# Tag Archives: arithmetic mean

## An inequality for the Consumer and Retail Price Indices

Since 1996, Britain has had two major ways of measuring inflation: but, when explaining the difference between the Retail Price Index (RPI) and Consumer Price Index (CPI), British newspapers typically mention that CPI is (generally) lower because it excludes housing costs, whereas RPI includes them. However, in 2013, the CPI is due to be updated, and may then also take these housing costs into account. This would cause CPI to rise closer to the level of RPI, but you would still expect inflation rates as given by RPI to be higher—why would this remain the case?

Let’s begin with some background. Both indices try to measure the rise in cost of an ‘average’ basket of goods bought by households or consumers across a year, and neither is an attempt to measure the cost of maintaining a minimum standard of living, which depends on how those minimum standards are set. Other methods exist: The Economist uses its partially tongue-in-cheek Big Mac index to double-check consumer inflation measures around the world—here the Big Mac burger is the physical basket of goods.

The Retail Price Index has the longer history—its predecessor is associated with price increases suffered by workers in World War 1—RPI officially began in 1956 (though an interim version started in 1947, after WW2). RPI tries to reflect the spending of the ‘average’ private household: it excludes the top 4% of households by income, and pensioners whose state pensions and benefits make up more than 3/4 of their incomes. It also excludes spending by overseas visitors (for instance university tuition fees paid by foreigners) and those living in institutions such as university accommodation or nursing homes. It also excludes, for instance, stockbroker fees.

The Consumer Price Index, on the other hand, was introduced in 1996 to harmonise inflationary measures across the European Union. For now, it excludes many housing costs such as mortgages, estate agent fees, council tax, as well as costs such as TV licences and trade union subscriptions. It also differs from its sister index in how it deals with car costs: whereas RPI imputes new car prices from those of second hand cars, CPI is obliged to use real data.

However, the most significant difference between the two, known as the formula effect, arises during the early stages of the calculation. The formula effect has contributed at least 0.4 percentage points difference each year since its inception (measured by recalculating each index with the other’s variables). In 2010, it contributed to a difference of 0.8 percentage points, compared with the 0.6 percentage point difference associated with housing costs.

The inequality

Essentially, this difference arises because the CPI uses a geometric mean, while RPI uses the more well-known average, the arithmetic mean. A famous inequality that links these two means, the AMGM inequality, tells us that the arithmetic mean is always greater than the geometric mean (but they will be equal if, and only if, all the numbers averaged are the same).

For a collection of non-negative numbers $x_1, x_2, \ldots, x_n$ we have:

$\frac{1}{n}(x_1 + x_2 + \cdots + x_n) \geq \sqrt[n]{x_1 \cdot x_2 \cdot \cdots \cdot x_n}$ ,

or in more succinct notation:

$\frac{1}{n} \sum_{i=1}^n x_i \geq (\prod_{i=1}^n x_i)^{1/n}$.