1. Index Numbers

2. Fro measuring changes in a variable or a group of related variables with respect to time, geographical location, or other characteristics is INDEX Numbers NEWS PAERS headlines the fact that prices are going up or down, that industrial production is rising or falling, that imports are increasing or decreasing, that crimes are rising in a particular period compared to previous period are disclosed by index numbers.

3. An Index Number (I.N.) is a number which is used as a device for comparison between the prices, quantities or values of a group of articles (related variables) in different situations, e.g. at a certain place or a period of time and that at another place or period of time. When the comparison in respect of prices, it is called index number of prices, when in respect of physical quantities, it is termed as Index number of quantities, other index no. are defined in the same manner.

4. Selection of Base Usually base is selected in three different ways and according to these three types of base periods, the following are the three methods of constructing index numbers: (a) Fixed Base Method (b) Chain Base Method (C) Average Base Method.

5. Fixed Base Method 1.Simple Index Numbers 2.Composite Index Numbers In fixed base method, a year is fixed as a base period and the prices during the base year are represented by 100. The price relatives of other years are the required index numbers.

6. Simple Index Numbers eg. YearPrice of rice per quintal (Rs.) Index Number with 1960 as the base year Index Number with 1957 as the base year 195740(40/50)*100=80100 195836(36/50)*100=72(36/40)*100=90 19594896(48/40)*100=120 196050100125 19614488110 196252104130 19634692115 The price index of a single commodity (rice) w.r.t. the base year is shown in the table

7. Composite Index Numbers In case of more than one item, their price relatives w.r.t. a selected base are determined separately. The statistical average of these relatives is called a composite Index Number

8. Composite Index Numbers eg. YearPrice of rice (Rs./qt) Price of wheat (Rs./qt) Price of Pulse (Rs./qt) Index NumberComposite Index Number RiceWheatPulse 1957402520100 300/3=100 1958362124908412098 1959482721120108105111 1960502622125104110113 1961442319110929599 1962522823130112115119 1963462417115968598.67 The price index of a single commodity (rice) w.r.t. the base year is shown in the table

9. Chain Base Methods or Link Index Numbers When year to year comparison is desired Index numbers are determined by Chain Base Method. In this method each year is taken as the base for the immediately next year, the first year itself is being its own index number.

10. Chain Base Method eg. YearPrice Index of Rice (Chain Base Method) Price of rice (Rs./qt) Chain Base Index Number 195740(40/40)*100=100 195836(36/40)*100=90 195948(48/36)*100=133.3 196050(50/48)*100=104.2 196144(44/50)*100=88 196252(52/44)*100=118.2 196355(55/52)*100=105.7

11. One of the great advantage of Chain Base Index Numbers are that new items may be readily included or old one dropped in their calculations; also such indices are more free from seasonal variations than the fixed base indices.

12. (C) Average Base Method The average of a number of years’ prices may be used as base price in determining index numbers. This has the effect of minimizing the abnormalities of any particular year.

13. Avg. Base Method eg. YearPrice Index of Rice (Chain Base Method) Price of rice (Rs./qt) Avg. Price (Rs./qt) Base Price Index Number 195740 (40+25+32+2 8.57+39.43)/5 = 33 (40/33)*100=121.2 195825(25/33)*100=75.8 195932(32/33)*100=97.0 196028.57(28.57/33)*100=86.6 196139.43(39.43/33)*100=119.5

14. Method of Constructing Index Number (Prices) (1) Fixed Base Method Method of Aggregates a)Simple Aggregate of Prices b)Weighted Aggregate of Prices c)Simple Arithmetic Mean of Prices Relatives d)Simple Geometric Mean of Price Relatives e)Weighted Arithmetic Mean of Price Relatives f)Geometric Mean of two weighted aggregates of Prices (special case) A)Laspeyres’ Index B)Paasche’s Index C)Fisher’s Index Number D)Dorbish & Bowley Method E)Marshall & Edgeworth Method F)Walsche’s Method G)Kelly’s Method

15. Method of Constructing Index Number (Prices)… (2) Chain Base Method Simple Arithmetic Mean Or Geometric Mean of Link Relatives (Chain Index) Notations: For the purpose of showing the above modes of construction by mathematical formulae, the following symbols will be used. Similar Notations for prices and quantities of items 1,2,3… resp. in the year Y 1,Y 2,Y 3,… etc. Will be used. However, p 0, q 0 and p 1, q 1 will generally refer to the prices & quantities at the base year Y 0 and at the current year Y 1 respectively without any specific mention about the different items. I 01 = Index no. for the year Y 1 with the year Y 0 as base I 12 = Index no. for the year Y 2 with the year Y1 as base ……

