Chapter 20

Introductory Mathematics
& Statistics

Chapter 20
Index Numbers
Copyright © 2010 McGraw-Hill Australia Pty Ltd
PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

20-1

Learning Objectives
• Interpret and use a range of index numbers commonly
used in the Australian business sector
• Define an index number and explain its use
• Perform calculations involving simple, composite and
weighted index numbers
• Understand the basic structure of the consumer price
index (CPI) and perform calculations involving its use
• Understand other indexes used in the Australian
business sector


20-2

20.1 Introduction
• An index number is a statistical value that measures
the change in a variable with respect to time
• Two variables that are often considered in this analysis
are price and quantity
• With the aid of index numbers, the average price of
several articles in one year may be compared with the
average price of the same quantity of the same articles
in a number of different years
• There are several sources of ‘official’ statistics that
contain index numbers for quantities such as food
prices, clothing prices, housing, wages and so on


20-3

20.2 Simple index numbers
• We will examine index numbers that are constructed
from a single item only
• Such indexes are called simple index numbers
• Current period = the period for which you wish to find
the index number
• Base period = the period with which you wish to
compare prices in the current period
• The choice of the base period should be considered
very carefully
• The choice itself often depends on economic factors
1. It should be a ‘normal’ period with respect to the relevant
index
2. It should not be chosen too far in the past


20-4

20.2 Simple index numbers (cont…)
• The notation we shall use is:
– pn = the price of an item in the current period
– po = the price of an item in the base period

• Price relative
– The price relative of an item is the ratio of the price of the
item in the current period to the price of the same item in the
base period
– The formal definition is:

pn
Price relative =
po

20-5

• Simple price index
– The price relative provides a ratio that indicates the
change in price of an item from one period to another
– A more common method of expressing this change is to
use a simple price index

Simple price index = price relative × 100
=

pn
× 100
po


20-6

• The simple price index finds the percentage change in the
price of an item from one period to another
– If the simple price index is more than 100, subtract 100 from
the simple price index. The result is the percentage increase
in price from the base period to the current period
– If the simple price index is less than 100, subtract the simple
price index from 100. The result is the percentage by which
the item cost less in the base period than it does in the
current period


20-7

20.3 Composite index numbers
• A composite index number is constructed from
changes in a number of different items
• Simple aggregate index
– the simple aggregate index has appeal because its nature
is simplistic and it is easy to find
Simple aggregate index =

∑p
∑p

n

× 100

o

Where
Σpn = the sum of the prices in the current period
Σpo = the sum of the prices in the base period

20-8

(cont…)
• Simple aggregate index (cont…)
– Even though the simple aggregate index is easy to calculate, it
has serious disadvantages:
1. An item with a relatively large price can dominate the
index
2. If prices are quoted for different quantities, the simple
aggregate index will yield a different answer
3. It does not take into account the quantity of each item sold
– Disadvantage 2 is perhaps the worst feature of this index,
since it makes it possible, to a certain extent, to manipulate the
value of the index


20-9

(cont…)
• Averages of relative prices
– This index also does not take into account the quantity of
each item sold, but it is still a vast improvement on the
simple aggregate index

Average of relative prices =

∑

 pn

 × 100 
p

 o

k

where
k = the number of items
pn = the price of an item in the current period
po = the price of an item in the base period

20-10

20.4 Weighted index numbers
• The use of a weighted index number or weighted index
allows greater importance to be attached to some items
• Information other than simply the change in price over time
can then be used, and can include such factors as quantity
sold or quantity consumed for each item

• Laspeyres index
– The Laspeyres index is also known as the average of
weighted relative prices
– In this case, the weights used are the quantities of each item
bought in the base period


20-11

20.4 Weighted index numbers (cont…)
– The formula is:

∑p q
Laspeyres index =
∑p q

n o

× 100

o o

Where:
qo = the quantity bought (or sold) in the base period
pn = price in current period
po = price in base period
– Thus, the Laspeyres index measures the relative change in
the cost of purchasing these items in the quantities
specified in the base period

