The overall CPI calculates the total expenditure in the current period required to purchase the same selection of goods and services that was surveyed in the reference or base period.
Calculating CPI
The ratio of this expenditure to that required in the reference period is then multiplied by 1000 to give the current index number.
The simplest formula for the CPI is the Laspeyres formula. The index for period t on reference period 0 is given by:
Where Qo is the quantity of an item at time 0, Po is the price of the item in the reference period, and Pt is the price in period t.
This is the total cost of the reference/base date basket of goods at current prices divided by the total cost of the same basket at reference/base date prices (the reference/base date basket is assumed to also represent current consumption).
Sometimes it is easier to calculate this in the following equivalent form using the relative prices in the two periods. This is a weighted sum of prices for all the goods and services.
EXAMPLE 1
In 2009 one third of a student's allowance was spent on food, one half on rent and one sixth on all other items. If between 2009 and 2012, the price of food has increased by 189%, the price of rent by 140% and the price of all other items by 120% then the Laspeyres Index, assuming the spending pattern remains the same is given by:
This can be interpreted as the cost of living has risen for this student by 153% over these three years.
EXAMPLE 2
In the following table the 'average' household expenditure on the four items in the first column has been derived from Statistics New Zealand's Household Expenditure Survey. The prices of these items have also been surveyed in both base and current periods, columns (3) and (4).
The price relatives in column (5) are found by dividing the current price (4) by the base price (3). For each item the quantity is multiplied by the price relative to give the current expenditure, column (7). These values are summed then divided by the total base expenditure to get the current Index Number.
| Price | Expenditure | ||||||
|---|---|---|---|---|---|---|---|
| Item (1) |
Base quantity E0/P0= Q0 (2) |
Base $ P0 (3) |
Current $ Pt (4) |
Price relative (5) |
Base $ P0Q0=E0 (6) |
Current $ PtQ0 (7) |
Total expenditure in base period % (8) |
| Milk | 7.5 | 1.20 | 1.70 | 1.417 | 9.00 | 12.75 | 38% |
| Butter | 3.3 | 1.90 | 1.70 | 0.895 | 6.27 | 5.61 | 26% |
| Yoghurt | 4.0 | 0.85 | 0.90 | 1.059 | 3.40 | 3.60 | 14% |
| Cheese | 0.7 | 7.50 | 8.00 | 1.067 | 5.25 | 5.60 | 22% |
| Total Expenditure | 23.92 | 27.56 | 100% | ||||
| Index Number | 1000 | 1152 | |||||
Source: Dairy Products Price Index Statistics New Zealand.
EXERCISE 1
The following table gives the prices paid and quantities consumed for three commodities in 2012 and 2013. Using 2012 as the reference/base year, calculate the Laspeyres Index for 2013. A spreadsheet for this exercise can be downloaded below.
Excel Download| Item | Year | A | B | C |
|---|---|---|---|---|
| Price ($) | 2012 | 5 | 2 | 12 |
| 2013 | 10 | 5 | 15 | |
| Quantity | 2012 | 40 | 90 | 10 |
| 2013 | 10 | 20 | 20 |
The Laspeyres Index for 2013 is:
EXERCISE 2
The increase price of petrol has caused a trend towards smaller family cars and better driving habits in some countries. Unit costs of four items and the quantities of these items used in the operation of a typical family car for years 2004 and 2010 are given in the table below.
| Item | Cost/Unit | $ | Annual | Usage |
|---|---|---|---|---|
| 2004 | 2010 | 2004 | 2010 | |
| Petrol litre | 0.82 | 0.97 | 2750 | 2500.0 |
| Oil litre | 1.68 | 2.72 | 20 | 18.0 |
| Tyre | 70.00 | 125.00 | 1 | 1.5 |
| Insurance/ Registration |
480.00 | 625.00 | 1 | 1.0 |
EXERCISE 3
Given the following information about price and quantities in 2004 and 2012, in the table below, construct a food price index for 2012 with 2004 as the reference/base year using Laspeyres's method.
| Item | Unit | Prices ($) | Quantities Weekly Family of 3 |
||
|---|---|---|---|---|---|
| 2004 | 2012 | 2004 | 2012 | ||
| Milk | 600ml | 0.15 | 0.45 | 10 | 8 |
| Eggs | dozen | 1.15 | 1.45 | 2 | 2 |
| Meat | kg | 2.20 | 3.30 | 3 | 2 |
| Bread | loaf | 0.40 | 0.44 | 3 | 4 |
END CALCULATING CPI
If price change comparisons are needed with a particular period (for example, when a natural disaster occurred) then an index number series can be re-referenced/rebased to another reference/base year by making the new reference/base have value 100 (or 1000) and re-calculating the other points.
