Mess-fee revision and choice of price index
A hostel mess buys rice, dal and cooking oil. In the base year 2022 it bought 100 kg rice at ₹40/kg, 50 kg dal at ₹80/kg and 20 litres oil at ₹120/litre. In 2023 it bought 90 kg rice at ₹48/kg, 55 kg dal at ₹90/kg and 18 litres oil at ₹150/litre. The mess secretary wants a price index to justify a mess-fee revision. The accountant proposes a Laspeyres index, while a student union member suggests a Paasche index; another suggests simply averaging the three price relatives. The actual 2023 mess bill was ₹11,970, while the 2022 bill was ₹10,400. The warden wants to know which index is appropriate for compensating students for the base-year basket, why the actual bill rose by less than the Laspeyres index suggests, and whether the simple average is reliable.
1. Compute the Laspeyres and Paasche price indices and interpret the difference.
Read the worked answer
Base-year expenditure: ΣP0Q0 = 100(40) + 50(80) + 20(120) = 4,000 + 4,000 + 2,400 = ₹10,400.
At 2023 prices with 2022 quantities: ΣP1Q0 = 100(48) + 50(90) + 20(150) = 4,800 + 4,500 + 3,000 = ₹12,300. So Laspeyres = 12,300/10,400 × 100 ≈ 118.27.
At 2023 prices and quantities: ΣP1Q1 = 90(48) + 55(90) + 18(150) = 4,320 + 4,950 + 2,700 = ₹11,970.
At 2022 prices with 2023 quantities: ΣP0Q1 = 90(40) + 55(80) + 18(120) = 3,600 + 4,400 + 2,160 = ₹10,160. So Paasche = 11,970/10,160 × 100 ≈ 117.81.
Laspeyres says the base basket costs about 18.27% more, while Paasche says the current basket costs about 17.81% more at current prices than at base prices. The difference comes from the change in quantities.
2. The secretary says prices rose about 18%, so the 2023 bill should be 10,400 × 1.18 = ₹12,272. The actual bill is ₹11,970. Explain the discrepancy and state which index is appropriate for compensating the base-year basket.
Read the worked answer
The figure ₹12,272 is only an approximation using 118 as the index. Using the Laspeyres index exactly gives 10,400 × 1.1827 = ₹12,300. That is the cost of buying the 2022 basket at 2023 prices.
The actual 2023 bill is lower because the mess did not buy the same basket. It bought less rice and less oil, and more dal. Rice and oil had relatively large price rises, while dal rose less. The actual current basket therefore costs ₹11,970, not ₹12,300.
For compensating students for the base-year consumption basket, the Laspeyres index is appropriate: it answers 'what would the old basket cost now?' The Paasche index is more relevant for deflating current expenditure, because it uses current quantities. The actual expenditure change is 11,970/10,400 × 100 ≈ 115.10, which is a value index, not a pure price index.
3. Another member computes a simple average of the three price relatives as 119.17. Compare this with the Laspeyres index and explain why it may mislead.
Read the worked answer
Price relatives: rice 48/40 = 1.20; dal 90/80 = 1.125; oil 150/120 = 1.25. Simple average = (1.20 + 1.125 + 1.25)/3 = 1.1917, or 119.17.
This is higher than the Laspeyres index of about 118.27. The simple average gives each commodity equal weight, regardless of its share in the mess budget. Oil has the largest price rise (25%) but a relatively small base-year weight: ₹2,400 out of ₹10,400, about 23%. Dal has the smallest price rise (12.5%) but a larger weight of ₹4,000, about 38%. Equal weighting therefore overstates the effect of oil and understates the effect of dal.
The simple average is a descriptive summary of price relatives, not a fixed-basket cost index. It should not be used to justify a mess-fee revision when quantity weights are available. The Laspeyres index is the more defensible choice for that purpose.