1. Definition
According to Morris Hamburg defined, an index number is nothing more than a relative number, which expresses are relationship between two figures, where one of the figure is used as a base.
Uses of Index Number:
They measure the relative change.
They are of better comparison.
They are economic barometers.
They compare the standard of living.
They provide guidelines to policy.
They measure the purchasing power of money.

2. Characteristics & Methods of Index Number
Characteristics: 1). Index numbers are specified averages. 2).Index numbers are expressed in percentage. 3). Index numbers measure changes. 4). capable of direct measurement. 5). Index numbers are for comparison.

3. Various Methods Index Numbers
Simple Aggregate Index Number
Simple Average Price Relative Index
Weighted Aggregate Index Number ; That is
* Laspeyre’s Method
* Paasche’s Method
* Fisher’s Ideal Method
* Bowley’s Method
* Marshall-Edgeworth Method
* Kelly’s Method

4. Types of Index Numbers Price Index:
Compares the prices for a group of commodities at a certain time as at a place with prices of a base period. The wholesale price index reveals the changes into general price level of a country, but the retail price index reveals the changes in the retail price of commodities such as consumption of goods, bank deposits, etc.
Quantity Index:
Is the changes in the volume of goods produced or consumed. They are useful and helpful to study the output in an economy.
Value Index:
Compare the total value of a certain period with total value in the base period. Here total value is equal to the price of commodity multiplied by the quantity consumed.

5. Notations
The following notations would be used through out the Presentation:
P1 = Price of current year
P0 = Price of base year
q1 = Quantity of current year
q0 = Quantity of base year

6. Problems in construction of Index Numbers
Purpose of the index numbers
Selection of base period
Selection of items
Selection of source of data
Collection of data
Selection of average
System of weighting

7. Methods of Index Number
Laspeyres Index Method
Paasche’s Method
Bowley Method
Fisher,s idle Method
Marshall Edworth Method
Kelly’s Method

8. Laspeyres Index Method
This method was devised by Laspeyres in 1871.In this method the weights are determined by the quantities base. Merits 1.Laspeyres index is simpler in calculation and can be computed 2) once the current year prices are known, as the weight are base year quantities in a price index. 3) This enables us an easy comparability of one index with another.

9. Laspeyres Index Method (contin)
∑P1Q0
Laspeyres Index Method = x 100
∑P0Q0
P1 = Prices in the current year
P0 = Prices in the base year
Q1 = Quantities in the current year
Q0 = Quantities in the base year

10. Paasche’s Method
This method was devised by a German statistician Paasche in The weights of current year are used as base year in constructing the Paasche's Index. The Paasche index can be calculated using the following formula.
∑P1Q1
Paasche Price Index = x100
∑P0Q1
Where,
P1 = Prices in the current year
P0 = Prices in the base year
Q1 = Quantities in the current year

11. Fisher,s idle Method
Fisher’s ideal index is the geometric mean of Laspeyre and paasche methods.
P01 F = √ [ P01L x P01P]
Fisher’ s ideal Method: It is known as ideal index number because:
(a) It is based on the geometric mean.
(b) It is based on the current year as well as the base year.
(c) It conform certain tests of consistency.
(d) It is free from bias

12. Illustration 1 - Weighted Aggregate index number
Compute the weighted aggregative price index numbers for  with as base year using (1) Laspeyre’s Index Number (2) Paashe’s Index Number (3) Fisher’s Ideal Index Number.
Commodity
Base year
Price
Current year
Quantities
Wheat 6 10 50
Sugar 2
100
120
Rice 4 60
Milk 12 30 25

13. Solution: Weighted Aggregate index number
Commodity
Price ($)(B.Y)
(P0)
Price
($)(C.Y)
(P1)
Qty
(B.Y)
(Q0)
(C.Y)
(Q1)
P0Q0
P1Q0
P0Q1
P1Q1
Wheat 6 10 50
300
500
Sugar 2
100
120
200
240
Rice 4 60
360
Milk 12 30 25
250
∑P0Q0
=1040
∑P1Q0
=1420
∑P0Q1
=1030
∑P1Q1
=1400

