Appendix 2C Index numbers

Mary O’Mahony

A2C.1 Introduction

index
An index shows the development of a data series over time relative to a base period. For simplicity, the reference value given to the base period is usually set to 100.

This appendix describes how index numbers can be constructed and used to identify changes over time for a data series. We cover the main types of indices – Laspeyres, Paasche, Fisher and Tornqvist – and finally, the procedure of chain-linking.

Indices can be used when:

The convention in index numbers is to set the value of the index in the “base period” to 100. An index based on a single variable x can be calculated using the following formula:

\[I\left(x\right)_{t} = 100 \times \left( \frac{x_{t}}{x_{0}} \right)\]

Where \(I\left(x\right)_{t}\) is the index value in time t compared to the base period 0 which is set to 100. \(x_{t}\) denotes the value of the variable in time t, and \(x_{0}\) the value of the variable in the base period.

Figure A2C.1 shows the application of index numbers to two simple time series: UK and US real GDP. The data is in the local currency of each country.

  2014 2015 2016 2017 2018
UK GDP (£bn) 1,720 1,760 1,792 1,824 1,850
US GDP ($bn) 16,243 16,710 16,972 17,349 17,844

Figure A2C.1 Real GDP in the UK and US, 2014 to 2018

Real GDP in the UK and US, 2014 to 2018

The numbers also differ because the US economy is several times larger than the UK economy. These factors make it more difficult to identify which country has grown faster. Using the formula above, we can turn each of the series into an index with base period 2014, as shown in Figure A2C.2. This allows for an easier comparison. For example, the index value for UK real GDP in 2017 is:

\[\text{UK}_{2017} = 100 \times \left( \frac{1,824}{1,720} \right) = 106.0\]
  2014 2015 2016 2017 2018
UK GDP 100.0 102.3 104.2 106.0 107.6
US GDP 100.0 102.9 104.5 106.8 109.9

Figure A2C.2 Index of real GDP in the UK and US, 2014 to 2018, 2014 = 100

Index of real GDP in the UK and US, 2014 to 2018, 2014 = 100

This is a simple use of index numbers where each index is based on a single variable. However, many economic series are built up by combining variables. Index numbers can be used to construct these composite indicators.

When producing an index from more than one data series, a method for how to combine them is required. The important decision is how much importance to give to each series, and this is done by attaching weights. These weights are usually empirically based or deduced from economic theory.

In this appendix, we only discuss the formation of price indices, but the same methods apply to quantity indices such as gross domestic product. We will illustrate using the concept of a consumer price index for expenditures, but the same analysis would carry through to a producer price index using sales instead of expenditures.

Suppose we observe consumer expenditures in time t on multiple goods or services. Suppose there are n such products each with their own price in time \(t \left(p_{i,t} \text{, for } i = 1, \ldots, n\right)\), and quantity purchased in time \(t (q_{i,t}\text{, for } i = 1, \ldots, n)\). Aggregate expenditures are found by summing the expenditure on each of the n products:

\[\text{Expenditures}_{t} = \sum_{i=1}^{n} \, p_{i,t}q_{i,t} = p_{1,t}q_{1,t} + p_{2,t}q_{2,t} + \ldots + p_{n,t}q_{n,t}\]

To construct a price index for these n items requires separating price and quantity movements over time, as we only want to capture the price change. We observe prices of each of the n products, but how do we combine these into a single index while keeping quantities purchased fixed? The solution is to weight the prices together using the quantities relating to a particular period. There are four approaches:

A2C.2 Laspeyres Index

Laspeyres price index
The change in prices in year t relative to a base year using base period weights.

