Index Numbers OPTIONAL - II Mathematics for Commerce, Economics and Business INDEX NUMBERS

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1 Index Numbers OPTIONAL - II 38 INDEX NUMBERS Of the imortant statistical devices and techniques, Index Numbers have today become one of the most widely used for judging the ulse of economy, although in the beginning they were originally constructed to gauge the effect of changes in rices. Today we use index numbers for cost of living, industrial roduction, agricultural roduction, imorts and exorts, etc. Index numbers are the indicators which measure ercentage changes in a variable (or a grou of variables) over a secified time. By saying that the index of exort for the year 200 is 25, taking base year as 2000, it means that there is an increase of 25% in the country's exort as comared to the corresonding figure for the year OBJECTIVES After studying this lesson, you will be able to : define index numbers and exlain their uses; identify and use the following methods for construction of index numbers : (i) aggregate method (ii) simle average of relative method; and exlain the advantages of different methods of construction. EXPECTED BACKGROUND KNOWLEDGE Knowledge of commercial mathematics Measures of Central Tendency 38. INDEX NUMBERS-DEFINITION Some rominent definitions, given by statisticians, are given below: According to the Siegel : "An index number is a statistical measure, designed to measure changes in a variable, or a grou of related variables with resect to time, geograhical location or other characteristics such as MATHEMATICS 27

2 OPTIONAL - II income, rofession, etc." According to Patternson : Index Numbers " In its simlest form, an index number is the ratio of two index numbers exressed as a ercent. An index is a statistical measure, a measure designed to show changes in one variable or a grou of related variables over time, with resect to geograhical location or other characteristics". According to Tuttle : "Index number is a single ratio (or a ercentage) which measures the combined change of several variables between two different times, laces or situations". We can thus say that index numbers are economic barometers to judge the inflation ( increase in rices) or deflationary (decrease in rices ) tendencies of the economy. They hel the government in adjusting its olicies in case of inflationary situations CHARACTERISTICS OF INDEX NUMBERS Following are some of the imortant characteristics of index numbers : Index numbers are exressed in terms of ercentages to show the extent of relative change Index numbers measure relative changes. They measure the relative change in the value of a variable or a grou of related variables over a eriod of time or between laces. Index numbers measures changes which are not directly measurable. The cost of living, the rice level or the business activity in a country are not directly measurable but it is ossible to study relative changes in these activities by measuring the changes in the values of variables/factors which effect these activities PROBLEMS IN THE CONSTRUCTION OF INDEX NUMBERS The decision regarding the following roblems/asect have to be taken before starting the actual construction of any tye of index numbers. (i) (ii) (iii) (iv) (v) Purose of Index numbers under construction Selection of items Choice of an aroriate average Assignment of weights (imortance) Choice of base eriod Let us discuss these one-by-one Purose of Index Numbers An index number, which is designed keeing, secific objective in mind, is a very owerful tool. For examle, an index whose urose is to measure consumer rice index, should not include wholesale rates of items and the index number meant for slum-colonies should not consider luxury items like A.C., Cars refrigerators, etc Selection of Items After the objective of construction of index numbers is defined, only those items which are 28 MATHEMATICS

3 Index Numbers related to and are relevant with the urose should be included Choice of Average As index numbers are themselves secialised averages, it has to be decided first as to which average should be used for their construction. The arithmetic mean, being easy to use and calculate, is referred over other averages (median, mode or geometric mean). In this lesson, we will be using only arithmetic mean for construction of index numbers Assignment of weights Proer imortance has to be given to the items used for construction of index numbers. It is universally agreed that wheat is the most imortant cereal as against other cereals, and hence should be given due imortance. OPTIONAL - II Choice of Base year The index number for a articular future year is comared against a year in the near ast, which is called base year. It may be ket in mind that the base year should be a normal year and economically stable year USES OF INDEX NUMBERS (i) (ii) (iii) Index numbers are economic barometers. They measure the level of business and economic activities and are therefore helful in gauging the economic status of the country. Index numbers measure the relative change in a variable or a grou of related variable(s) under study. Consumer rice indices are useful in measuring the urchasing ower of money, thereby used in comensating the emloyes in the form of increase of allowances TYPES OF INDEX NUMBERS Index numbers are names after the activity they measure. Their tyes are as under : Price Index : Measure changes in rice over a secified eriod of time. It is basically the ratio of the rice of a certain number of commodities at the resent year as against base year. Quantity Index : As the name suggest, these indices ertain to measuring changes in volumes of commodities like goods roduced or goods consumed, etc. Value Index : These ertain to comare changes in the monetary value of imorts, exorts, roduction or consumtion of commodities CONSTRUCTION OF INDEX NUMBERS Suose one is interested in comaring the sum total of exenditure on a fixed number of commodities in the year 2003 as against the year 998. Let us consider the following examle. MATHEMATICS 29

