Statistics: Index NumberConceptualization 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 Key Terms - Slide I of III Index Number Price Index * Whole Price Index * Retail Price Index Quantity Index Value Index Base Period Current Period Conceptualization By Soumen Roy 3 Key Terms - Slide II of III Simple Aggregate Index Number Simple Average Price Relative Index Weighted Aggregate Index Number * Laspeyre’s Method * Paasche’s Method * Fisher’s Ideal Method * Bowley’s Method * Marshall-Edgeworth 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 . In general. Conceptualization By Soumen Roy 6 . cloth. geographic location or other characteristic. .Index Number What is Index Number?…. . rice.We want one figure to indicate the changes of different commodities as a whole. we shall have to consider a group of variables such as price of wheat. vegetables.is a statistical measure designed to show changes in variable or a group of related variables with respect to time. This is called an Index number.For example.. house rent etc. . if we want to compare the price level of 2009 with what it was in 2008. index numbers are used to measure changes over time in magnitude which are not capable of direct measurement. Conceptualization By Soumen Roy 7 . Index numbers are for comparison.Characteristics of Index Number Index numbers are specified averages Index numbers are expressed in percentage Index numbers measure changes not capable of direct measurement. Conceptualization By Soumen Roy 8 .Uses of Index Numbers They measure the relative change. They measure the purchasing power of money. They are of better comparison. They compare the standard of living. They are economic barometers. They provide guidelines to policy. Value Index: Compare the total value of a certain period with total value in the base period. Conceptualization By Soumen Roy 9 .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. Quantity Index: Is the changes in the volume of goods produced or consumed. but the retail price index reveals the changes in the retail price of commodities such as consumption of goods. They are useful and helpful to study the output in an economy. bank deposits. etc. The wholesale price index reveals the changes into general price level of a country. Here total value is equal to the price of commodity multiplied by the quantity consumed. 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 Selection 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 Simple Aggregate Index Numbers Simple Average of Price Relative Conceptualization By Soumen Roy Weighted Weighted Aggregate Index Number Weighted Average of Price Relative 12 . Commodity A B C D Price Per Unit (In Rupees) Year: 2000 80 50 90 30 Year: 2004 95 60 100 45 13 .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 Example 1:Calculate index numbers from the following data by simple aggregate method taking prices of 2000 as base. 14 .Simple Aggregate Index Number Solution 1: Commodity A B C D Total Price Per Unit (In Rupees) Year: 2000 (P0) 80 50 90 30 250 Year: 2004 (P1) 95 60 100 45 300 Simple aggregate price index = P01 = ∑P1 / ∑P0 * 100 = 300 / 250 * 100 = 120. Simple average of price relative by Geometric Mean: P01 = Antilog [ ∑ log (P1 / P0 *100)] / n 15 .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. where n is the number of commodities. Simple average of price relative by Arithmetic Mean: P01 = [∑ P1 / P0 *100] / n. Commodity A B C D Price in 2000 50 40 80 20 Price in 2004 70 60 100 30 16 .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. 25 17 .Simple Average Price Relative Index Solution: (a) Price relative index number using Arithmetic Mean Commodity A B C D Price in 2000 (P0) 50 40 80 20 Price in 2004 (P1) 70 60 100 30 Total P1 / P0 * 100 140 150 125 150 565 Simple average of price relative index = (P01) = [∑ P1 / P0 *100] / n = 565 / 4 = 141. 