10 Time Series & Index Numbers · 10.1 Time series and index numbers
10.1.2 Index numbers: types and methods (Laspeyres, Paasche, Fisher, Marshall-Edgeworth), tests of adequacy
An index number measures the relative change in a group of related variables (prices, quantities, values) between a base period (= 100) and a current period; the weighted formulas of Laspeyres (base-year quantities), Paasche (current-year quantities) and Fisher (their geometric mean) are judged by the unit, time reversal, factor reversal and circular tests.
- APPSC Paper I
- Advanced level
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Core facts
- Index numbers are called economic barometers. Base period index = 100.
- Notation: p₀, q₀ = base-year price and quantity; p₁, q₁ = current-year price and quantity. P₀₁ = price index for year 1 on base 0.
- Types: price index, quantity (volume) index, value index (Σp₁q₁ ÷ Σp₀q₀ × 100).
- Simple aggregative: P₀₁ = Σp₁ ÷ Σp₀ × 100.
- Price relative: R = p₁ ÷ p₀ × 100. Simple average of relatives: ΣR ÷ n (AM) or antilog (Σlog R ÷ n) (GM).
- Laspeyres: P₀₁ = Σp₁q₀ ÷ Σp₀q₀ × 100 (weights = base-year quantities).
- Paasche: P₀₁ = Σp₁q₁ ÷ Σp₀q₁ × 100 (weights = current-year quantities).
- Fisher: P₀₁ = √(Laspeyres × Paasche) — the geometric mean of L and P; called the ideal index.
- Marshall-Edgeworth: P₀₁ = Σp₁(q₀ + q₁) ÷ Σp₀(q₀ + q₁) × 100 (weights = sum, i.e. average, of both years' quantities).
- Dorbish-Bowley: (L + P) ÷ 2 — the arithmetic mean of L and P.
- Kelly (fixed weight): Σp₁q ÷ Σp₀q × 100, q = fixed quantities of some chosen period.
- Quantity indices: swap p and q, e.g. Laspeyres Q₀₁ = Σq₁p₀ ÷ Σq₀p₀ × 100.
- Tests of adequacy (Fisher):
- Unit test: index unaffected by units of measurement. Failed only by the simple aggregative index.
- Time reversal test: P₀₁ × P₁₀ = 1. Passed by Fisher, Marshall-Edgeworth, simple aggregative, simple GM of relatives, Kelly. Failed by Laspeyres, Paasche, simple AM of relatives.
- Factor reversal test: P₀₁ × Q₀₁ = Σp₁q₁ ÷ Σp₀q₀ (value ratio). Passed only by Fisher.
- Circular test: P₀₁ × P₁₂ × P₂₀ = 1. Passed by the simple GM of relatives, the simple aggregative and Kelly's fixed-weight index; not by Fisher.
- Fisher is "ideal" because it passes the time reversal and factor reversal tests (not because it passes all tests).
- Geometric mean is theoretically the best average for index numbers.
Example 1. Laspeyres index = 120 and Paasche index = 118.75. Find Fisher's index. Solution. √(120 × 118.75) = √14,250. Answer: 119.37 (about)
Example 2. Which index satisfies the factor reversal test? (a) Laspeyres (b) Paasche (c) Marshall-Edgeworth (d) Fisher Solution. Only Fisher's formula gives P₀₁ × Q₀₁ = value ratio. Answer: (d)
Explained
1. Problems in constructing an index number.
- Purpose: decides items, prices and weights (a CPI for workers uses retail prices; WPI uses wholesale prices).
- Base period: a normal year, free from wars, floods, booms; not too distant. Fixed base or chain base.
- Selection of items: representative, standard quality, enough in number.
- Price quotations: wholesale or retail, from representative markets.
- Choice of average: AM is easy; GM gives equal weight to equal ratios and passes the time reversal test.
- Weights: quantity weights (aggregative) or value weights (relatives); weighting is needed because items differ in importance.
2. Simple (unweighted) indices and the weighted formulas: one data set.
| Item | p₀, q₀ | p₁, q₁ | Relative p₁/p₀ |
|---|---|---|---|
| A | 10, 4 | 12, 5 | 120 |
| B | 8, 10 | 10, 8 | 125 |
| C | 5, 6 | 6, 10 | 120 |
| D | 20, 2 | 22, 3 | 110 |
Products:
| Item | p₀q₀ | p₁q₀ | p₀q₁, p₁q₁ |
|---|---|---|---|
| A | 40 | 48 | 50, 60 |
| B | 80 | 100 | 64, 80 |
| C | 30 | 36 | 50, 60 |
| D | 40 | 44 | 60, 66 |
| Total | 190 | 228 | 224, 266 |
Example 3. Find the simple aggregative index and the simple AM of relatives. Solution. Σp₁ = 12 + 10 + 6 + 22 = 50; Σp₀ = 10 + 8 + 5 + 20 = 43. Simple aggregative = 50 ÷ 43 × 100 = 116.28. AM of relatives = (120 + 125 + 120 + 110) ÷ 4 = 475 ÷ 4 = 118.75. Answer: 116.28 and 118.75
Example 4. Find the Laspeyres, Paasche, Fisher, Marshall-Edgeworth and Dorbish-Bowley price indices. Solution.
- Laspeyres = 228 ÷ 190 × 100 = 120.00
- Paasche = 266 ÷ 224 × 100 = 118.75
- Fisher = √(120 × 118.75) = 119.37
- Marshall-Edgeworth = (228 + 266) ÷ (190 + 224) × 100 = 494 ÷ 414 × 100 = 119.32
- Dorbish-Bowley = (120 + 118.75) ÷ 2 = 119.375
Answer: L 120, P 118.75, F 119.37, M-E 119.32, D-B 119.38
3. Weighted average of relatives. P₀₁ = Σ(R × W) ÷ ΣW. With value weights W = p₀q₀ it gives exactly the Laspeyres index: (120 × 40 + 125 × 80 + 120 × 30 + 110 × 40) ÷ 190 = 22,800 ÷ 190 = 120.
