1. Applied Quantitative Methods MBA course Montenegro
Peter Balogh
PhD

2. 7. Index numbers
It is often necessary to describe and interpret changes in economic, business and social variables over time.
Information on change may come from different types of data, recorded in different ways.
Index numbers can provide a simple summary of change by aggregating the information available and making a comparison to a starting figure of 100.
A typical index then could take the form of 100, 105, 107, where 100 is the starting point, 105 shows the relative increase one year later and 107 shows the relative increase two years on.
Index numbers, therefore, are not concerned with absolute values but rather the movement of values.
The retail prices index (RPI) is one the better known indices and is a general mea­sure of how the prices of goods and services change, rather than as an indicator of the absolute amounts we actually spend each week.

4. 7.1 The interpretation of index numbers
Indices provide a measure of change over time, making reference to a base year value of 100.

5. 7.1.1 Percentage changes
An index is a scaling of numbers so that a start is made from a base figure of 100.
Suppose, for example, the price for bread over the past 4 years was as shown in Table 7.1.

6. 7.1.1 Percentage changes
We would first need to decide which year should be used for the base year, and then scale all figures accordingly.
If year 1 was chosen for the base year, we would divide all the prices by 0.50 and multiply by 100, as shown in Table 7.2.

7. 7.1.1 Percentage changes
The index numbers given for years 2-4 all measure the change from the base year.
The index number of 120 shows that there was a 20% increase from year 1 to year 2, and the index number 160 shows that there was a 60% increase from year 1 to year 3.
To calculate a percentage increase, we first find the difference between the two figures, divide by the base figure and then multiply by 100.

8. 7.1.1 Percentage changes The percentage increase from 100 to 188 is
In the same way, the percentage increase in the price of bread from £0.50 to £0.94 is

9. 7.1.1 Percentage changes
One important feature of index numbers is that by starting from 100, the percentage increase from the base year is found just by subtraction.
However, the differ­ences thereafter are referred to as percentage points.
It can be seen from Table 7.2 that there was a 28 percentage point increase from year 3 to year 4.
The percentage increase, however, is

10. 7.1.2 Changing the base year
There are no hard and fast rules for the choice of a base year and, as shown in Table 7.3, any year can be made into the base year from a purely mathematical point of view.

11. 7.1.2 Changing the base year
Each of the indices measures the same change over time (to two decimal places).
The percentage increase from year 1 to year 2 using index 2, for example, is
To change the base year (move the 100) requires only a scaling of the index up or down.
If we want index 1 to have year 2 as the base year (construct index 2), we can use the equivalence between 110 and 100 and multiply index 1 by this scaling factor 100/110.

12. 7.1.2 Changing the base year
In practice, there are a number of important considerations in the choice of a base year.
As the index gets larger the same percentage change is represented by a larger increase in percentage points.
A change from 100 to 120 is the same as a change from 300 to 360 but the impression created can be very different.
If, for example, our index were used as a measure of inflation, like the Retail Prices Index, we would not want the index to move very far from 100.
We would like the seen change (points) to be close to the actual change (percentages).

13. 7.1.2 Changing the base year
An index number is typically a summary of what is happening to a group of items (often referred to as a basket of goods).
From time to time we may review and change the items to be included and this is often when the index is again started at 100.
Footnotes or other forms of referencing may indicate these changes.
Suppose a manufacturer constructed a productivity index using as a measure of productivity the times taken to make the most popular products.

14. 7.1.2 Changing the base year
As new products appear and established products disappear, the manufacturer would need to reconsider the basis of the index.
The manufacturer would also need to consider the compatibility of the indices produced as the new products may be adding a different level of value and involve different methods of production.
An index can be unadjusted or adjusted.
To show the general (underlying) trend in unemployment, the index can be adjusted to allow for predictable changes through the year, like the number of school leavers.

15. 7.1.2 Changing the base year
A change in base year is shown in Table 7.4.