14. Solution ( continu) ∑ P1Q0 1420
Laspeyres Index Method = x x 100 =
∑P0Q
∑P1Q
Paasche Price Index = x x 100 =
P0Q
Fisher’s Ideal Index Method = √ L x P = √ x = √ =

15. Illustration 2 Year 2000 Year 2006 Commodity Price Per Unit
From the data given below , construct index number of prices for 2006with 2000 as base using 1) Laspeyre’s Index 2 ) Paache ‘ s Method and Fisher ideal Index Method.
Year 2000
Year 2006
Commodity
Price Per Unit
Expenditure in Rupees A 2 10 4 16 B 3 12 6 18 C 1 8 14 D 20 32

16. Solution
Since we are given price and total expenditure for the year 2000 and 2006 , we shall first calculate the quantities for two years by dividing the expenditure by price , and then we shall calculate the index numbers is as follows:
Commodity P0 q0 P1 q1
Po qo
poq1
p1q0
p1q1 A 2
10/2=5 4
16/4=4 10 8 20 16 B 3
12/3=4 6
18/6=3 12 9 24 18 C 1
8/1=8
14/2=7 7 14 D
20/4=5
32/8=4 40 32
∑Po qo
= 50
∑poq1
= 40
∑p1q0
= 100
∑p1q1
= 80

17. Continue
∑ P1Q
Laspeyres Index Method = x 100 = x 100 = 200
∑P0Q
∑P1Q
Paasche Price Index = x x 100 = 200
P0Q
Fisher’s Ideal Index Method = √ L x P = √ 200 x = √ = 200

18. Additional Problem
Calculate the Index number of prices and quantity index for the following data, using Laspeyres & Fisher’s Ideal formula.
Commodity
Unit (Per)
Unit Price
Quantity
Year : 2000
Year : 2010 A
Meter 25 35
2500
2000 B Kg 7
10.50
1000
900 C
Hundred 4 5
600 D
Ton
1100
100
120 E
Cubic foot
2.00
2.50
9000
10000

19. Additional Problem in Weighted Aggregate index number

20. University Exam Question:
Problem: Bill Simpson, owner of a California Vineyard , has collected the following information describing the prices and quantities of harvested crops for the years
Construct a Laspeyres Index and Fisher’s Index for each of this 4 years using 1992 as the base period.
Price Per Ton
Quantity Harvested (tons)
1992
1993
1994
1995
Ruby cabernet
$108
$109
$113
$111
1280
1150
1330
1360
Barbera
$93
$96
$101
830
860
850
890
Chenin blanc
$97
$99
$106
$107
1640
1760
1630
1660

21. Answer: 1992-93 Commodity p1q0 p0q0 p1q 0 Ruby cabemet 139520 138240
144640
142080
Barbera
79680
77190
83830
Chenin blanc
159080
173840
175480
∑p1q0
378280
∑p0q0
374510
398160
401390

22. Solution: Help to Find out Fisher’s Ideal Index
Commodity
p1q0
p0q0
p1q1
p0q1
p1q 0
Ruby cabemet
139520
138240
125350
124200
144640
150290
143640
142080
150960
146880
Barbera
79680
77190
82560
79980
81600
79050
83830
89890
82770
Chenin blanc
159080
174240
170720
173840
172780
158110
175480
177620
161020
∑p1q0
378280
∑p0q0
374510
∑p1q1
382150
374900
398160
404670
∑p0q1
380800
401390
418470
390670

23. Continue
year Laspeyres Index : ∑p1q Laspeyres Index Method = x x 100 = ∑p0q year Laspeyres Index : ∑p1q Laspeyres Index Method = x x 100 = ∑p0q year Laspeyres Index : ∑p1q Laspeyres Index Method = x x 100 =

24. Index Number Test Time Reversal Test :
A test used with index numbers that is satisfied when the new index is the reciprocal of the original index if the functions of the base period and given period are interchanged; the advantage of index numbers meeting the criteria of the test is that a symmetric comparison of the two periods is obtained and the results are consistent whether one or the other period is used as a base.
Factor Reversal Test:
A test for index numbers in which an index number of quantity, obtained if symbols for price and quantity are interchanged in an index number of price, is multiplied by the original price index to give an index of changes in total value.

