Index Numbers

Measuring relative change over time: price relatives, simple and weighted indices, Laspeyres and Paasche, CPI, WPI, IIP and the Sensex, real wages and inflation.

Key ideas

An index number measures the relative change in a group of items between a base period (taken as 100) and a current period.
A good base year is normal, without extreme values, and not too far in the past.
A simple aggregative index treats all items as equally important; a weighted index weights them by importance.
Laspeyres weights prices by base period quantities and Paasche by current period quantities; only the weights differ.
The CPI (cost of living index) tracks retail prices for a group of consumers; the WPI tracks wholesale prices of goods; the IIP tracks industrial output; the Sensex tracks the shares of 30 companies.
When prices rise, the purchasing power of money falls, so wages must be adjusted to keep the same standard of living.

Remember

Price relative	p₁ ÷ p₀ × 100
Simple aggregative	Σp₁ ÷ Σp₀ × 100
Laspeyres	Σp₁q₀ ÷ Σp₀q₀ × 100
Paasche	Σp₁q₁ ÷ Σp₀q₁ × 100
Weighted index of relatives	ΣWR ÷ ΣW
Real wage	money wage ÷ CPI × 100
Salary to keep up	base salary × CPI ÷ 100
Inflation rate	(new index − old index) ÷ old index × 100
IIP weights (book)	mining 14.4, manufacturing 77.6, electricity 8.0 (base 2011–12)

Watch out

Dividing the base price by the current price (the current price goes on top).
Swapping the weights: Laspeyres uses q₀, Paasche uses q₁.
Finding inflation by dividing by the new index instead of the old one.

Worked example

Find the price index by the simple average of price relatives: sugar ₹18 → ₹24, oil ₹23 → ₹22, milk ₹16 → ₹16, tea ₹10 → ₹12.

Working: relatives 133.33, 95.65, 100, 120; their mean

Answer: 112.25