16. 7.1.2 Changing the base year
We can use the equivalence of 150 in the 'old' index with 100 in the 'new' index at year 5.
We either scale down the 'old' index using a multiplication factor of 100/150 as shown in Table 7.5 or scale up the 'new' index using 150/ 100 as shown in Table 7.6

18. 7.1.3 Nominal and real change
Index numbers allow us to distinguish between nominal and real values.
Suppose your annual entertainment allowance had increased from £500 to £510.
This £10 increase is referred to as nominal value (and is given in the original units of measurement, in this case £'s).
However, you may be more concerned with the purchasing power of the new £510 allowance and how this compares with the £500 allowed in the previous year.
Suppose that you are now told that the cost of entertainment has increased by 5%. To maintain your purchasing power you would need £525 (£500 plus the extra 5%, which is £25).
We would now say that in real terms your purchasing power has decreased. Indices can measure change in real or nominal terms.

19. 7.1.3 Nominal and real change
Case 2: the use and construction of indices
Managers are always likely to be concerned with measures of change over time.
Case 2 data gives the number of business enquiries received, the value of new business and an index of inflation over a 3-year period.
It is relatively clear from the figures that the Important business performance measurements of the number of enquiries and the value of new business are declining.
It is less clear that the rate of inflation is about 3% and what impact the inflation rate is having on the 'real' value of new business.

20. 7.1.3 Nominal and real change
Case 2: the use and construction of indices
Table 7.7 shows the construction of indices to illustrate the changes using year 1, quarter 1 for the base.
The columns in Table 7.7 for the number of enquiries and the value of new business both show the downward trend and a quarterly variation.
The indices confirm this trend and show the greater variation (in percentage terms) in the number of enquiries.
The spreadsheet shown as Table 7.8 can also be constructed to show how the Veal' value of new business has declined.
It can be seen that if we are working with the purchasing power of the £ in year 1, quarter 1 (real as opposed to nominal pounds), the drop in the value of new business is even greater.
The spreadsheet has also been used to show that the rate of inflation and how that has been declining.

22. stakeholders, about the position of the company.

23. 7.1.3 Nominal and real change
The figures 'on the surface' look bad, but need to be interpreted within their business context (business significance rather than statistical significance).
Trading conditions might have become particularly difficult and the company may still have done better than other rivals (the business could consider benchmarking against best practice wherever that is to be found).
The figures may reflect a change in company strategy where new business of this kind has not been sought and existing business has been consolidated.
What is important is that the analysis is able to inform a debate and highlight the realities that the company may face (which is better understood through analysis and debate).
Analysis should also inform policy formulation and change.
It is important to explore the data for improved insight; you could, for example, calculate the ratio of the value of new business to the number of enquiries and consider what these figures mean.

24. 7.2 The construction of index numbers
Index numbers are perhaps best known for measuring the change of price or prices over time.
To illustrate the methods of calculation, we will use the infor­mation given in Table 7.9.
The price can be taken as the average amount paid in pence for a cup and the quantity as the average number of cups drunk per person per week.

25. 7.2.1 The simple price index
If we want to construct an index for the price of one item only we first calculate that ratio of the 'new' price to the base year price, the price relative, and then multiply by 100.
In terms of a notation
where P0 is the base year price and Pn is the 'new' price.
A simple index for the price of tea, taking year 0 as the base year, can be calculated as in Table 7.10.

26. The doubling of the price of tea from 32p to 64p over the 3-year period gives a 100% increase in the index, from 100 to 200.
The increase from 48p to 64p is a 50 percentage point increase (the index increases from 150 to 200) or a percentage increase of 334%.
In reality we are likely to drink more than just tea. When constructing an index of beverage prices, for example, we may wish to include coffee and chocolate drinks.

27. 7.2.2 The simple aggregate price index
To include all items, we could sum the prices year by year and construct an index from this sum. If the sum of the prices in the base year is and the sum of the prices in year n is then the simple aggregate price index is
The calculations are shown in Table 7.11.

