Definition According to Morris Hamburg defined, an index number is nothing more than a relative number, which expresses are relationship between two figures, where one of the figure is used as a base. Uses of Index Number: They measure the relative change. They are of better comparison. They are economic barometers. They compare the standard of living. They provide guidelines to policy. They measure the purchasing power of money.

Characteristics & Methods of Index Number Characteristics: 1). Index numbers are specified averages. 2).Index numbers are expressed in percentage. 3). Index numbers measure changes. 4). capable of direct measurement. 5). Index numbers are for comparison.

Various Methods Index Numbers Simple Aggregate Index Number Simple Average Price Relative Index Weighted Aggregate Index Number ; That is * Laspeyre’s Method * Paasche’s Method * Fisher’s Ideal Method * Bowley’s Method * Marshall-Edgeworth Method * Kelly’s Method

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 commodities such as consumption of goods, bank deposits, etc. 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.

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

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

Methods of Index Number Laspeyres Index Method Paasche’s Method Bowley Method Fisher,s idle Method Marshall Edworth Method Kelly’s Method

Laspeyres Index Method This method was devised by Laspeyres in 1871.In this method the weights are determined by the quantities base. Merits 1.Laspeyres index is simpler in calculation and can be computed 2) once the current year prices are known, as the weight are base year quantities in a price index. 3) This enables us an easy comparability of one index with another.

Laspeyres Index Method (contin) ∑P1Q0 Laspeyres Index Method = -------- x 100 ∑P0Q0 P1 = Prices in the current year P0 = Prices in the base year Q1 = Quantities in the current year Q0 = Quantities in the base year

Paasche’s Method This method was devised by a German statistician Paasche in 1874. The weights of current year are used as base year in constructing the Paasche's Index. The Paasche index can be calculated using the following formula. ∑P1Q1 Paasche Price Index = --------- x100 ∑P0Q1 Where, P1 = Prices in the current year P0 = Prices in the base year Q1 = Quantities in the current year

Fisher,s idle Method Fisher’s ideal index is the geometric mean of Laspeyre and paasche methods. P01 F = √ [ P01L x P01P] 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. (d) It is free from bias

Illustration 1 - Weighted Aggregate index number Compute the weighted aggregative price index numbers for  with as base year using (1) Laspeyre’s Index Number (2) Paashe’s Index Number (3) Fisher’s Ideal Index Number. Commodity Base year Price Current year Quantities Wheat 6 10 50 Sugar 2 100 120 Rice 4 60 Milk 12 30 25

Solution: Weighted Aggregate index number Commodity Price ($)(B.Y) (P0) Price ($)(C.Y) (P1) Qty (B.Y) (Q0) (C.Y) (Q1) P0Q0 P1Q0 P0Q1 P1Q1 Wheat 6 10 50 300 500 Sugar 2 100 120 200 240 Rice 4 60 360 Milk 12 30 25 250 ∑P0Q0 =1040 ∑P1Q0 =1420 ∑P0Q1 =1030 ∑P1Q1 =1400

Solution ( continu) ∑ P1Q0 1420 Laspeyres Index Method = ------- x 100 --------------x 100 = 136.54 ∑P0Q0 1040 ∑P1Q1 1400 Paasche Price Index = ---- ----- x 100 ------------- x 100 = 135.92 P0Q1 1030 Fisher’s Ideal Index Method = √ L x P = √ 136.54 x 135.92 = √ 185322.57 = 136.13

Illustration 2 Year 2000 Year 2006 Commodity Price Per Unit From the data given below , construct index number of prices for 2006with 2000 as base using 1) Laspeyre’s Index 2 ) Paache ‘ s Method and Fisher ideal Index Method. Year 2000 Year 2006 Commodity Price Per Unit Expenditure in Rupees A 2 10 4 16 B 3 12 6 18 C 1 8 14 D 20 32

