index 1
TRANSCRIPT
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IBA-JU Course Instructor: Dr Swapan Kumar Dhar
Index Numbers An index number is a statistical measure of relative changes in a variable or a group of related variables with respect to time, place or other characteristics. It is a pure number and expressed in percentage. It is used to compare the level of price, quantity or value of a group related articles at different times or at different geographical locations at the same time. Example:
Prices of Food Items
Commodity Unit Price per unit (Taka)
2001 2010
Rice Kg 25.00 48.00
Wheat Kg 15.00 25.00
Fish Kg 200.00 300.00
Bread Pound 12.00 24.00
Milk Liter 24.00 40.00
If I ask someone “ How does the overall food price in 2010 compare with that in 2001 – how many times or what percent? A number that provides an answer to this question will be called an “ index number”. Simple Index Numbers If an index number is used to measure the relative change in just one variable – such as hourly wages in manufacturing we refer to it as a simple index number. Example: The average hourly wage in manufacturing in 1980 was $ 7.27. In March 1994 it was $ 12.01. What is the index of hourly earnings in manufacturing for March 1994 based on 1980? Solution:
Hourly earnings in March,1994Index 100
Hourly earnings in 1980
$12.01100 165.2
$7.27 .
Thus hourly earnings in manufacturing in 1994 compared with 1980 were 165.2%, that is, they increased 65.2% during that period. Example: It is reported that the farm population dropped from 30,529,000 in 1930 to an estimated 5,100,000 in 1994. What is the index of farm population for 1994 based on 1930? Solution:
1994 PopulationIndex = ×100
1930 Population
5,100, 000100
30, 529, 000 16.7 .
This indicates that the farm population in 1994 compared with 1930 was 16.7% or it decreased 83.3% during that period found by (100-16.7). Types of Index Numbers An index can be classified as a price index, a quantity index, a value index, a special purpose index, cost of living index.
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Methods of Construction of Index Numbers The following chart gives the various methods of constructing Index numbers. Aggregative method Relative method Simple Average Weighted Average of Relatives of Relatives A.M. G.M. A.M. G.M. method method method method Simple Aggregative Weighted Aggregative method method Laspeyres’ Passche’s Edgeworth- Fishder’s Bowley’s formula formula Marshall’s “Ideal” formula formula formula Example: [Simple Aggregative method] Find Price Index number by the method of Aggregative from the following data:
Commodity Base Price Current Price
Rice 36 42
Wheat 30 35
Fish 42 38
Pulse 25 33
Solution: Let 0p and np denote the Base Price and Current Price.
Table: Calculations for Price Index Number
Commodity Base Price
)( 0p
Current Price
)( np
Rice 36 42
Wheat 30 35
Fish 42 38
Pulse 25 33
Total 0p =133 n
p =148
Using Simple Aggregative method, we have
Price Index Number n
0n
0
p 148(P )= ×100= ×100
p 133
=111.28 .
INDEX NUMBERS
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Example: [Weighted Aggregative method] Find by the weighted aggregative method, the Price Index number from the following data:
Commodity Base Price
(1961) Current Price
(1965) Weight
A 32 50 8
B 25 25 6
C 120 140 3
D 35 40 5
E 90 100 7
Solution: Let npp ,0 and w denote the Base Price, Current Price and weight respectively.
Table: Calculations of Price Index Numbers
Commodity Base Price
)( 0p
Current Price
)( np
Weight
)(w wp0 wpn
A 32 50 8 256 400
B 25 25 6 150 150
C 120 140 3 360 420
D 35 40 5 175 200
E 90 100 7 630 700
Total - - - 1571
0p w
1870
np w
Using weighted aggregative method, we have
Price Index Number n
0n
0
p w 1870(P )= ×100= ×100
p w 1571
=119.03 .
Example: Calculate the price index number from the following data using weighted aggregative formula.
