INDEX NUMBERS. Introduction

Transcription

1 INDEX NUMBERS Introduction Index numbers are the indicators which reflect changes over a secified eriod of time in rices of different commodities industrial roduction (iii) sales (iv) imorts and exorts (v) cost of living etc. Definition Index numbers are statistical devices designed to measure the relative change in the level of a henomenon variable or a grou of variables) with resect to time, geograhical location or other characteristics such as income, rofession etc. That is the index numbers are some secial averages of the value of the variable at any given data called the given eriod as the ercentage of the value of the variable at some standard date called the base eriod. The variable may be (iii) the rice of a articular commodity examle; silver, iron, etc. or a grou of commodities lie consumer goods, food stuffs, etc. the volume of trade, exorts and imorts, agricultural or industrial roduction, sales in a deartmental store, etc. the national income of a country or cost of living of ersons belonging to articular income grou/ rofession etc. Construction of index numbers The roblems involved in the construction of index numbers deserves the careful study of the following roblems. 1. Purose of index numbers 2. Selection of items. 3. Data for index numbers. 4. Selection of base eriod. 5. Selection of the average and system of waiting. 1. The urose of index numbers The first and foremost roblem in construction of index numbers is to define the urose for which index number is going to construct. The recise definition of the urose hels us to select the commodities, base year, weights, etc. 2. The Selection of items The selection of item is made deending uon the nature and urose of the index numbers. For examle if we desired to construct a secific urose index number such as the cost of living index numbers of secial categories of eole, eg: the industrial class or the middle class or the clers of the government etc., then the selection of commodities reuires a careful thought. However the following oints must be noted in selecting any items for the construction of the index numbers.

2 2 The items should be strictly reresentative. That is the item should be true indicator of the consumtion habits of the eole and that a change in their rices truly reflects the change in the rice of the grou under study. The items should be standard uality. That is the item selected should not vary in uality from time to time, or from lace to lace. For examle suose rice is an item which effect the general level of the rice, we should select such a variety of rice which is in common demand. Non-tangible items should not be included in the construction of index number. Things for which the rices or values are difficult to ascertain are not included in the construction of the index number (e.g. goodwill, ersonal service). The number of items should be small. The number of items included must not be very large since this result in greater comlication, delay and exense in construction of index numbers. Also the number of items must not be too small, because even a slight change in the rice of one of the item will reflect in the index number. In short the number of items should be moderate. The best solution of the roblem of selection of the items for any index number is to slit the whole grou of commodities into various homogeneous subgrous and to select an adeuate number of reresentative items from each sub-grou. Now the uestion is how many varieties of an items are to be selected from each of the sub-grou, the first criteria is select all those varieties which are in common use and liely to effect the index number. Secondly, if it is desired to attach more imortance to a commodity on account of its larger use or for any other reason, then more than one varieties of the commodity should be selected. 3. Data for index numbers The data, usually the set of rices and of uantities consumed of the selected commodities for different eriods, laces etc., are either obtained from a secondary source lie standard trade journals, official ublications, eriodical secial reorts from the roducers, exorters etc. or through the rimary data collection. If the data is adoted from the secondary source one must have to chec for the accuracy, comarability, samle reresentative etc. On the other hand if the second method is used one has to collect the data relating to the items under consideration. This done by obtaining the rice uotations. The following things must be considered while obtaining the rice uotations. 1. The selection of the lace from which uotations have to be obtained is an imortant consideration. The method is choose the imortant marets for various commodities and select from them a moderate number (Pre-determined) of marets by the method of random samle. It is not necessary to select the same maret of centres for all the commodities. 2. The second ste is to selection of the dealers from whom the rice uotations are to be obtained. Again this is done by the method of random samles, if ossible select those dealers who suly most of the reuired items before the random samle is taen. 3. After deciding the maret and dealers, the next things to secify the volume of the uotations. There will not have any general rule for this. The freuency of the uotation is generally decided uon by the nature of the index number we are going to construct. For e.g. if we are going to construct a weely index number one uotation a wee would be all right. If it is a monthly index then at least one uotation er wee should be obtained. It is therefore clear that in no case a uotation at intervals of more than one wee should be obtained for one commodity. After obtaining all the uotations they are averaged out to form a single uotation. 4. The method of uoting rices should be clearly secified. There are two methods in ractice, one is to uote the in terms of units or commodity (money rice) as Rs. 5 er Kg. And the other in terms of unit of money (commodity rices), as 8 gms a ruee. The former one is more logical and is uniformly adoted.

