# Statistics - Index Number

June 2018

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## 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 * 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 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