ÇAĞ UNIVERSITY
FACULTY OF ECONOMICS & ADMINISTRATIVE SCIENCES
MATHEMATICS FOR BUSINESS
LECTURE NOTES-10
Dr. Elma Satrovic|
elmasa1991@gmail.com
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Index numbers
Economic data often take the form of a time
series; values of economic indicators are available
on an annual, quarterly or monthly basis, and we
are interested in analysing the rise and fall of these
numbers over time.
Index numbers enable us to identify trends and
relationships in the data.
The following example shows you how to calculate
index numbers and how to interpret them.
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Example 1
Table 3.1 shows the values of household spending
(in billions of dollars) during a 5-year period.
Calculate the index numbers when 2000 is taken as
the base year and give a brief interpretation.
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Solution 1
When finding index numbers, a base year is chosen and the
value of 100 is allocated to that year.
So, in this example, the index number of 2000 is 100.
To find the index number of the year 2001 we work out the
scale factor associated with the change in household
spending from the base year, 2000 to 2001, and then multiply
the answer by 100.
In this case, we get
This shows that the value of household spending in 2001 was
103.8% of its value in 2000. In other words, household
spending increased by 3.8% during 2001.
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Solution 1
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Index numbers themselves have no units. They merely
express the value of some quantity as a percentage of a
base number.
This is particularly useful, since it enables us to compare
how values of quantities, of varying magnitudes, change in
relation to each other.
The following example shows the rise and fall of two share
prices during an 8-month period.
The prices (in dollars) listed for each share are taken on the
last day of each month.
Share A is exceptionally cheap. Investors often include this
type of share in their portfolio, since they can occasionally
make spectacular gains. This was the case with many
dot.com shares at the end of the 1990s.
The second share is more expensive and corresponds to a
larger, more established firm.
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Practice 1 (for students)
Find the index numbers of each share price shown
in Table 3.3, taking April as the base month.
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Inflation
Over a period of time, the prices of many goods and
services usually increase.
The annual rate of inflation is the average percentage
change in a given selection of these goods and services,
over the previous year.
Seasonal variations are taken into account, and the
particular basket of goods and services is changed
periodically to reflect changing patterns of household
expenditure.
The presence of inflation is particularly irritating when
trying to interpret a time series that involves a monetary
value. It is inevitable that this will be influenced by inflation
during any year, and what is of interest is the fluctuation of
a time series ‘over and above’ inflation.
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Inflation
Economists deal with this by distinguishing between
nominal and real data.
Nominal data are the original, raw data such as those
listed in tables in the previous sub-section. These are
based on the prices that prevailed at the time.
Real data are the values that have been adjusted to
take inflation into account.
The standard way of doing this is to pick a year and
then convert the values for all other years to the level
that they would have had in this base year.
This may sound rather complicated, but the idea and
calculations involved are really quite simple as the
following example demonstrates.
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Example 2
Table below shows the price (in thousands of pounds) of an
average house in a certain town during a 5-year period.
The price quoted is the value of the house at the end of
each year. Use the annual rates of inflation given in table
below to adjust the prices to those prevailing at the end of
1991. Compare the rise in both the nominal and real values
of house prices during this period.
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Solution 2
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Solution 2
Table above presents the nominal and the ‘constant 1991’
values of the house (rounded to the nearest thousands) for
comparison. As can be seen from the table, apart from the
gain during 1991, the increase in value has, in fact, been quite
modest.
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Practice 2 (for students)
Table 3.10 shows the average annual salary (in
thousands of dollars) of employees in a small firm,
together with the annual rate of inflation for that
year. Adjust these salaries to the prices prevailing
at the end of 1991 and so give the real values of
the employees’ salaries at constant ‘1991 prices’.
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Exercises
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10.
Table 3.10 gives the annual rate of inflation during
a 5-year period.
Table 3.10
2000 2001 2002 2003 2004
Annual rate of
inflation 1.8% 2.1% 2.9% 2.4% 2.7%
If a nominal house price at the end of 2000 was
$10.8 million, find the real house price adjusted to
prices prevailing at the end of the year 2003.
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11.
Table 3.11 shows the index numbers associated with transport
costs during a 20-year period. The public transport costs reflect
changes to bus and train fares, whereas private transport costs
include purchase, service, petrol, tax and insurance costs of cars.
Table 3.11
Year
1985 1990 1995 2000 2005
Public transport 100 130 198 224 245
Private transport 100 125 180 199 221
(1) Which year is chosen as the base year?
(2) Find the percentage increases in the cost of public transport from
a) 1985 to 1990 b) 1990 to 1995 c) 1995 to 2000 d) 2000 to 2005
(3) Repeat part (2) for private transport.
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12.
Table 3.12 shows the number of items (in thousands)
produced from a factory production line during the course of
a year. Taking the second quarter as the base quarter,
calculate the associated index numbers.
Table 3.12
Quarter
Q1 Q2 Q3 Q4
Output 13.5 1.4 2.5 10.5
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13.
Table 3.13 shows the prices of a good for each year between 1999
and 2004.
Table 3.13
Year 1999 2000 2001 2002 2003 2004
Price ($) 40 48 44 56 60 71
(a) Work out the index numbers, correct to 1 decimal place, taking
2000 as the base year.
(b) If the index number for 2005 is 135, calculate the corresponding
price. You may assume the base year is still 2000.
(c) If the index number in 2001 is approximately 73, find the year that
is used as the base year.
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14. For students
Table 3.5 shows the index numbers of the output of a
particular firm for the years 2004 and 2005.
Calculate the percentage change in output:
a) from 05Q1 to 05Q3
b) from 04Q2 to 05Q2
c) from 04Q3 to 05Q1
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15.
The prices of a good at the end of each year between 2003 and
2008 are listed in the table 3.16, which also shows the annual rate
of inflation.
Table 3.16
Year 2003 2004 2005 2006 2007 2008
Price 230 242 251 257 270 284
Inflation 4% 3% 2,5% 2% 2%
Find the values of the prices adjusted to the end of year 2004,
correct to 2 decimal places. Hence, calculate the index numbers of
the real data with 2004 as the base year. Give your answers correct
to 1 decimal place.
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Thank you for
attention!
