Chapter 6 (index) student

Chapter 6
Index numbers


Index Numbers


Index numbers allow relative comparisons
over time



It measures the percentage change in the
value of some economic commodity over time



Index numbers are reported relative to a Base
Period Index



Base period index = 100 by definition


Notes






‘Economic commodity’: used to describe
anything measurable which has some economic
relevance.
‘Economic commodity’ can be: price, quantity,
wage, productivity….
Index numbers must always be related to some
time period (i.e. base time period and another
time period)


Examples


If the index number values are:
130
95
250


Index Relatives


Index relative or simple index number is
an index number which measures the
change in a single distinct commodity


Index Relatives


Formula:

yt
I y  �100
y0
where
Iy = index number of commodity ‘y’
yt = value of commodity ‘y’ at time t
y0 = value of commodity ‘y’ in the base period


Index Numbers: Example


Company orders from 1995 to 2003:

Year

Number
of Orders

1995

272

1996

288

1997

295

1998

311

1999

322

2000

320

2001

348

2002

366

2003

384

Index

(base
year =
2000)


Index Numbers: Interpretation


Price relatives


Formula

pt
I p  �100
p0
Where:

pt: price at time t
p0: price in the base period

I p (2003/ 2000)

The
price
index
for
2003,
 120
based on 2000(as 100), is 120

Quantity relatives


Formula

qt
I q  �100
q0
Where:

qt: quantity at time t
q0: quantity in the base period

I q (2003/2000)  150

The quantity index for 2003,
based on 2000 (as 100), is 150


Example
Calculate price index relatives and quantity index
relatives

Item Uni
s
t

Price (USD)

Quantity

(p0)

(pt)

(q0)

(qt)

A

kg

3,0

4,5

1000

1100

B

m

5,0

6,0

2000

2400

C

l

2,0

2,2

4000

4200


Time series of relatives



Describe how the values of an index relative
change overtime.
There are two distinct ways in which relatives can
be calculated
- Fixed base relatives: each relative is calculated
based on the same fixed time point
- Chain base relatives: each relative is calculated
with respect to the immediately preceding time

point


Fixed base relatives
Year

2000

2001

2002

2003

2004

2005

Sales
(1000 USD)

10.0

10.2

11.0

11.8

13.0

14.8

Fixed base
relatives
(2000=100)
Fixed base
relative
(2002=100)


Chain base relatives
Year

2000

2001

2002

2003

2004

2005

Sales
(1000 USD)

10.0

10.2

11.0

11.8

13.0

14.8

Chain base
relative


Changing the base of
fixed base relative






Used when the old base time point is no longer
relevant (usually it is too far in the past or out of
date)
Step 1: Choose the required new base time
point and identify the corresponding relative.
Step 2: Divide each relative in the set by the
value of the relative identified above and multiply

the result by 100.


Changing the base of
fixed base relative
Year

2000

2001

2002

2003

2004

2005

Old Index
(1980=100)

244

260

270

285

300

315

New Index
(2000=100)

100

106.56 110.66 116.80

122.95 129.10


Time series deflation


A technique used to obtain index relatives that measure
the changes in the real value of some commodity with
respect to a given indicator
Year

2000

2001

2002

2003

2004

2005

Average daily
wage (USD)

10

15

17

19

22

25

CPI

104.3

106.8

107.8

109.8

110.5

112.6

What is the real wage index for 2005?


Time series deflation: procedure


Step 1: Choose a base for the index of real
values of the series



Step 2: For each time point, find the ratio of the
current value to the base value


Time series deflation: procedure


Step 3: Multiply by the ratio of the base indicator
to the current indicator



Step 4: Multiply the result by 100


Real value Index (RVI)



Formula:

xt I 0
RVI  � �100
x0 I t

Where:

xt :
x0 :
It :
I0 :

The value at time t of a given time series
The value at base time point of a given time series
The indicator at time t
The indicator at base time point


Time series deflation: example
Year

2000

2001

2002

2003

2004

2005

Average
daily wage
(USD)

10

15

17

19

22

25

CPI

104.3

106.8

107.8

109.8

110.5

112.6

Real wage
index


Composite index numbers




A composite index number is an index number
obtained by combining the information from a set
of economic commodities.
It measure the percentage changes of a group of
items (not one item)


Weighting of components






Usually, a composite index can be calculated by

weighting each component.
A weighting factor is an indicator of the
importance of each component in calculating the
composite index
The need for weights (read in the textbook)


Types of composite index number
1.

Weighted average of relatives
Weighting the index relative calculated for
each component

2.

Weighted aggregates
Multiplying each component value by its
corresponding weight and adding these
products to form an aggregate


Weighted average of relatives
Formula:

I AR

wI
�


�w

Where:
w: weighting factor of each component
I: index relative of each component