# 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:

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

The inequality straightforward to show in the case $n=2$, ie. by rearranging $\frac{x + y}{2} \geq \sqrt{x\cdot y}$, and if you have trouble remembering which side is greater, think about what happens when one of the numbers $x_i$ is zero.

So how does this affect our inflation indices?

Weighing up the methods

Price inflation is the rise in cost of goods or services over time: if you look at a single product at a single shop, you can measure exactly how it’s price changes: $p^{new}/p^{old}$ (subtract one and multiply by 100 to get a percentage change). So, for instance, if a loaf of bread cost £1 last year and £1.09 this year, which gives a %9 inflation rate. But once you introduce a second product, a chair, a service such as getting your cat spayed, 0r the same product at a different shop, it isn’t so clear how to combine the percentage changes. Loaves of bread are bought more frequently than chairs, and so a change in their price has more impact on the population’s spending, and so perhaps we should give it more weight in the average.

So now, let’s say the inflation is $I = \sum_{i = 1}^n w_i \frac{p_i^{new}}{p_i^{old}}$, where $w_i$ is some weight. How do we pick the weights? Let’s say chairs cost £50, and I buy one a year, compared with 100 loaves of bread. If the price of chairs stays the same then I spent £150 last year, and with the increase in food prices, £159 this year, giving a %6 rise. So what we did was take the total spent on each item this year divided by the amount I spent last year

$\frac{\sum_{i} p_i^{new} q_i}{\sum_{i} p_i^{old} q_i} = \sum_{i} \left( \frac{p_i^{old} q_i}{\sum_{j} p_j^{old} q_j}\right) \frac{p_i^{new} }{ p_i^{old} }$,   (1) Lasprayes index

where $q_i$ is the quantity of item $i$ which makes $w_i=\frac{p_i^{old} q_i}{\sum_{j} p_j^{old} q_j}$. Perfect! Unfortunately, the reality is not so straightforward. It’s far too difficult to collect information about the quantities of every type of good sold, and so the RPI does not try to estimate at the lowest levels.

Firstly, it creates a hierarchy for nearly every type of good: eg. Household goods > Consumables > Envelopes > Envelopes in the South-East region. The last of these categories is known as the elementary aggregate. Then shops are chosen at random (it’s complicated) and prices are collected for comparable goods and services.  For most elementary aggregates each item is weighted equally (ie. $w_i = 1/n$). This gives us the arithmetic mean, an average of relatives:

$\frac{1}{n} \sum_{i=1}^n \frac{p_i^{new}}{p_i^{old}}$.   (2) Carli index

Elsewhere, for more uniformly priced products such as energy and fuel , a ratio of averages is taken:

$\frac{1/n \sum_{i=1}^n p_i^{new}}{1/n \sum_{i=1}^n p_i^{old}}= \frac{\sum_i p_i^{new}}{\sum_i p_i^{old}}$,   (3) Dutot index

that is, $w_i=\frac{p_i^{old}}{\sum_i p_i^{old}}$. The weight is therefore proportional to the price, meaning high goods contribute more, and so is less distorted if some items are on sale. Now an item index is compiled using Lasprayes index (1) above, but with our elementary indices used instead of item prices, and so on back up the hierachy.

In the CPI, the EU prohibits the use of the average of relatives, the Carli index (2) above. Instead, the geometric mean is used (note that, unlike (2) and (3), here the ratio of the geometric means and the geometric mean of relatives are equal, as shown in the equation below—changing the power $1/n$ to different weights would give a geometric alternative):

$\frac{\left(\prod_i p_i^{new}\right)^{1/n}}{\left(\prod_i p_i^{old}\right)^{1/n}}=\left(\prod_i \frac{p_i^{new}}{p_i^{old}} \right)^{1/n}$.   (4) Jevons index

The geometric mean gives greater weight to lower priced goods, which is viewed as a positive property, due to consumer substitution: if one shop starts selling loaves of bread for £0.01, people will swap away from more expensive brands. For food and clothing this is regarded as reasonable, whereas for petrol, say, where substitution is not possible, the Dutot index (3) is used as before. Then the hierarchy of Laspreys indices are used to combine the highly-specific aggregate indices, but with differing weights to the RPI.

Price bounce

The EU has a second reason for outlawing the use of the arithmetic mean as found in the Carli index (2): the RPI suffers a ‘price bounce’ and overestimates. This is due to chain-linking the index across months and years. The old and new prices are found regularly every month, the elementary aggregate index calculated on a monthly basis. To get the index over a year or two years, these monthly figures are then multiplied together. This makes calculations simpler, and individual products may only be available or included for certain pairs of months. This makes sense for the Dutot (3) and Jevons (4) indices above, as they are of the form $\frac{f(t_{new})}{f(t_{old})}$, and when successive intervals are mulitplied we get cancellation:

$\frac{f(t_3)}{f(t_2)} \cdot \frac{f(t_2)}{f(t_1)} = \frac{f(t_3)}{f(t_1)}$.

However, this is not the case for the mean of relatives. For example, if you take a basket of only two goods, whose prices change after one month and return to the originals the next, we would expect no inflation. Instead, writing $p_1^{new} = x \cdot p_1^{old}$, and similarly for $y$ and $p_2$, we get an expression of the form:

$\frac{1}{2}(x+y)\cdot \frac{1}{2}(\frac{1}{x} + \frac{1}{y})$.

Multiplying out, we get:

$\frac{1}{4xy}(x^2+2xy+y^2)$.

To see how this differs from our expected 0% inflation rate for two months, we subtract 1.00:

$\frac{1}{4xy}(x^2+2xy+y^2) - 1 = \frac{1}{4xy}(x^2-2xy+y^2) = \frac{1}{4xy}(x-y)^2 \geq 0$.

This inequality shows us, in this example, that no matter what the prices did in the middle month, we always calculated a higher inflation rate. For instance, if one price increased by 50% (or decreased by 33%) and then returned to normal, while another stayed constant, our inflation rate would be about 4% (instead of 0%). This upward trend is generally the case, especially occurring when certain price correlations are observed.

Miscellany

A few miscellaneous facts:

• Smuggling, for instance, of alcohol and tobacco is taken into account.
• Shops with closing down sales are excluded, as they won’t exist any more; shops only selling second-hand goods, such as most charity shops, are also excluded.
• More individual prices are collected for items where the price differs more, such as clothing (as opposed to, say, petrol or cigarrettes).
• Gambling, 0.5% of total expenditure is not included (how would you measure the increase in the cost of a bet?).
• Pension contributions are not included, as they are treated as deferred consumption: money to be spent later and measured then.
(Thanks to the BBC’s “More or Less” which talked  about the difference between RPI and CPI earlier in 2011.)