5952 / 4 = Antilog [2.Simple Average Price Relative Index Solution: (b) Price relative index number using Geometric Mean Commodi ty A B C D Price in 2000 (P0) 50 40 80 20 Price in 2004 (P1) 70 60 100 30 P1 / P0 * 100 140 150 125 150 Total log(P1/P0 *100) 2.0969 2.1761 2.5952 Simple average of price relative index = (P01) = Antilog [ ∑ log (P1 / P0 *100)] / n = Antilog 8.9 18 .1488] = 140.1761 8.1461 2. Bowley’ s Method 5. Paasche’ s method 3. Kelly’ s Method 19 . Laspeyre’ s method 2. 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.Edgeworth method 6. Marshall.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. Fisher’ s ideal Method 4. 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 . where the weights are determined by quantities in the base period and is given by: P01 L = [∑P1q0 / ∑P0q0 ] *100 Paasche’ s method: The Paasche’ s price index is a weighted aggregate price index in which the weight are determined by the quantities in the current year.Weighted Aggregate Index Numbers Laspeyre’ s method: The Laspeyre’s price index is a weighted aggregate price index. Symbolically: P01 ME = [ ∑ (q0 + q1) p1 / ∑ (q0 + q1) p0] * 100 21 . Symbolically: P01 B = [P01L + P01P] / 2 Marshall.Weighted Aggregate Index Numbers Fisher’ s ideal Method: It is known as ideal index number because: (a) It is based on the geometric mean. Bowley’ s Method: Bowley’ s price index number is the arithmetic mean of Laspeyre’ s and Paasche’ s method. (d) It is free from bias. (b) It is based on the current year as well as the base year. (c) It conform certain tests of consistency.Edgeworth method: This method also both the current year as well as base year prices and quantities are considered. 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 applying (i) Laspeyre’s. 8 12 15 18 Price 4 6 5 4 2001 Qty 5 10 12 20 22 . Symbolically: P01 K = [∑P1q / ∑P0q ] *100 . (ii) Paasche’s and (iii) Fisher’s Ideal Method. Commodity Price A B C D 2 5 4 2 2000 Qty.Weighted Aggregate Index Numbers Kelly’s Method: The following formula is suggested for constructing the index number. 93 23 .Weighted Aggregate Index Numbers Solution 3: Commodity A B C D p0 q0 2 5 4 2 8 12 15 18 p1 4 6 5 4 q1 5 10 12 20 p0q0 16 60 60 36 172 p0q1 10 50 48 40 148 p1q0 32 72 75 72 251 p1q1 20 60 60 80 220 (i) Laspeyre’ s Price Index = P01 L = [∑P1q0 / ∑P0q0 ] *100 = 251 / 172 * 100 = 145. 93 * 148.49 = 147.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] = √ [145.93 in the current year to buy the same amount of the commodities as per the Laspeyre’ s formula. 24 . Other values give similar meaning.28 Interpretation: The results can be interpreted as follows: If 100 rupees were used in the base year to buy the given commodities.64] = √ 21692. we have to use Rs 145. Weighted Aggregate Index Numbers Example 4: Calculate a suitable price index from the following data Commodity A B C 20 15 8 Quantity 2006 2 5 3 Price 2007 4 6 2 Solution 4: Here the as quantities are given in common we can use Kelly’ s index price number. 25 . where q = (q0 + q1) / 2 i.e.. P01 K = [∑P1q / ∑P0q ] *100 .Weighted Aggregate Index Numbers Commodity Q A B C 20 15 8 p0 2 5 3 p1 4 6 2 Total p0q 40 75 24 139 p1q 80 90 16 186 Now. P01 K = 186/139*100 = 133.81 26 . . the weighted index number is calculated by the formula: ∑pw / ∑w.Weighted Average of Price Relative Index When the specific weights are given for each commodity. When the weights are taken as W=P0q1. then the above becomes Paasche’s formula 27 .e. W= P0q0. then the above becomes Laspeyre’s formula. Note: When the base year value P0q0 is taken as weight. i. where W= Weight of the commodity P = the price relative index = (P1 / P0 * 100). Weighted Average of Price Relative Index Example 5: Compute the Weighted Average index number for the following data : Commodity