This subsection concentrates on how the RPI is calculated. Generally the CPI is calculated in a similar way, though some of the details differ. To measure price changes in general, it is sufficient to select a limited number of representative items to indicate the price movements of a broad range of similar items. For each section of the RPI, a number of representative items are selected for pricing. The selection is made in such a way that the price movements of the representative items, when combined using a weighted mean, provide a good estimate of price movements in the section as a whole.
For example, in 2004 the representative items in the ‘Bread? section (which is contained in the ‘Food and catering? group) were: large white sliced loaf, large white unsliced loaf, small brown loaf, large wholemeal loaf, bread rolls, pitta bread, French stick. Changes in the prices of these types of bread are assumed to be representative of changes in bread prices as a whole. Note that although the price ratio for bread is based on this sample of seven types of bread, the calculation of the appropriate weight for bread is based on all kinds of bread. This weight is calculated using data collected in the Expenditure and Food Survey.
The representative items are selected at the beginning of the year and remain the same throughout the year.
One aim of the RPI is to make it possible to compare prices in any two months, and this involves calculating a value of the price index itself for every month.
The bulk of the data on price changes required to calculate the RPI is collected by staff of a market research company and forwarded to the Office for National Statistics for processing. Collecting the prices is a major operation: well over 100 000 prices are collected each month for over 650 different items. The prices being charged at a large range of shops and other outlets throughout the UK are mostly recorded on a predetermined Tuesday near the middle of the month. (Because it is not always possible to collect all shop prices on one day, some are collected on the Monday or Wednesday either side.)
Calculating the RPI, then, involves two kinds of data:
the price data, collected every month,
the weights, representing expenditure patterns, updated once a year.
Once the price data have been collected each month, various checks, such as looking for unbelievable prices, are applied and corrections are made if necessary. Checking data for obvious errors is an important part of any data analysis.
Then an averaging process is used to obtain a price ratio for each item that fairly reflects how the price of the item has changed across the country. The exact details are quite complicated and are not described here.
For each item a price ratio is calculated, comparing its price with the previous January. For instance, for April 2004, the resulting price ratio for an item is an average value of
The next steps in the process combine these price ratios, using weighted means, to obtain 14 subgroup price ratios, and then the group price ratios for the five groups. Finally the group price ratios are combined to give the all-item price ratio. This is the price ratio, relative to the previous January, for the ‘basket? of goods and services as a whole that make up the RPI.
The all-item price ratio tells us how, on average, the RPI ‘basket? compares in price with the previous January. The value of the RPI for a given month is found by multiplying the value of the RPI for the previous January by the all-item price ratio for that month (relative to the previous January). So, to calculate the RPI for April 2004, the final step is to multiply the value of the RPI in January 2004 by the all-item price ratio for April 2004. In general:
If you want to know more details, they are given in the Consumer Price Indices Technical Manual, available via www.statistics.gov.uk[accessed 12 October 2006]. At the time of writing, the exact address was http://www.statistics.gov.uk/downloads/theme_economy/CPLTechnical_Manual_2005.pdf [accessed 13 October 2006].
Here are the details of the last two stages of calculation of the RPI for April 2004, after calculating the group price ratios, relative to January 2004. The appropriate data are in Table 21.
| Group | Price ratio r | Weight w | Ratio × weight rw |
|---|---|---|---|
| Food and catering | 1.003 | 160 | 160.480 |
| Alcohol and tobacco | 1.016 | 97 | 98.552 |
| Housing and household expenditure | 1.028 | 367 | 377.276 |
| Personal expenditure | 1.010 | 93 | 93.930 |
| Travel and leisure | 1.004 | 283 | 284.132 |
| Sum | 1000 | 1014.370 |
Although Table 21 gives the individual rw values, there is no need for you to write down these individual products when finding a weighted mean (unless you are asked to do so, in an assignment for example). You can calculate the weighted mean directly. See Section 2.2 of the Calculator Book.
