Transcript
Statistics: Index Number
Conceptualization By: Soumen Roy, B.Com (H), AICWA.
Learning Objectives Acquaintance with Key Terms Introduction to overall concept Solving of basic problems
Conceptualizations By: Soumen Roy
2
Learning Objectives Acquaintance with Key Terms Introduction to overall concept Solving of basic problems
Conceptualizations By: Soumen Roy
2
Key Ke y Ter Terms ms  Sl Slid idee I of II IIII Index Number Price Index * Whole Price Index eta
r ce n ex
Quantity Index Value Index Base Period Current Period Conceptualization By Soumen Roy
3
Key Ke y Ter Terms ms  Sl Slid idee II II of of III III Simple Aggregate Index Number Simple Average Price Relative Index
ei hted A
re
te Index
mber
* Laspeyre’s Method * Paasche’s Method * Fisher’s Ideal Method * Bowley’s Method * MarshallEdgeworth Method * Kelly’s Method 4
Key Terms  Slide III of III Quantity / Volume Index Number Test of Consistency * Unit Test * Time Reversal Test * Factor Reversal Test
Consumer Price Index Number
Conceptualization By Soumen Roy
5
Index Number What is Index Number?….is a statistical measure designed to show changes in variable or a group of related variables with respect to time, geographic location or other characteristic.  For example, if we want to compare the price level of 2009 with what it was in 2008, we shall have to consider a group of variables such as price of wheat, rice, vegetables, cloth, house rent etc., 
We want one figure to indicate the changes of different commodities as a whole. This is called an Index number.

In general, index numbers are used to measure changes over time in magnitude which are not capable of direct measurement. Conceptualization By Soumen Roy
6
Characteristics of Index Number Index numbers are specified averages
Index n mbers re ex ressed in ercent
e
Index numbers measure changes not
capable of direct measurement. Index numbers are for comparison.
Conceptualization By Soumen Roy
7
Uses of Index Numbers They measure the relative change. They are of better comparison.
The
re economic b rometers.
They compare the standard of living. They provide guidelines to policy. They measure the purchasing power of
money. Conceptualization By Soumen Roy
8
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 , , .
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. Conceptualization By Soumen Roy
9
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
Conceptualization By Soumen Roy
10
Problems in construction of Index Numbers Purpose of the index numbers Selection of base period
election of items Selection of source of data Collection of data Selection of average System of weighting Conceptualization By Soumen Roy
11
Method of construction of Index Numbers: Un Weighted
Weighted
Aggregate
Aggregate
Index
Index
Numbers
Number
Simple
Weighted
Average
Average
of Price
of Price
Relative
Relative
Conceptualization By Soumen Roy
12
Simple Aggregate Index Number The price of the different commodities of the current year are added
and the sum is divided by the sum of the prices of those commodities by 100. Symbolically: Simple aggregate price index = P01 =
∑P1 / ∑P0 * 100
simple aggregate method taking prices of 2000 as base. Commodity
Price Per Unit (In Rupees)
Year: 2000
Year: 2004
A
80
95
B
50
60
C
90
100
D
30
45 13
Simple Aggregate Index Number Solution 1: Commodity
Price Per Unit (In Rupees)
Year: 2000 (P0)
Year: 2004 (P1)
A
80
95
B
50
60
C
90
100
D
30
45
Total
250
300
Simple aggregate price index = P01 =
∑P1 / ∑P0 * 100
= 300 / 250 * 100 = 120. 14
Simple Average Price Relative Index First calculate the price relative for the various commodities and then
average of these relative is obtained by using arithmetic mean and geometric mean.