3 3 5. The tye of rices to be uoted, whether wholesale or retail, is another imortant consideration and must be decided with reference to the tye of index number which we are going to construct. Wholesale rices are more uniform and sensitive to slightest changes in the movement of economic forces effecting suly and demand. Retail rices shows a certain time lag articularly in their downward movement. Therefore wholesale rices are better choice for the general uroses index numbers and retail rices may be used in secific cases lie the construction of cost of living index number, because it is the retail rice which effects cost of living. 6. Finally one must be vary careful in choosing the enumerators for the collection wor. Since they form the rimary reorting agency, great care should be taen in their aointment, because the uality of the index number would ultimately deend uon the uality of the wor done by the enumerators. They should be ersons free from bias of any ind and must aroach their tas with objectivity. 4. Selection of Base Period The eriod with which the comarisons of relative changes in the level of a henomenon are made is termed as base year and the index for this eriod is always taen as 1. There are two tye of base eriods 1. Fixed base 2. Chain base 1. Fixed base: under this method a date is fixed as the base which is adhered to throughout in the construction of the index number. The following are basic criteria for the choice of the base eriod. The base eriod should be a normal eriod. ie. A eriod free from all sorts of abnormalities or chance fluctuations such as economic boom or deression, labour stries, wars, floods, earthuae etc. The base eriod should not be too distant from the given eriod. 2. Chain base: If comarison is desired from date to date a system of chain base is used. It simly consists in assuming the ervious date as the base and then calculate the indices on that basis. 5. Selection of Average and System of Weighting The following are the averages commonly used for the construction of the index numbers 1. Arithmetic mean (A.M.): Simle and Weighted 2. Geometric Mean (G.M.): Simle and Weighted 3. Median Use of Mean as an average. The arithmetic mean is widely used in the construction of index numbers. It is so simle that even a common man can easily understand it. But it has the following disadvantages a. It is effected too much by the extreme items and gives larger weights to the bigger items. If one commodity increased or decreased too much in rice the entire index is liely to be effected thereby. b. It is an absolute measure and in index numbers we are concerned with relative changes. Hence, such a measure cannot be satisfactorily used in the construction of index numbers.

4 4 Use of Median as an average. The median though easiest to calculate and fairly easy to understand, comletely ignores the extreme observations. Also it has the following disadvantages a. If the number of items is small it is not a reresentative average. In index numbers the number of commodities is generally small, hence median is liable to be erratic. b. Lie arithmetic mean the median also is irreversible and hence does not reflect tyical movements of rices or uantities. Use of Geometric mean as average. From the theoretical consideration geometric mean is the most aroriate average to be used. The following are some merit of the G.M. a. It maes the index number reversible so far as time is concerned. b. The index constructed on G.M. osses circular roerty. c. It enables the index number to give eual imortance to eual ratio changes. Weighting Generally, various items or commodities included in the index are not of eual imortance. So we attached weights to the items deending on their relative imortance while calculating the index numbers. Thus we have two tyes of indices: unweighted Indices, in which no secific weights are attached to various commodities and Weighted Indices, in which aroriate weights are assigned to various items. Weighting an index number maes it free from bias. Basis of Weighting:- In scientific study we cannot assign weights arbitrary or by chance. That is we have to adot some system of rational weighting based on some logic, which usually done on the basis of the urose of the index numbers. The following are some of the uantities used as weights 1. The value (or uantity) roduced or manufactured 2. The value (or uantity) sold or ut for sale, or demanded 3. The value (or uantity) consumed. Imlicit or Exlicit weights. Weights may be either imlicit weights or Exlicit weights In imlicit weighting the weights are not exressly laid, but they are imlied by the nature of commodities selected. Under this system, a commodity which is to be given a greater weight is counted as many time as it is necessary to give the weight. Generally, the ractice is that several varieties of that commodities are taen into account. In exlicit weighting, the weights are exressly laid down on the basis of some outward evidence of imortance of items, e.g. the value of roduce, or sale, or consumtion, etc. Fixed and Fluctuating Weights. The next roblem now is whether weights should be fixed or fluctuating. If weights are allowed to vary from eriod to eriod, they give better measure of relative imortance of items. That is an index number with fluctuating weight not only gives changes in the rice but also of shifts in emhasis.

