Index Numbers
Core rules
- An index number measures relative change from a base period, usually with base = 100.
- Simple aggregate index = (Σp1/Σp0)×100.
- Simple average of price relatives = [Σ(p1/p0×100)]/n.
- Laspeyres = (Σp1q0/Σp0q0)×100; Paasche = (Σp1q1/Σp0q1)×100.
- Fisher ideal = √(Laspeyres × Paasche); it is a geometric mean.
- Chain indices link relatives; base shifting and splicing convert series to a common base.
Worked example
Base prices p0 = 10, 20; current p1 = 12, 24; base quantities q0 = 5, 2. Laspeyres = (12×5 + 24×2)/(10×5 + 20×2)×100 = (60+48)/(50+40)×100 = 108/90×100 = 120. Prices rose 20% on the base-weighted basket. Paasche would use current quantities q1, so it needs those weights.
Traps
- Swapping base and current quantities between Laspeyres and Paasche.
- Treating Fisher as an arithmetic mean; it is the geometric mean of the two indices.
- Ignoring that an index of 120 means a 20% rise, not a 120% rise.
- Using a cost-of-living index without checking its weight source.
- Forgetting base year = 100 when splicing.