The all-item price ratio is a weighted average of the group price ratios given in the table. If the price ratios are denoted by the letter r, and the weights by w, then the weighted mean of the price ratios is the sum of the five values of rw divided by the sum of the five values of w. The formula, from Subsection 3.2, is
The sums are given in Table 21. (The sum of the weights is 1000, because the RPI weights are chosen to add up to 1000.) Now the all-item price ratio for April 2004 (relative to January 2004) can be calculated as
This tells us that, on average, the RPI basket of goods cost 1.014370 times as much in April 2004 as in January 2004.
The published value of the RPI for January 2004 was 183.1. So, using the formula on the previous page,
The final result is rounded to the same number of decimal places as the group price ratios. To this level of rounding, it matches the published value of the RPI for April 2004.
Published RPI figures are always rounded to one decimal place. The Government statisticians perform their calculations and store the price ratios involved to a greater degree of accuracy than has been used here, but they round their results to one decimal place before publishing them.
The same 2004 weights are used to calculate the RPI for every month from February 2004 to January 2005 inclusive. For each of these months, the price ratios are calculated relative to January 2004, and the RPI is finally calculated by multiplying the RPI for January 2004 by the all-item price ratio for the month in question. In February 2005, however, the process begins again. A new set of weights, the 2005 weights, comes into use. Price ratios are calculated relative to January 2005, and the RPI is found by multiplying the RPI value for January 2005 by the all-item price ratio. This procedure is used until January 2006. and so on.
The process of calculating the RPI can be summarised as follows.
Data used are prices, collected monthly, and weights, based on the EFS, changed annually.
Each month, for each item, a price ratio is calculated, which gives the price of the item that month divided by its price the previous January.
Weighted means are then used to calculate the all-item price ratio. Denoting the group price ratios by r and the group weights by w, the all-item price ratio is
The value of the RPI for that month is found by multiplying the value of the RPI for the previous January by the all-item price ratio:
The weights for a particular year are used in calculating the RPI for every month from February of that year to January of the following year.
The group price ratios are themselves weighted means.
Find the value of the RPI in June 2004, by completing the table and the formulas below. The value of the RPI in January 2004 was 183.1. (The base date was January 1987.)
| Group | Price ratio for June 2004 relative to January 2004 (r) | 2004 weights (w) | (r × w) |
|---|---|---|---|
| Food and catering | 1.001 | 160 | |
| Alcohol and tobacco | 1.019 | 97 | |
| Housing and household expenditure | 1.043 | 367 | |
| Personal expenditure | 1.006 | 93 | |
| Travel and leisure | 1.008 | 283 | |
| Total |
The published value for the RPI in June 2004 was 186.8, slightly different from the value you should have obtained in this activity (186.9). The discrepancy arises because the Government statisticians use more accuracy during their RPI calculations, and round only at the end before publishing the results.
Now consider the calculation of the value of the CPI for June 2004. In general terms, you do this in the same way as calculating the RPI, but there are some differences in the details. For the CPI, the price ratios are calculated relative to the previous December, instead of January. Also, the broadest groupings of items in the CPI are called Divisions instead of Groups, and there are twelve of them rather than just five. To summarise, the procedure is as follows.
Data used are prices, collected monthly, and weights, based on the UK National Accounts and changed annually.
Each month, for each item, a price ratio is calculated, which gives the price of that item that month divided by its price the previous December.
Weighted means are then used to calculate the all-item price ratio. Denoting the division price ratios by r and the division weights by w, the all-item price ratio is
The division price ratios are themselves weighted means.
The value of the CPI for that month is found by multiplying the value of the CPI for the previous December by the all-item price ratio:
The weights for a particular year are used in calculating the CPI for every month from January to December of that year.