16. 1(a): Method of Aggregates: Simple Aggregate of Prices Simple Aggregate Price Index = I 01 = Eg. Determination of Simple Aggregative Index Numbers. Simple Aggregate Index Number = I 01 = CommodityPrice (Rs/qt) =p 0 (Base 2001) Price (Rs/qt) =p 1 (current 2011) Rice3250 Wheat25 Oil (edible)90100 Fish120140 Potato3540 Total = 302 = 355

17. 1(b): Weighted aggregate of prices Weighted Aggregate of prices= I 01 = Eg. Find by the Weighted Aggregate Method, the Index Numbers Weighted Index Number = I 01 = CommodityPrice (Rs/qt) =p 0 (Base 2001) Price (Rs/qt) =p 1 (current 2011) Weight Rice32508 Wheat25 6 Oil (edible)901007 Fish1201403 Potato35403

18. A. Laspeyres’ Index. Named after the name of German economist Etienne Laspeyres who formulated it in 1871, we have an Index Number known as Laspeyres Index which is equal to

19. B. Paasche’s Index: Named after German statistician Paasche who formulated it in 1874. We have,

20. Example From the following data, construct the index number for 1988 with 1985 as base using Laspeyre’s and Paasche’s formula. CommodityPricesQuantity 1985198819851988 A20251012 B18321610 C354888 D28401210

21. Example…. Comm odity Base Year (1985) Current Year (1988)p0q0p0q0 p1q0p1q0 P0q1P0q1 p1q1p1q1 Price (p 0 ) Quant ity (q 0 ) Prices (p 1 ) Quantity (q 1 ) A20102512200250240300 B18163210288512180320 C358488280384280384 D28124010336480280400 Total110416269801404

22. C. Fisher’s Index Number: It is obtained by the (GM) of Laspeyres’ Index and Paasche’s Index. It is named after Prof Irving Fisher. This is also called Fisher’s Ideal Index. Because: 1.Geometric Mean is useful in averaging % ’s and ratios. Index no. indicates % changes and Fisher’s Index no is a G.M. between Paspeyres’ & Paasche’s Index nos.

23. Because……..: 2.It takes into account both current year and base year quantities. 3.It satisfies Time Reversal test and Factor Reversal test. 4.It is free bias upward as well as downward.

24. D. Dorbish and Bowley Method: To take into account the influence of both the base as well as current periods, Dorbish and Bowley suggested the arithmetic average of the Laspeyre’s and Paasche’s indices.

25. E. Marshall and Edgeworth Method: In this method, both the current year and base year prices and quantities are considered. F. Walsche’s Method:

26. G. Kelly’s Method: This method is also known as fixed weight aggregative index and is currently in great favor of index number series. The formula is as Here weights are the quantities which may refer to some period and anre kept constant for all periods. The AM or GM of the quantities of 2, 3 or more years can be used as weights. The important advantage of Kelly’s method over Laspeyres’, index is that in this index, the cahnge in the base period does not necessitate a corresponding change in the weights which can be kept constant until new data become available for revising the index.

27. Example: Construct Index number of Price from the following data by applying CommodityBase Year (1983)(1984) PriceQuantityPricesQuantity A2846 B51065 C414510 D219213

28. Example… Comm odity Base Year (1983) (1984) p1q0p1q0 P0q0P0q0 p1q1p1q1 P0q1P0q1 (p 0 ) (q 0 )(p 1 )(q 1 ) A284632162412 B5106560503025 C41451070565040 D21921338 26 Total200160130103

29. Fixed Based Method: Methods of Relatives 1(C). Simple Arithmetic Mean of Price Relatives Expressed in symbols The Index number calculated by this method is given by

30. 1(d). Simple Geometric Mean of Price Relatives The Index number calculated by this method is given by 1(e). Weighted Arithmetic Mean of Price Relatives But for all practical purposes, the weights adopted in this method are the values (=price X quantity) of items.

31. 1(f). Geometric Mean of Two Aggregative Price Index Numbers (Special case)

32. Test of Adequacy of Index numbers 1.Unit Test 2.Time Reversal Test 3.Factor Reversal Test 4.Circular Test

33. Unit Test This test requires that the formula should be independent of the unit in which or for which prices and quantities are quoted. Except for the simple (unweighted) aggregative index, all other formula satisfy this test.