29. 7.2.2 The simple aggregate price index
This particular index ignores the amounts consumed of tea, coffee and chocolate drinks.
In particular, the construction of this index ignores both consumption patterns and the units to which price refers.
If, for example, we were given the price of tea for a pot rather than a cup, the index values would differ.

30. 7.2.3 The average price relatives index
To overcome the problem of units, we could consider price ratios of individual commodities instead of their absolute prices and treat all price movements as equally important.
In many cases, the goods we wish to include will be measured in very different units.
Breakfast cereal could be in price per packet, potatoes price per kilo and milk price per pint bottle.

31. 7.2.3 The average price relatives index
As an alternative to the simple aggregate price index we can use the average price relatives index:
where k is the number of goods.
Here the price relative, for a stated commodity will have the same value whatever the units.
The calculations are shown in Table 7.12

33. 7.2.3 The average price relatives index
Comparing Tables 7.12 and 7.11 we can see that the average price relatives index, in this case, shows larger increases than the simple aggregate price index.
To explain this difference we could consider just one of the items: tea.
The value of tea is low in comparison to other drinks so it has a smaller impact on the totals in Table 7.11.
In contrast, the changes in the price of tea are larger than any of the other drinks and this makes a greater impact on the totals in Table 7.12.
To construct a price index for all goods and sections of the community we need to take account of the quantities bought.

34. 7.2.3 The average price relatives index
It is not just a matter of comparing what is spent year by year on drinks, food, transport or housing.
If prices and quantities are both allowed to vary, an index for the amount spent could be constructed but not an index for prices.
If we want a price index we need to control quantities.
In practice, we consider a typical basket of goods in which the quantity of goods of each kind is fixed and we find how the cost of that basket has changed over time.
To construct an index for the price of beverages we need the quantity information for a selected year as given in Table 7.9.

35. 7.2.4 The Laspeyre index
This index uses the quantities bought in the base year to define the typical basket.
It is referred to as a base-weighted index and compares the cost of this basket of goods over time.
This index is calculated as
where is the cost of the base year basket of goods in the base year and is the cost of the base year basket of goods in any year (thereafter) n.

36. 7.2.4 The Laspeyre index
It can be seen from Table 7.13 that we only require the quantities from the chosen base year (Q0 in this case).
The index implicitly assumes that whatever the price changes, the quantities purchased will remain the same.
In terms of economic theory, no substitution is allowed to take place. Even if goods become relatively more expensive it assumes that the same quantities are bought. As a result, this index tends to overstate inflation.

38. 7.2.5 The Paasche index
This index uses the quantities bought in the current year for the typical basket.
This current year weighting compares what a basket of goods bought now (in the current year) would cost, with cost of the same basket of goods in the base year.
This index is calculated as
where is the cost of the basket of goods bought in the year n at year n prices and is the cost of the year n basket of goods at base year prices.
The calculations are shown in Table 7.14.

40. 7.2.5 The Paasche index
As the basket of goods is allowed to change year by year, the Paasche index is not strictly a price index and as such, has a number of disadvantages.
Firstly, the effects of substitution would mean that greater importance is placed on goods that are relatively cheaper now than they were in the base year.
As a consequence, the Paasche index tends to understate inflation.
Secondly, the comparison between years is difficult because the index reflects both changes in price and the basket of goods.
Finally, the index requires information on the current quantities and this may be difficult or expensive to obtain.

41. 7.2.6 Other indices
The Laspeyre and Paasche methods of index construction can also be used to measure quantity movements with prices as the weights.
Laspeyre quantity index using base year prices as weights:
Paasche quantity index using current year prices as weights:

42. 7.2.6 Other indices
To measure the change in value the following 'value' index can be used:
These calculations are shown, along with the Laspeyre and Paasche index, in Table 7.15.