Commodity Unit Weight Price per unit
Base period Current period
A Quintal 14 90 120
B Kilogram 20 10 17
C Dozen 35 40 60
D Litre 15 50 95
Solution: Let nppw ,, 0 denote the Weight, Base Price and Current Price respectively.
Table: Calculations of Price Index Number
Commodity Base Price
)( 0p
Current Price
)( np
Weight
)(w wp0 wpn
A 90 120 14 1260 1680
B 10 17 20 200 340
C 40 60 35 1400 2100
D 50 95 15 750 1425
Total - - - 3610
wp0
5545
wpn
Weighted Aggregative Price Index Number n
0
p w 5545= ×100= ×100=153.6
p w 3610
.
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Example: Calculate the Price Index Number from the following data, by using Laspeyres’ formula.
Commodity 1990 1995
Price (Taka) Price (Taka) Quantity (Kg)
Rice 300 10 450
Oil 400 6 500
Wheat 250 4 400
Sugar 200 2 350
Fish 125 2 200
Solution: Let us take 1990 as Base year and 1995 as Current year. Let npp ,0 be the prices at
base and current year and 0q be quantity at base year.
Table: Calculations of Laspeyres’ Price Index Number
Commodity 0p np 0q 00qp 0qpn
Rice 300 450 10 3000 4500
Oil 400 500 6 2400 3000
Wheat 250 400 4 1000 1600
Sugar 200 350 2 400 700
Fish 125 200 2 250 400
Total - - - 7050 10200
In Laspeyres’s formula, base year quantity )( 0q is taken as weight.
Laspeyres’ Price Index Number 0 0
0 0
p q 10200= ×100= ×100=144.68
p q 7050
.
Example: Calculate the price index number by (a) Paasche’s method, (b) Laspeyre’s method and hence find Fisher’s ideal price index number and Bowley’s price index number from the following data:
Commodity 1980 1990
Price Quantity Price Quantity
A 8 5 12 6
B 12 4 16 5
C 10 6 12 8
Solution: Let 00 , qp be the price and quantity at the base year and np and nq be the price and
quantity at the current year. Table: Calculations of Price Index Number
Commodity 0p 0q np nq 00qp 0qpn nqp0 nnqp
A 8 5 12 6 40 60 48 72
B 12 4 16 5 48 64 60 80
C 10 6 12 8 60 72 80 96
Total - - - - 148 196 188 248
(a) Paasche’s Price Index Number n n
0 n
p q 248= ×100= ×100=131.91
p q 188
(b) Laspeyres’ Price Index Number n 0
0 0
p q 196= ×100= ×100=132.43
p q 148
(c) Fisher’s Ideal Price Index Number Laspeyres' Index No.×Paasche's Index No.
131.91×132.43= 17468.84=132.17
(d) Bowley’s Price Index Number 1
=2
(Laspeyres’ Index No. + Paasche’s Index No.)
1
= (131.91+132.43)=132.172
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Example: Find the Fisher’s Ideal Index Number for the year 1990 with 1980 as base year:
Commodities Unit 1980 1990
Quantity Price (Taka) Quantity Price (Taka)
A Kilogram 6 4 8 6
B Quintal 7 3 10 5
C Dozen 5 9 6 7
D Kilogram 3 2 5 3
Solution: Let npp ,0 be the prices and nqq ,0 be the quantities for the base year and current year
respectively. Table: Calculations for Price Index Number
Commodity 0p 0q np nq 00qp 0qpn nqp0 nnqp
A 4 6 6 8 24 36 32 48
B 3 7 5 10 21 35 30 50
C 9 5 7 6 45 35 54 42
D 2 3 3 5 6 9 10 15
Total - - - - 96 115 126 155
Fisher’s Ideal Price Index Number
n 0 n n
0 0 0 n
p q p q 115 155= × ×100 = × ×100
p q p q 96 126
17825= ×100
12096
=121.62.