25. Fisher’s Index–Time& Factor Reversal test
Calculate Fisher’s Index number from the following data and prove that it satisfied both the time reversal test and factor reversal test.
Commodity
Price $
2006
Quantity
2007
Food 6 50 10 56
Fuel & Lighting 2
100
120
Clothing 4 60
House Rent 30 12 24
Miscellaneous 8 40 36

26. Solutions in Weighted Aggregate index number
Commodity
Price ($)(B.Y)
(P0)
Price
($)(C.Y)
(P1)
Qty
(B.Y)
(Q0)
(C.Y)
(Q1)
P1Q0
P0Q0
P1Q1
P0Q1
Food 6 10 50 56
500
300
560
336
Fuel 2
100
120
200
240
Clothing 4 60
360
House Rent 12 30 24
288
Miscellaneous 8 40 36
480
320
432
∑p1q0
=1900
∑p0q0
=1360
∑p1q1
= 1880
∑p0q1
= 1344

27. Continue : Time Reversal Test
∑p1q
Laspeyres Index Method = x x =
∑p0q
∑p1q
Paasche Price Index = x x 100 =
p0q
Fisher Ideal Index = √LxP = √ x = √ =
Time Reversal Test: P01 x P10 = 1
∑p1q ∑p1q ∑p0q ∑ p0qo
P01 x P10 = √ x x x
∑p0q ∑ p0q ∑ p1q ∑p1q0
P01 x P10 = √ x x x = √1 =1
Hence time reversal test is satisfied.

28. Continue ; Factor Reversal Test
Factor reversal test is satisfied when ∑p1q1 P01 x q01 = ∑p0q0 ∑p1q0 ∑p1q1 ∑p0q1 ∑ p1q1 P01 x q01 = √ x x x ∑p0q0 ∑ p0q1 ∑ p0q0 ∑p1q ∑ p1q1 P01 x P10 = √ x x x = √ x = = ∑ p0q0 Hence, factor reversal test is satisfied by the given data.

29. Illustration.2
From the data given below Calculate the Price Index by Fisher’s Ideal formula and then verify the Fisher’s Ideal formula satisfies both time reversal test and factor reversal test.
Commodity
Base Year
Current Year
Price (R.s)
Qty (000) Tonnes
Rice 56 71 50 26
Pulses 32
107 30 83
Sugar 41 62 28 48

30. Solution: Calculation of Fisher Ideal Index
Fisher Ideal Index price: ∑p1q0 ∑p1q P01 x P10 = √ x x 100 = √ x x 100 ∑p0q0 ∑ p0q √ x = √ = 84.92
Commodity p0 q0 p1 q1
p0q0
p0q1
p1q0
p1q1 A 56 71 50 26
3976
1456
3550
1300 B 32
107 30 83
3424
2656
3210
2490 C 41 62 28 48
2542
1968
1736
1344
∑ p0q0
9942
∑ p0q1
6080
∑p1q0
8496
∑p1q1 5134

31. Time and Factor Reversal Test
Time Reversal Test: P01 x P10 = 1 ∑p1q0 ∑p1q1 ∑p0q1 ∑ p0qo P01 x P10 = √ x x x ∑p0q0 ∑ p0q1 ∑ p1q1 ∑p1q P01 x P10 = √ x x x = √1 = Hence time reversal test is satisfied. Factor reversal test is satisfied when ∑p1q1 P01 x q01 = ∑p0q0 ∑p1q0 ∑p1q1 ∑p0q1 ∑ p1q1 ∑p1q1 P01 x q01 = √ x x x = ∑p0q0 ∑ p0q1 ∑ p0q0 ∑p1q0 ∑p0q ∑ p1q1 P01 x P10 = √ x x x = √ x = = ∑ p0q0 Hence, factor reversal test is satisfied by the given data.