43. Having constructed a spreadsheet you can experiment by making changes to price or quantity information and observing the overall effect.
(This spreadsheet (sp715.XLS) is available on the website).

44. 7.3 The weighting of index numbers
Weights can be considered as a measure of importance.
The Laspeyre index and the Paasche index both refer to a typical basket of goods.
The prices are weighted by the quantities in these baskets.
In measuring a diverse range of items, it is often more convenient to use amount spent as a weight rather than a quantity.
If we consider travel, for example, it could be more meaningful to define expenditure on public transport than the number of journeys.

45. 7.3 The weighting of index numbers
In the same way, we would enquire about the expenditure on meals bought and consumed outside the home rather than the number of meals and their price.
Expenditure on public transport, meals outside the home and other items are additive since money units are homogeneous; the number of journeys, number of meals and number of shirts are not.

46. 7.3 The weighting of index numbers
In constructing a base-weighted index we can use
where Pn/P0 are the price relatives (see Section 7.2.1) and w are the weights.
Each weight is the amount spent on the item in the base year.
Consider again our example from Section 7.2 (Table 7.16).
It is no coincidence that this base-weighted index is identical to the Laspeyre index of Table 7.13.
The identity is proven below:

47. 7.3 The weighting of index numbers
The identity is proven below:

48. 7.3 The weighting of index numbers
The weights only need to represent the relative order of magnitude and in practice are scaled to sum to 1000.
(If we were to multiply each of the weights in Table 7.16 by 1000/748, the value of the index would not change but the sum of weights would add to 1000.)
The items included in the Retail Prices Index are assigned weights in this way.

50. 7.4 The Retail Prices Index (RPI)
The general Retail Prices Index (RPI) is the main index used to measure of inflation in the UK.
It measures the average change on a monthly basis of the prices of goods and services purchased by most households.
It is the measure of inflation reported in the media, debated by politicians and used to revise benefits and pensions.
Increases in wages are often justified in terms of the RPI, with recent or anticipated changes often forming the basis of a wage claim.
In many cases, savings and pensions are index-linked; they increase in line with the index.
All forms of economic planning take some account of inflation, and economists will use both real and nominal values in their analysis.

51. 7.4 The Retail Prices Index (RPI)
The RPI covers a range of goods and services bought by a typical household.
It is useful to think of the RPI as representing the changing cost of a large 'basket of goods and services' reflecting the full range of things that people buy including leisure goods, fuel, food and footwear.
In many ways, the RPI is not a 'cost of living' index as it does not attempt to provide a measure of the cost of staying alive.
A 'cost of living' index would imply some definition or knowledge of what were essential purchases.
Who could make such a judgement?

52. 7.4 The Retail Prices Index (RPI)
The index reflects what people choose to buy; for example, some people buy cigarettes and alcohol, so these are included in the index.
Coverage includes housing and travel but excludes items like savings, investments, charges for credit, betting and cash gifts.
The expenditure of certain higher income households and of pensioner households mainly dependent on state benefit is excluded.

53. 7.4 The Retail Prices Index (RPI)
The 'basket of goods' is kept fixed for a year at a time, so that only changes in prices are recorded that year.
The basket is reviewed each year to keep it as up to date as possible.
Changes made in 'year 2000' basket include broccoli, prepacked salad and takeaway/delivered pizza in the 'food and catering category', and the introduction of PC printers in the 'leisure goods' category.
It has also been decided that the RPI should begin to reflect Internet prices, and some books and toys typically bought on the Internet have been included.