Example: Calculate the Price index number for the year 1991 with 1981 as base using Laspeyres’ or Paasche’s formula, whichever will be applicable, on the basis of following data:
Commodity Price (in Taka) Money value (‘000 Taka)
1991 1981 1991
A 22 30 240
B 16 18 72
C 20 25 150
D 8 12 36
(Here money value means total value of a commodity)
Solution: Let 0p and np be the prices at base year and current year and nq be the quantity at the
current year. Since the values at the current year are given and value = price quantity, we have
Value ,)( nnqpv i.e.,
n
np
vq .
Hence, we find the price index number by using Paasche’s formula, since the quantity at the current year is available.
[Note: Here Laspeyres’ formula is not applicable since 0q is not available].
Table: Calculations for Paasche’s Price Index Number
Commodity 0p np v )( 0qpn nq
np
v nqp0 nnqp
A 22 30 240 8 176 240
B 16 18 72 4 64 72
C 20 25 150 6 120 150
D 8 12 36 3 24 36
Total - - - - 384 498
Paasche’s Price Index Number
n n
0 n
p q 498= ×100= ×100=129.69
p q 384
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Example: [Simple Price Relative method] Find Prince Index Number by the method of relatives (a) using arithmetic mean, and (b) using geometric mean from the following:
Commodity Base Price Current Price
Wheat 5 7
Milk 8 10
Fish 25 32
Sugar 6 12
Solution: Let npp ,0 denote the Base Price and Current Price.
Table: Calculations for Price Index Number
Commodity 0p np Price Relative n
0
p×100
p=
Wheat 5 7 7
×100=1405
Milk 8 10 10
×100=1258
Fish 25 32 32
×100=12825
Sugar 6 12 12
×100=2006
Total 44 61 (Price Relatives)=593
(a) Simple A.M. of Price Relative Index Number(Price Relatives) 593
= = =148.25k 4
.
(b) Simple G.M. of Price Relative Index Number
4k (Product of Price Relatives) 140 125 128 200= = 145.50
Example: (Weighted Price Relative method) Find, by the method of relatives (using arithmetic mean), the Index number from the following data:
Commodity Weight Current Price (1995) Base Price (1991)
Cloth 12 90 60
Wheat 18 250 180
Rice 30 400 320
Potato 6 180 120
Solution: Table: Calculations for Price Index Number
Commodity Base Price
0p
Current price
np Weight
w Price Relative n
0
pI= ×100
p Iw
Cloth 60 90 12 90
×100=15060
1800
Wheat 180 250 18 250
×100=138.9180
2500.2
Rice 320 400 30 400
×100=125320
3750
Potato 120 180 6 180
×100=150120
900
Total - - 66 - 8950.2
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Weighted A.M. of Price Relative Index Number
Iw 8950.2= = =135.61
w 66.
Example : Find by the arithmetic mean method, (a) simple and (b) weighted price index numbers, from the following data:
Commodity Base price Current Price Weight
Rice 25 31 10
Wheat 20 27 7
Fish 100 115 4
Coal 2.5 3 3
Solution: Let npp ,0 denote base price and current price and w be the weight.
Table: Calculations for Price Index Number
Commodity op np w n
0
pI= ×100
p Iw
Rice 25 31 10 31
×100=12425
1240
Wheat 20 27 7 27
×100=13520
945
Fish 100 115 4 115
×100=115100
460
Coal 2.5 3 3 3
×100=1202.5
360
Total - - 24 494 3005
(a) Simple arithmetic mean of Price Relatives Index number Price Relative 494
= = =123.5k 4
.
