Index Numbers

INDEX NUMBERS •SPECIAL AVERAGE DESIGNED TO MEASURE THE CHANGE IN A GROUP OF RELATED VARIABLE OVER A PERIOD OF TIME. FEATURES:1] Special type of averages. 2] Measured in percentages 3] Change may be single variable or a group of variables. 4] Used for comparison. 5] Capable of measuring special changes;-cost of living and prices. 1

IMPORTANCE OF INDEX NUMBER :1] IN FRAMING ECO POLICIES-In Industries,its directors would wish to know about supply and wage distribution. Govt has to study price n cost of living index for future planning of country. 2] MEASURING PRICE LEVEL OVER A PERIOD OF TIME. Value of money is inversely proportional to price level. Purchasing power is value of money. 2

3]USEFUL IN DEFLATING:-they r used to adjust the original data for price changes or to adjust the original data for cost of living changes n thus transform nominal wages to real wages.

3

CLASSIFICATION OF INDEX NUMBERS:In eco n business the classifications are: 3. Price. 4. Quantity. 5. Value. 6. Special purpose.

4

PROBLEMS IN CONSTRUCTION OF INDEX NUMBERS • PURPOSE OF INDEX NUMBER-Must be clear. What the index is to measure n why? • SELECTION OF BASE PERIODBASE PERIOD-Period with which other periods r to be compared. Thus, prices in BP r to be compared. Thus, prices in BP r taken as standard prices and are taken as 100. 5

Following two points must be taken into care:2. Should not be too far from the current period.-thus we can calculate number with base for a period of about 10 years, after which it may not serve the purpose we desire due to difference in taste, change in price, growth of population. 3. Should not be an abnormal period:-BP should be free from any type of turmoil or abnormality etc. period, containing wars, famines, booms, depressions,strikes,earthquake should not be taken as base.

6

Two methods of selecting base period :2. Fixed base method:-Here, one period is taken as base with its index as 100.other values are compared with it only to get other indices. Price Relative For Current Year:-Current Year’s Price*100/Base Year’s Price. 2.Chain Base Method:-By taking preceding period as base, we prepare the link relatives. This method removes the shortcomings of 1st method i.e far from the current period ad normal periods. Price Relative For Current Year;-Current Year' Price*100/PrecedingYear’sPrice. 7

3.SELECTION OF ITEMS AND THEIR NUMBERS-Which terms are to be included to compare in order to get Index number is extremely important. The items to be included must have following properties:• Should be stable in quality, should be graded or standardized . • Should not be variable in character so that it is ensured that we are comparing the same items over a period of time. • Item should be true representatives of habits ,tastes,customs,traditions and needs of people. 8

• Number of items should neither be too large nor too small. • Non tangible items should not be included. 4. CHOICE OF PRICE QUOTATION:Price differ from place to place,from wholesale to retail sale.So great care has to be taken while taking quotations. • Price can be quoted in 2 ways.Eg:Sugar=14/- per Kg or Potatoes=50 gms a Rupee. 9

14/- per Kg IS MONEY PRICE. 1/2 Kg a Rupee IS QUANTITY OR INVERSE PRICE. • All the commodities are sold at differet places at different prices.So we should select a place where price is most stable and hence will be representative. • Whether wholesale or retail sale price:- Retail sale price often differ at different places but wholesale price are not much variable,hence we should include wholesale price. 10

• Should not be affected by the personal biases.So persons must be fully qualified to perform this duty. • How many quotations to be considered:Number of quotations must be fairly large and moreover average of these quotations should be taken.

11

5. CHOICE OF AVERAGES:Which average is to be used for price relatives as we have too many averages such as:A.M=Affected by extreme values,so not suitable. MEDIAN AND MODE=Are positional measures and don’t include all the items for calculations,hence not suitable. GEOMETRIC MEAN=Such average removes all the shortcomings mentioned above and hence considered good for this purpose. 12

6.WEIGHTING SYSTEM:-There are two types of index numbers:2. Weighted Indices. 3. Unweighted Indices. 7.SELECTION OF APPROPIATE FORMULA:The choice of formula would depend not only on the purpose of the indices but also on the data available. 13

Two types of formula:2. Unweighted Indices Formula a. Simple aggregative method. b. Simple Average of relatives method 2. Weighted average of relatives method . a. Weighted average of relatives. b.Weighted aggregative. 14

INDEX NUMBER

UNWEIGHTED OR SIMPLE

SIMPLE AGGREGATE METHOD

SIMPLE AVERAGE OF RELATIVES METHOD

WEIGHTED

WEIGHTED AVERAGE OF RELATIVES

WEIGHTED AGGREGATIVE

15

UNWEIGHTED AGGREGATES INDEX SIMPLE AGGREGATIVE METHOD:-SIMPLEST. STEPS:4. ADD THE CURRENT YEAR PRICES FOR VARIOUS COMMODITIES i.e OBTAIN ∑P1. 5. ADD BASE YEAR PRICES FOR SAME COMMODITIES i.e OBTAIN ∑Po. 6. DIVIDE ∑P1 BY ∑Po AND MULTIPLY BY 100.

