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# Stats 7

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### Stats 7

1. 1. INDEX NUMBERS Index number is an indicator of the level of a phenomenon at a specific point of time in comparison with its level at some other specific point of time. Index numbers may be of varying price, production, growth rate, imports, exports, cost of living, etc. Generally, index numbers of various economic activities are found useful. For Economists, index numbers are of use at every stage of planning, policy making, decision making etc. and so, index numbers may very be called ‘Economic Barometers’. Just as Barometers measure atmospheric pressure, index numbers measure changes occurring in economic field. An index number is a statistical device designed to measure relative level of a group of related variables over a period of time and space. In other words it is a number which expresses the overall level of a group of related variables at a given time called ‘Current Period’ as compared to the level a some other time called ‘Base Period’. Generally, index numbers are expressed in percentage. Thus, if index number of wholesale prices of food articles in 1995 as compared to 1990 is 150, the implication is that overall level of wholesale prices of food articles I 1995 is 150% of the level in 1990. Here, 1995 is the current year and 1990 is the base year. Index number can very well be calculated for individual variables. For instance, if price of a commodity is Rs. 5 in 1992 and Rs. 8 in 1995, the index number of price for the year 1995 with respect to the base 1992 is P = (8/5)* 100 = 160. That is, the price of the commodity in 1995 is 160% of its price in 1992. Here, since only a single variable is considered, the index number is called ‘Relative’. In this particular case, it is the ‘Price Relative’. Price Relative is the price in the current year expressed as a percentage of the price in the base year. If p 0 and p1 are the prices of a commodity in the base year and the current year respectively, the price relative is P = (p1/p0)* 100. This is an indicator which reflect the relative changes in the level of certain phenomenon in any given period (or over a specified period of time) called the current period with respect to its values in some fixed period, called base period selected for comparison
2. 2. DEFINITION “Index Numbers are statistical devices designed to measure the relative change in the level of a phenomenon (variable or group of variables) with respect to time, geographical location or other characteristics such as income, profession etc.” Generally index numbers are of three types. 1. Price index number 2. Quantity index number 3. Value index number Various price index numbers which are in use are wholesale price index number, consumer price index number, etc. The price index number may be of different groups of commodities – food articles, laboratory equipments etc. Price Index Numbers indicate the general level of prices of articles in the current period as compared to that of the base period. Quantity Index Numbers are index numbers of quantity of goods imported or exported, quantity of agricultural produce etc. Value Index Numbers are the index numbers of the total money value of transaction taking place. Note 1: price index is 125 means price level in the current year is 125% of price level in the base year. Note 2: Average price level in 1990 is double the average price level in 1980 means index numbers of price for 1990 with base 1980 is 200. Note 3: index number for 1995 with base 1970 is 325 means average price level has increased by 225% from 1970 to 1995. PROBLEMS IN CONSTRUCTION o The Purpose of Index Numbers
3. 3. o Selection of Commodities or Items o Data for Index Numbers o Selection of Base Period o Type of Average to be used o System of Weighting o Choice of formula IMPORTANT NOTATIONS o p0: Price of the Commodity in the Base Period o p1: Price of the Commodity in the Current Period o q0: Quantity of a Commodity consumed or purchased during the Base Period o q1: Quantity of a Commodity consumed or purchased in the Current Period o w: Weight assigned to a commodity according to its relative importance in the group. o I: Simple Index Number or Price Relative obtained on expressing current year price as a percentage of the base year price and is given by: I = Price Relative = (p1/p0)*100 o P01: Price Index Number for the Current Year w.r.t. the Base Year o P10: Price Index Number for the Base Year w.r.t. the Current Year o Q01: Quantity Index Number for the Current Year w.r.t. the Base Year o Q10: Quantity Index Number for the Base Year w.r.t. the Current Year o V01: Value Index Number for the Current Year w.r.t. the Base Year o p0j: Price for the jth commodity in the Base Year, j = 1,2,3 … n. o p1j: Price for the jth commodity in the Current Year USES OF INDEX NUMBER 1. Index numbers are useful to governments in formulating policies regarding economic activities such as taxation, imports and exports, grant of license to new firms, bank rate. 2. Index number are useful in comparing variation in production , price etc. 3. Index numbers help industrialist and businessman in planning their activities such as production of goods, their stock etc.