Suppose we want to calculate the price change from some base year, denoted by 0, to the current year, denoted by t. The formula for a Laspeyres price index is given by:

\[\text{Laspeyres}\left(p\right)_{t} = \frac{\sum\limits_{i=1}^{n} \, p_{i,t}q_{i,0}}{\sum\limits_{i=1}^{n} \, p_{i,0}q_{i,0}}\]

Note that in this formula the prices for the n goods in the numerator are for the current year (t) and the prices in the denominator are those in the base year (0). The quantity weights, which are for the base year (0), remain the same in both the numerator and denominator.

It is useful at this stage to show a hypothetical example. Suppose there are three goods produced by an economy: bread (B), shoes (S), and games (G). The prices and quantities for each of four years (0, 1, 2, 3) are given in Figure A2C.3. The prices are very different for the three products and behave differently through time. Bread prices rise the most over the four years, whereas the price for games falls over the four years. The quantities also vary across the three goods. Although the quantities are not comparable across the three types of goods – bread is measured in loaves, shoes in pairs and games in numbers of units – their prices are comparable as they are all measured in money.

  Bread   Shoes   Games  
Years pB qB pS qS pG qG
0 0.90 550 12.50 60 9.50 120
1 1.00 560 14.50 55 10.10 135
2 1.10 565 16.00 57 9.80 160
3 1.35 530 16.50 64 8.70 210
Ratio pi,3/pi,0 1.50   1.32   0.90  

Figure A2C.3 Prices for three consumer products: the basic data

Prices for three consumer products: the basic data

Next, in Figure A2C.4, we apply the Laspeyres formula to these data where \(n = 3\) and \(i \in \left(\text{B, S, G}\right)\). First, we calculate the product of prices in each year and the quantities in the base year for each of the three goods. We then sum these across the rows, calculate the sums relative to the base year, and multiply by 100 (so that the reference value for year 0 = 100). Over the four-year period, The Laspeyres price index shows that the consumer prices increased by 16.4%.

Bread Shoes Games
Years pB qB pS qS pG qG Sum Index
0 0.90 550 12.50 60 9.50 120
1 1.00 560 14.50 55 10.10 135
2 1.10 565 16.00 57 9.80 160
3 1.35 530 16.50 64 8.70 210
pi,0 × qi,0 495 750 1,140 2,385 100.0
pi,1 × qi,0 550 870 1,212 2,632 110.4
pi,2 × qi,0 605 960 1,176 2,741 114.9
pi,3 × qi,0 743 990 1,140 2,777 116.4

Figure A2C.4 Prices for consumer products: Laspeyres price index calculations

Prices for consumer products: Laspeyres price index calculations

The calculation of the price index shown in Figure A2C.4 requires observations for both prices and quantities for all products. An alternative approach if we do not directly know expenditure quantities, but use expenditure shares instead, is as follows:

\[\text{Laspeyres} \left(p\right)_{t} = \sum_{i=1}^{n} \, s_{i,0}\frac{p_{i,t}}{p_{i,0}}\]

where the share of good i in total expenditure in the base year is

\[s_{i,0} = \frac{p_{i,0} \, q_{i,0}}{ \sum\limits_{i}^{n} \, p_{i,0} \, q_{i,0}}\]

We can see this in the example in Figure A2C.5, by calculating the shares of each of the products in total expenditures in year 0, and then using these shares to weight the respective price ratios in the lower panel. The resulting price index, where the base year is set to 100, is identical to that calculated in Figure A2C.4.

Bread Shoes Games
Years pB qB pS qS pG qG Sum
0 0.90 550 12.5 60 9.5 120
1 1.00 560 14.5 55 10.1 135
2 1.10 565 16.0 57 9.8 160
3 1.35 530 16.5 64 8.7 210
pi,0 × qi,0 495 750 1,140 2,385
si,0 0.208 0.314 0.478 1.000
pB,t /pB,0 pS,t /pS,0 pG,t /pG,0
0 1.00 1.00 1.00
1 1.11 1.16 1.06
2 1.22 1.28 1.03
3 1.50 1.32 0.92
sB,0 × (pB,t /pB,0) sS,0 × (pS,t /pS,0) sB,0 × (pG,t /pG,0) Sum × 100
0 0.208 0.314 0.478 100.0
1 0.231 0.365 0.508 110.4
2 0.254 0.403 0.493 114.9
3 0.311 0.415 0.438 116.4