P01 = [∑ P1 / P0 *100] / n, where n is the number of commodities. Simple average of price relative by Geometric Mean:
P01 = Antilog [ ∑ log (P1 / P0 *100)] / n
15
Simple Average Price Relative Index Example 2: From the following data, construct an index for 2004
taking 2000 as base by the average of price relative using (a) arithmetic mean and (b) Geometric mean. Commodity
Price in 2000
Price in 2004
A
50
70
B
40
60
C
80
100
D
20
30
16
Simple Average Price Relative Index Solution: (a) Price relative index number using Arithmetic Mean Commodity
Price in 2000 (P0)
Price in 2004 (P1)
P1 / P0 * 100
A
50
70
140
B
40
60
150
C
80
100
125
D
20
30
150
Total
565
Simple average of price relative index = (P01) = [ ∑ P1 / P0 *100] / n
= 565 / 4 = 141.25 17
Simple Average Price Relative Index Solution: (b) Price relative index number using Geometric Mean Commodi ty
Price in 2000 (P0)
Price in 2004 (P1)
P1 / P0 * 100
log(P1/P0 *100)
A
50
70
140
2.1461
B
40
60
150
2.1761
C
80
100
125
2.0969
D
20
30
150
2.1761
Total
8.5952
Simple average of price relative index = (P01) = Antilog [
∑ log (P1 /
P0 *100)] / n = Antilog 8.5952 / 4 = Antilog [2.1488] = 140.9 18
Weighted Aggregate Index Numbers In order to attribute appropriate importance to each of the items used in
an aggregate index number some reasonable weights must be used. There are various methods of assigning weights and consequently a large number of formulae for constructing index numbers have been devised of which some of the most important ones are: 1. Laspeyre’ s method 2. Paasche’ s method 3. Fisher’ s ideal Method 4. Bowley’ s Method 5. Marshall Edgeworth method 6. Kelly’ s Method 19
Weighted Aggregate Index Numbers Laspeyre’ s method: The Laspeyre’s price index is a weighted
aggregate price index, where the weights are determined by quantities in the base period and is given by: P01 L = [∑P1q0 / ∑P0q0 ] *100 ’ ’ aggregate price index in which the weight are determined by the quantities in the current year. This is given by: P01 P = [∑P1q1 / ∑P0q1 ] *100 Fisher’ s ideal Method: Fisher’ s Price index number is the geometric
mean of the Laspeyres and Paasche indices Symbolically: P01 F = √[ P01L * P01P]
20
Weighted Aggregate Index Numbers 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. t s ree rom
as.
Bowley’ s Method: Bowley’ s price index number is the arithmetic
mean of Laspeyre’ s and Paasche’ s method. Symbolically: P01 B = [P01L + P01P] / 2 Marshall Edgeworth method: This method also both the current
year as well as base year prices and quantities are considered. Symbolically: P01 ME = [ ∑ (q0 + q1) p1 / ∑ (q0 + q1) p0] * 100 21
Weighted Aggregate Index Numbers Kelly’s Method: The following formula is suggested for constructing
the index number. Symbolically: P01 K = [∑P1q / ∑P0q ] *100 , where q = (q0 + q1) / 2 Here the average of the quantities of two years is used as weights. Example 3: Construct price index number from the following data by
app ying (i Laspeyre’s, (ii Paasche’s and (iii Fisher’s Idea Method. Commodity
2000
2001
Price
Qty.
Price
Qty
A
2
8
4
5
B
5
12
6
10
C
4
15
5
12
D
2
18
4
20
22
Weighted Aggregate Index Numbers Solution 3: Commodity
p0
q0
p1
q1
p0q0
p0q1
p1q0
p1q1
A
2
8
4
5
16
10
32
20
C
4
15
5
12
60
48
75
60
D
2
18
4
20
36
40
72
80
172
148
251
220
(i) Laspeyre’ s Price Index = P01 L = [∑P1q0 / ∑P0q0 ] *100
= 251 / 172 * 100 = 145.93
23
Weighted Aggregate Index Numbers (ii) Paasche’ s Price Index = P01 P = [∑P1q1 / ∑P0q1 ] *100
= 220 / 148 * 100 = 148.64 (iii) Fisher’s Ideal Index = P01 F = √[ P01L * P01P]
= =
.
.
√ 21692.49
= 147.28 Interpretation: The results can be interpreted as follows: If 100
rupees were used in the base year to buy the given commodities, we have to use Rs 145.93 in the current year to buy the same amount of the commodities as per the Laspeyre’ s formula. Other values give similar meaning. 24
Weighted Aggregate Index Numbers Example 4: Calculate a suitable price index from the following data Commodity
Quantity
Price
2006
2007
A
20
2
4
B
15
5
6
C
8
3
2
Solution 4: Here the as quantities are given in common we can use
Kelly’ s index price number.