(b) Weighted Arithmetic Mean of Price Relatives Index Number
Iw 3005= = =125.21.
w 24
(i) Fisher’s Ideal Quantity Index Numbern 0 n n
0 0 0 n
q p q p= × ×100
q p q p
248 756× ×100
320 980 =77
Cost of Living Index Numbers: Cost of Living Index Number C.L.I) are those index numbers which are designed to measure the relative change in the cost of living in various sections or classes of the society for maintaining same standard of living in two different time periods due to the rise or fall of price of consumer goods during these two periods. Cost of living index numbers are also called Consumer Price Index numbers because they help in determining the change in the retail prices of commodities consumed by the people. Construction of cost of living index number The following steps are necessary in constructing a Cost of Living Index Number of a specific class in a specific locality. 1. The first step is to decide the specific class of people for whom the index number is intended. The class selected should consist of homogeneous group of people with respect to income. 2. The second step is to conduct a `family enquiry’, i.e. to collect information regarding the nature, quality and amount of goods consumed by an `average family’ of that class. This helps to select the items to be included in the construction of index number and also to fix the weight of each item. 3. The commodities are then classified into five major groups (a) Food, (b) Clothing, (c) Fuel and Lighting, (d) House rent and (e) Miscellaneous. Each of these groups are given subdivided into smaller groups, termed as subgroups. For example, the `Food group’ consists of several subgroups, viz. cereals (wheat, rice, etc.); pulses, meat, fish and poultry, milk and milk products; fats and oils, fruits and vegetable, etc. 4. The next step is to collect the retail prices of the commodities from the standard shops and cooperative stores in local market. Price quotations should be taken at least once a week. If there is a
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different price quotation for one commodity, the price relative of each commodity is then computed by taking simple average of price relatives of different price quotations. 5. Cost of Living Index Numbers are generally constructed for each week. The average of the weekly index numbers is taken as the index number for a month. The average of monthly index numbers gives the cost of living index number for the whole year. Method of construction 1. First, we determine the price index number of each group using the method of weighted arithmetic mean of the price relatives of the commodities included in that group.
Group Index Number (Price Relative)×w
(I)=w
=
n
0
p×100 ×w
p=
w
.
Weight of each item in the group is calculated as the percentage expenditure on that item in relation to the total expenditure of that group.
0 0
0 0
p qw= ×100
p q and obviously w=100.
2. Finally, we determine the weighted A.M. of the five group index numbers with weight as the percentage expenditure on a group in relation to the total expenditure by an average family.
Cost of Living Index Number (C.L.I)
IW=
W
where W is the weight of a group and W=100 as the weight of each group is expressed in
percentage. Applications and uses 1. Cost of Living Index Numbers are used to calculate the dearness allowance (D.A) of workers and employees in order to maintain the same standard of living as in the base year. 2. Purchasing power of money can be measured with the help of Cost of Living Index Numbers since,
Purchasing power of money 1
Cost of Living Index Number .
3. Real wage may be measured using Cost of Living Index Number since
Real wage Actual wage
×100Cost of Living Index Number
.
Similarly, Real income Actual income
×100Cost of Living Index Number
.
Example: Find the general cost of living index of 1983 from the following table:
Class Food Dress House rent Fuel Miscellaneou
s
Class Index 620 575 325 255 280
Weight 30 20 25 15 10
Solution: Table: Calculations for Cost of Living Index
Class Class Index (I) Weight (W) IW
Food 620 30 18600
Dress 575 20 11500
House rent 325 25 8125
Fuel 255 15 3825
Miscellaneous 280 10 2800
Total - W=100 IW=44850
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Cost of Living Index
IW 44850= =448.5
W 100
Example: With the base year 1960 as the base, the C.L.I. in 1972 stood at 250. X was getting a
monthly salary of Taka 500 in 1960 and Taka 750 in 1972. How much should X have received as extra allowance in 1972 to maintain his standard of living as in 1960? Solution: The cost of living index number in 1972 was 250 with respect to the base 1960.
The price in 1972 was Taka 250 when the price in 1960, was Taka 100. Hence, to maintain the same standard of living as in 1960, his salary should be 250 in 1972 if he was getting Taka 100 in 1960. Salary
1960 1972
100 250
500 250×500=1250
100
So, his salary should be Taka 1250 in 1972 and therefore, the amount of extra allowance = Taka 1250 – Taka 750 = Taka 500. Example: The following data show the cost of living indices for different groups with their respective weights, for middle class people of Dhaka in 1957. Obtain the general cost of living index number.