Price Current Year A B C 5 3 2 Base Year 4 2 1 60 50 30 Weight 28 . Weighted Average of Price Relative Index Solution 5: Commodity A B C P1 5 3 2 P0 4 2 1 W 60 50 30 140 P=P1 / P0 * 100 125 150 200 PW 7500 7500 6000 21000 Weighted Average of Price Relative Index = ∑pw / ∑w = 21000 / 140 = 150 Conceptualization By Soumen Roy 29 . employment and etc. The most common type of the quantity index is that of : Laspeyre’ s quantity index number = Q01 L = ∑q1p0 / ∑ q0p0 *100 Paasche’s quantity index number = Q01 P = ∑q1p1 / ∑ q0p1 * 100 Fisher’s quantity index number = Q01 F = √ [ Q01 L * Q01 P ] 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 The quantity index numbers measure the physical volume of production. 2000 Commodity A B C Price 10 12 15 Total Value 100 240 225 Price 12 15 17 2002 Total Value 180 450 340 Conceptualization By Soumen Roy 31 . (ii) Paasche’ s method and (iii) Fisher’ s method.Quantity / Volume Index Number Example 6: From the following data compute quantity indices by (i) Laspeyre’ s method. e.Quantity / Volume Index Number Solution 6: Here instead of quantity. i.. Com.36 Conceptualization By Soumen Roy 32 . Quantity = Total Value / Price. A B C P0 10 12 15 q0 10 20 15 P1 12 15 17 q1 15 30 20 P0q0 100 240 225 565 P0q1 150 360 300 810 P1q0 120 300 255 675 P1q1 180 450 340 970 (i) Laspeyre’ s quantity index number = Q01 L = ∑q1p0 / ∑ q0p0 *100 = 810 / 565 *100 = 143. total values are given. Hence first we find the quantities of base year and current year. 53.70.36 * 143.70] = 143.Quantity / Volume Index Number (ii) Paasche’s quantity index number = Q01 P = ∑q1p1 / ∑ q0p1 * 100 = 970 / 675 *100 = 143. (iii) Fisher’s quantity index number = Q01 F = √ [ Q01 L * Q01 P ] = √ [ 143. Conceptualization By Soumen Roy 33 . (2) Time Reversal test: …. Conceptualization By Soumen Roy 34 . no matter which of the two is taken as base. requires that the formula for constructing an index should be independent of the units in which prices and quantities are quoted. A number of tests been developed and the important among these are: (1) Unit test: …. Except for the simple aggregate index (unweighted) . The question arises as to which formula is appropriate to a given problems.Test of Consistency of Index Numbers Several formulae have been studied for the construction of index number.the formula for calculating the index number should be such that it gives the same ratio between one point of comparison and the other. all other formulae discussed here satisfy this test. 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.: P01 * P10 = 1. the following relation should be satisfied. Fisher’ s ideal index satisfies the time reversal test. If the product is not unity. Proof: P01 F = √ [ ∑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 Therefore Fisher ideal index satisfies the time reversal test. Conceptualization By Soumen Roy 35 .Test of Consistency of Index Numbers Symbolically. there is said to be a time bias is the method. Proof: P01 F= √ [∑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 Therefore Fisher ideal index satisfies the time reversal test. then P01 *q01 = ∑P1q1 / ∑P0q0. if P01 represent the changes in price in the current year and Q01 represent the changes in quantity in the current year.holds that the product of a price index and the quantity index should be equal to the corresponding value index. In other word. Conceptualization By Soumen Roy 36 . Fisher’ s ideal index satisfies the factor reversal test.Test of Consistency of Index Numbers (3) Factor Reversal test: …. Test of Consistency of Index Numbers Example 7: Construct Fisher’ s ideal index for the following data. Commodity A B C 12 15 5 Base Year Quantity Price 10 7 5 Quantity 15 20 8 Current Year Price 12 5 9 Conceptualization By Soumen Roy 37 . Test whether it satisfies time reversal test and