Find the value of the CPI in June 2004, given the information in the table below. The value of the CPI in December 2003 was 110.7. (The base date was 1996.)
| Division | Price ratio for June 2004 relative to December 2003 | 2004 weights |
|---|---|---|
| Food and non-alcoholic beverages | 0.992 | 106 |
| Alcoholic beverages and tobacco | 1.024 | 46 |
| Clothing and footwear | 0.957 | 62 |
| Housing, water, electricity, gas and other fuels | 1.025 | 103 |
| Furniture, household equipment and maintenance | 0.990 | 75 |
| Health | 1.011 | 22 |
| Transport | 1.023 | 151 |
| Communication | 1.006 | 26 |
| Recreation and culture | 0.996 | 150 |
| Education | 1.000 | 16 |
| Restaurants and hotels | 1.016 | 137 |
| Miscellaneous goods and services | 1.008 | 106 |
The following activity is intended to help you to draw together many of the ideas you have met in this section, both about what the RPI is and how it is calculated. Take your time over it; it will require some careful thought. You may find that you need to refresh your memory about some of the details about weights and price changes: take the time to do this.
Between July 2003 and July 2004, the price of clothing and footwear fell on average by 2.8%, while the price of rail fares rose by 3.8%. Answer the following questions about the likely effects of these changes on the value of the RPI. (No calculations are required.)
(a) Looked at in isolation (that is, supposing that no other prices changed), would the change in the price of clothing and footwear lead to an increase or a decrease in the value of the RPI?
Would the change in the price of rail fares (looked at in isolation) lead to an increase or a decrease in the value of the RPI?
(b) In each case, is the size of the increase or decrease likely to be large or small?
(c) Using what you know about the structure of the RPI, decide which of ‘Clothing and footwear? and ‘Rail fares? has the larger weight.
(d) Which of the price changes mentioned in the question will have a larger effect on the value of the RPI? Briefly explain your answer.
(e) Can you list some different strategies you have used to learn about the RPI?
More detail has been included in this comment than you would have been expected to produce. When you have read it, make sure you understand all the points we have included. If your explanations were much shorter, ask yourself whether your explanations were really sufficient. If you think not, note particularly any additional points that you did not include.
(a) The RPI is calculated using the price ratio and weight of each item. Since the weights of items change very little from one year to the next, the price ratio alone will normally tell you whether a change in price is likely to lead to an increase or a decrease in the value of the RPI. If a price rises, then the price ratio is greater than one, so the RPI is likely to increase as a result. If a price falls, then the price ratio is less than one, so the RPI is likely to decrease. Therefore, since the price of clothing and footwear fell, this is likely to lead to a decrease in the value of the RPI. But since the price of rail fares rose, this is likely to lead to an increase in the value of the RPI.
(b) Both changes are likely to be small for two reasons. First, the price changes are themselves fairly small. Second, clothing, footwear and rail fares form only part of a household's expenditure: no single group, subgroup or section will have a large effect on the RPI on its own, unless there is a very large change in its price.
(c) The weight of ‘clothing and footwear? was 51 in 2004. (See Table 19). Since ‘Rail fares? is only one section in the subgroup ‘Fares and other travel costs? which had weight 21 in 2004, the weight of ‘Rail fares? is much smaller than 21. (In fact it was 5.) So the weight of ‘Clothing and footwear? is much larger than the weight of ‘Rail fares?.
(d) Since the weight of ‘Clothing and footwear? is much larger than the weight of ‘Rail fares?, and the percentage change in the prices are not too different in size, the change in the price of clothing and footwear is likely to have a much larger effect on the value of the RPI as a whole.
(e) Strategies include: doing calculations indicated in the activities to get more of a sense of how it might go ‘in general?; writing down ideas and earlier understandings; reading and re-reading the text.
After studying this section, you should be able to:
♦ understand the meaning of the weights in the RPI and CPI, and say where they come from (Activities 23, 24, 25 and 28.);
♦ identify and summarise the key points in a piece of text in your own words and constructively criticise your account (Activity 25);
♦ calculate the value of the RPI and the CPI given the relevant information; for example, price ratios (Activities 26 and 27);
♦ describe the likely effect of particular price changes on the value of a price index (Activity 28);
♦ explain the principles behind the calculation of the RPI to someone else (Activities 25 and 28).
Before reading on, check that you have a record of each new concept or technique from this section.
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