Index Numbers 📊

Laspeyre's, Paasche's, Fisher's — price change measure karna

1. Index Numbers — Parichay

Index number ek statistical measure hai jo kisi variable ki relative change show karta hai ek base period ke comparison mein. Price, quantity, ya cost of living measure karne ke liye use hota hai.

Price Relative = (P₁/P₀) × 100 P₁ = Current period price P₀ = Base period price

2. Types of Index Numbers

2.1 Simple Aggregative Method

P₀₁ = (ΣP₁/ΣP₀) × 100

2.2 Laspeyre's Price Index (Base period weights)

Lp = (ΣP₁Q₀/ΣP₀Q₀) × 100

2.3 Paasche's Price Index (Current period weights)

Pp = (ΣP₁Q₁/ΣP₀Q₁) × 100

2.4 Fisher's Ideal Index (Geometric Mean)

Fp = √(Lp × Pp) Fisher's is called "Ideal" because it satisfies Time Reversal Test aur Factor Reversal Test!

Udaharan: Complete Index Number Calculation

Item	P₀	Q₀	P₁	Q₁	P₁Q₀	P₀Q₀	P₁Q₁	P₀Q₁
A	5	10	7	12	70	50	84	60
B	8	15	9	18	135	120	162	144
C	3	20	4	25	80	60	100	75
Total	—	—	—	—	285	230	346	279

Laspeyre's = (285/230)×100 = 123.9
Paasche's = (346/279)×100 = 124.0
Fisher's = √(123.9 × 124.0) = √15363.4 = 123.9

3. Consumer Price Index (CPI)

CPI measure karta hai ki ek average consumer ke liye cost of living kitna change hua hai.

CPI = (ΣP₁W/ΣP₀W) × 100 (W = weight ya quantity consumed) Real Wages = (Nominal Wages / CPI) × 100 Purchasing Power = 100 / CPI

Udaharan: CPI Calculation

Item	Weight (W)	P₀	P₁	P₁W	P₀W
Food	40	100	120	4800	4000
Clothing	20	100	115	2300	2000
Housing	30	100	110	3300	3000
Others	10	100	105	1050	1000
Total	100	—	—	11450	10000

CPI = (11450/10000)×100 = 114.5
Inflation rate = 14.5%
If salary ₹50,000 hai: Real salary = (50000/114.5)×100 = ₹43,668

4. Tests for Index Numbers

Test	Formula	Pass karne wale
Time Reversal Test	P₀₁ × P₁₀ = 1	Fisher's ✅, Laspeyre's ❌
Factor Reversal Test	P₀₁ × Q₀₁ = V₀₁	Fisher's ✅ only
Circular Test	P₀₁ × P₁₂ × P₂₀ = 1	Simple aggregative ✅

💡 Isliye Fisher's index ko "Ideal" kehte hain — yeh time reversal aur factor reversal dono tests pass karta hai!

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