Price indices

The House of Lords has a report on the problems with the RPI (Report: Measuring Inflation (PDF).  They have a neat box explaining some of the formulae effects, Box 4.

The data are below.  The purpose of a price index is to find out if prices have risen or fallen over a period.  (Digression: note 9 explains this wonderfully: 

The Bishop had been tasked with determining whether a 15th century stipulation by an Oxford college—that a fellow of the college must vacate his fellowship if his annual income exceeded five pounds—should still be adhered to. He investigated the change in the value of money over the period by assessing changing price levels and found that five pounds in the mid-15th century was worth 25 to 30 pounds in the late 17th century/early 18th century. He concluded that the intention behind the 15th-century stipulation was to allow for the changing value of money. Robert O’Neill et al (2017), Inflation: History and Measurement, Palgrave Macmillan.)

 Returning to the Box, as can be seen the price ratios are a mix of rises and falls.

Box 4.

For the purposes of illustration, it is assumed that there are four varieties of potato available in shops: Charlotte, King Edward, Maris Piper and Vivaldi. The table below shows the price of each variety in October 2017 and October 2018, and the percentage increase/decrease in price, expressed as a ratio, between the two periods.

Variety	Price in Oct 2017 (£)	Price in Oct 2018 (£)	Price ratio
Charlotte	2.00	1.90	0.95
King Edward	2.00	1.60	0.80
Maris Piper	1.80	1.89	1.05
Vivaldi	2.00	2.50	1.25

And when you use different indices, you get different results

• Carli: the price ratios are added together and divided by the number of products: (0.95 + 0.80 + 1.05 + 1.25) ÷ 4 = 1.0125
• Dutot: the ratio of price averages in October 2017 and October 2018 is calculated:
  Average price in Oct 2018: ((1.90 + 1.60 + 1.89 + 2.50) ÷ 4) = 1.9725
  Average price in Oct 2017: ((2.00 + 2.00 + 1.80 + 2.00) ÷ 4) = 1.95
  Ratio of average prices: 1.9725 ÷ 1.95 = 1.0115
• Jevons: the price ratios are multiplied together and the 4th root is taken: (0.95 × 0.80 × 1.05 × 1.25)1/4 = 0.9900

In this example, the Carli and Dutot formulas produce a 1 per cent average price increase in potatoes over the period whereas the Jevons formula produces a 1 per cent decrease.