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Index Numbers

1. Use of Index Numbers

Uses:

1. To show how an economic variable (e.g. price) is changingover time.

2. To make comparisons. This is achieved by taking a typicalyear’s figure as base and epressing the figures for otheryears as a ratio or percentage of the base year figure.

Advantages:

1. !acilitates comparisons.

2. "nables analysis of percentage or relative changes ratherthan absolute changes.

Problems:

1. There are many types of inde number. #hich to usedepends on circumstances.

2. $oss of information. %ne figure represents a mass of data.

Note: As index numbers are used for comparisons it isconventional to work to 1 decimal place only.

2. Unweighted Index Numbers

2.1 Simple Index Number

&imple price inde 100

0

×= P

P nwhere 'n is the price in year n

nd ' is the price in the base year.

#orked "ample:



The price of a Tyson 1 vacuum cleaner for each year from1**+,2 is given below. -onstruct a simple inde using 1! as the base year.

Year Price1**+ 11**/ 1+1**0 1*+1** 1**1*** 202 21

&olution:

Year Price Index

1995 180 100195

180× = 92.3

1996 185 9.94100195

185=×

1997 195 100

1998 199 1.102100195

199=×

1999 207 2.106100195

207=×

2000 210 7.107100

195

210=×

"xamples:

The inde for thebase year isA#$A%S 100



1. The price of +g of a brand of instant coffee for each yearsince 2 is given below. -onstruct a simple inde using 2 as thebase year.

%ear Pri&e Index

2000 2.00 100

2001 2.20

2002 2.40

2003 2.50

2. -onstruct a simple inde for the instant coffee using 22 as thebase year.

%ear Pri&e Index

2000 2.00

2001 2.20

2002 2.40 100

2003 2.50

sually we are concerned with more comprehensive changes than ust oneitem. There is a need to combine price changes of several items to producean aggregate inde.

2.2 Simple Aggregate Index



-onsiders a 3basket4 of items.

&imple aggregate inde 100

0

×=∑∑

P

P n

$or'ed "xample:

The table below shows the cost5 in 6’s5 of running an averagefamily car in 1**5 2 and 22. !ind the simple aggregateinde for 2 and 22 using 1** as the base year.

1** 2 22

7oad ta 1 11 11+8nsurance + ++ +0!uel 9 *+7epairsmaintenance 2 22 29

Σ 1/ 101 1/+

&olution:

;ear 8nde1** 1

20009.1061001600

1710

=×

226.116100

1600

1865=×

"xamples:

1. The table below gives the prices5 in <’s5 of seats at the ston'alladium Theatre.



1998 1999 2000

!ront stalls 2 21 217ear stalls 1/ 10 1

!ront circle 2 2 217ear circle 12 1 1=o 2+ 2 9

Totals

!ind the simple aggregate inde for 1*** and 2 using 1**as the base year.

;ear 8nde

1998

1999

2000



2. The price of a typical "nglish breakfast in 1*5 1** and2 is given below:

1980(! 1990(! "000(!

1 egg .+ .0 .17asher of bacon .9 .9+ .>1 sausage .2 .1+ .21 gm bakedbeans

.+ .+ .+

Totals

!ind the simple aggregate inde for 1* and 1** using2 as the base year.

;ear 8nde

1*

1**

2

?isadvantage:The value of the inde does not depend onthe number of items used or on the @uantities chosen to@uote the price in.



2.( Simple )ean of Pri&e *elatives

&imple mean of price relatives 1001

0

×= ∑ P

P

k

nwhere k is the

number of items inthe basket.

#orked "ample:

P+ P1 0

1

P

P

P2 0

2

P

P

10 15 15/10 = 1.5 20 20/10 = 2

100 120 120/100 = 1.2 150 150/100 = 1.5

5 10 10/5 = 2 10 10/5 = 2

30 40 40/30 = 1.3 60 60/30 = 2

Σ P1/P0 = 6

Σ P2/P0 = 7.5

k A the number of items so in this case k A >.

%ear

0

1=××=×∑ 1006

4

1100

1

0

1

P

P

k

1+

2 =××=×∑ 1005.7

4

1100

1

0

2

P

P

k

10.+



"xample:The table below gives the prices in <’s of seats at the ston 'alladium Theatrefor the 9 years 1**5 1*** and 2. !ind the &imple Bean of 'rice 7elativesinde for 1*** and 2 using 1** as the base year.