5 5 Method of Weighting In the case of exlicit weighting, where weights have to be laid down secifically, one of the following method is adoted. Weighted Average of Relatives Method or the Family Budget method The Aggregate Exenditure Method. (iii) Weighting by Fishers ideal Method. Construction of Various Tyes of Index numbers Let X 1, X 2,,X n be set of variables and let our interest be in studying the change in the values of these variables taen as whole at two oint of time. Let X 1, X 2,, X n be the value of X 1, X 2,, X n in the base eriod and X 1, X 2,, X n be the values of the variable in the current X i eriod. Then the ratio, j 1,2,..., n is called a relative. A weighted or unweighted X jo average of this relative is called an Index Number. Let denote the rice of commodity in the base year and that in the current year. Then is the relative. We also assume that and as the uantities of goods in the current year and the base year. Simle Index Number The following are some simle index numbers in common use 1) Simle Arithmetic Index Number The simle A.M. of the rice relative exressed as a ercentage is called simle A.M. Index number Simle A.M. Index Number = 1 n x 1 2) Simle Geometric Index Number The simle G.M. of the rice relative exressed as a ercentage is called simle G.M. Index number Simle G.M. Index Number = n x 1 3) Simle Aggregate Index Number This is defined as the ratio of the sum of the rices in the current year and base year exressed as a ercentage. P Simle aggregate index number = x 1. P Weighted Index Numbers Weighted index numbers are obtained by taing weighted. The following are some of the commonly used weighted index numbers.

6 6 1) Laseyre s Index number In this case we find the weighted A.M. of the rice relatives taing as the weight. Laseyre s Index number = x 1. 2) Paasche s Index number In this case we find the weighted A.M. of the rice relatives taing as the weight. Paasche s Index number = x 1 3) Marshall and Edgeworth Index number In this case we find the weighted A.M. of the rice relatives taing the average of and as the weight. ( ) Marshall and Edgeworth Index number = x 1 ( ) 4) Fisher s Ideal Index number The geometric mean of Laseyre s and Paasche s Index number is called the Fisher s Ideal Index number. Fisher s Ideal Index number = x x 1 Test to be Satisfied by a good Index Number We have observed that there are many Index Numbers. Now the uestion is which one is better. To judge this we chec weather the index numbers satisfies the following mathematical tests. 1. The Commodity reversal test 2. Unit test 3. Time reversal test 4. Circular test 5. Factor reversal test 1. The Commodity reversal test An index number is said to satisfy this test if it remains unchanged even if the order in which the commodities are considered is changed. All the index numbers satisfies this test. 2. Unit test If the index number is indeendent of the units in which the rices and uantities are exressed it is said to satisfy the unit test. All index numbers considered so far satisfies this test. 3. Time reversal test Let I o denote the index number calculated with the eriod denoted by as the base eriod and the eriod denoted by as the current and I the index number calculated with the eriods interchanged. The index number is said to satisfy the time reversal test if I x I = 1. Only the simle G.M. index number and Fishers Ideal Index number satisfies this test.

7 7 4. Circular test This is another test for the adeuacy of an index number. This test is based on the shiftability of the base and is an extension of the time reversal test. Let, 1 and 2 are three years and I 1, I 12 and I 2 are the indices for year 1 with as base year, year 2 with 1 as base year and year 2 with as base resectively. The circular test is said to be satisfied if, I 1 x I 12 = I 2 or I 1 x I 12 x I 2 =1. The index number considered so far only simle G.M. index number, aggregate index number and Fishers ideal number satisfies this test. 5. Factor reversal test This test is alicable only to weighted index number. Let I be the index number calculated with denoting the rice and denoting the uantity and I denote the index number obtained by interchanging and. The index number is said to satisfy this test if I x I =. Only Fisher s ideal index number satisfies this test. Classification of Index Numbers 1. Price index numbers which measure the general changes in the retail or wholesale rice level of a articular commodity or grou of commodities. 2. Cost of living index numbers are intended to study the effect of change in the rice level on the cost of living of different classes of eole. 3. Quantity index numbers which are indices to measure the changes in the uantity of goods manufactured in a factory, e.g. the indices of industrial roduction or agriculture roduction. Limitations of Index Number The following are some of the limitations of the index numbers 1. Since index numbers are constructed taing into account only a reresentative set of variates and also based on a set of values of the variates collected by samling index numbers are subject to samling errors. 2. Only uantitative characteristics can be considered in the construction of index numbers. So ualitative changes in the items are not reflected by index numbers. 3. There are different index numbers and the choice in any articular case is arbitrary. So errors can haen due to wrong choice. 4. Interested arties can misuse the index numbers and may maniulate so as to suort their ersonal interests. 5. Errors may enter the collection of data.