Hostel Mess Fee and a Fixed-Basket Index
The mess committee of a government hostel wants to revise the monthly mess fee. In the base year 2023, a student's monthly basket was rice 15 kg at ₹40 per kg, dal 4 kg at ₹90 per kg, cooking oil 2 litres at ₹120 per litre, and vegetables 10 kg at ₹30 per kg. In 2024 the respective prices are ₹52, ₹108, ₹150 and ₹45. The committee finds that a typical student now consumes 16 kg rice, 4 kg dal, 2.5 litres of oil and 12 kg vegetables. The warden suggests simply averaging the four price-relative percentages, which gives 131.25. A student representative says vegetables alone rose by 50%, so the index must be 150. The committee must justify the fee increase to the principal. It wants a fixed-basket index that reflects the 2023 consumption pattern, plus a current-pattern index for comparison. The mess fee in 2023 was ₹1,500 per month.
1. Construct the Laspeyres price index using 2023 quantities as weights. What monthly fee does it imply, and how should the committee explain the increase?
Read the worked answer
Laspeyres index = (Σp1q0 / Σp0q0) × 100.
| Item | p0 | q0 | p1 | p1q0 | p0q0 |
|---|---|---|---|---|---|
| Rice | 40 | 15 | 52 | 780 | 600 |
| Dal | 90 | 4 | 108 | 432 | 360 |
| Oil | 120 | 2 | 150 | 300 | 240 |
| Vegetables | 30 | 10 | 45 | 450 | 300 |
| Total | 1,962 | 1,500 |
Index = 1,962/1,500 × 100 = 130.8. The 2023 basket now costs ₹1,962, so the fee would rise from ₹1,500 to ₹1,962, an increase of 30.8%. The committee should say this is the cost of buying the same 2023 quantities at 2024 prices, not a claim that every student buys exactly that basket.
2. Why is the simple average 131.25 not the same as the weighted index, and why is 150 wrong?
Read the worked answer
The simple average treats each item equally: (130 + 120 + 125 + 150)/4 = 131.25. But rice alone is 600/1,500 = 40% of the 2023 basket, while oil is only 240/1,500 = 16%. A 50% rise in vegetables, weighted 300/1,500 = 20%, cannot by itself make the whole index 150. The 150 figure would be right only if vegetables were the entire basket. A weighted index is needed because price changes matter in proportion to budget share.
3. Compute the Paasche index using 2024 quantities and compare it with Laspeyres. Which index is more appropriate for revising the mess fee?
Read the worked answer
Paasche index = Σp1q1/Σp0q1. With 2024 quantities: p1q1 = 52×16 + 108×4 + 150×2.5 + 45×12 = 832 + 432 + 375 + 540 = 2,179. p0q1 = 40×16 + 90×4 + 120×2.5 + 30×12 = 640 + 360 + 300 + 360 = 1,660. Index = 2,179/1,660 × 100 = 131.27. It is close to Laspeyres 130.8, so the conclusion is robust. For revising the mess fee, Laspeyres is usually preferred because it holds the base basket fixed and measures the cost of maintaining the same standard; Paasche reflects current consumption and substitution. The committee can report both and use Laspeyres as the main justification.