Laspeyres, Paasche and Fisher's ideal index with TRT and FRT - Business Statistics

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On this page

1. Formulas
2. The tests
3. Worked: January 2026
4. Worked: June 2025
5. Worked: expenditure given instead of quantity (January 2025)
How this is examined

Cases checked to exist. Read the judgment before quoting a holding.

Every case cited here has been checked to exist — the citation resolves and names the right parties. The propositions attributed to each case have not been verified line by line against the judgment. Case names link through to the full judgment: read it before quoting a holding in an examination answer, and never carry a proposition from notes into court on trust.

The examinable core. This is the most predictable problem in the paper: a table of p₀, q₀, p₁, q₁ and “calculate Laspeyres, Paasche and Fisher and show that Fisher satisfies TRT and FRT”. Build four product columns (p₀q₀, p₁q₀, p₀q₁, p₁q₁); everything else follows from their four totals.

1. Formulas

Laspeyres (base-year quantities as weights): P₀₁^L = Σp₁q₀ / Σp₀q₀ × 100.
Paasche (current-year quantities): P₀₁^P = Σp₁q₁ / Σp₀q₁ × 100.
Fisher’s ideal (geometric mean of the two): P₀₁^F = √(L × P) = √[(Σp₁q₀ / Σp₀q₀) × (Σp₁q₁ / Σp₀q₁)] × 100.
Others for reference: Bowley = (L + P)/2; Marshall-Edgeworth = Σp₁(q₀ + q₁) / Σp₀(q₀ + q₁) × 100.

Laspeyres needs base quantities only, so it is easy to compute every year, but it tends to overstate price rises because consumers shift away from items that become dearer. Paasche needs fresh quantities every year and tends to understate. Fisher lies between them.

Why Fisher is “ideal”: it uses both base and current quantities; it is based on the geometric mean, theoretically the best average for ratios; it is free from the upward bias of Laspeyres and the downward bias of Paasche; and it satisfies both the time reversal and the factor reversal tests. Its drawbacks: heavy computation, and the need for current quantities each year.

2. The tests

Time reversal test (TRT): if the base and current periods are interchanged, the two indices should be reciprocals: P₀₁ × P₁₀ = 1 (without the factor 100). For Fisher:

P₀₁ × P₁₀ = √[(Σp₁q₀/Σp₀q₀)(Σp₁q₁/Σp₀q₁)] × √[(Σp₀q₁/Σp₁q₁)(Σp₀q₀/Σp₁q₀)] = √1 = 1.

Factor reversal test (FRT): the price index multiplied by the quantity index (obtained by interchanging p and q) should equal the value ratio: P₀₁ × Q₀₁ = Σp₁q₁ / Σp₀q₀. For Fisher:

P₀₁ × Q₀₁ = √[(Σp₁q₀/Σp₀q₀)(Σp₁q₁/Σp₀q₁)] × √[(Σp₀q₁/Σp₀q₀)(Σp₁q₁/Σp₁q₀)] = √[(Σp₁q₁)² / (Σp₀q₀)²] = Σp₁q₁ / Σp₀q₀.

Laspeyres and Paasche satisfy neither test.

3. Worked: January 2026

Item	p₀	q₀	p₁	q₁	p₀q₀	p₁q₀	p₀q₁	p₁q₁
A	12	10	20	12	120	200	144	240
B	4	20	4	24	80	80	96	96
C	9	12	12	15	108	144	135	180
D	12	15	24	2	180	360	24	48
Total					488	784	399	564

Laspeyres = 784/488 × 100 = 160.66.
Paasche = 564/399 × 100 = 141.35.
Fisher = √(160.66 × 141.35) = √22,709 = 150.70.

TRT: P₀₁ × P₁₀ = √(784/488 × 564/399) × √(399/564 × 488/784) = √1 = 1. Satisfied.

FRT: Q₀₁ = √(399/488 × 564/784). P₀₁ × Q₀₁ = √(784/488 × 564/399 × 399/488 × 564/784) = √(564²/488²) = 564/488 = Σp₁q₁/Σp₀q₀. Satisfied. (Numerically, 564/488 = 1.1557.)

The large gap between L and P comes from item D, whose price doubled while its quantity fell from 15 to 2: Laspeyres weights it heavily, Paasche hardly at all.

4. Worked: June 2025

Item	p₀	q₀	p₁	q₁	p₀q₀	p₁q₀	p₀q₁	p₁q₁
A	8	5	10	11	40	50	88	110
B	8.5	6	9	9	51	54	76.5	81
C	9	4	12	6	36	48	54	72
Total					127	152	218.5	263

Fisher = √(152/127 × 263/218.5) × 100 = √(1.19685 × 1.20366) × 100 = √1.44060 × 100 = 120.03. TRT and FRT are shown exactly as above; the value ratio for the FRT is 263/127 = 2.0709.

5. Worked: expenditure given instead of quantity (January 2025)

When the paper gives price and expenditure, recover quantity as q = expenditure ÷ price. The expenditures themselves are p₀q₀ and p₁q₁.

Item	p₀	q₀ = E₀/p₀	p₁	q₁ = E₁/p₁	p₀q₀	p₁q₀	p₀q₁	p₁q₁
A	8	80/8 = 10	10	120/10 = 12	80	100	96	120
B	10	12	12	8	120	144	80	96
C	5	8	5	10	40	40	50	50
D	4	14	3	20	56	42	80	60
E	20	5	25	6	100	125	120	150
Total					396	451	426	476

Fisher = √(451/396 × 476/426) × 100 = √(1.13889 × 1.11737) × 100 = 112.81.

Other papers, totals only (for self-checking): 100-mark January 2026 (Food, Transportation, Clothing, Others): 450, 790, 367, 640; Fisher = 174.97. August 2024 (six items A to F): 7,240, 10,160, 8,012, 10,712; L = 140.33, P = 133.70, Fisher = 136.98. August 2024 (ten food items, 2009 base) follows the same layout with larger tables.

Trap. For the FRT the quantity index interchanges p and q: Q₀₁ = √[(Σq₁p₀/Σq₀p₀)(Σq₁p₁/Σq₀p₁)]. The same four totals are used, rearranged. No new columns are needed.

How this is examined

Laspeyres, Paasche and Fisher with TRT and FRT: January 2026; Fisher with both tests: June 2025, January 2025 (expenditure form), the 100-mark January 2026 paper and August 2024 (six marks). L, P and F for ten items: August 2024 (ten marks). “Write a short note on time reversal and factor reversal tests”: June 2025. Write the formula, the four-column table, each index to two decimals and a one-line statement that each test is satisfied.