54. 7.4 The Retail Prices Index (RPI)
The prices of more than 600 separate goods and services are collected each month.
The movements in these prices are taken as representative of all price movement in the goods and services covered by the index.
There are six price indicators for beef, for example (January 2000), which are combined together to estimate the overall change in beef price.
The base period is January of each year and current prices are compared to this base period.
The RPI for any month is calculated by weighting (averaged) price relatives.
Essentially, the RPI is a Las-peyres base-weighted index.
For a more detailed description of the RPI refer to http// or http//

55. CPI and RPI
CPI and RPI measure change in the prices charged for goods and services bought for consumption in the UK. Prices are recorded monthly for a typical selection of products (referred to as the ‘basket of goods’) , using a large sample of shops and other outlets.
Each month price collectors record about 120,000 prices for over 650 goods and services. These prices are ‘weighted’ to ensure they reflect the relative importance of the items in the average shopping basket. The ‘basket’ is updated on an annual basis to ensure that the indices reflect UK consumer spending patterns.
After the price data have been processed, the Office for National Statistics (ONS) calculates an overall average price change, which forms the basis of the monthly CPI and RPI. These data are published each month on the National Statistics website in the form of a First Release, which contains relevant tables, texts and charts.

56. 7.4 The Retail Prices Index (RPI)
Weights are used to allow for the relative importance of the various categories of goods and services.
The weights are derived mainly from the Family Expenditure Survey (see Figure 7.3).
The Family Expenditure Survey is based on a set sample size of about households each year ( in 1998/99 – see Figure 11.3) and uses addresses from the postcode address file.
Selected house­holds are asked to keep records of what they spend over a 2-week period and are also asked to give details of their major purchases over a longer period.
The response rate given in Figure 7.3 is 59%, with a more recent response rate of 63% being reported.
Analysis is based on about 7000 households.

58. The weights used in the RPI for 1990, 1995 and 2000 are shown in Table 7.17.

59. 7.4 The Retail Prices Index (RPI)
As far as the RPI is concerned, using the weights given in Table 7.17, food accounted for 15.8% of the typical basket in 1990, 13.9% in 1995 and 11.8% in 2000.
You could also note the recent decrease in the relative expenditure on alcoholic drink (from 7.7% in 1990 to 6.5% in 2000) and the recent increase on the relative expenditure on motoring (from 13.1% in 1990 to 14.6% in 2000).
The weightings therefore provide a useful guide to changing patterns of expendi­ture.
The weighting can be used to demonstrate the effect of a price change in one category on the overall RPI.
If, for example, the 'price' of fuel and light increased by 10% in 2000, the overall impact would be 0.32% (under A%), as the category accounts for 3.2% of expenditure (a weight of 32 out of 1000).
To show this using weights, the weight for fuel and light would increase from 32 to 35.2 (a 10% increase).

60. 7.4 The Retail Prices Index (RPI)
The sum of the weight would become and the percentage increase would be:
In practice, calculations can be more complex because a number of changes take place at the same time.
The typical basket of goods and services indicated by the weights shown in Table 7.17 will only reflect completely the expenditure of a proportion of households.
Some families will spend more on some items and less on others, particu­larly on an annual basis.
The RPI, like all aggregated statistics, will have an averaging-out effect and will reasonably describe most families most of the time.

62. 7.5 Conclusions
Index numbers play an important role in describing the economy, managing the economy and measuring the performance of business.
Percentage increases from the base year can be seen at a glance, and that the numbers provide a manageable and understandable sequence.
As we have seen, we are able to aggregate a wide range of different items into a single index series, which will enhance our comprehension of an overall situation, for example, the level of inflation in an economy.
In the presentation of accounting information allowance needs to be made for inflation.
Historic cost accounting (with no allowance for inflation) only works well in periods of stable prices.
In current cost accounting (CCA), adjust­ments are made in proportion to relevant indices.
The Office for National Statistics publishes price index numbers for current cost accounting.

63. 7.5 Conclusions Index numbers can be misleading if care is not taken.
When an index is rebased it is important to compare the last value of the previous series to the starting value of the new series and make any necessary adjustments.
When items are excluded, or new items included in an index, there may be drastic movements in the series, which do not reflect major changes in prices or quantities, but merely the changed composition of the index.
Crime statistics, in particular, are statistically (and politically) very sensitive to changes in definition and reporting.