16

Po1=∑P1*100/∑Po. PROBLEM:-FROM THE FOLLOWING DATA CALCULATE AN INDEX FOR 2007 TAKING 2006 AS BASE:COMMODIT Y A

PRICE 06(/PER Kg) 36

PRICE 07(/PER Kg) 39

B

40

40

C

40

46

D

46

50 17

SOLN:∑Po=162, ∑P1=175 175*100/162=108.02. IF 2007 IS TAKEN AS BASE YEAR 162*100/175=92.57. INDEX NUMBERS ARE NOT MEANT FOR PRICES ONLY,IT CAN ALSO BE APPLIED TO QUATITIES OR VOLUMES ALSO. 18

MERITS:2. SIMPLEST METHOD TO FOLLOW. DEMERITS:5. ALL COMMODITIES ARE GIVEN EQUAL IMPORTANCE,WHICH IS NOT POSSIBLE. 6. UNITS USED IN PRICE OR QUANTITY CAN EXERT A BIG INFLUENCE ON THE VALUE OF INDEX

19

SIMPLE AVERAGE OF RELATIVE METHOD USING A.M:- P01=∑P*100/N P=P1/P0. P1=PRICE OF INDIVIDUAL COMMODITY IN CURRENT YEAR. Po=PRICE OF INDIVIDUAL COMMODITY IN BASE YEAR. •

20

USING G.M:P01=ANTILOG{∑ LOG(P1*100/P0)/N}. USING MEDIAN:-P01 CAN ALSO BE CALCULATED. PROBLEM:-USING SIMPLE AVERAGE OF RELATIVES METHOD,COMPUTE INDEX NUMBER FOR THE YEAR 1997 USING 1990 PRICES AS BASE. 21

COMMODIT Y I

PRICES IN 1990(Po) 15

PRICES IN 1997(P1) 30

II

18

24

III

16

20

IV

14

21

V

25

35

VI

40

30 22

COMMODIT PRICES IN Y 1990(Po) I

15

PRICES IN 1997(P1) 30

II

18

24

133.33

III

16

20

125

IV

14

21

150

V

25

35

140

VI

40

30

75

N

P=(P1/P0) *100 200

23

∑=823.33

P01=823.33/6=137.32. USING MEDIAN:ARRANGE THE PRICE RELATIVE IN ASCENDING ORDER. P01(MEDIAN)=133.33.

24

COMPUTE QUANTITY INDEX FOR THE FOLLOWING BYSIMPLE AGGREGATIVE METHOD AND AVERAGE OF QUANTITY RELATIVES METHOD BY USING BOTH ARITHMETIC AND GEOMETRIC MEAN.

25

COMMODITY A

B

C

D

E

F

PROD IN1971(Rs)

20

30

10

25

40

50

PROD IN1981(Rs)

25

30

15

35

45

55

26

COMMODIT Y

Q0 1971(Rs)

Q1 1981(Rs)

Q1/Q0*10 0=Q

LOG Q

A

20

25

125

2.0969

B

30

30

100

2

C

10

15

150

2.1761

D

25

35

140

2.1461

E

40

45

112.5

2.0511

F

50

55

110

2.0414

TOTAL

175

205

737.5

12.5116

27

SIMPLE AGGREGATIVE METHOD:Q01 =205*100/175=117.143 ARITHMETIC METHOD:=737.5/6=122.92 G.M=ANTILOG(12.5116/6)=121.7

28

MERITS:• SIMPLE TO CALCULATE. • EXTREME VALUES DON’T INFLUENCE AS MUCH IN 1ST METHOD. DEMERITS:• NO PARTICULAR RULE IS APPLIED,WHEN MEASURE OF CENTRAL TENDENCY IS APPLIED. • NO CONSIDERATION IS GIVEN TO IMPORTANCE OF COMMODITIES. 29

WEIGHTED INDEX NUMBER WEIGHTED AVERAGE OF RELATIVES

WEIGHTED AGGREGATIVE METHOD 30

• WEIGHTED AVERAGE OF RELATIVES METHOD:-HERE WE

FIND THE PRICE RELATIVES FOR THECURRENT PERIOD TAKING THE GIVEN YEAR AS BASE YEAR.LET IT BE DENOTED BY ‘P’.WEIGHT(W) IS OBTAINED BY MULTIPLYING THE BASE YEAR QUANTITY WITH BASE YEAR PRICES.