32. Additional Problem
From the data given below Calculate the Price Index by Fisher’s Ideal formula and then verify the Fisher’s Ideal formula satisfies both time reversal test and factor reversal test.
Commodity
2005
2010
Price (R.s)
Qty (000) Tonnes
Bricks 20 8 40 6
Timber 50 10 60 5
Sand 15
Cement

33. Index Numbers Anna University – Jan -2010
The table below gives the prices of four items- A, B,C, D sold at a store in 2000 and 2006.
Using 2000 as the base year , calculate the price relative index for the four items.
Calculate an unweighted aggregate price index for these items.
Find the Laspeyres weighted aggregate index for these items.
Find the passche index for these items.
Find a weighted aggregate quantity index using 2000 as the base year and price as the weight.
Solution: Calculation of Price Relative Index: =Po1/ n= 148/ 4=37.00
Item
Price $ 2000
Price 2006
Quantity 2000
Quantity2006 A B C D 40 55 95
250 10 25 90
1000
1900
600 50
800
5000
3000
200
Item
Price in (Po)
Price in 2006 (P1)
Price Relative
Po1 =P1/ Po x 100
Log P A B C D 40 55 95
250 10 25 90 45 42 36
1.3979
1.6532
1.6232
1.5797
440
165
148
6.254

34. Continued
ii) Unweighted aggregate price = ∑P1/ ∑Po = 165/ 440 x 100=37.5 iii) Laspeyres weighted aggregate Index L = ∑P1qo/ ∑Poqox100 = / x100 = x 100 = iv) Passche’s Method: ∑P1q1/ ∑Poq1x100 = / x 100 = Weighted aggregate Quantity Index or Fisher Index Method. √L x P = √ 40.2 x 16.3 = 15.9
Item qo Po q1 p1
p1qo
poq0
p1q1
poq1 A B C D
1000
1900
600 50 40 55 95
250
800
5000
3000
200 10 25 90
10000
47500
24000
4500
40000
104500
57000
12500
8000
125000
120000
18000
32000
27500
285000
50000
86000
214000
271000
394500

35. Index Numbers Anna University – June -2012
You are given the following information regarding four items.
Using 2000 as the base year , calculate the price relative index for the four items.
Calculate an unweighted aggregate price index for these items.
Find the Laspeyres weighted aggregate index for these items.
Find the passche index for these items.
Find a weighted aggregate quantity index using 2000 as the base year and price as the weight.
Solution: Calculation of Price Relative Index: =Po1/ n= / 4=204.28
Item
Price $ 2000
Quantity 2000
Price $ 2006
Quantity2006 A B C D
2.00
.80
22.00
7.00 3 20 4 2
5.75
1.25
35.00
15.00 8 10 1
Item
Price in (Po)
Price in 2006 (P1)
Price Relative
Po1 =P1/ Po x 100
Log P A B C D
2.00
.80
22.00
7.00
5.75
1.25
35.00
15.00
287.50
156.25
159.09
214.28
31.80
57.00
817.13

36. Continued
ii) Unweighted aggregate price = ∑P1/ ∑Po x 100=57.00/31.80 x 100= iii) Laspeyres weighted aggregate Index L = ∑P1qo/ ∑Poqox100 = / 124x100 = iv) Passche’s Method: ∑P1q1/ ∑Poq1x100 =178.50/ 97x 100 = v)Weighted aggregate Quality Index or Fisher Index Method. √L x P = √ x =
Item qo Po q1 p1
p1qo
poq0
p1q1
poq1 A B C D 3 20 4 2
2.00
.80
22.00
7.00 8 10 1
5.75
1.25
35.00
15.00
17.25
25.00
140.00
30.00 6 16 88 14
46.00
12.5
105.00 66 7
212.25
124
178.5 97