4. Fixed base and chain base.
- Fixed base: every year compared with one base year.
- Chain base: each year compared with the previous year. Link relative = current value ÷ previous year's value × 100; chain index = link relative × previous chain index ÷ 100.
Example 5. Link relatives of three years are 110, 120 and 90 (first year's base = 100). Find the chain indices. Solution. Year 1: 110. Year 2: 120 × 110 ÷ 100 = 132. Year 3: 90 × 132 ÷ 100 = 118.8. Answer: 110, 132, 118.8
| Point | Fixed base | Chain base |
|---|---|---|
| Compares with | one base year | previous year |
| New items, weights | hard to add | easy to add |
| Long-run comparison | direct | via chaining |
| Error | does not build up | can accumulate |
5. Uses of index numbers. Measure changes in price level and the value of money; guide policy (wages, dearness allowance, monetary policy); deflate money values into real values; show trends in production, trade and business; compare places and periods.
Deeper
1. Checking the tests with the data of Example 4.
- Time reversal for Laspeyres: P₀₁ = 228 ÷ 190 = 1.2; P₁₀ (year 1 as base, its quantities q₁ as weights) = Σp₀q₁ ÷ Σp₁q₁ = 224 ÷ 266. Product = 1.2 × 224 ÷ 266 = 1.0105 ≠ 1. Fails.
- Time reversal for Fisher: P₀₁ × P₁₀ = √(228/190 × 266/224) × √(224/266 × 190/228) = 1. Passes.
- Factor reversal for Fisher: quantity indices are Laspeyres Q = 224 ÷ 190 = 117.89, Paasche Q = 266 ÷ 228 = 116.67, Fisher Q = √(117.89 × 116.67) = 117.28. P₀₁ × Q₀₁ = 1.1937 × 1.1728 = 1.40 = 266 ÷ 190. Passes.
2. Proof sketch of factor reversal for Fisher. P₀₁ × Q₀₁ = √(Σp₁q₀/Σp₀q₀ × Σp₁q₁/Σp₀q₁) × √(Σq₁p₀/Σq₀p₀ × Σq₁p₁/Σq₀p₁) = √((Σp₁q₁)² ÷ (Σp₀q₀)²) = Σp₁q₁ ÷ Σp₀q₀.
3. Bias. Laspeyres uses old quantities and ignores consumers shifting away from goods whose prices rise, so it tends to overstate price rises (upward bias); Paasche tends to understate them (downward bias). Fisher lies between them. This holds when prices and quantities move in opposite directions; it is not a mathematical law, and the order can reverse.
4. Fisher's limitations. Needs current-year quantities every year (costly; that is why official CPIs and WPIs use the Laspeyres type), is harder to compute and to explain, and fails the circular test.
| Index | Unit | Time reversal | Factor reversal |
|---|---|---|---|
| Simple aggregative | fails | passes | fails |
| Laspeyres / Paasche | passes | fails | fails |
| Marshall-Edgeworth | passes | passes | fails |
| Fisher | passes | passes | passes |
Circular test: passed by the simple aggregative, simple GM of relatives and Kelly's fixed-weight index; failed by all four weighted indices above.
Example 6. If Σp₀q₀ = 200, Σp₁q₀ = 250, Σp₀q₁ = 240 and Σp₁q₁ = 300, find Laspeyres, Paasche and Fisher. Solution. L = 250 ÷ 200 × 100 = 125; P = 300 ÷ 240 × 100 = 125; F = √(125 × 125) = 125. Answer: all 125 (when L = P, Fisher equals both)
Common traps
- ✗ "Laspeyres uses current-year quantities." ✓ Laspeyres = base-year (q₀); Paasche = current-year (q₁).
- ✗ "Fisher's index is the arithmetic mean of L and P." ✓ It is the geometric mean; the AM is Dorbish-Bowley.
- ✗ "Fisher's index passes every test." ✓ It fails the circular test; it is ideal for passing time reversal and factor reversal.
- ✗ "Marshall-Edgeworth passes the factor reversal test." ✓ It passes time reversal only; factor reversal is Fisher alone.
- ✗ "Time reversal test: P₀₁ + P₁₀ = 1." ✓ The product P₀₁ × P₁₀ = 1 (without the factor 100).
- ✗ "Simple aggregative index passes the unit test." ✓ It fails it: changing kg to quintal changes the result.
- ✗ "Base-year index = 1 or 0." ✓ Base-year index = 100.
Quick revision
- Laspeyres = Σp₁q₀/Σp₀q₀ × 100 (base quantities).
- Paasche = Σp₁q₁/Σp₀q₁ × 100 (current quantities).
- Fisher = √(L × P), ideal index.
- Marshall-Edgeworth = Σp₁(q₀ + q₁)/Σp₀(q₀ + q₁) × 100.
- Dorbish-Bowley = (L + P)/2.
- Unit test: fails only simple aggregative.
- Time reversal: P₀₁ × P₁₀ = 1 (Fisher, M-E pass; L, P fail).
- Factor reversal: P₀₁ × Q₀₁ = Σp₁q₁/Σp₀q₀ (Fisher only).
- Circular test: simple aggregative, simple GM and Kelly pass; Fisher fails.
- Chain index = link relative × previous chain index ÷ 100.
Practise this topic. Questions on Index numbers: construction, tests & cost of living, with answers explained.
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