Solution Since we are given price and total expenditure for the year 2000 and 2006 , we shall first calculate the quantities for two years by dividing the expenditure by price , and then we shall calculate the index numbers is as follows: Commodity P0 q0 P1 q1 Po qo poq1 p1q0 p1q1 A 2 10/2=5 4 16/4=4 10 8 20 16 B 3 12/3=4 6 18/6=3 12 9 24 18 C 1 8/1=8 14/2=7 7 14 D 20/4=5 32/8=4 40 32 ∑Po qo = 50 ∑poq1 = 40 ∑p1q0 = 100 ∑p1q1 = 80

Continue ∑ P1Q0 100 Laspeyres Index Method = ------- x 100 = --------------x 100 = 200 ∑P0Q0 50 ∑P1Q1 80 Paasche Price Index = ---- ----- x 100 ------------- x 100 = 200 P0Q1 40 Fisher’s Ideal Index Method = √ L x P = √ 200 x 200 = √ 40000 = 200

Additional Problem Calculate the Index number of prices and quantity index for the following data, using Laspeyres & Fisher’s Ideal formula. Commodity Unit (Per) Unit Price Quantity Year : 2000 Year : 2010 A Meter 25 35 2500 2000 B Kg 7 10.50 1000 900 C Hundred 4 5 600 D Ton 1100 100 120 E Cubic foot 2.00 2.50 9000 10000

Additional Problem in Weighted Aggregate index number

University Exam Question: Problem: Bill Simpson, owner of a California Vineyard , has collected the following information describing the prices and quantities of harvested crops for the years 1992-1995. Construct a Laspeyres Index and Fisher’s Index for each of this 4 years using 1992 as the base period. Price Per Ton Quantity Harvested (tons) 1992 1993 1994 1995 Ruby cabernet $108 $109 $113 $111 1280 1150 1330 1360 Barbera $93 $96 $101 830 860 850 890 Chenin blanc $97 $99 $106 $107 1640 1760 1630 1660

Answer: 1992-93 Commodity p1q0 p0q0 p1q 0 Ruby cabemet 139520 138240 144640 142080 Barbera 79680 77190 83830 Chenin blanc 159080 173840 175480 ∑p1q0 378280 ∑p0q0 374510 398160 401390

Solution: Help to Find out Fisher’s Ideal Index 1992-93 1992-94 1992-95 Commodity p1q0 p0q0 p1q1 p0q1 p1q 0 Ruby cabemet 139520 138240 125350 124200 144640 150290 143640 142080 150960 146880 Barbera 79680 77190 82560 79980 81600 79050 83830 89890 82770 Chenin blanc 159080 174240 170720 173840 172780 158110 175480 177620 161020 ∑p1q0 378280 ∑p0q0 374510 ∑p1q1 382150 374900 398160 404670 ∑p0q1 380800 401390 418470 390670

Continue 1992-93 year Laspeyres Index : ∑p1q0 378280 Laspeyres Index Method = ------- x 100 -------------- x 100 = 101.01 ∑p0q0 374510 1992-94 year Laspeyres Index : ∑p1q0 398160 Laspeyres Index Method = ------- x 100 -------------- x 100 = 106.31 ∑p0q0 374510 1992-95 year Laspeyres Index : ∑p1q0 401390 Laspeyres Index Method = ------- x 100 -------------- x 100 = 107.18

Index Number Test Time Reversal Test : A test used with index numbers that is satisfied when the new index is the reciprocal of the original index if the functions of the base period and given period are interchanged; the advantage of index numbers meeting the criteria of the test is that a symmetric comparison of the two periods is obtained and the results are consistent whether one or the other period is used as a base. Factor Reversal Test: A test for index numbers in which an index number of quantity, obtained if symbols for price and quantity are interchanged in an index number of price, is multiplied by the original price index to give an index of changes in total value.