1998

1999

2000

P0 P1 P1/P0 P2 P2/P0

!rontstalls

20 21 21

7ear stalls 16 17 18

!rontcircle

20 20 21

7ear circle 12 10 10

=o 25 28 30

Totals (Σ)

k = _______

Year Index

1998

1999

2000

,isadvantages: ssumes all goods have e@ual importance

although we do not spend e@ual amounts on allitems.



3. $eighted Index Numbers

Introdu&tion

,isadvantage of unweighted index numbers: 'rices are @uotedin different units and give no indication of the relative importance of each item.

)ethod of weighting: Bultiply price by weight that will adust theitem’s siCe in proportion to its importance.

Advantages:

1. =oth the importance of the item and the unit in whichthe price is epressed are taken into consideration.

2. #eighted indices are directly comparable.

(.1 #aspe-re Pri&e Index

=ase year weighted inde.

$aspeyre price inde 100

00

0 ×=∑∑

q p

q pn

Worked Example

p0 !0 p0!0 p1 p1!0 p2 p2!0

10 1 10110 =× 15 15115 =× 20 20120 =×

100 2 2002100 =× 120 2402120 =× 150 3002150 =×

5 10 50105 =× 10 1001010 =× 10 1001010 =×

30 5 150530 =× 40 200540 =× 60 300560 =×

Σ p0!0 = 410 Σ p1!0 = 555 Σ p2!0 = 720



;ear 8nde

0

1001

=×=×∑∑

100410

555100

00

10

q p

q p 135.4

2=×=×

∑∑

100410

720100

00

2 0

q p

q p

175.6

"ample:

The table below gives the prices in <’s of seats at the ston'alladium Theatre for 9 years. !ind the $aspeyre 'rice 8nde for1*** and 2 using 1** as the base year.

1** 1*** 2

p @ p@ p1 p1@ p2 p2@

!ront&talls

2 12 21 21

7ear &talls

1/ 1+ 10 1

!ront-ircle

2 2 21

7ear -ircle

1 1 12 1>

=o 2+ 2 2 9

Σ Σ Σ

;ear 8nde



1**

1***

2

(.2 Paas&he Pri&e Index

-urrent year weighted inde.

'aasche price inde 100

0

×=∑∑

n

nn

q p

q p

#orked "ample:

p p1 @1 p1@1 p@1 p2 @2 p2@2 p@2

1 1+ 2 30215 =× 20210 =× 2 > 80420 =× 40410 =×

1 12 > 4804120 =× 4004100 =× 1+ / 9006150 =× 6006100 =×

+ 1 + 50510 =× 2555 =× 1 1 1001010 =× 50105 =×

9 > 2 8002040 =× 6002030 =× / 1 6001060 =× 3001030 =×

Σp1@1A 19/ D D Σp@1A 1>+ Σp2@2A 1/ Σp@2 A **



;ear 8nde

1

1 =×1001045

1360 19.1

2 =×100990

1680 1/*.0

"ample:

The table below gives the prices in <’s of seats at the ston

'alladium Theatre for 9 years. !ind the 'aasche 'rice 8nde for1*** and 2 using 1** as the base year.

p A price in 1** 5 p1 A price in 1***5 p2 A price in 2@1 A number of seats in 1*** 5 @2 A number of seats in 2

p p1 @1 p1@1 p@1 p2 @2 p2@2 p@2

!ront&talls

2 21 12 21 19

7ear

&talls

1/ 10 1/ 1 1/

!ront-ircle

2 2 * 21 1

7ear-ircle

12 1 * 1

=o 2+ 2 22 9 2>

Σ

;ear 8nde

1**

1***

2



#aspe-re Paas&he

1. -ompares cost of buying 1. -ompares cost of buyingbase year @uantities at current year @uantitiescurrent year prices with base at current year prices withyear prices. base year prices.

2. ssumes that if prices had 2. ssumes that if prices hadrisen would still purchase same risen would have bought same@uantities as in base year. Euantities as in current year.(%verestimates inflation) (nderestimates inflation)

9. "asy to calculate as weights 9. ?ifficult5 epensive and timefied. consuming to keep re,

calculating weights.