20-12

• Paasche index
– The Paasche index uses the consumption in the current
period
– It measures the change in the cost of purchasing items, in
terms of quantities relating to the current period
– The formal definition of the Paasche index is:

∑p q
Paasche index =
∑p q

n n

× 100

o n

Where:
pn = the price in the current period
po = the price in the base period
qn = the quantity bought (or sold) in the current period

20-13

• Comparison of the Laspeyres and Paasche
indexes
– The Laspeyres index measures the ratio of expenditures on
base year quantities in the current year to expenditures on
those quantities in the base year
– The Paasche index measures the ratio of expenditures on
current year quantities in the current year to expenditures on
those quantities in the base year
– Since the Laspeyres index uses base period weights, it may
overestimate the rise in the cost of living (because people
may have reduced their consumption of items that have
become proportionately dearer than others)

20-14

• Comparison of the Laspeyres and Paasche
indexes (cont…)
– Since the Paasche index uses current period weights, it may
underestimate the rise in the cost of living
– The Laspeyres index is usually larger than the Paasche
index
– With the Paasche index it is difficult to make year-to-year
comparisons, since every year a new set of weights is used
– The Paasche index requires that a new set of weights be
obtained each year, and this information can be expensive to
obtain
– Because of the last 2 points above, the Laspeyres index is
the one most commonly used


20-15

• Fisher’s ideal index
– Fisher’s ideal index is the geometric mean of the Laspeyres
and Paasche indexes
– Although it has little use in practice, it does demonstrate the
many different types of index that can be used

Fisher' s index =

( Laspeyres index )( Paasche index )

∑p q ∑p q
∑p q ∑p q
n o

n n

o o

=

o n

× 100


20-16

20.5 The Consumer Price Index (CPI)
•

The measure most commonly used in Australia as a general
indicator of the rate of price change for consumer goods and
services is the consumer price index

•

The Australian CPI assumes the purchase of a constant ‘basket’
of goods and services and measures price changes in that
basket alone

•

The description of the CPI commonly adopted by users is in
terms of its perceived uses; hence there are frequent references
to the CPI as
– a measure of inflation
– a measure of changes in purchasing power, or
– a measure of changes in the cost of living


20-17

(cont…)
•
•
•

The CPI has been designed as a general measure of price
inflation for the household sector.
The CPI is simply a measure of the changes in the cost of a
basket, as the prices of items in it change
From the September quarter 2005 onwards, the total basket has
been divided into the following 11 major commodity groups:
– food
– alcohol and tobacco
– clothing and footwear
– housing
– household contents and services
– health
– transportation
– communication
– research
– education
– financial and insurance services


20-18

(cont…)
• Historical details of the CPI
– Retail prices of food, other groceries and average rentals of
houses have been collected by the ABS for the years
extending back to 1901
– From its inception in 1960, the CPI covered the six state
capital cities. In 1964 the geographical coverage of the CPI
was extended to include Canberra. From the June quarter in
1982 geographic coverage was further extended to include
Darwin
– Index numbers at the ‘Group’ and ‘All groups’ levels are
published for each capital city and for the weighted average
of eight capital cities


20-19

(cont…)
• The conceptual basis for measuring price changes
– The CPI is a quarterly measure of the change in average
retail price levels
– In measuring price changes, the CPI aims to measure only
pure price changes
– The CPI is a measure of changes in transaction prices, the
prices actually paid by consumers for the goods and
services they buy
– It is not concerned with nominal, recommended or list prices
– The CPI measures price change over time and does not
provide comparisons between relative price levels at a
particular date


20-20

(cont…)
• The index population
– Because the spending patterns of various groups in the
population differ somewhat, the pattern of one large group,
fairly homogeneous in its spending habits, is chosen for the
purpose of calculating the CPI
– The CPI population group is, in concept, metropolitan
employee households
– For this purpose, employee households are defined as those
households that obtain the major part of their household
income from wages and salaries