25
Weighted Aggregate Index Numbers Commodity
Q
p0
p1
p0q
p1q
A
20
2
4
40
80
B
15
5
6
75
90
C
8
3
2
24
16
Total
139
186
Now, P01 K = [∑P1q / ∑P0q ] *100 , where q = (q0 + q1) / 2
i.e., P01 K = 186/139*100 = 133.81
26
Weighted Average of Price Relative Index When the specific weights are given for each commodity, the weighted
index number is calculated by the formula:
∑pw / ∑w, where
W= Weight of the commodity P = the price relative index . Note: When the base year value P0q0 is taken as weight, i.e., W= P0q0, then
the above becomes Laspeyre’s formula. When the weights are taken as W=P0q1, then the above becomes
Paasche’s formula
27
Weighted Average of Price Relative Index Example 5: Compute the Weighted Average index number for the
following data : Commodity
Price
Weight
Current Year
Base Year
A
5
4
60
B
3
2
50
C
2
1
30
28
Weighted Average of Price Relative Index Solution 5: Commodity
P1
P0
W
P=P1 / P0 * 100
PW
A
5
4
60
125
7500
B
3
2
50
150
7500
C
2
1
30
200
6000
140
21000
Weighted Average of Price Relative Index = ∑pw / ∑w = 21000 / 140
= 150
Conceptualization By Soumen Roy
29
Quantity / Volume Index Number The quantity index numbers measure the physical volume of
production, employment and etc. The most common type of the quantity index is that of : Laspeyre’ s quantity index number = Q01 L =
∑q1p0 / ∑ q0p0
*100
∑q1p1 / ∑ q0p1 * 100 Q01 F = √ [ Q01 L * Q01 P ]
Paasche’s quantity index number = Q01 P = Fisher’s quantity index number =
These formulae represent the quantity index in which quantities of the
different commodities are weighted by their prices.
Conceptualization By Soumen Roy
30
Quantity / Volume Index Number Example 6: From the following data compute quantity indices by
(i) Laspeyre’ s method, (ii) Paasche’ s method and (iii) Fisher’ s method.
Commodity
Price
Total Value
Price
Total Value
A
10
100
12
180
B
12
240
15
450
C
15
225
17
340
Conceptualization By Soumen Roy
31
Quantity / Volume Index Number Solution 6: Here instead of quantity, total values are given. Hence first
we find the quantities of base year and current year, i.e., Quantity = Total Value / Price. Com.
P0
q0
P1
q1
P0q0
P0q1
P1q0
P1q1
A
10
10
12
15
100
150
120
180
B
12
20
15
30
240
360
300
450
C
15
15
17
20
225
300
255
340
565
810
675
970
(i) Laspeyre’ s quantity index number = Q01 L =
∑q1p0 / ∑ q0p0
*100 = 810 / 565 *100 = 143.36 Conceptualization By Soumen Roy
32
Quantity / Volume Index Number (ii) Paasche’s quantity index number = Q01 P = ∑q1p1 / ∑ q0p1 *
100 = 970 / 675 *100 = 143.70. (iii) Fisher’s quantity index number = Q01 F =
√ [ Q01 L * Q01 P ]
= √ [ 143.36 * 143.70] = 143.53.
Conceptualization By Soumen Roy
33
Test of Consistency of Index Numbers Several formulae have been studied for the construction of index
number. The question arises as to which formula is appropriate to a given problems. A number of tests been developed and the important among these are: (1) Unit test: …. re uires that the formula for constructin an index
should be independent of the units in which prices and quantities are quoted. Except for the simple aggregate index (unweighted) , all other formulae discussed here satisfy this test. (2) Time Reversal test: ….the formula for calculating the index
number should be such that it gives the same ratio between one point of comparison and the other, no matter which of the two is taken as base. Conceptualization By Soumen Roy
34
Test of Consistency of Index Numbers Symbolically, the following relation should be satisfied.:
P01 * P10 = 1, Where P01 is the index for time ‘ 1’ as time ‘ 0’ as base and P10 is the index for time ‘ 0’ as time ‘ 1’ as base. If the product is not unity, there is said to be a time bias is the method. ’ Proof:
√ [ ∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1] P10 F = √ [ ∑P0q1 / ∑P1q1 * ∑P0q0 / ∑P1q0] Then P01 F * P10 F = √ [ ∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1* ∑P0q1 / ∑P1q1 * ∑P0q0 / ∑P1q0] = √1=1
P01 F =
Therefore Fisher ideal index satisfies the time reversal test. Conceptualization By Soumen Roy
35
Test of Consistency of Index Numbers (3) Factor Reversal test: ….holds that the product of a price index
and the quantity index should be equal to the corresponding value index. In other word, if P01 represent the changes in price in the current year and Q01 represent the changes in quantity in the current year, then P01 *q01 = ∑P1q1 / ∑P0q0. Fisher’ s ideal index satisfies the factor reversal test. Proof:
√ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1] Q01F = √ [∑q1P0 / ∑q0P0 * ∑q1P1 / ∑q0P1] Then P01 F * q01F = √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q01* ∑q1P0 / ∑q0P0 * ∑q1P1 / ∑q0P1] = √ [∑P1q1 / ∑P0q0 ]² = ∑P1q1 / ∑P0q0
P01 F=
Therefore Fisher ideal index satisfies the time reversal test. Conceptualization By Soumen Roy
36
Test of Consistency of Index Numbers Example 7: Construct Fisher’ s ideal index for the following data. Test
whether it satisfies time reversal test and factor reversal test. Commodity
Base Year
Current Year
Quantity
Price
Quantity
Price
A
12
10
15
12
B
15
7
20
5
C
5
5
8
9
Conceptualization By Soumen Roy
37
Test of Consistency of Index Numbers Solution 7: Com
q0
P0
q1
P1
P0q0
P0q1
P1q0
P1q1
A
12
10
15
12
120
150
144
180
C
5
5
8
9
25
40
45
72
250
330
264
352
Fisher’s Ideal Index = P01F =
√ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1] *100
= √[ 264 / 250 * 352 / 330] * 100
=
√1.056 * 1.067] *100 = 106.12
Conceptualization By Soumen Roy
38
Test of Consistency of Index Numbers Time Reversal Test: This is satisfied when P01 * P10 = 1.
