Fisher’s ideal index calculator

The formula

P₀₁ = √( Σp₁q₀/Σp₀q₀ × Σp₁q₁/Σp₀q₁ ) × 100 = √(L × P)

Work out the four aggregates Σp₀q₀, Σp₁q₀, Σp₀q₁ and Σp₁q₁, form the Laspeyres and Paasche ratios, multiply them and take the square root. The quantity index has the same shape with p and q swapped:

Q₀₁ = √( Σq₁p₀/Σq₀p₀ × Σq₁p₁/Σq₀p₁ ) × 100

A worked example

The calculator's default data:

1. L = 970 / 770 = 1.25974, P = 925 / 750 = 1.23333.
2. P₀₁ = √(1.25974 × 1.23333) × 100 = √1.55368 × 100 = 124.65.

Time reversal test

P₁₀ = √(Σp₀q₁/Σp₁q₁ × Σp₀q₀/Σp₁q₀) = √(750/925 × 770/970) = 0.80227 P₀₁ × P₁₀ = 1.24647 × 0.80227 = 1.0000 ✓

Factor reversal test

Q₀₁ = √(Σq₁p₀/Σq₀p₀ × Σq₁p₁/Σq₀p₁) = √(750/770 × 925/970) = 0.96376 P₀₁ × Q₀₁ = 1.24647 × 0.96376 = 1.2013 V₀₁ = Σp₁q₁ / Σp₀q₀ = 925 / 770 = 1.2013 ✓

Total spending rose by 20.13%. Fisher splits that exactly into a price change (+24.65%) and a quantity change (−3.62%): 1.24647 × 0.96376 = 1.2013. Laspeyres cannot do this: its price and quantity indices multiply to 1.2597 × 0.9740 = 1.2270, which overstates the change in value.

Why it is called “ideal”

Irving Fisher judged index formulas by whether they behave consistently. An index that says prices rose 25% from 2020 to 2025 should say they fell back by the same factor from 2025 to 2020 (time reversal), and the price index times the quantity index should give the actual change in spending (factor reversal). Laspeyres, Paasche and Dorbish–Bowley fail both; Marshall–Edgeworth and Kelly pass only time reversal. Fisher's formula passes both, which is the reason for the name and the reason it is the favourite of exam questions on the tests.

Does it over- or under-state inflation?

Laspeyres overstates the cost-of-living change (it ignores substitution away from goods that became dear) and Paasche understates it (its basket has already substituted). The geometric mean of the two sits between them, and economic theory shows that for common patterns of consumer behaviour it is very close to the true cost-of-living index. Fisher is therefore close to unbiased with respect to substitution. It still needs current quantities, so in practice it is published later than a Laspeyres CPI, and like every formula it cannot correct for new goods or quality changes by itself.

Common mistakes

• Averaging L and P arithmetically. (L + P) / 2 is the Dorbish–Bowley index; Fisher is the geometric mean √(L × P). The two are close but only Fisher passes the reversal tests.
• Rounding too early. Keep L and P to four or five decimals as ratios (1.25974, not 1.26) before multiplying, or the reversal tests will not come out to exactly 1 and V₀₁.
• Using Laspeyres in the factor reversal check. Only the Fisher Q₀₁ makes the product equal V₀₁.