34. 2. Time Reversal Test An Index no formula satisfies this test if works both ways, forward and backward with respect to time. In other words, an index no I 01 for the year Y 1 with base year Y 0, Symbolically I 01 X I 10 =1 omitting the factor 100 from both the indices. This test satisfied by – (i) Simple aggregative Index (ii) Marshall-Edgeworth’s Index (iii) Fisher’s Ideal Index (iv) Simple GM of price relatives (v) Walsch Formula (vi) Kelly’s fixed weight formula (vii) Weighted GM of price relative formula with fixed weights

35. 3. Factor Reversal Test An Index no formula satisfies this test if the product of the price index and the quantity index gives the TRUE VALUE RATIO, omitting the factor 100 from both indices. Symbolically an index no formula satisfies this test if Where, I 01 = Price index for Y 1 with base year Y 0 Q 01 = Quantity ideal index for Y 1 with base year Y 0 Fisher’s Ideal Index is the only formula which satisfies this test.

36. 4. Circular Test This test is based on shifting the base in a circular fashion. It may be considered as an extension of Time Reversal Test. An index number is said to satisfy the circular test if it satisfies This test is concerned with the measurement of price changes over a period of years when the shifting of base is desirable. This test is satisfied by (i)Simple aggregate index (ii) Simple GM of price relatives (iii) Weighted aggregative formula with fixed weights

37. Example: Using following data, show that the Fisher’s Ideal formula satisfies the Factor Reversal Test. CommodityPrice per unit (in Rs.)Number of Units Base PeriodCurrent periodBase PeriodCurrent period A6105056 B22100120 C4660 D10123024 E8124036

38. Solution: Omitting to factor 100, Fisher’s price Index I 01 is given by This shows that Fisher’s Ideal Index satisfies Factor Reversal Test

39. Chain Index Numbers

40. Chain Index Number In fixed base Index no., the index no of a given year on a given fixed base was not affected by changes in the relevant values of any other year. But in the chain base method, the value of each period is related with that of the immediately preceding period and not with any fixed period. For constructing Index no by chain base method, a series of Index nos. are computed for each year with preceding year as the base year. These index nos. are known as Link Index Number or Link relatives.

41. Chain Index Number…. The link relatives I 01, I 12, I 23, I 34,……I (n-1)n when multiplied successively known as the chaining process gives the relatives of a common base. Thus I 01 = First Link I 02 = I 01 * I 12 I 03 = (I 01 * I 12 )*I 23 …………………………………… ……………………………………. I 0n = I 0(n-1) * I (n-1)n

42. Construction of Chain Indicies Step (1): Express the figures for each period as a % of the preceding period to obtain Link Relatives (L.R.) Step (2): Chain base indices (CBI) are obtained by multiplying successively the link relatives as explained above. Chain Base Index (CBI) = NOTE: 1. Chain relatives differ from fixed base relatives in computation, chain relatives are computed from link relatives whereas fixed base relatives are computed directly from the original data.

43. Construction of Chain Indicies NOTE: 2. Link Relative Price Relative 3. Conversion of Chain base Index no to Fixed base Index no Current year FBI The FBI for the Ist period being same as the CBI for the first period

44. Example: From the following data of wholesale prices of a certain commodity, construct Index Numbers by Chain Base Method. Year1979808182838485868788 Price750500650600720700690750840800

45. Solution: YearPriceLink RelativesChain Base Index No.Fixed base Index no. (1979=100) 1979750100 1980500(500/750)*100=66.67 1981650(650/500)*100=130 1982600(600/650)*100=92.31 1983720120 198470097.22 198569098.57 1986750108.69 1987840112 198880095.24 It may be noted that chain base index nos are the same as the fixed base index nos.

46. Base Shifting Base shifting refers to the preparing of a new series with a new or more recent base period than the original one. This method requires the taking of a new base year as 100 and express the given series of index nos as a % of the index no of the time period selected. The series of index no with a new base is obtained by the formula

47. Base Shifting Eg. The following are the index nos of prices based on 1977. Shift the base from 1977 to 1982 YearIndex Nos 1977100 1978110 1979120 1980200 1981320 1982400 1983410 1984400 1985380 1986370 1987350 1988366

48. Base Shifting Eg. The following are the index nos of prices based on 1977. Shift the base from 1977 to 1982 Old Base YearYearIndex Nos (old) 1977=100 Index Nos (new) 1982 =100 1977100(100/400)*100=25 1978110(110/400)*100=27.5 1979120(125/400)*100=30 1980200(200/400)*100=50 1981320(320/400)*100=80 New Base Year1982400100 1983410(410/400)*100=102.5 1984400100 198538095 198637092.5 198735087.5 198836691.5