Fisher’s Index–Time& Factor Reversal test Calculate Fisher’s Index number from the following data and prove that it satisfied both the time reversal test and factor reversal test. Commodity Price $ 2006 Quantity 2007 Food 6 50 10 56 Fuel & Lighting 2 100 120 Clothing 4 60 House Rent 30 12 24 Miscellaneous 8 40 36

Solutions in Weighted Aggregate index number Commodity Price ($)(B.Y) (P0) Price ($)(C.Y) (P1) Qty (B.Y) (Q0) (C.Y) (Q1) P1Q0 P0Q0 P1Q1 P0Q1 Food 6 10 50 56 500 300 560 336 Fuel 2 100 120 200 240 Clothing 4 60 360 House Rent 12 30 24 288 Miscellaneous 8 40 36 480 320 432 ∑p1q0 =1900 ∑p0q0 =1360 ∑p1q1 = 1880 ∑p0q1 = 1344

Continue : Time Reversal Test ∑p1q0 1900 Laspeyres Index Method = ------- x 100 -------------- x 100 = 139.70 ∑p0q0 1360 ∑p1q1 1880 Paasche Price Index = ---- ----- x 100 ---------- x 100 = 139.89 p0q1 1344 Fisher Ideal Index = √LxP = √ 139.70 x 139.89 = √ 19541.37 = 139.79 Time Reversal Test: P01 x P10 = 1 ∑p1q0 ∑p1q1 ∑p0q1 ∑ p0qo P01 x P10 = √ ---------- x ---------- x ------ x --------- ∑p0q0 ∑ p0q1 ∑ p1q1 ∑p1q0 1900 1880 1344 1360 P01 x P10 = √------ x ------ x ------------x -------- = √1 =1 1360 1344 1880 1900 Hence time reversal test is satisfied.

Continue ; Factor Reversal Test Factor reversal test is satisfied when ∑p1q1 P01 x q01 = ---------- ∑p0q0 ∑p1q0 ∑p1q1 ∑p0q1 ∑ p1q1 P01 x q01 = √ ---------- x ---------- x ------ x ------------------- ∑p0q0 ∑ p0q1 ∑ p0q0 ∑p1q0 1900 1880 1344 1880 1880 1880 1880 ∑ p1q1 P01 x P10 = √--------- x -------- x -------x --------- = √ --------- x -------- = ------- = ------------ 1360 1344 1360 1900 1360 1360 1360 ∑ p0q0 Hence, factor reversal test is satisfied by the given data.

Illustration.2 From the data given below Calculate the Price Index by Fisher’s Ideal formula and then verify the Fisher’s Ideal formula satisfies both time reversal test and factor reversal test. Commodity Base Year Current Year Price (R.s) Qty (000) Tonnes Rice 56 71 50 26 Pulses 32 107 30 83 Sugar 41 62 28 48

Solution: Calculation of Fisher Ideal Index Fisher Ideal Index price: ∑p1q0 ∑p1q1 8496 5134 P01 x P10 = √ ---------- x ---------- x 100 = √ -------- x ---------- x 100 ∑p0q0 ∑ p0q1 9942 6080 √0.8544 x 0.844 = √0.7211 = 84.92 Commodity p0 q0 p1 q1 p0q0 p0q1 p1q0 p1q1 A 56 71 50 26 3976 1456 3550 1300 B 32 107 30 83 3424 2656 3210 2490 C 41 62 28 48 2542 1968 1736 1344 ∑ p0q0 9942 ∑ p0q1 6080 ∑p1q0 8496 ∑p1q1 5134

Time and Factor Reversal Test Time Reversal Test: P01 x P10 = 1 ∑p1q0 ∑p1q1 ∑p0q1 ∑ p0qo P01 x P10 = √ ---------- x ---------- x ------ x --------- ∑p0q0 ∑ p0q1 ∑ p1q1 ∑p1q0 8496 5134 6080 9942 P01 x P10 = √------ x ------ x -------x -------- = √1 =1 9942 6080 5134 8496 Hence time reversal test is satisfied. Factor reversal test is satisfied when ∑p1q1 P01 x q01 = ---------- ∑p0q0 ∑p1q0 ∑p1q1 ∑p0q1 ∑ p1q1 ∑p1q1 P01 x q01 = √ ---------- x ---------- x ------ x ----------- = ----------- ∑p0q0 ∑ p0q1 ∑ p0q0 ∑p1q0 ∑p0q0 8496 5134 6080 5134 5134 5134 5134 ∑ p1q1 P01 x P10 = √------ x ------ x -------x -------- = √ --------- x -------- = ------- = --------- 9942 6080 9942 8496 9942 9942 9942 ∑ p0q0 Hence, factor reversal test is satisfied by the given data.

