Is 28% Turnover Growth Real? Laspeyres, Paasche and Deflating a Wholesaler's Sales
A stationery wholesaler in Guwahati wants to know whether the business actually grew between 2022-23 and 2024-25 or whether the higher turnover is only price inflation. Three items account for almost all sales, and the records give base-year and current-year prices and quantities:
| Item | Price 2022-23 (₹) | Qty 2022-23 | Price 2024-25 (₹) | Qty 2024-25 |
|---|---|---|---|---|
| Long notebooks (dozen) | 30 | 400 | 36 | 500 |
| Ball pens (pack of 10) | 10 | 1,200 | 11 | 1,000 |
| Geometry boxes | 80 | 100 | 100 | 120 |
Turnover rose from ₹32,000 to ₹41,000, and the owner has told his bank that the business grew by 28%. The loan officer wants a price index and a volume measure before sanctioning an enhancement. The wholesaler has never built an index and is unsure whether to weight by the old basket, the new basket, or simply average the three price ratios, which come to 20%, 10% and 25%.
1. Compute the Laspeyres and Paasche price indices for 2024-25 with 2022-23 as base.
Read the worked answer
First compute the four required aggregates (base year 0 = 2022-23, current year 1 = 2024-25).
- Σp₀q₀ = 30(400) + 10(1,200) + 80(100) = 12,000 + 12,000 + 8,000 = 32,000
- Σp₁q₀ = 36(400) + 11(1,200) + 100(100) = 14,400 + 13,200 + 10,000 = 37,600
- Σp₀q₁ = 30(500) + 10(1,000) + 80(120) = 15,000 + 10,000 + 9,600 = 34,600
- Σp₁q₁ = 36(500) + 11(1,000) + 100(120) = 18,000 + 11,000 + 12,000 = 41,000
Laspeyres (base-year quantities): P₀₁ = 37,600/32,000 × 100 = 117.5. Prices are up 17.5% on the old basket.
Paasche (current-year quantities): P₀₁ = 41,000/34,600 × 100 = 118.5. Prices are up 18.5% on the basket actually sold now.
The simple average of the three price ratios, 18.33, is not a usable index: it weights a geometry box the same as a pack of pens, although spending on pens is comparable to spending on boxes and notebooks combined.
2. Explain why the two indices differ and what the direction of the gap reveals about the shop's sales mix.
Read the worked answer
The two indices weight differently. Laspeyres holds the 2022-23 basket fixed, so it measures the price change of what customers used to buy. Paasche uses the 2024-25 basket, so it measures the price change of what customers buy now. Both answers to the same question are legitimate; they differ because the basket moved.
The shift. Notebook prices rose 20%, geometry boxes 25%, pens only 10%. Yet the notebook share of spending rose from 12,000/32,000 = 37.5% to 18,000/41,000 = 43.9%, and quantities of pens fell from 1,200 to 1,000 packs.
Interpretation. Paasche (118.5) is above Laspeyres (117.5) because the mix drifted toward the items whose prices rose fastest. Normally we expect the opposite - customers substitute away from price-risers, which pulls Paasche below Laspeyres. Here they did not: the shop sold relatively more of the expensive-rising goods, probably because school demand for notebooks and geometry boxes is compulsory while pens were bought less.
The practical consequence is that the gap of one index point is not a statistical detail. It says part of the turnover growth is a mix effect, and that customers absorbed the increases rather than resisting them.
3. Using a suitable deflator, work out how much of the turnover growth was price and how much was real volume. Assess the owner's claim of 28% growth.
Read the worked answer
Choice of deflator. A current-period value (₹41,000 at 2024-25 prices) must be deflated by an index built on current-period quantities, i.e. the Paasche index of 118.5 (1.185). Laspeyres would only be appropriate for deflating a base-period quantity basket.
Real value. 41,000/1.185 = ₹34,600 - the current volumes valued at base-period prices (this is exactly Σp₀q₁ computed earlier, a good check).
Volume growth. 34,600/32,000 = 1.0813, i.e. +8.1% real. Nominal growth is 41,000/32,000 = 1.2813, i.e. +28.1%. Splitting the two: roughly 18.5 percentage points are price and 8.1 points are volume (1.185 × 1.081 ≈ 1.281).
Assessment of the claim. The owner's 28% is a turnover figure, not a growth figure. Four-fifths of it is the shop simply charging more for a similar physical volume. Using Laspeyres (1.175) instead would imply 9.1% real growth, so the choice of index does move the answer by about a point - but neither version supports 28%.
The bank should be told: real volume growth is about 8%, driven almost entirely by notebooks and geometry boxes, while pen volumes fell 17% and may signal a lost line. Ask why pens declined before treating the business as expanding.