Weighted Index Numbers ⚖️

In real life, all items are not equally important. Wheat is more important than Salt. To reflect this, we assign "weights" (importance) to items.

There are two major methods for Weighted Aggregative Index Numbers:

1. Laspeyres' Method
2. Paasche's Method

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1. Laspeyres' Method 👨‍🏫

Concept: Uses Base Year Quantity (q0) as weights. This is the most popular method because base year quantities are fixed and easier to know.

Formula

L = (∑ p1 q0 / ∑ p0 q0) * 100

• Bias: It tends to have an upward bias (overestimates inflation) because people shift to cheaper substitutes when prices rise, but this formula assumes they buy the old quantities.

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2. Paasche's Method 👨‍🎓

Concept: Uses Current Year Quantity (q1) as weights. This reflects current consumption habits.

Formula

P = (∑ p1 q1 / ∑ p0 q1) * 100

• Bias: It tends to have a downward bias (underestimates inflation).

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3. Fisher's Ideal Index Method 🏆

Since Laspeyres overestimates and Paasche underestimates, Irving Fisher proposed taking the Geometric Mean of both.

Formula

F = √(L * P)
F = √[ (∑ p1 q0 / ∑ p0 q0) * (∑ p1 q1 / ∑ p0 q1) ] * 100

• Why "Ideal"?
  1. It considers both base and current year quantities.
  2. It satisfies Time Reversal and Factor Reversal Tests.
  3. It is free from bias.

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Example Calculation 📝

Data:

Item	p0	q0	p1	q1
X	10	5	20	2
Y	5	10	10	8

Step 1: Calculate Columns

• p1q0: (20 * 5) = 100, (10 * 10) = 100. Sum = 200
• p0q0: (10 * 5) = 50, (5 * 10) = 50. Sum = 100
• p1q1: (20 * 2) = 40, (10 * 8) = 80. Sum = 120
• p0q1: (10 * 2) = 20, (5 * 8) = 40. Sum = 60

Step 2: Apply Formulas

Laspeyres (L):

L = (200 / 100) * 100 = 200

Paasche (P):

P = (120 / 60) * 100 = 200

Fisher (F):

F = √(200 * 200) = 200

(In this simpler example, all match. In reality, they differ slightly).

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Comparison Table 📊

Method	Weights Used	Formula Code	Bias
Laspeyres	Base Year (q0)	10 / 00	Upward
Paasche	Current Year (q1)	11 / 01	Downward
Fisher	Both (q0, q1)	√(L * P)	None (Ideal)

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