>. &ame basket of goods so >. ?ifficult to make comparisonsas different years are directly changes in inde reflect bothcomparable. changes in price and weights.

(.( /uantit- Indi&es.

Beasure changes in @uantity rather than price. !or eample the8nde of 8ndustrial 'roduction.

#aspe-re 0uantit- index 100

00

×=∑∑

pq

pq on

Paas&he 0uantit- index 100

0

×=∑∑

n

nn

pq

pq



"ample 1:

$aspeyre Euantity 8nde

p @ p@ @1 p@1 @2 p@2

1 1 1 2

1 2 9 +

+ 1 1+ 1

9 + 1 2

;ear 8nde

1

2

"ample 2: 'aasche Euantity 8nde

!0 p1 !1 p1!1 p1!0 p2 !2 p2!2 p2!0

0 15 2 20 4

2 120 4 150 6

5 10 5 10 10

30 40 20 60 10



;ear 8nde

0

1

2

. Use of Index Numbers

.1 onstru&ting Indi&es

hoi&e of index: depends on the purpose of constructing the inde.

Sele&ting items: must be representative5Bust be unambiguous and their absolute valuesascertainable.

hoi&e of weights:changing weights does not appear to have much effecton an inde.

hoi&e of base -ear: should be 3normal45&hould not be too distant.

.2 ,eflators

-an be used to produce comparisons in real rather than monetary terms. 8.e.eliminates the effects of inflation.

#orked "ample: -alculate the profit in 1**/ terms for the following data.

;ear ctual 'rofits"#000$000%&'

8nde of inflation

'rofit in 1**/ terms"#000$000%&'

1996 12 120 12

1997 15 1254.14

125

12015 =×

1998 20 1401.17

140

12020 =×



"ample: -alculate the profit in "001 terms for the following data:.;ear ctual 'rofits

(<5’s)8ndeof 8nflation

'rofit in 21terms (<5’s)

1999 44 115

2000 46 120

2001 50 129

2002 54 131

.( #in'ing Series 3ogether

The base year needs to be updated every few years as:• The inde becomes out of date

• The weights become out of date.

Year 1980=100 1985=100 1987=100

1983 103.3

1984 106.7

1985 110.7 100

1986 102.1

1987 105.8

-ompleted table:

Year 1980 = 100 1985 = 100 1987 = 100

1983 103.3.93

7.110

1003.103 =× 2.88

8.105

1003.93 =×

1984 106.7.96

7.110

1007.106 =× 1.91

8.105

1004.96 =×

1985 110.7 1005.94

8.105

100100 =×

1986.113

100

7.1101.102 =×

102.15.96

8.105

1001.102 =×

1987.117

100

7.1108.105 =×

105.8 100

"ample:



-omplete the following table

Year 1990 = 100 1996 = 100 2000 = 100

1994 104.4

1995 107.8

1996 111.8 100

1997 103.2

1998 106.9

1999 108.8

2000 110



"nd of 3opi& 3est.

1. The base year has an inde of

2. simple inde is where item(s) isare monitored.

9. The $aspeyre inde is an eample of a

weighted inde.

>. The 'aasche inde is an eample of a

weighted inde.

+. The weights used in the 7'8 are derived from the

ependiture survey.

/. The 7'8 is updated each

7. The $aspeyre inde is easier to calculate than the 'aasche inde.

T7"!$&".

. ;ou cannot directly compare years with the 'aasche inde.

T7"!$&".

*. 8t is not possible to have an inde below 1. T7"!$&".

1. The 7'8 in 1** was 11+.2 and by 1**2 it was 19.+. This representsa rise in the price of goods by:

(a) 1+F (b) 2.2F (c) 29.9F (d) <11.

11. The turnover by a company in 1**0 was <+>.+m and in 1*** it was

<0+m. 8f the 7'8 has increased from 11.* in 1**0 to 11+.2 by 1***5the real change in turnover has been:

(a) <2.+m (b) <1/m (c) <11.m

12. 8f the price inde of bananas was 1 in 1**9 and that for apples was115 then the price of a kg of bananas was:

(a) $ess than for apples(b) Bore than for apples

(c) 8mpossible to compare Uploaded By Iftikhar Changazi

indexnumbers-120308133801-phpapp02

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