31

A.M=P01=∑P01=∑PW/∑W. G.M=P01=A.L{∑W*LOGP)/∑W}. COMPUTE PRICE INDEX USING WEIGHTED AVERAGE OF RELATIVE METHOD USING 6. ARITHMETIC MEAN. 7. GEOMETRIC MEAN. 32

BASE YEAR QUANTITY

BASE YEAR PRICE

CURRENT YEAR PRICE

10

20

32

20

12

18

30

8

10

40

4

8 33

Q0

W=P0* P=P1*100 Q0 /P0 P0 P1

10

20 32

200

20

12 18

30

8

40 TOTA L

4

10 8

PW

LOGP

W LOGP

160

32000

2.2041

440.823

240

150

36000

2.1760

522.261

240

125

30000

2.0969

503.258

200

32000 0

2.3010

368.164

160 840

13000 0

1834.5 34

USING A.M:-130000/840=154.76. USING G.M:-A.L(1834.5/840)=152.7.

35

WEIGHTED AGGREGATIVE METHOD:-HERE WE HAVE NOT

ONLY TO FIND PRICE INDEX NUMBER BUT ALSO QUANTITY INDEX NUMBER. P0,Q0=PRICE AND QUANTITY FOR BASE YEAR. P1,Q1=PRICE AND QUANTITY FOR CURRENT YEAR. 36

KELLY FORMULA MARSHALL EDGE WORTH FORMULA

LASPEYRE’S FORMULA

IMPORTANT FORMULAES

FISHER’S FORMULA

PAASCHE’S FORMULA

BOWLEY’S FORMULA

37

O

METHOD

PRICE INDEX NUMBER:P01

QUANTITY INDEX NUMBER:Q01

1

LASPEYRE'S FORMULA

∑(p1*q0/∑p0*q0)*100

∑(p0*q1/∑p0*q0)*100

2

PAASCHE FORMULA

∑(p1*q1/∑p0*q1)*100

∑(p1*q1/∑p1*q0)*100

3

BOWLEY FORMULA

[{∑(p1*q0/∑p0*q0)}+{∑(p1*q1/∑(p0*q1)}]*100

[{∑(p0*q1/∑p0*q0)}+{∑(p1*q1/∑(p1*q0)}]*100

4

FISHER FORMULA

√[{∑(p1*q0/∑p0*q0)}*{∑(p1*q1/∑(p0*q1)}]*100

√[{∑(p0*q1/∑p0*q0)}*{∑(p1*q1/∑(p1*q0)}]*100

5

M-E FORMULA

{∑(p1*(q0+q1/2)/∑p0*(q0+q1/2)}*100

{∑(q1*(p0+p1/2)/∑q0*(p0+p1/2)}*100

6

KELLY'S FORMULA

{∑p1*q/∑p0*q}*100

{∑p*q1/∑p*q0}*100

38

PRICE INDEX NUMBERS ARE CALCULATED ONLY IF NOTHING SPECIAL IS MENTIONED.WE CALCULATE QUANTITY INDEX NUMBERS ONLY IF MENTIONED. PROBLEM:-USING ALL THE ABOVE FORMULA,CALCULATE INDICES FOR 1995 TAKING 1994 AS BASE.

39

COMMODIT 1994, Y PRICE

1994,QTY

1995, 1 PRIC 995, E QTY

A

8

10

10

9

B

10

12

15

12

C

12

8

18

7

D

15

6

16

8

40

SOLUTION:COMMO DITY

p0*q p0*q p1*q p1*q 0 1 0 1

p0

q0

p1

q1

A

8

10

10

9

80

72

100

90

B

10

12

15

12

120

120

180

180

C

12

8

18

7

96

84

144

126

D

15

6

16

8

90 ∑ =38 6

120 ∑ =39 6

96 ∑ =52 0

128 ∑ =52 4

41

LASPEYRE'S FORMULA PAASCHE FORMULA

BOWLEY FORMULA

∑(p1*q0/∑p0*q0)*100=520/386*100=134.7 ∑(p1*q1/∑p0*q1)*100=524/396*100=132.3 [{∑(p1*q0/∑p0*q0)}+{∑(p1*q1/∑(p0*q1)}] *100= (134.7+132.3)/2=133.52

42

FISHER FORMULA

√[{∑(p1*q0/∑p0*q0)}*{∑(p1*q1 /∑(p0*q1)}]*100=√134.7*133.3 =133.5

{∑(p1*(q0+q1/2)/∑p0*(q0+q1/2 M-E FORMULA )}*100=133.50

43

KELLY’S FORMULA IS USED WHERE WEIGHT OF ONLY BASE PERIOD ARE GIVEN.

44

45