Index Numbers – Numerical Problems 🧮
Practice calculating Laspeyres, Paasche, and Fisher's Index Numbers with these examples.
Problem 1: Weighted Index Numbers
Given:
| Commodity | Base Price (p0) | Base Qty (q0) | Current Price (p1) | Current Qty (q1) |
|---|---|---|---|---|
| A | 2 | 8 | 4 | 6 |
| B | 5 | 10 | 6 | 5 |
| C | 4 | 14 | 5 | 10 |
| D | 2 | 19 | 2 | 13 |
Calculate: Laspeyres, Paasche, and Fisher's Ideal Index.
Solution: Weighted Index Calculation 💡
Step 1: Create Table
| Item | p0 | q0 | p1 | q1 | p1q0 | p0q0 | p1q1 | p0q1 |
|---|---|---|---|---|---|---|---|---|
| A | 2 | 8 | 4 | 6 | 32 | 16 | 24 | 12 |
| B | 5 | 10 | 6 | 5 | 60 | 50 | 30 | 25 |
| C | 4 | 14 | 5 | 10 | 70 | 56 | 50 | 40 |
| D | 2 | 19 | 2 | 13 | 38 | 38 | 26 | 26 |
| Sum | 200 | 160 | 130 | 103 |
Step 2: Apply Formulas
1. Laspeyres (L):
L = (200 / 160) * 100 = 125
2. Paasche (P):
P = (130 / 103) * 100 = 126.21
3. Fisher (F):
F = √(125 * 126.21) = 125.60
Problem 2: Real Wages (Deflation)
Given:
| Year | Monthly Wages | Index Number |
|---|---|---|
| 1970 | 5000 | 100 |
| 1971 | 6000 | 150 |
| 1972 | 8000 | 200 |
Calculate: Real Wages for each year.
Solution: Real Wages Calculation 💡
Real Wages = (Money Wages / Index Number) * 100
- 1970:
(5000 / 100) * 100 = 5000 - 1971:
(6000 / 150) * 100 = 4000 - 1972:
(8000 / 200) * 100 = 4000
Conclusion: Although wages increased, purchasing power actually decreased after 1970.
Problem 3: Splicing
Given:
- Old Series (Base 2000): 2010 Index = 250
- New Series (Base 2010): 2010 Index = 100, 2011 Index = 110.
Task: Find the specific index for 2011 with Base 2000.
Solution: Splicing Index Series 💡
Spliced Index = (New Index[2011] * Old Index of Common Year[2010]) / 100
Spliced Index = (110 * 250) / 100 = 275
Answer: The index for 2011 is 275.
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