Additional Problem From the data given below Calculate the Price Index by Fisher’s Ideal formula and then verify the Fisher’s Ideal formula satisfies both time reversal test and factor reversal test. Commodity 2005 2010 Price (R.s) Qty (000) Tonnes Bricks 20 8 40 6 Timber 50 10 60 5 Sand 15 Cement

Index Numbers Anna University – Jan -2010 The table below gives the prices of four items- A, B,C, D sold at a store in 2000 and 2006. Using 2000 as the base year , calculate the price relative index for the four items. Calculate an unweighted aggregate price index for these items. Find the Laspeyres weighted aggregate index for these items. Find the passche index for these items. Find a weighted aggregate quantity index using 2000 as the base year and price as the weight. Solution: Calculation of Price Relative Index: =Po1/ n= 148/ 4=37.00 Item Price $ 2000 Price 2006 Quantity 2000 Quantity2006 A B C D 40 55 95 250 10 25 90 1000 1900 600 50 800 5000 3000 200 Item Price in 2000 (Po) Price in 2006 (P1) Price Relative Po1 =P1/ Po x 100 Log P A B C D 40 55 95 250 10 25 90 45 42 36 1.3979 1.6532 1.6232 1.5797 440 165 148 6.254

Continued ii) Unweighted aggregate price = ∑P1/ ∑Po = 165/ 440 x 100=37.5 iii) Laspeyres weighted aggregate Index L = ∑P1qo/ ∑Poqox100 = 86000 / 214000x100 = 0.402 x 100 = 40.2. iv) Passche’s Method: ∑P1q1/ ∑Poq1x100 = 642000/ 394500 x 100 = 16.3. Weighted aggregate Quantity Index or Fisher Index Method. √L x P = √ 40.2 x 16.3 = 15.9 Item qo Po q1 p1 p1qo poq0 p1q1 poq1 A B C D 1000 1900 600 50 40 55 95 250 800 5000 3000 200 10 25 90 10000 47500 24000 4500 40000 104500 57000 12500 8000 125000 120000 18000 32000 27500 285000 50000 86000 214000 271000 394500

Index Numbers Anna University – June -2012 You are given the following information regarding four items. Using 2000 as the base year , calculate the price relative index for the four items. Calculate an unweighted aggregate price index for these items. Find the Laspeyres weighted aggregate index for these items. Find the passche index for these items. Find a weighted aggregate quantity index using 2000 as the base year and price as the weight. Solution: Calculation of Price Relative Index: =Po1/ n= 817.13/ 4=204.28 Item Price $ 2000 Quantity 2000 Price $ 2006 Quantity2006 A B C D 2.00 .80 22.00 7.00 3 20 4 2 5.75 1.25 35.00 15.00 8 10 1 Item Price in 2000 (Po) Price in 2006 (P1) Price Relative Po1 =P1/ Po x 100 Log P A B C D 2.00 .80 22.00 7.00 5.75 1.25 35.00 15.00 287.50 156.25 159.09 214.28 31.80 57.00 817.13

Continued ii) Unweighted aggregate price = ∑P1/ ∑Po x 100=57.00/31.80 x 100=179.25 iii) Laspeyres weighted aggregate Index L = ∑P1qo/ ∑Poqox100 = 212.25 / 124x100 = 171.16 iv) Passche’s Method: ∑P1q1/ ∑Poq1x100 =178.50/ 97x 100 = 184.02. v)Weighted aggregate Quality Index or Fisher Index Method. √L x P = √171.16 x184.02 = 177.47 Item qo Po q1 p1 p1qo poq0 p1q1 poq1 A B C D 3 20 4 2 2.00 .80 22.00 7.00 8 10 1 5.75 1.25 35.00 15.00 17.25 25.00 140.00 30.00 6 16 88 14 46.00 12.5 105.00 66 7 212.25 124 178.5 97