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Index Number Notation for Price and Quantity

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Lecture 6

This lecture note was originally written by a student and partly modified and certified by the course instructor.

INDICES

An index is a number that compares the value of a variable at any given point in time with its value at a fixed reference point.

The fixed reference point is usually called “Base period” and the associated value is called the “Base value”.

In business, an index measures percentage (%) change in the value of some commodity over a period of time.

Simple Index Number Construction

Suppose the price of a 330ml can of coke was GH¢2.0 in January and rose to GH¢2.2 in March, then;

% increase = = 10%

The price of 330ml can of coke increased (↑) by 10% from January to March.

In Index Number Form, the 10% increase is added to the base of 100 giving 110. So, the price index of 330ml can of coke in March was 110.

NB: Any increase or decrease must be related to some time period, otherwise, it is meaningless.

Index Number Notation for Price and Quantity

Most index numbers in business relates to either price or quantity. We will use the following notations for price and quantity.

P0 ≡ Price at the base time point

Pn ≡ Price at some other time point

Q0 ≡ Quantity at base time point

Qn ≡ Quantity at some other time point

Ip, Iq are the price index and quantity index respectively.

Ip for any time period n = ;

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Example:

The price of a quantity of a product A from 2010 to 2014 is as shown in the table below. Using the year 2010 as the base year, calculate the price index from 2010 to 2014.

Year 2010 2011 2012 2013 2014

Price 50 60 55 65 70

Solution

Index

An Index Relative

An Index Relative (sometimes called “relative”) is the name given to an index number which measures the change in a single distinct commodity.

Example:

Item 2013 2014

Price (P0) No Sold (Q0) Price (P0) No Sold (Q0)

Video recorder GH¢500 40 GH¢542 28

30 inch television GH¢410 25 GH¢465 42

Solution

For Video Recorder; Price Index (Ip)

Quantity Index (Iq)

For 30 inch television; Price Index (Ip)

Quantity Index (Iq)

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Comparing Values at Different Times

Index at period n

Index at period m

Then,

E.g.

Year 1 2 3 4 5

Value 200 209.5 300 228.6 250

index 105 110 157.5 120 131.5

Time Series for Relative Indices

The base period for indices over many time periods could be based on a non-changing base period or the so-called chain based approach where the base period is changed every period using the immediate previous year’s value.

Example

Month Jan Feb(base) March April May June

Production 4563 4245 4841 4644 5290 5156

Fixed Base Relative Index

100% 114% 109% 124.6% 121.5%

Chain Base Relative Index

113.9

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Changing Base Period

Sometimes it becomes imperative to change a base period to a more recent one. The following are some of the reasons for changing a base period:

1. Changing circumstances: an index should be reset whenever there is a significant change that makes comparison with earlier periods meaningless.

2. An index becomes too large - say an index rises to 5000 then a 10% increase raises it by 500 points, which is a huge jump than a jump of 10% from 100 to 110.

However, note that changing the base period introduces discontinuity that makes comparison over periods more difficult.

Percentage Change Period

The difference between two percentages is termed as percentage point. For example, if inflation in July 2013 was 10% and inflation in July 2013 was 9%, we can say there was a change of 1% percentage point in inflation.

Percentage point however is different from percentage increase. For example, suppose a presidential candidate was polling at 30% last month and this month is polling at 40%, we cannot say the change in number of potential vote is 10%. Why?

Suppose there are 10,000 eligible voters. A 30% poll translates to 3000 votes. A 40% vote translates to 4000 votes.

In this case the change is (4000-3000)/3000 = (1000/3000)*100% = 33.33%. However, when we want to talk of a change from one percentage to another, then we have (40-10) = 10 percentage point.

Example

Period 1 2 3 4 5

Index 100 120 142 108 135

% change point = index at point t1 – index at point t0

At period 2, there is a 20 point increase over the index at period 1.

Changing the Base of a Fixed Base Relative

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Example 1

The example below starts with a base period of 1975. In 2004, it was observed that a lot has changed and the 1975 base period is no longer relevant for comparison. Row 3 in the table below shows the new indices if the year 2004 is chosen as the new base period.

Year 2004 2005 2006 2007 2008 2009 2010

Old index

(1975=100) 324 351 377 384 391 404 428

New index (2004=100)

Example 2

The following indices monitor the annual profits of Tigo Company Ghana;

Year 1 2 3 4 5 6 7 8

Index 1 100 140 168 199 230

Index 2 100 125 142 160

a. What are the base years for the indices?

b. If the company had not changed to index 2, what values would index 1 have in years 6 to 8?

c. What values does index 2 have in years 1 to 4?

d. If the company made a profit of GH¢5M in year 3, how much did it make in other years?

Solution

a. Base year for Index 1 is year 1 Base year for Index 2 is year 5 b. From example 1 above,

Therefore

Year 1 2 3 4 5 6 7 8

Index 1 100 140 168 199 230

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Similary,

c.

d. Let profit in year be . We know that

Then

…..

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Indices for More Than One Variable

Sometimes an entity (eg. Company) might be involved in more than one commodity. In such a case, an interest could center on composite index to represent the company. There are two ways of generating a composite indices;

1. Mean (Average) Relative Index 2. Simple Aggregate Relative Index

Mean Relative Index at a period n

Simple Aggregate Relative Index at a period n

Illustration

Year 2013 2014 2015 Fixed indices

Item Price Price Price 2014 2015

Coffee

55p 62p 64p

112.7

=116.6

Tea

28p 32p 30p

114.3

107.1

Hot

Chocolate 72p 74p 50p

102.8

Relative Mean Index

Simple Aggregate Relative Index

92.9

References

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