Index Numbers
Index number questions ask you to construct a price or quantity index from a small data set, to compare Laspeyres and Paasche, and to apply the reversal tests. Marks come from using the right weights and remembering the ×100.
Construction
An index number expresses change in the current year relative to a base year, usually as a percentage. The price relative is (p₁/p₀) × 100. Simple aggregative index = (Σp₁/Σp₀) × 100; simple average of price relatives = [Σ(p₁/p₀ × 100)]/n.
Weighted formulae
- Laspeyres: L = (Σp₁q₀/Σp₀q₀) × 100, using base-year quantities.
- Paasche: P = (Σp₁q₁/Σp₀q₁) × 100, using current-year quantities.
- Fisher's ideal: F = √(L × P), a geometric mean, not an arithmetic mean.
- Dorbish–Bowley: (L + P)/2.
Tests and use
Unit test: the index is independent of units. Time reversal test: P₀₁ × P₁₀ = 1. Factor reversal test: P₀₁ × Q₀₁ equals the value ratio. Fisher satisfies both reversal tests; Laspeyres and Paasche fail the time reversal test. Real income = (money income/cost of living index) × 100.
Worked example
Base year: p₀ = 4 and 6 with q₀ = 10 and 5. Current year: p₁ = 5 and 9. Laspeyres = (5 × 10 + 9 × 5)/(4 × 10 + 6 × 5) × 100 = (50 + 45)/(40 + 30) × 100 = 95/70 × 100 = 135.71. Prices rose about 35.7% on base-year consumption.
Traps
- Mixing weights: Laspeyres uses q₀ in both numerator and denominator, Paasche uses q₁.
- Reporting 1.3571 and forgetting the ×100.
- Calling the arithmetic mean of L and P Fisher's index.
- Swapping base and current years in the time reversal test.