024 with 2020 as base using:

Commodity	Price 2020 (p₀)	Price 2024 (p₁)
A	10	15
B	20	25
C	30	35
D	40	50

Answer (Simple Aggregate Method):

P₀₁ = Σp₁/Σp₀ × 100 = (15+25+35+50)/(10+20+30+40) × 100 = 125/100 × 100 = 125

Interpretation: Prices increased by 25% from 2020 to 2024.

Question 2

Calculate the price index using the simple average of price relatives method:

Commodity	p₀	p₁	Price Relative (p₁/p₀ × 100)
A	10	15	150
B	20	25	125
C	30	35	116.67
D	40	50	125

AnswerP₀₁ = (Σ Price Relatives)/n = (150 + 125 + 116.67 + 125)/4 = 516.67/4 = 129.17

Question 3

Calculate Laspeyres index from the following data:

Commodity	p₀	q₀	p₁	q₁
A	10	50	15	40
B	20	30	25	35
C	30	20	35	25

Answer

Commodity	p₁q₀	p₀q₀
A	15×50 = 750	10×50 = 500
B	25×30 = 750	20×30 = 600
C	35×20 = 700	30×20 = 600
Total	2200	1700

Lasp = Σp₁q₀/Σp₀q₀ × 100 = 2200/1700 × 100 = 129.41

Question 4

Calculate Paasche index from the same data (Question 3).

Answer

Commodity	p₁q₁	p₀q₁
A	15×40 = 600	10×40 = 400
B	25×35 = 875	20×35 = 700
C	35×25 = 875	30×25 = 750
Total	2350	1850

Paas = Σp₁q₁/Σp₀q₁ × 100 = 2350/1850 × 100 = 127.03

Question 5

Calculate Fisher's Ideal Index from the Laspeyres and Paasche indices calculated above.

Answer

Fisher = √(Lasp × Paas) = √(129.41 × 127.03) = √16439.06 = 128.27

Question 6

Calculate the Consumer Price Index (CPI) from the following data:

Group	Weight (W)	Price Relative (I)
Food	40	150
Clothing	15	130
Fuel	10	120
Rent	20	140
Misc	15	110

Answer

Group	W	I	WI
Food	40	150	6000
Clothing	15	130	1950
Fuel	10	120	1200
Rent	20	140	2800
Misc	15	110	1650
Total	100		13600

CPI = ΣWI/ΣW = 13600/100 = 136

Interpretation: The cost of living has increased by 36% compared to the base year.

Question 7

What is the time reversal test? Which index numbers satisfy it?

AnswerThe Time Reversal Test states that if we calculate the index for period 1 with period 0 as base (P₀₁), and then calculate the index for period 0 with period 1 as base (P₁₀), the product should equal 1.

Condition: P₀₁ × P₁₀ = 1

Fisher's Ideal Index satisfies this test because it is the geometric mean of Laspeyres and Paasche indices. Neither Laspeyres nor Paasche alone satisfies this test.

Question 8

What is the factor reversal test?

AnswerThe Factor Reversal Test states that the product of the price index and the quantity index for the same period should equal the value index (value relative).

Condition: P₀₁ × Q₀₁ = V₀₁

Where P₀₁ = price index, Q₀₁ = quantity index, V₀₁ = (p₁q₁)/(p₀q₀) (value relative)

Fisher's Ideal Index satisfies this test, which is why it is called the "ideal" index.

Question 9

Explain the uses of index numbers.

Answer

1. Measure of inflation: CPI and WPI measure the rate of inflation in the economy. 2. Dearness Allowance: CPI is used to calculate DA for government employees and workers. 3. Deflation of national income: Index numbers convert nominal income into real income. 4. Comparison of cost of living: CPI helps compare living costs across cities and time. 5. Economic policy-making: Government uses index numbers for fiscal and monetary policy. 6. Business decisions: Firms use price indices for pricing, wage negotiations, and inventory management. 7. Stock market analysis: Sensex and Nifty are index numbers tracking stock prices. 8. Agricultural analysis: Agricultural price indices track farm product prices.

Question 10

What are the limitations of index numbers?

Answer

1. Based on samples: Not all items are included; index is based on a representative basket. 2. Choice of base year: A non-normal base year distorts the index. 3. Quality changes: Index does not account for changes in product quality over time. 4. New products: New items entering the market are not immediately included. 5. Weighting problems: Weights may become outdated as consumption patterns change. 6. International comparison: Different countries use different baskets and base years. 7. Not exact: Index numbers are approximate measures, not precise.

Question 11

Explain the difference between CPI and WPI.

Answer

Basis	CPI	WPI
Full form	Consumer Price Index	Wholesale Price Index
Level	Retail prices	Wholesale prices
Coverage	Goods and services consumed by households	Goods traded at wholesale level
Services	Includes services	Generally excludes services
Use	Calculating DA, cost of living	Indicator of inflation
Frequency	Monthly	Weekly/Monthly
Published by	Labour Bureau / CSO	Office of Economic Adviser

Question 12

Calculate the index number using weighted average of price relatives:

Commodity	p₀	p₁	q₀
A	5	8	10
B	10	12	5
C	15	18	8

Answer

Commodity	p₀	p₁	q₀	Weight W = p₀q₀	Price Relative I = (p₁/p₀)×100	WI
A	5	8	10	50	(8/5)×100 = 160	8000
B	10	12	5	50	(12/10)×100 = 120	6000
C	15	18	8	120	(18/15)×100 = 120	14400
Total				220		28400

P₀₁ = ΣWI/ΣW = 28400/220 = 129.09

Internal Links

[Chapter 8 Guide — Index Numbers](/school/cbse/class/11/economics/cbse-index-numbers)
[All Class 11 Economics Solutions](/school/cbse/class/11/economics/solutions)
[Class 11 Economics Subject Guide](/school/cbse/class/11/economics)

Frequently Asked Questions

Laspeyres index uses base year quantities as weights (P₀₁L = Σp₁q₀/Σp₀q₀ × 100). Paasche index uses current year quantities as weights (P₀₁P = Σp₁q₁/Σp₀q₁ × 100). Laspeyres tends to overstate price changes; Paasche tends to understate them.