Figure A2C.5 Prices for consumer products: Laspeyres price index, share weighted calculation

Prices for consumer products: Laspeyres price index, share weighted calculation

The Laspeyres formula relies on base year weights, which could make it a weaker approximation the further we move forward in time. Economic theory suggests that consumers will substitute away from relatively expensive goods and consume cheaper ones, so we would expect the weights to change over time in response to changing prices and preferences. The Laspeyres index does not allow for this substitution.

A2C.3 Paasche index

Paasche price index
The change in prices in year t relative to a base year using current period (year t) weights.

An alternative to a Laspeyres price index which uses base period weights is a Paasche price index which uses current period weights and is calculated by:

\[\text{Paasche}\left(p\right)_{t} = \frac{\sum\limits_{i=1}^{n} \, p_{i,t}q_{i,t}}{\sum\limits_{i=1}^{n} \, p_{i,0}q_{i,t}}\]

Note in this formula again only prices change, and the quantities are fixed at the current (year t) amounts. Using the same data as in Figure A2C.3, the Paasche price index is calculated in Figure A2C.6.

Bread Shoes Games
Years pB qB pS qS pG qG Sum Index
0 0.90 550 12.50 60 9.50 120
1 1.00 560 14.50 55 10.10 135
2 1.10 565 16.00 57 9.80 160
3 1.35 530 16.50 64 8.70 210
pi,0 × qi,3 477 800 1,995 3,272 100.0
pi,1 × qi,3 530 928 2,121 3,579 109.4
pi,2 × qi,3 583 1,024 2,058 3,665 112.0
pi,3 × qi,3 716 1,056 1,827 3,599 110.0

Figure A2C.6 Prices for consumer products: Paasche price index calculations

Prices for consumer products: Paasche price index calculations

The Paasche price index in this example has a different time pattern to the Laspeyres price index. Price growth is lower for each year, and in the final year the price index falls due to the much higher quantity weights for games whose price is falling. While it might be tempting to prefer the Paasche index on the grounds that the quantity weights are more up to date, they face the problem of being correspondingly less representative of the bundle of goods and services demanded in the earlier periods.

Just as the Laspeyres price index can be calculated with expenditure shares, the same is true of the Paasche index. The Laspeyres price index tends to be more commonly used over the Paasche price index by statistical offices for the practical reason that price data tend to be available more rapidly than expenditure data.

A2C.4 Fisher index

Fisher price index
The geometric mean of the Laspeyres price index and Paasche price index.

Given that the Laspeyres price index allows for no substitution between goods and services, while the Paasche price index implies complete substitution, neither is likely to be an exact measure of price growth for all years between the base and current years. An alternative is the Fisher price index given by:

\[\text{Fisher} \left(p\right)_{t} = \sqrt {\text{Laspeyres}\left(p\right)_{t} \, × \, \text{Paasche} \left(p\right)_{t}}\]

This is an index derived from economic theory, which says that a “true” price index that measures the cost of maintaining consumers’ utility must lie between the Laspeyres and Paasche price indices. Figure A2C.7 shows the simple calculation of the Fisher price index as the geometric mean of the Laspeyres price index calculated in Figure A2C.4 and the Paasche price index calculated in Figure A2C.6.

Year Laspeyres price index Paasche price index Fisher calculation Fisher price index
0 100.0 100.0 \(\sqrt{\left(100.0 \times 100.0\right)}\) 100.0
1 110.4 109.4 \(\sqrt{\left(110.4 \times 109.4\right)}\) 109.9
2 114.9 112.0 \(\sqrt{\left(114.9 \times 112.0\right)}\) 113.5
3 116.4 110.0 \(\sqrt{\left(116.4 \times 110.0\right)}\) 113.2

Figure A2C.7 Calculating the Fisher price index

Calculating the Fisher price index

A2C.5 Tornqvist index

Tornqvist price index
A price index constructed using log growth rates and average weights.