4 OPTIONAL - II Index Numbers Commodity Price (er unit) (in Ruees) Wheat Petrol Pulses Sugar 0 8 Cooking Oil Cloth Since all the commodities are in different units and their rices are not enlarged roortionally, we just cannot get an average for comarison. For that reason, we exress the rates of all commodities in 998 as 00 each and roortionally increase for the corresonding commodities for Commodity rice Index Price Index Wheat Petrol = = Pulses = Sugar Cooking Oil Cloth = = = Average 00 Average 94 6 = We find that the average number (Index) for 2003 is as against 00 for the year 998. We can say that the rices have gone u by 56.83% in the year 2003 as against 998. This method is used for finding rice index numbers METHODS OF CONSTRUCTING INDEX NUMBERS Construction of index numbers can be divided into two tyes : (a) Unweighted indices 30 MATHEMATICS

5 Index Numbers (b) Weighted indices In this lesson, we will discuss only the unweighted indices: The following are the methods of constructing unweighted index numbers : (i) (ii) Simle Aggregative method Simle average of rice relative method Simle Aggregative Method This is a simle method for constructing index numbers. In this, the total of current year rices for various commodities is divided by the corresonding base year rice total and multilying the result by 00. \ Simle Aggregative Price Index Σ P 0 = 00 Σ Where P 0 = Current rice Index number 0 Σ = the total of commodity rices in the current year Σ 0 = the total of same commodity rices in the base year. Let us take an examle to illustrate : Examle 38. Construct the rice index number for 2003, taking the year 2000 as base year Commodity Price in the year Price in the year A B C D E Solution : Calculation of simle Aggregative index number for 2003 (against the year 2000) Commodity Price in 2000 Price in 2003 OPTIONAL - II (in Rs) 0 (in Rs.) A B C D E Total 0 = 400 = 550 MATHEMATICS 3

6 OPTIONAL - II Here 0 = 400, = 550 Σ 550 Po = 00 = 00 Σ = = Index Numbers i.e. the rice index for the year 2003, taking 2000 as base year, is 37.5, showing that there is an increase of 37.5% in the rices in 2003 as against Examle 38.2 Comute the index number for the years 200, 2002, 2003 and 2004, taking 2000 as base year, from the following data : Year Price Solution : Price relatives for different years are = = = = = \ Price index for different years are : Year Price-Index Examle 38.3 Preare simle aggregative rice index number from the following data : Commodity Rate Unit Price (995) Price (2004) Wheat er 0 kg Rice er 0 kg Pulses er 0 kg Sugar er kg 4 20 Oil er litre MATHEMATICS

7 Index Numbers Solution : Calculation of simle aggregative index number. Commodity Rate Unit Price (995) Price (2004) Wheat er 0 kg Rice er 0 kg Pulses er 0 kg Sugar er kg 4 20 Oil er litre Simle Aggregative index number 80 = 00 = OPTIONAL - II CHECK YOUR PROGRESS 38.. Write the characteristics and uses of index numbers. 2. Enumerate the roblems /asects in the construction of index numbers. 3. Find the simle aggregative index number for each of the following : (i) For the year 2000 with 980 as base year Commodity Price in 980 Price in 2000 A B 0 50 C D E (ii) (iii) For the years 999, 2000, 200, 2002, 2003 taking 998 as base year Year Price For the years 200 and 2002 taking 999 as base year. Commodity A B C D E F rice in MATHEMATICS 33