20-21

(cont…)
• Details of the 15th series CPI
– Since 1960, when the CPI was first compiled, the ABS has
maintained a program of periodic reviews of the CPI
– The main objective of these reviews is to update item
weights, but they also provide an opportunity to reassess the
scope and coverage of the index
– The latest (15th series) review has resulted in three main
outcomes:
1. updating the weighting pattern for the CPI
2. incorporating a price index for financial services into the
CPI
3. introducing an Australian hedonic price index for
computers


20-22

(cont…)
• Collecting prices
– This involves collecting prices from many sources, including
supermarkets, department stores, footwear stores,
restaurants, motor vehicle dealers and service stations,
dental surgeries, etc.
– In total, around 100 000 separate price quotations are
collected each quarter
– Prices of the goods and services included in the CPI are
generally collected quarterly
– The prices used in the CPI are those that any member of the
public would have to pay on the pricing day to purchase the
specified good or service
– Any sales or excise taxes that the consumer must pay when
purchasing specific items are included in the CPI price


20-23

(cont…)
• Periodic revision of the CPI
– The CPI is periodically revised in order to ensure it continues
to reflect current conditions
– CPI revisions have usually been carried out at approximately
5-yearly intervals

• Changes in quality
– it is necessary to ensure that identical or equivalent items are
priced in successive time periods
– This involves evaluating changes in the quality of goods and
services


20-24

(cont…)
• Long-term linked series
– A single series of index numbers has been constructed by
linking together selected retail price index series
– The index numbers are expressed on a reference base 1945
= 100

• International CPIs
– A comparison of the CPIs for a number of countries,
including Australia, as measured in September quarters
– The base year is 2000
– During the 8-year period up to 2008–09, of the countries
listed, Japan had the smallest increase in CPI (0.4%), while
Indonesia had the largest (111.0%)


20-25

20.6 Using the CPI
• There are a number of situations where the CPI is used
to make adjustments to prices charged and payments
made
• Examples include changes in rent, pension payments
and child support payments
• Such adjustments are often made by the relevant
government agency, but if members of the community
wish to use the CPI for such purposes, it is their
responsibility to ensure that this index is suitable


20-26

20.7 Index numbers as a measure of
deflation
• One of the uses for price indexes is to measure the changes
in the purchasing power of the dollar
• This is known as deflation
• In order to eliminate the effect of inflation and obtain a clear
picture of the ‘real’ change, the values must be deflated
• For example, to deflate an actual salary and express it in
terms of ‘real’ salary (of the base year), use:
' Real' salary = actual salary in current year ×

CPI in base year
CPI in current year


20-27

20.8 Other types of Australian indexes
• Producer price indexes
– Several producer price indexes (PPIs) are
produced and published
– PPIs can be constructed as either output
measures or input measures
– Output indexes measure change in the prices of
goods and/or services sold by a defined sector
of the economy
– Input indexes measure changes in the prices of
goods and/or services purchased by a particular
economic sector

20-28

(cont…)
• International trade price indexes
– The international trade price indexes are
intended to broadly measure changes in the
prices of goods imported into Australia (the
export price index, EPL)
– As the prices used in the indexes are expressed
in Australian currency, changes in the relative
value of the Australian dollar and overseas
currencies can have a direct impact


20-29

(cont…)
• The All-Ordinaries index
– The All-Ordinaries index (also know as the ‘All-Ords’) is
an Australian Stock Exchange (ASX) measure of the
share-price movements of the largest 500 Australian
companies
– The market capitalisation of these companies totals
about 95% of the value of shares listed on the exchange

• The trade-weighted index
– The trade-weighted index (TWI) provides an indication of
movements in the average value of the Australian dollar
against the currencies of our main trading partners
– The method of calculating the TWI has been basically
unchanged since 1974, but new weights were announced
by the Reserve Bank of Australia as from 1 October 2008

20-30

Summary
• We have interpreted and used a range of index
numbers commonly used in the Australian business
sector
• We defined an index number and explained its use
• We performed calculations involving simple, composite
and weighted index numbers
• We understood the basic structure of the consumer
price index (CPI) and performed calculations involving
its use
• We understood other indexes used in the Australian
business sector


20-31

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