Mr. X was getting as salary Taka 250 in 1939 and Taka 429 in 1957. State how much he ought to have received as extra allowance in 1957 to maintain his standard of living at 1939 level. Base 1939=100
Group Group Index Group Weight
Food 411.8 61.4
Clothing 544.8 4.5
Fuel and Light 388.0 6.5
House rent 116.9 8.9
Miscellaneous 284.5 18.7
Solution: Table: Calculations for Cost of Living Index Number
Group Group Index (I) Group Weight (W) IW
Food 411.8 61.4 25284.5
Clothing 544.8 4.5 2451.6
Fuel and Light 388.0 6.5 2522.0
House rent 116.9 8.9 1040.4
Miscellaneous 284.5 18.7 5320.1
Total - 100 = W 36618.6 IW
Cost of Living Index
IW 36618.6= =366.2
W 100.
The price in 1975 was Taka 366.2 with respect to the base 1939 when the price was Taka 100. Salary
1939 1957
100 366.2
250 366.2×250=915.50
100
So, to maintain the same standard of living as in 1939, his salary should be Taka 915.50 in 1957. Hence he ought to have received as an extra allowance in 1957 an amount (Taka 915.50 – Taka 429) = Taka 486.50.
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Example: An enquiry into the budgets of middle class families in a certain city gave the following information.
Expenses on Food 35%
Fuel 10%
Clothing 20%
Rent 15%
Miscellaneous 20%
Prices (1983) in Taka
150 25 75 30 40
Prices (1984) in Taka
145 23 65 30 45
What is the cost of living index number of 1984 as compared that of 1983? Solution: The percentage expense on each group is taken as the weight of that group. Table: Calculations for Cost of Living Index Number
Group Weight
(W)
Price Price Relative
0
1
pI= ×100
p IW 1983
0(p )
1984
1(p )
Food 35 150 145 96.67 3383.45
Fuel 10 25 23 92.00 920.00
Clothing 20 75 65 86.67 1733.40
Rent 15 30 30 100.00 1500.00
Miscellaneous 20 40 45 112.50 2250.00
Total 100 W - - - 9786.85
IW
Cost of Living Index Number for 1984 with base 1983 IW 9786.85
= =97.87W 100
Example 21: The table below shows the average wages in Taka for a day of a group of industrial workers during the year 1976 to 1987. The consumer price indices for these years with 1976 as base are also shown:
Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987
Average Wage
11.19 11.33 11.44 11.57 11.75 11.84 11.89 11.94 11.97 12.13 12.28 12.45
CPI(1976=100)
100 107.6 106.6 107.6 116.2 118.8 119.8 120.2 119.9 121.7 125.9 129.3
(a) Determine the real wages of the workers during the years 1976 – 87 as compared with their wages in 1976. (b) Determine the purchasing power of the Taka for the year 1976. What is the significance of this result? Solution: (a) For finding the real wages we have to divide average wage of workers by the consumer price index.
Year Average Wage CPI Real Wage
1976 11.19 100.0 (11.19/100)%=11.19
1977 11.33 107.6 10.53
1978 11.44 106.6 10.73
1979 11.57 107.6 10.75
1980 11.75 116.2 10.11
1981 11.84 118.8 9.97
1982 11.89 119.8 9.92
1983 11.94 120.2 9.93
1984 11.97 119.9 9.98
1985 12.13 121.7 9.97
1986 12.28 125.9 9.75
1987 12.45 129.3 9.63
(b) If we divide Taka 1 by the price index of 1987, we get the purchasing power of Taka in 1976. Thus the purchasing power of Taka in 1987 shall be 100/129.3 = 0.77. This means that the purchasing of money has gone down. In 1987 the Taka could buy only 77% of what it could buy in 1976.