Measuring Price Change

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:

\textsf{Index}_t = \frac{\Sigma P_t Q_o}{\Sigma P_o Q_o} \times 1000

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.

\textsf{Index}_t = \frac{\Sigma \frac{P_t}{P_o}\times Q_o}{\Sigma Q_o} \times 1000

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:

L=\frac{\frac{1}{3}(189) + \frac{1}{2}(140) + \frac{1}{6}(120)}{\frac{1}{3}+\frac{1}{2}+\frac{1}{6}} = 153\%

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.

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
• a. State the Laspeyres price index for 2010 taking 2004 as the reference/base year (100). The Laspeyres price index is
• b. Give the percentage increase in vehicle operating expenses over the period 2004 to 2010. The percentage increase is %.

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
• a. State the Laspeyres index. The Laspeyres price index is
• b. Give the percentage increase in food prices. The percentage increase is %.

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:

\frac{601}{586} \times 100 = 102.6
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:

\frac{105.8}{105.8} \times 100 = 100

The index for the 1988 period (second column above) is:

\frac{100}{105.8} \times 100 = 94.5

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.

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.

• a. Re-reference/rebase the index so that the reference/base year is the first quarter (Q1) of 2012. Click 'Check' after you have entered all values in each table, any correct responses will be highlighted with a green border.

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
• b. What is the percentage change between Q1 of 2012 and Q3 of 2014.