Fixing Dearness Allowance When the Basket and the Base Year Both Move
A workers' welfare association at a textile unit in Hubballi is negotiating dearness allowance. The 2011 base-year monthly basket for a family is rice 20 kg at ₹40 per kg, dal 10 kg at ₹80 per kg, edible oil 5 litres at ₹120 per litre and sugar 8 kg at ₹30 per kg. A 2024 market survey of the same shops records rice ₹50, dal ₹100, oil ₹150 and sugar ₹36. Consumption in the sampled families has meanwhile shifted to 22 kg, 9 kg, 4 litres and 9 kg. Basic pay is ₹18,000; at the last revision the index stood at 108 and DA of 8 per cent (₹1,440) was sanctioned, so a worker now draws ₹19,440. Management has offered a flat ₹2,500 extra per month. The union wants DA computed so that the 2011 basket stays affordable, and separately insists the new percentage be applied to the total pay of ₹19,440 rather than to basic pay. From next year the unit will publish the index with 2024 = 100, and last year's reading of 110 (on 2011 = 100) must be carried into the new series.
1. Compute the Laspeyres and Paasche price index numbers for 2024 on 2011 = 100 from the given data, and explain in one paragraph why the two numbers differ.
Read the worked answer
Build the four required sums item by item (rice, dal, oil, sugar):
- Σp₀q₀ = 800 + 800 + 600 + 240 = 2440
- Σp₁q₀ = 1000 + 1000 + 750 + 288 = 3038
- Σp₀q₁ = 880 + 720 + 480 + 270 = 2350
- Σp₁q₁ = 1100 + 900 + 600 + 324 = 2924
Laspeyres: P₀₁ = (Σp₁q₀ / Σp₀q₀) × 100 = (3038/2440) × 100 = 124.51. Paasche: P₀₁ = (Σp₁q₁ / Σp₀q₁) × 100 = (2924/2350) × 100 = 124.43.
The two differ because Laspeyres freezes 2011 quantities and therefore ignores the household's substitution away from items whose relative price rose most (oil quantity fell from 5 to 4 litres, sugar rose from 8 to 9 kg). Holding the dear old basket makes Laspeyres a slight over-statement of the true cost-of-living change; Paasche uses current quantities but its weights are the very quantities that prices have already distorted.
2. Using the Laspeyres index, determine the additional DA payable now and evaluate the union's proposal to apply the full percentage to ₹19,440. Is the management offer adequate?
Read the worked answer
Full DA on basic pay at index 124.51 = 24.51% of ₹18,000 = ₹4,411.80. Already compensated at index 108: 8% of ₹18,000 = ₹1,440. Additional DA due = 4,411.80 − 1,440 = ₹2,971.80, which equals 16.51 index points × ₹180 per point - the correct base for the increment is the rise in the index (124.51 − 108), not the whole 124.51.
Union's method: 24.51% of ₹19,440 = ₹4,764.74. That double counts, because ₹1,440 of that total pay already reflects earlier price rise, and it applies 24.51 points instead of 16.51. Overstatement = ₹1,792.94 per worker per month.
Management's flat ₹2,500 falls short of the ₹2,971.80 due by ₹471.80 per worker per month. A flat rupee amount also breaks the link with the index: the next revision will reopen the same dispute unless the formula is written as points above 108 × ₹180.
3. The series is to be rebased to 2024 = 100. Convert the old-base value 110 into the new series and state what the conversion factor is doing.
Read the worked answer
The linking factor is (100 / index of the new base on the old base) = 100/124.51 = 0.8032. New-base value = old-base value × 0.8032 = 110 × 0.8032 = 88.35.
Interpretation: 2024 becomes the new 100 and 2011 is no longer 100 but 100/1.2451 = 80.32. Any old-base index number is multiplied by 0.8032 to sit on the new base.
Check that rebasing is only a change of origin, not of economic content: from 110 to 124.51 the old series rose by 13.19%; on the new base the same movement is 88.35 to 100, again 13.19%. Rebasing therefore leaves percentage changes between two dates intact and alters only the level. In the report, state the linking factor and the overlap-year values explicitly, otherwise a reader comparing 110 (old) with 100 (new) will wrongly conclude that prices fell.