8 OPTIONAL - II Simle Average of Price Relatives Method Index Numbers In this method, the rice relatives for all commodities is calculated and then their average is taken to calculate the index number. Σ 00 Thus, P 0 0 =, if A.M. is used as average where P 0 is the rice index, N is the N number of items, 0 is the rice in the base year and of corresonding commodity in resent year (for which index is to be calculated) Let us take an examle. Examle 38.4 Construct by simle average of rice relative method the rice index of 2004, taking 999 as base year from the following data : Commodity A B C D E F Price (in 999) Price (in 2004) Solution : Commodity Price (in 999) Price (in 2004) Price Relatives (in Rs.)[ 0 ] (in Rs.) [ ] 00 P0 A B C D E F \ P 0 = Σ 00 0 N = = \ Price index for 2004, taking 999 for base year = Examle 38.5 Using simle average of Price Relative Method find the rice index for 200, taking 996 as base year from the following data : 34 MATHEMATICS

9 Index Numbers Commodity Wheat Rice Sugar Ghee Tea Price (in 996) er unit Price (in 200) er unit Solution : Commodity Price (in 996) Price (in 200) Price Relatives (in Rs.)[ 0 ] (in Rs.) [ ] P 0 00 OPTIONAL - II Wheat 2 6 Rice Sugar 2 6 Ghee Tea = = = = = \ P 0 = Σ 00 0 N = = \ Price Index for 200, taking 996 as base year, = CHECK YOUR PROGRESS 38.2 Using Simle Average of Relatives Method, find rice index for each of the following : (i) For 2004, taking 2000 as base year Commodity A B C D E Price in Price in (ii) For 200, taking 999 as base year MATHEMATICS 35

10 OPTIONAL - II Index Numbers Commodity Wheat Rice Sugar Ghee Tea Price (er unit) in Price (er unit) in An index number is a statistical measure, designed to measure changes in a variable(s) with time/geograhical location/other criteria Index Numbers are of three tyes : (i) Price-Index Numbers LET US SUM UP Method of construction of Index numbers (ii) Quantity Index Numbers (iii) Value-index Numbers (i) Simle Aggregative method P 0 =Σ 00 0 where P 0 is the rice index 0 is the rice of a commodity in base year (ii) is the rice of the commodity in resent year Simle Average of Price Relatives Method Σ 00 P 0 0 = N Where N is the number of commodities and all others as in (i) above. SUPPORTIVE WEB SITES htt :// htt :// mathworld.wolfram.com TERMINAL EXERCISE. Use Simle Aggregative Method, find the rice index for each of the following : 36 MATHEMATICS

11 Index Numbers (i) For the year 2000, taking 990 as base year Commodity A B C D E Price (in Rs.) in Price (in Rs.) in (ii) For the year 2004, taking 998 as base year Commodity A B C D E F Price in Price in OPTIONAL - II (iii) For the year 996, 997, 998, 999, Taking 990 as base year Commodity A B C D Price in Using Simle Average of Price Relative Method, find the rice index for each of the following: (i) For 2000, taking 998 as base year Commodity A B C D E Price (in Rs.) Price (in Rs.) (ii) For the year 2004, taking 999 as base year Commodity A B C D E F Price (in Rs.) in Price (in Rs.) in (iii) For the years 2003 and 2004 Taking 998 as base year Commodity A B C D Price (in Rs.) in Price (in Rs.) in Price (in Rs.) in MATHEMATICS 37

12 OPTIONAL - II ANSWERS CHECK YOUR PROGRESS (i) (ii) 999:25; 2000:40; 200:50; 2002:75; 2003:200 (iii).74 ; CHECK YOUR PROGRESS (i) (ii) Index Numbers TERMINAL EXERCISE. (i) 5.43 (ii) (iii) 996: ; 997: ; 998 : 60.0 ; 999: (i) (ii) (iii) 2003:23.75 ; 2004 : MATHEMATICS

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