The Tornqvist price index is another possibility for using the average of the base and current year weights. This approach is based on the natural logarithm growth rates and average weights:

\[\text{Tornqvist} \left(p\right)_{t} = exp \left\lbrack \sum_{i=1}^{n} \; 0.5\left( s_{i,0} + s_{i,t} \right) \ln \left(p_{i,t}/p_{i,0}\right) \right\rbrack\]

The process of calculating the Tornqvist price index for our data is set out in Figure A2C.8. The first step is to calculate the expenditure shares of the three goods in the first and last years (or the base and current years if you prefer). These are then averaged. The second step is to use these average shares to weight together the natural logarithm of price changes in each year relative to the base year. The third and final step is to sum these weighted price relatives together and to construct the index in the final column using the exponential of this sum. We set the value in year 0 to 100.

Bread Shoes Games
Years pB qB pS qS pG qG Sum Index
0 0.90 550 12.50 60 9.50 120
1 1.00 560 14.50 55 10.10 135
2 1.10 565 16.00 57 9.80 160
3 1.35 530 16.50 64 8.70 210
pi,0 × qi,0 495 750 1,140 2,385
si,0 0.208 0.314 0.478 1.00
pi,3 × qi,3 715.5 1056 1,827 3,599
si,3 0.199 0.293 0.508 1.00
0.5(si,0 + si,3) 0.203 0.304 0.493 1.00
Bread Shoes Games
$$\ln\left(p_{i,t}/p_{i,0}\right)$$
0 0.000 0.000 0.000
1 0.105 0.148 0.061
2 0.201 0.247 0.031
3 0.405 0.278 −0.088
$$\ln\left(p_{i,t}/p_{i,0}\right) × \left(0.5\left(s_{i,0} + s_{i,3}\right)\right)$$
0 0.000 0.000 0.000 0.000 100.0
1 0.022 0.046 0.030 0.097 110.2
2 0.041 0.076 0.015 0.133 114.2
3 0.083 0.086 −0.043 0.126 113.5

Figure A2C.8 Prices for consumer products: Tornqvist price index calculations

Prices for consumer products: Tornqvist price index calculations

The calculations for the Fisher and Tornqvist indices merely use expenditure shares for years 0 and t, and not in the years in between. So, just like Laspeyres and Paasche price indices, they are examples of “fixed base” indices.

Figure A2C.9 shows the four fixed base price index measures calculated for our example data.

  Laspeyres Paasche Fisher Tornqvist
0 100.0 100.0 100.0 100.0
1 110.4 109.4 109.9 110.2
2 114.9 112.0 113.5 114.2
3 116.4 110.0 113.2 113.5

Figure A2C.9 Summary of fixed base price indices

Summary of fixed base price indices

The Laspeyres price index implies the greatest price growth and the Paasche the lowest, with Fisher and Tornqvist in between. The conventional wisdom is that the Laspeyres price index tends to overestimate inflation by using base year weights, while the Paasche price index using current period weights has a tendency to underestimate. The Fisher and Tornqvist indices are based on an average of base and current year weights, so usually sit between the Laspeyres and Paasche price indices.

In the examples of price indices given in this appendix, we have taken year 0 as the reference or base year. This has been merely for convenience, and the reference year can easily be changed by a simple process of rebasing the index, so a different year is set at 100.

How it’s done Rebasing series

Suppose we have a series defined relative to a reference year, for example, 1990 = 100, and want to consider the same series but relative to another reference year. For instance, this might be to focus on trends in more recent years. Simple rebasing merely multiplies each entry in the series by 100, divided by the entry in the new reference year.

