Textile Mill Wage Talks: Laspeyres, Paasche, Fisher and a Trend Forecast
The workers' union at a Coimbatore textile mill is negotiating dearness allowance. The agreed basket has three items with 2022 base prices and quantities and 2024 prices. Rice: Rs 30 per kg, 20 kg in the base basket; edible oil: Rs 80 per litre, 5 litres; cloth: Rs 40 per metre, 10 metres. In 2024 the prices are Rs 36, Rs 100 and Rs 44 respectively, while the union's own consumption survey gives quantities of 22 kg, 4 litres and 12 metres. Management has offered DA on a current-weighted (Paasche) basis, and the union is quoting a base-weighted (Laspeyres) figure. The basic wage is Rs 18,000 a month across 1,200 workers. Separately, the union cites the mill's output of 42, 45, 47, 50 and 54 lakh metres in 2019 to 2023 as evidence that the mill can afford to pay, while management calls this a straight-line fit over five years and disputes any forecast from it.
1. Compute the Laspeyres, Paasche and Fisher price indices for 2024 on a 2022 base, showing the four value sums used.
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Base-year value sum: sum p0q0 = 30(20) + 80(5) + 40(10) = 600 + 400 + 400 = 1400.
Current prices on base quantities: sum p1q0 = 36(20) + 100(5) + 44(10) = 720 + 500 + 440 = 1660.
Current-year value sum: sum p1q1 = 36(22) + 100(4) + 44(12) = 792 + 400 + 528 = 1720.
Base prices on current quantities: sum p0q1 = 30(22) + 80(4) + 40(12) = 660 + 320 + 480 = 1460.
Laspeyres = (1660/1400) x 100 = 118.57 Paasche = (1720/1460) x 100 = 117.81 Fisher = square root of (118.57 x 117.81) = square root of 13,968.7 = 118.19
Note that Laspeyres exceeds Paasche here because the item whose price rose most (edible oil, up 25 per cent) is the one whose quantity fell in the current basket, so the base-weighted index gives it extra weight. Fisher sits between the two, which is exactly why it is called ideal.
2. Fit a straight-line trend to the output figures by least squares and forecast 2024 output. State the trend equation and the arithmetic clearly.
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Code the years t = 1 to 5 and output y in lakh metres: (1,42), (2,45), (3,47), (4,50), (5,54).
n = 5, sum t = 15, sum y = 238, sum ty = 42 + 90 + 141 + 200 + 270 = 743, sum t^2 = 55. Means: t-bar = 3, y-bar = 47.6.
b = [n(sum ty) - (sum t)(sum y)] / [n(sum t^2) - (sum t)^2] = [5(743) - 15(238)] / [5(55) - 225] = (3715 - 3570)/50 = 145/50 = 2.9
a = y-bar - b(t-bar) = 47.6 - 2.9(3) = 38.9
Trend: y = 38.9 + 2.9t, origin at 2019, t in years.
Forecast for 2024 (t = 6): y = 38.9 + 17.4 = 56.3 lakh metres.
The interpretation is a steady rise of about 2.9 lakh metres a year. The fit is a straight line through only five points, so the forecast is a working figure, not a certainty; a check against actual 2024 output should be made before it is used in bargaining.
3. Recommend which index should anchor the DA settlement, and quantify the money at stake for the workforce.
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Start with the money. On a basic of Rs 18,000, the DA at each index is: Laspeyres 18.57 per cent = Rs 3,342.60; Paasche 17.81 per cent = Rs 3,205.80; Fisher 18.19 per cent = Rs 3,274.20.
The union-management gap is 3,342.60 - 3,205.80 = Rs 136.80 per worker per month, which is Rs 1,64,160 per month across 1,200 workers, or nearly Rs 19.7 lakh over a year. That is too large to leave to which formula each side happens to prefer.
Recommendation: anchor DA at the Fisher index, which is the geometric mean of the two and therefore accepts both weighting systems instead of one. Its Rs 3,274.20 is a defensible midpoint, and the parties should agree in advance that Fisher will be used in future settlements so the formula itself never becomes a bargaining chip again.
On the output argument: the trend of +2.9 lakh metres a year supports ability to pay, so management's poverty plea is weak. But the forecast of 56.3 lakh metres for 2024 is an extrapolation from five points and should not be treated as audited output; the union should use the direction of the trend, not the exact forecast, as its argument.