Re-referencing the Index Time Series
This is called re-referencing or rebasing the index time series. Below is an example taken from 'Chance Encounters', C.Wild & G.Seber, Chapter 14.
EXAMPLE
The reference period in this example is 1988. The index for the reference period is 100. The index for the 1989 period (highlighted in the table) is:
| Year | Price | Index Number Reference Year = 1988 |
Index Number Reference Year = 1999 |
|---|---|---|---|
| 1988 | 586 | 100 | 94.5 |
| 1989 | 601 | 102.6 |
|
| 1990 | 632 | 107.8 | |
| 1991 | 620 | 105.8 | 100 |
| 1992 | 645 | 110.1 | |
| 1993 | 668 | 114.0 | |
| 1994 | 699 | 119.3 |
Table 1: Annual Prices of an Appliance with Index Number for Two Reference/Base Years
The time series plot below gives us an idea of the change relative to 1988. To look at the change relative to 1991 (also highlighted in the table) divide each index number by that for 1991 and multiply by 100.
When re-referencing, it is necessary to retain unrounded index numbers to preserve exactly the same percentage changes as are obtained from the time series on the original reference period (otherwise they may be different, even in the first decimal place.)
The index number for the 1991 period is:
The index for the 1988 period (second column above) is:
EXERCISE 1
Now calculate the index numbers relative to 1991 for other years in the table. Two index numbers have already been provided. The answers should be to one decimal place.
| Year | Price | Index Number Base Year = 1988 |
Index Number Base Year = 1999 |
|---|---|---|---|
| 1988 | 586 | 100 | 94.5 |
| 1989 | 601 | 102.6 | |
| 1990 | 632 | 107.8 | |
| 1991 | 620 | 105.8 | 100 |
| 1992 | 645 | 110.1 | |
| 1993 | 668 | 114.0 | |
| 1994 | 699 | 119.3 |
The correct answers are: 94.5, 97.0, 101.9, 100, 104.1, 107.8, 112.8
NOTE: The time series plot does not change in shape when it is re-referenced.
Re-referencing Exercises
Attempt these re-referencing exercises using the previous description to guide you.
EXERCISE 2
Change the reference period from 2004 to 2008 in the table below, plot both the original and the re-referenced time series on the same graph.
You can use the spreadsheet download below for this exercise, the spreadsheet contains a completed answer.
Excel Download| Year | Reference 2004 |
|---|---|
| 2004 | 100.0 |
| 2005 | 110.3 |
| 2006 | 119.6 |
| 2007 | 132.1 |
| 2008 | 158.6 |
| 2009 | 162.2 |
| 2010 | 167.8 |
EXERCISE 3
The following table is extracted from Statistics New Zealand's time series for the New Zealand CPI. This has a base quarter of the second quarter of 2006.
CPI All Groups for New Zealand
| Qrtly Mar/Jun/Sep/Dec | All Groups | Re-referenced /Rebased Index |
|---|---|---|
| 2012Q1 | 1164 | |
| 2012Q2 | 1168 | |
| 2012Q3 | 1171 | |
| 2012Q4 | 1169 | |
| 2013Q1 | 1174 | |
| 2013Q2 | 1176 | |
| 2013Q3 | 1187 | |
| 2013Q4 | 1188 | |
| 2014Q1 | 1192 | |
| 2014Q2 | 1195 | |
| 2014Q3 | 1199 |
The completed table should look like the one below:
| Qrtly Mar/Jun/Sep/Dec | All Groups | Re-referenced /Rebased Index |
|---|---|---|
| 2012Q1 | 1164 | |
| 2012Q2 | 1168 | |
| 2012Q3 | 1171 | |
| 2012Q4 | 1169 | |
| 2013Q1 | 1174 | |
| 2013Q2 | 1176 | |
| 2013Q3 | 1187 | |
| 2013Q4 | 1188 | |
| 2014Q1 | 1192 | |
| 2014Q2 | 1195 | |
| 2014Q3 | 1199 |
The correct answer is %
END RE-REFERENCING