Choosing a Cost-of-Living Index for a Wage Dispute
A labour office in Dibrugarh studies the cost of living for a typical family using two commodities, rice and mustard oil. In the base year 2020, rice costs ₹20/kg with quantity 10 kg, and oil costs ₹50/litre with quantity 10 litres. In 2023, rice costs ₹30/kg with quantity 4 kg, and oil costs ₹55/litre with quantity 16 litres. A worker's monthly wage rose from ₹20,000 to ₹24,000 over the same period. The office must compute the Laspeyres and Paasche price indices, choose one for a cost-of-living adjustment, and decide whether the worker's real income improved. A union argues that the government should use the index that gives the larger wage increase; the employer argues for the smaller one.
1. Compute the Laspeyres and Paasche price indices with 2020 as base. Why do they differ?
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For Laspeyres, use base-year quantities:
[ \sum p_0q_0=20(10)+50(10)=700 ] [ \sum p_1q_0=30(10)+55(10)=850 ] [ L=\frac{850}{700}\times100\approx121.43 ]
For Paasche, use current-year quantities:
[ \sum p_1q_1=30(4)+55(16)=1000 ] [ \sum p_0q_1=20(4)+50(16)=880 ] [ P=\frac{1000}{880}\times100\approx113.64 ]
The gap arises because the household substituted away from rice, whose price rose by 50%, toward oil, whose price rose by only 10%. Laspeyres keeps the old fixed basket and so overstates the cost increase when consumers substitute. Paasche uses the current basket and can understate the cost of maintaining the old standard of living.
2. Which index should the office use for a cost-of-living adjustment? Discuss the union's and employer's positions.
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There is no purely statistical answer; the choice depends on purpose. Laspeyres (121.43) measures the cost of buying the base-year basket and is common in consumer price indices and wage indexation because it is fixed and easy to explain. Paasche (113.64) reflects what households actually buy now and is better for current consumption patterns. A defensible compromise is the Fisher ideal index:
[ F=\sqrt{L\times P}=\sqrt{121.43\times113.64}\approx117.47 ]
The union prefers Laspeyres because it is larger and supports a bigger compensation claim; the employer prefers Paasche because it is smaller. Neither position is neutral. The office should state which basket it is protecting and report both indices, or use Fisher, rather than choosing the index after seeing which side it favours.
3. Using each index, determine whether the worker's real income improved. What should the office report?
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The nominal wage ratio is
[ \frac{24000}{20000}=1.20 ]
Deflating by Laspeyres:
[ \frac{1.20}{1.2143}\approx0.988 ]
so real income fell by about 1.2% if the base basket is used. Deflating by Paasche:
[ \frac{1.20}{1.1364}\approx1.056 ]
so real income rose by about 5.6% if the current basket is used. The direction of the conclusion therefore depends on the index. The office should not report only one figure without qualification. It should state the base basket, note the substitution, and report a range or the Fisher result (1.20/1.1747 ≈ 1.022, a small gain). The substantive message is that the wage rise roughly kept pace with prices, with the sign of the real change sensitive to the index chosen.