For example, suppose our original series is given by:

   
1990 100
1995 106
2000 111
2005 114
2010 121
2015 126

In order to rebase to 2010 = 100, we simply multiply the original index values by (100/121):

  Old New
1990 100 = 100 × 100/121 = 83
1995 106 = 106 × 100/121 = 88
2000 111 = 111 × 100/121 = 92
2005 114 = 114 × 100/121 = 94
2010 121 = 121 × 100/121 = 100
2015 126 = 126 × 100/121 = 104

This calculation simply moves the data onto a different reference year, but the time-series pattern does not change. Comparing growth in the index between 1995 and 2005 provides the same ratios in both the old series, 114/106 = 1.075, as in the new series, 94/88 = 1.075.

A2C.6 The Laspeyres chain-linked index

An alternative to using fixed base indices is to allow the weights to vary more frequently. In principle, a more frequent updating of weights should lead to more accurate calculation of aggregate price changes. An annual chained index aims to achieve this by updating the weights every year, so the length of time between the current period and base period is always only one year. Chain linking then involves joining together a series of comparisons across pairs of years.

For example, to form a Laspeyres chain-linked index between 2015 and 2017, we would carry out the following two sets of calculations.

For a comparison between 2015 and 2016, the weights from 2015 are used, for the comparison between 2016 and 2017, weights from 2016 are used, and so on. Each year is compared with the previous year using fixed weights.

\[\text{Laspeyres}\left(p\right)_{2016} = \frac{\sum\limits_{i=1}^{n} \; p_{i,2016}q_{i,2015}}{\sum\limits_{i=1}^{n}\; p_{i,2015}q_{i,2015}}\] \[\text{Laspeyres}\left(p\right)_{2017} = \frac{\sum\limits_{i=1}^{n} \; p_{i,2017}q_{i,2016}}{\sum\limits_{i=1}^{n} \; p_{i,2016}q_{i,2016}}\]

The two numbers formed in the first step then need to be chain-linked together. If we set 2015 = 100, then the value of the index for 2016 is \(100 × \text{Laspeyres}\left(p\right)_{2016}\). Likewise, the calculation of the chain-linked index for 2017 is the index value just calculated for 2016 multiplied by \(\text{Laspeyres}\left(p\right)_{2017}\).

The calculation of a chain-linked Laspeyres price index for the data we have been using in this Appendix is shown in Figure A2C.10.

    Bread Shoes Games Sum Index
Year            
1 pi,1 /pi,0 1.11 1.16 1.06    
2 pi,2 /pi,1 1.10 1.10 0.97    
3 pi,3 /pi,2 1.23 1.03 0.89    
             
1 si,1 0.208 0.314 0.478    
2 si,2 0.206 0.293 0.501    
3 si,3 0.200 0.294 0.506    
             
0           100.0
1 si,0 × pi,1 /pi,0 0.231 0.365 0.508 1.104 110.4
2 si,1 × pi,2 /pi,1 0.226 0.323 0.486 1.036 114.4
3 si,2 × pi,3 /pi,2 0.246 0.303 0.449 0.998 114.2

Figure A2C.10 Prices for consumer products: chain-linked Laspeyres price index calculations

Prices for consumer products: chain-linked Laspeyres price index calculations

The first panel shows the price ratios, year-on-year. These are then multiplied by the relevant shares in the lower part, where si,0 are the expenditure shares for each of the three goods in year 0, and so on. For each year, the share weighted price ratios are summed across products, shown in the column “Sum”. To construct the index each year, the sum is multiplied by the previous year’s index shown in the column “Index”. So, for example, the index value for year 1 is 100 × 1.104 = 110.4. For year 2 the index value is 110.4 × 1.036 = 114.4. And finally, for year 3 the index value is 114.4 × 0.998 = 114.2.

