Rebasing the Price Index and Two Formulas for the Same Basket
The directorate of economics and statistics of a state publishes a monthly consumer price index and an annual index of industrial production. In 2018-19 the consumer price index was rebased from 2011-12 = 100 to 2017-18 = 100. On the old base the index stood at 145.0 for 2017-18; on the new base it reads 100.0 for 2017-18, 108.0 for 2018-19 and 115.0 for 2019-20. A newspaper headline announces that 'prices fell 45 points in a single year'. The directorate also publishes a weighted price index for three commodity groups, with base-year prices and quantities and current-year prices and quantities: group A, p0 = 20, p1 = 22, q0 = 10, q1 = 14; group B, p0 = 50, p1 = 45, q0 = 4, q1 = 6; group C, p0 = 10, p1 = 20, q0 = 30, q1 = 20. A users' committee asks why the Laspeyres and Paasche versions of this basket differ, which one should be quoted for a cost-of-living statement and which for deflating nominal expenditure, and whether the base should now be shifted to 2019-20.
1. Assess the newspaper's claim and place the new series on the old base so the two can be compared. State the 2019-20 inflation rate.
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The claim compares two numbers on different bases: 145.0 is 2017-18 measured with 2011-12 = 100, while 100.0 is the same year 2017-18 measured with 2017-18 = 100. They are the same price level, so 'a fall of 45 points' is an artefact of rebasing, not a price movement.
Linking factor = old index for the link year / new index for the link year = 145.0/100.0 = 1.45. Multiplying the new series by 1.45 splices it onto the old base:
- 2017-18: 100.0 × 1.45 = 145.0
- 2018-19: 108.0 × 1.45 = 156.6
- 2019-20: 115.0 × 1.45 = 166.75
On the old base, prices rose 145.0 → 156.6 (8.0%) in 2018-19 and 156.6 → 166.75 (6.5%) in 2019-20.
2019-20 inflation = (115/108 − 1) × 100 = 6.48%. The ratio of two index numbers on a common base is invariant to the base, so the growth rate is the same whether it is computed from 115/108 or 166.75/156.6; only the levels are affected by rebasing. The correct headline is 'prices rose about 6.5%, slower than last year's 8.0%'.
2. Compute the Laspeyres and Paasche price indices, explain why they diverge, and say which one belongs in a cost-of-living statement and which in deflating nominal expenditure.
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Sums: Σp0q0 = 20(10) + 50(4) + 10(30) = 200 + 200 + 300 = 700 Σp1q0 = 22(10) + 45(4) + 20(30) = 220 + 180 + 600 = 1,000 Σp0q1 = 20(14) + 50(6) + 10(20) = 280 + 300 + 200 = 780 Σp1q1 = 22(14) + 45(6) + 20(20) = 308 + 270 + 400 = 978
Laspeyres = (Σp1q0/Σp0q0) × 100 = 1000/700 × 100 = 142.86 Paasche = (Σp1q1/Σp0q1) × 100 = 978/780 × 100 = 125.38 Fisher = √(142.86 × 125.38) = √17,912 ≈ 133.8; Marshall–Edgeworth = (Σp1(q0+q1))/(Σp0(q0+q1)) = 1978/1480 × 100 = 133.6.
Why they differ: group C's price doubled while its quantity fell from 30 to 20, and group B's price fell while its quantity rose - households shifted away from what had become expensive. Laspeyres prices the old basket, so it ignores that substitution and overstates the cost of maintaining today's standard of living; Paasche prices the current basket, which has already adapted. For a fixed-weight index with rising relative prices, Laspeyres ≥ Paasche, as here (the gap of 17.5 points is unusually large because the basket is only three groups).
Which to use: for a cost-of-living or dearness-allowance statement the question is what a fixed basket costs now, so a Laspeyres-type index (or a superlative alternative such as Fisher) is appropriate, and it is also cheaper to compile because only base quantities are needed. For deflating nominal expenditure, Paasche is the right deflator: current-value expenditure divided by a Paasche price index yields a Laspeyres volume index, and a deflator must weight by the current period's composition of spending - the same reasoning that makes the implicit GDP deflator a Paasche-type measure.
3. The committee proposes shifting the base to 2019-20. Advise on whether that is the right step, distinguishing a base shift from a rebasing.
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Two different operations are being run together. Shifting the base is pure presentation: dividing every index number by the 2019-20 value and multiplying by 100. Every growth rate is unchanged, the 2019-20 figure becomes 100, and comparability with earlier years is preserved only if the series is spliced by a linking factor, exactly as in the first task. It buys nothing analytically.
Rebasing is substantive: it means replacing the weighting diagram and item list with those from a fresh household expenditure survey, so that the basket reflects where households now spend, and adding items that did not exist or did not matter in 2011-12 while dropping those that no longer do. This is worth doing when the weights have gone stale - the relative price movements in the table above (C doubling, B falling) are precisely how a fixed 2011-12 basket drifts away from actual expenditure patterns. Proper rebasing requires a link year in which both the old and new series are computed, back-casting of the new series so that a continuous history survives, clear documentation of the break, and an explicit rule that index levels are never compared across the break (growth rates may be, once spliced).
Shifting the base every year, or even every few years, merely to look current would fragment the long time series that wage indexation, deflation and research all depend on. The recommendation: do not shift; rebase once the new survey is available, publish the spliced back series and the linking factor, and warn users in the release note.