factor reversal test. 056 * 1.Test of Consistency of Index Numbers Solution 7: Com A B C q0 12 15 5 P0 10 7 5 q1 15 20 8 P1 12 5 9 P0q0 120 105 25 250 P0q1 150 140 40 330 P1q0 144 75 45 264 P1q1 180 100 72 352 Fisher’s Ideal Index = P01F = √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1] *100 = √[ 264 / 250 * 352 / 330] * 100 = √1.12 Conceptualization By Soumen Roy 38 .067] *100 = 106. Now.Test of Consistency of Index Numbers Time Reversal Test: This is satisfied when P01 * P10 = 1. P01 F * q01F = √ [264 / 250 * 352 / 330 * 330 / 352 * 250 / 264] = √ 1 = 1. Conceptualization By Soumen Roy 39 . P01 F = √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1] = √ [264 / 250 * 352 / 330] And P101 F = √ [ ∑P0q1 / ∑P1q1 * ∑P0q0 / ∑P1q0] = √ [330 / 352 * 250 / 264] Then. Hence Fisher ideal index satisfy the time reversal test. Conceptualization By Soumen Roy 40 .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] And Q01F = √ [∑q1P0 / ∑q0P0 * ∑q1P1 / ∑q0P1] =√ [330 / 250 * 352 / 264] Then. P01 *q01 = √ [264 / 250 * 352 / 330 * 330 / 250 * 352 / 264] = √ [ (352 / 250)² ] = 352 / 250 = ∑P1q1 / ∑P0q0 Hence Fisher ideal index number satisfy the factor reversal test. It represent the average change over time in the prices paid by the ultimate consumer of a specified basket of goods and services. working class. Conceptualization By Soumen Roy 41 . The scope of consumer price is necessary.Consumer Price Index Also called the cost of living index. city etc. etc and the geographical areas must be covered as urban. richer class. poor class. middle class. town. rural. For example. A change in the price level affects the costs of living of different classes of people differently. to specify the population group covered. Index numbers are also used for analyzing market price for particular kinds of goods and services. price policy. Conceptualization By Soumen Roy 42 . taxation and general economic policies. rent control.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. Change in the purchasing power of money and real can be measured. Method of Constructing Consumer Price Index Methods of Construction of CPI Aggregate Expenditure Method / Aggregate Method Family Budget Method / Method of Weighted Relative Conceptualization By Soumen Roy 43 . w = value weight i. Where P = (P1 / P0 * 100) for each item. Note: “Weighted average price relative method” which we have studied before and “Family Budget method” are the same for finding out consumer price index. The formula is Consumer Price Index number = ∑P1q0 / ∑P0q0 Family Budget method or Method of Weighted Relatives: This method is estimated aggregate expenditure of an average family on various items and it is weighted. The quantities of commodities consumed by a particular group in the base year are the weight.. The formula is Consumer Price index number = ∑Pw / ∑w.Method of Constructing Consumer Price Index Aggregate Expenditure method: This method is based upon the Laspeyre’ s method. Conceptualization By Soumen Roy 44 . It is widely used. P0q0.e. Items Food Rent Clothing Fuel & Lighting Miscellaneous Weights 35 20 10 15 20 Price in 2000 (Rs) 150 75 25 50 60 Price in 2004 (Rs) 140 90 30 60 80 Conceptualization By Soumen Roy 45 .Consumer Price Index Example 8: From the following calculate the cost of living index using Family Budget Method taking 2000 s base year. Consumer Price Index Solution (8): Items Food Rent Clothing Fuel & Lighting Miscellaneous W 35 20 10 15 20 100 P0 P1 P = P1/P0 *100 PW 3266.00 1200.33.00 120. Conceptualization By Soumen Roy 46 .60 11333.00 120.00 133.00 2666.55 2400.33 75 25 50 60 90 30 60 80 120.15 150 140 93.15 / 100 = 113.33 Consumer price index by Family Budget method = ∑Pw / ∑w = 11333.00 1800. Acknowledgements Various sources from internet free from IPR restriction. Conceptualization By Soumen Roy 47 .