Question 5 (a)
Cost-of-living index of a household by Laspeyres', Paasche's and Fisher's methods
9 marks
- Time to write
- About 18 minutes
- Topic
- Index numbers
Question
Question 5 (a) · 9 marksThe following data record the yearly consumption of a household for the current year (2023) and the base year (2022).
| Item | 2022 Quantity | 2022 Price | 2023 Quantity | 2023 Price |
|---|---|---|---|---|
| Wheat (Kg) | 160 | 25 | 170 | 30 |
| Rice (Kg) | 55 | 40 | 60 | 44 |
| Pulses (Kg) | 55 | 66 | 58 | 82 |
| Edible Oil (Liters) | 25 | 88 | 27 | 94 |
| Sugar (Kg) | 44 | 50 | 50 | 52 |
| Cooking Gas (Kg) | 55 | 38 | 62 | 42 |
| Clothing (Meters) | 66 | 100 | 75 | 108 |
Compute the cost of living Index number by Laspeyres', Paasche's and Fisher's methods.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- ComputeFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Working table of price-quantity products 3 marks 5 min
- 2 Laspeyres' index 2 marks 4 min
- 3 Paasche's index 2 marks 4 min
- 4 Fisher's index and interpretation 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Method. A cost-of-living index measures the change in what a fixed basket of goods and services costs a group of consumers. With prices and quantities of both years given, it is computed as a weighted aggregative price index: Laspeyres' method weights by base-year (2022) quantities, Paasche's by current-year (2023) quantities, and Fisher's index is the geometric mean of the two. Here p₀, q₀ are the 2022 prices and quantities and p₁, q₁ those of 2023.
| Item | q₀ | p₀ | q₁ | p₁ | p₀q₀ | p₁q₀ | p₀q₁ | p₁q₁ |
|---|---|---|---|---|---|---|---|---|
| Wheat (Kg) | 160 | 25 | 170 | 30 | 4,000 | 4,800 | 4,250 | 5,100 |
| Rice (Kg) | 55 | 40 | 60 | 44 | 2,200 | 2,420 | 2,400 | 2,640 |
| Pulses (Kg) | 55 | 66 | 58 | 82 | 3,630 | 4,510 | 3,828 | 4,756 |
| Edible Oil (Liters) | 25 | 88 | 27 | 94 | 2,200 | 2,350 | 2,376 | 2,538 |
| Sugar (Kg) | 44 | 50 | 50 | 52 | 2,200 | 2,288 | 2,500 | 2,600 |
| Cooking Gas (Kg) | 55 | 38 | 62 | 42 | 2,090 | 2,310 | 2,356 | 2,604 |
| Clothing (Meters) | 66 | 100 | 75 | 108 | 6,600 | 7,128 | 7,500 | 8,100 |
| Total | 22,920 | 25,806 | 25,210 | 28,338 |
P₀₁ (L) = Σp₁q₀ / Σp₀q₀ × 100 = 25,806 / 22,920 × 100 = 112.59.
P₀₁ (P) = Σp₁q₁ / Σp₀q₁ × 100 = 28,338 / 25,210 × 100 = 112.41.
P₀₁ (F) = √[(Σp₁q₀ / Σp₀q₀) × (Σp₁q₁ / Σp₀q₁)] × 100 = √(112.59 × 112.41) = 112.50.
Interpretation. On Fisher's ideal measure, the household's cost of living rose by about 12.5% from 2022 to 2023: the 2022 basket costs 12.59% more at 2023 prices (Laspeyres) and the 2023 basket 12.41% more than at 2022 prices (Paasche). The two weightings give nearly the same answer because the household's quantities changed little between the years. To keep the same standard of living, its income would have to rise by about 12.5%. The largest rises are in pulses (66 to 82, about 24%) and wheat (25 to 30, 20%).
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Working table of price-quantity products | 3 |
| 2 | Laspeyres' index | 2 |
| 3 | Paasche's index | 2 |
| 4 | Fisher's index and interpretation | 2 |
| Total | 9 |
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