Now, P01 F = √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1] = √ [264 / 250 * 352 / 330] And P101 F = √ [ ∑P0q1 / ∑P1q1 * ∑P0q0 / ∑P1q0] = Then, P01 F * q01F = √ [264 / 250 * 352 / 330 * 330 / 352 * 250 / 264] = √ 1 = 1. Hence Fisher ideal index satisfy the time reversal test.
Conceptualization By Soumen Roy
39
Test of Consistency of Index Numbers Factor Reversal Test: This is satisfied when P01 *q01 =
∑P1q1 /
∑P0q0. Now, P01 F= √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1] = √ [264 / 250 * 352 / 330] =√ [330 / 250 * 352 / 264]
√ [264 / 250 * 352 / 330 * 330 / 250 * 352 / 264] = √ [ (352 / 250)² ] = 352 / 250 = ∑P1q1 / ∑P0q0
Then, P01 *q01 =
Hence Fisher ideal index number satisfy the factor reversal test.
Conceptualization By Soumen Roy
40
Consumer Price Index Also called the cost
of living index.
It represent the average change over time in the prices paid by the
ultimate consumer of a specified basket of goods and services. A change in the price level affects the costs of living of different
classes of people differently. The scope of consumer price is necessary, to specify the population
group covered. For example, working class, poor class, middle class, richer class, etc and the geographical areas must be covered as urban, rural, town, city etc. Conceptualization By Soumen Roy
41
Use of Consumer Price Index Very useful in wage negotiations, wage contracts and
dearness allowance adjustment in many countries. At government level, the index numbers are used for wage
policy, price policy, rent control, taxation and general economic policies. Change in the purchasing power of money and real can be
measured. Index numbers are also used for analyzing market price for
particular kinds of goods and services.
Conceptualization By Soumen Roy
42
Method of Constructing Consumer Price Index Methods of Construction of CPI ggregate xpen ture Method / Aggregate Method Family Budget Method / Method of Weighted Relative Conceptualization By Soumen Roy
43
Method of Constructing Consumer Price Index Aggregate Expenditure method: This method is based upon the
Laspeyre’ s method . It is widely used. The quantities of commodities consumed by a particular group in the base year are the weight. The formula is Consumer Price Index number = ∑P1q0 / ∑P0q0 Famil Bud et method or Method of Wei hted Relatives: This
method is estimated aggregate expenditure of an average family on various items and it is weighted. The formula is Consumer Price index number = ∑Pw / ∑w, Where P = (P1 / P0 * 100) for each item. w = value weight i.e., P0q0. Note: “Weighted average price relative method” which we have
studied before and “Family Budget method” are the same for finding out consumer price index. Conceptualization By Soumen Roy
44
Consumer Price Index Example 8: From the following calculate the cost of living index using
Family Budget Method taking 2000 s base year. Items
Weights
Price in 2000 (Rs)
Price in 2004 (Rs)
Food
35
150
140
Rent
20
75
90
Clothing
10
25
30
Fuel & Lighting
15
50
60
Miscellaneous
20
60
80
Conceptualization By Soumen Roy
45
Consumer Price Index Solution (8): Items
W
P0
P1
P = P1/P0 *100
Food
35
150 140 93.33
3266.55
Rent
20
75
90
120.00
2400.00
Clothing
10
25
30
120.00
1200.00
Fuel & Lighting
15
50
60
120.00
1800.00
Miscellaneous
20
60
80
133.33
2666.60
100
Consumer price index by Family Budget method =
PW
11333.15
∑Pw / ∑w
= 11333.15 / 100 = 113.33. Conceptualization By Soumen Roy
46