A2C.7 The Tornqvist chain-linked index

An alternative approach is to construct a chain-linked Tornqvist index. Although not generally used for constructing price indices, it is frequently used in productivity analysis, as it is linked to the economic theory of production. The Tornqvist chain-linked index for the price index between two adjacent periods t and t − 1 is given by the formula:

\[\text{Tornqvist chain index} = exp \left( \sum_{i=1}^{n} \; 0.5\left(s_{i}^{t} + s_{i}^{t − 1}\right)\ln\left(p_{i}^{t}/p_{i}^{t − 1}\right)\right)\]

Using the same example data as used throughout in this appendix, the calculations for a Tornqvist chain-linked price index are shown in Figure A2C.11.

    Bread Shoes Games Sum Index
Year            
1 \(\ln\left(p_{i,1}/p_{i,0}\right)\) 0.11 0.15 0.06    
2 \(\ln\left(p_{i,2}/p_{i,1}\right)\) 0.10 0.10 −0.03    
3 \(\ln\left(p_{i,3}/p_{i,2}\right)\) 0.20 0.03 −0.12    
             
1 \(s_{i,\left(0,1\right)} = 0.5\left[s_{i,0} + s_{i,1}\right]\) 0.207 0.304 0.490    
2 \(s_{i,\left(1,2\right)} = 0.5\left[s_{i,1} + s_{i,2}\right]\) 0.203 0.294 0.503    
3 \(s_{i,\left(2,3\right)} = 0.5\left[s_{i,2} + s_{i,3}\right]\) 0.200 0.294 0.507    
             
0           100.0
1 \(s_{i,\left(0,1\right)} \ln\left(p_{i,1}/p_{i,0}\right)\) 0.022 0.045 0.030 0.097 110.2
2 \(s_{i,\left(1,2\right)} \ln\left(p_{i,2}/p_{i,1}\right)\) 0.019 0.029 −0.015 0.033 113.9
3 \(s_{i,\left(2,3\right)} \ln\left(p_{i,3}/p_{i,2}\right)\) 0.041 0.009 −0.060 −0.010 112.8

Figure A2C.11 Prices for consumer products: chain-linked Tornqvist price index calculations

Prices for consumer products: chain-linked Tornqvist price index calculations

The steps are similar to the calculation of the Laspeyres chain-linked price index in Figure A2C.10. The first step is to calculate the natural logs of the relative price ratios for each of the three goods in successive time periods. These are then multiplied by the average of the expenditure share of each goods in those two time periods. These values are then summed together.

If the starting value for the price index is year 0 is set to 100, then the value for year 1 is given by 100 × exp(0.097) = 110.2. For year 2 the price index is calculated as 110.2 × exp(0.033) = 113.9. Finally, for year 3 the price index is calculated as 113.9 × exp(−0.010) = 112.8.

A2C.8 Summary

We have demonstrated how several different types of price indices can be constructed using a simple set of data. Let’s summarise the calculations carried out in this appendix in Figure A2C.12. It clearly shows that the index we use can have an important effect on the statistic we report.

Fixed base price indices Chain-linked price indices
Year Laspeyres Paasche Fisher Tornqvist Chain-linked Laspeyres Chain-linked Tornqvist
0 100.0 100.0 100.0 100.0 100.0 100.0
1 110.4 109.4 109.9 110.2 110.4 110.2
2 114.9 112.0 113.5 114.2 114.4 113.9
3 116.4 110.0 113.2 113.5 114.2 112.8

Figure A2C.12 Summary of the price indices calculated in this appendix (year 0 = 100)

Summary of the price indices calculated in this appendix (year 0 = 100)

The indexes all give similar numbers for year 1. By year 2, however, they vary from 12% to 14.9% relative to year 0, and by year 3 from 10% to 16.4%. The fixed-base Laspeyres price index is the only index that shows a rise in price from year 2 to year 3.

Note that in all the above we have calculated price indices so that we can report inflation. The same methods can also be used to calculate quantity indices.