Price relatives and weighted aggregate index numbers

Last time

Before we start: last time, in Crude and standardised death rates, fertility rates and other rates, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Measuring price changes

An index number shows how a price or a cost has changed compared with a base year, which is given the value 100. This lesson covers price relatives and weighted aggregate indices, changing the base year, and using an index to find costs.

Price relatives

A price relative is the price in the current year divided by the price in the base year, times 100. A price that rises from 40 dollars to 50 dollars has a price relative of 125. 125 means a 25 per cent increase since the base year; 90 would mean a 10 per cent fall.

Why weights?

A family spends far more on food than on clothing, so a rise in food prices matters more to it. Weights show the importance of each item, often based on how much is spent on it. A weighted aggregate index combines the price relatives using these weights.

The weighted aggregate index

The weighted aggregate index is the sum of weight times price relative, divided by the sum of the weights. Food: weight 5, relative 110. Transport: weight 3, relative 120. Clothing: weight 2, relative 105. 550 plus 360 plus 210 is 1,120; divided by 10, the index is 112.

The index, worked

The weighted aggregate index worked in a table.

Using the index

An index of 112 means costs are 12 per cent higher than in the base year. If the family spent 8,000 dollars in the base year, the same goods now cost 8,000 times 112 over 100: 8,960 dollars. To find an unknown weight or price relative, write the index formula with the unknown in it, and solve.

Changing the base year

To compare with a different year, divide by that year's index and multiply by 100. If the index is 120 in 2022 and 150 in 2024, with base year 2020, then 2024 compared with 2022 is 150 over 120 times 100: 125. Index numbers cannot simply be subtracted to give a percentage: 150 minus 120 is 30 points, but a 25 per cent rise.

Limits of an index

Weights from the base year may become out of date as spending patterns change. An index shows an average change; single items may rise by much more or much less. And the quality of goods can change, which prices alone do not show.

Mistakes that lose marks

Mistake one: dividing the base-year price by the current price. Mistake two: dividing by the number of items instead of the sum of the weights. Mistake three: reading a rise of 30 index points as 30 per cent when the base has changed. The fix: current over base, Σw underneath, and rebasing by division.

The full-marks tip

The full-marks tip: set out a table with the weight, the relative and w times I, and total each column. Interpret the final index in words: an index of 112 means costs rose by 12 per cent.

On one card

So, on one card: index numbers. Keep it for revision.