Dearness allowance, real wages and two price indices
A trade union in Aurangabad is negotiating wages with a mid-sized engineering firm. The agreed base year is 2020 = 100, and the union's basket contains three items with these prices per unit and base-year quantities:
| Item | Price 2020 (₹) | Quantity 2020 | Price 2023 (₹) |
|---|---|---|---|
| Rice | 20 | 10 | 25 |
| Clothing | 50 | 5 | 60 |
| Transport | 10 | 8 | 14 |
A worker's nominal monthly wage was ₹18,000 in 2020 and is ₹24,000 in 2023. The union's notice claims workers have 'gained 33% in real terms'. Management replies with a price index computed on 2023 quantities instead: 12 units of rice, 4 of clothing and 10 of transport. Both sides want the arithmetic checked, and the Labour Commissioner wants a defensible statement of how much of the ₹6,000 rise merely restores the 2020 standard of living.
1. Compute the Laspeyres price index for 2023 on base 2020 = 100 and interpret it.
Read the worked answer
Laspeyres: L = (Σp₁q₀ / Σp₀q₀) × 100.
Σp₀q₀ = 20(10) + 50(5) + 10(8) = 200 + 250 + 80 = 530. Σp₁q₀ = 25(10) + 60(5) + 14(8) = 250 + 300 + 112 = 662.
L = (662 / 530) × 100 = 124.91.
Interpretation: buying the 2020 basket in 2023 costs 24.91% more than it did in 2020. A worker needs ₹1.2491 in 2023 for every ₹1 spent in 2020 to command the same quantities.
2. Using the Laspeyres index, compute the real wage in 2023 at 2020 prices and the real wage index. Assess the union's '33% real gain' claim and state how much of the ₹6,000 rise is pure compensation.
Read the worked answer
Wage index = (24,000 / 18,000) × 100 = 133.33.
Real wage index = (Wage index / Price index) × 100 = (133.33 / 124.91) × 100 = 106.74.
Real wage = 24,000 / 1.2491 = ₹19,213.8 at 2020 prices.
So the real gain over 2020 is ₹19,213.8 − ₹18,000 = ₹1,213.8, i.e. about 6.7%, not 33%. The 33% figure is purely nominal; about four-fifths of the apparent rise was eaten by prices.
Compensation requirement: 18,000 × 1.2491 = ₹22,483.8. Of the ₹6,000 increase, ₹4,483.8 merely restores 2020 purchasing power (Dearness Allowance) and only ₹1,516.2 is a real increase. The union's claim, as worded, should be withdrawn; the defensible claim is a real gain of roughly 6.7%.
3. Compute the Paasche index on 2023 quantities and compare it with the Laspeyres figure. Which index should be quoted to the Commissioner, and what does the comparison reveal about the usual textbook warning?
Read the worked answer
Paasche: P = (Σp₁q₁ / Σp₀q₁) × 100, with q₁ = (12, 4, 10).
Σp₁q₁ = 25(12) + 60(4) + 14(10) = 300 + 240 + 140 = 680. Σp₀q₁ = 20(12) + 50(4) + 10(10) = 240 + 200 + 100 = 540.
P = (680 / 540) × 100 = 125.93.
Paasche real wage = 24,000 / 1.2593 = ₹19,058, a real gain of about 5.9%.
Which to quote. Present both: real gain lies between roughly 5.9% (Paasche) and 6.7% (Laspeyres). Either way the union's '33% real' claim fails, and the DA arithmetic of ₹22,483.8 is unaffected in substance.
The textbook warning is a tendency, not a law. The standard caution is that Laspeyres overstates because it freezes quantities while consumers substitute away from items whose prices rose most. Here the data do the opposite: rice rose 25% and transport 40%, and their 2023 quantities rose (10 → 12 and 8 → 10) while clothing, up only 20%, was cut (5 → 4). With quantities shifting toward the dearer items, Paasche comes out above Laspeyres. The direction must be checked against the actual q₁ pattern, not assumed.