4. 4. 4. Consumer price index number is used for the fixation of salary and grant of allowance to employees. 5. Consumer price index numbers are used for the evaluation of purchasing power of money. LIMITATIONS OF INDEX NUMBERS 1. While constructing index numbers, some representative items alone are made use of. The index number so obtained may not indicate the changes in the concerned fields accurately. 2. As customs and habits change from time to time the use of commodities also varies. And so, it is not possible to assign proper weights to various items. 3. Many formulae are used for the construction of index numbers. These formulae give different values for the index. 4. There is ample scope for bias in the construction of index numbers. By altering the price quotation or by improper selection of items, index numbers can be manipulated. STEPS IN THE CONSTRUCTION OF INDEX NUMBERS The various steps in the construction of index numbers are – o Defining (Stating) the purpose of the index number. o Selecting the base period o Selecting the items o Obtaining price quotations o Selecting the appropriate systems of weights. o Selecting the appropriate formula. 1. Defining (Stating) the purpose of the index number. At the very outset, the purpose of the index number should be decided. As different index numbers are useful for different purposes, the purpose on hand may need a particular index number. A clear definition of purpose will help in the selection of the right index number. While
5. 5. constructing the index number, the selection of items, base periods, weights, etc, depend mainly on the purpose. Absence of clear definition of purpose often leads to construction of an unsuitable index number. 2. Selecting the base period. While constructing an index number, appropriate base period should be selected. The base period should be selected. The base period should be economically stable. There should not be abnormal variations. The period should be free of wars, floods, famines, etc. it should not be too distant from the current period. Again, the consumption pattern during the two periods should not differ much. Depending on the situation, fixed base index number or chain base index number may be preferred. 3. Selecting the items. Selection of items is mainly based on the purpose of the index number. Items differ with the purpose. For example, a wholesale price index number requires items which are transacted at the wholesale market. A consumer price index number requires items which are consumed by the particular group of people. However, in a consumer price index number, items differ with the habits, customs and standard of living. Generally, there are many items that could be included in the index number. But the list can be reduced by selecting representative items only. 4. Obtaining price quotations. After selecting the items for constructing an index number, price quotations for these items should be obtained. Since price is likely to vary from place to place, it is better to obtain price quotations from different places. Also, it is advisable to obtain price quotations from different agencies. Then, the prices should be averaged. Again prices are likely to vary during the span of the base period and also during the span of the current period. Hence, it is better to collect price quotations at regular intervals. These quotations should be averaged and the average should be used in the construction. 5. Selecting the appropriate systems of weights.
6. 6. The items considered in constructing index numbers often have varied importance; weights are attached to the items. Mostly, these weights are quantities in the base period, those in the current period or these in any other period. Sometimes, a combination of quantities in different periods may be considered as weights. 6. Selecting the appropriate formula The selection of formula is based mainly on the availability of data regarding quantities, Laspeyre’s, Paasche’s, fisher’s or any other index number is calculated. While selecting the formula care should be taken to see that maximum use of available data is made. PRICE INDEX NUMBER The various price index numbers in common use are – o Laspeyre’s index number o Paasche’s index number o Marshall – Edgeworth index number o Fischer’s ideal index number. QUANTITY INDEX NUMBERS Generally, quantity index numbers are calculated by adopting price as weights. Some of the quantity index numbers are - o Laspeyre’s Quantity index number o Paasche’s Quantity index number o Marshall – Edgeworth Quantity index number o Fischer’s ideal Quantity index number. Tests for an Index Number A good index number should satisfy the following tests.
7. 7. 1. Time reversal test 2. Factor reversal test. Time reversal test. This test is proposed by Irving Fisher. According to him, an index number (formula) should be such that when the base year and current year are interchanged (reversed) the resulting index number should be the reciprocal of the earlier. The time reversal test requires that the index number computed backwards should be the reciprocal of the index number computed forwards, except for the constant of proportionality. Let P01 be the index number (based on certain formula) for the period ‘1’ with respect to the base period ‘0’. Let P10 be the index number (based on the same formula) for the period ‘0’ with respect to the base period ‘1’. Then, the particular index number (formula) satisfies time reversal test if - P01 x P10 = 1 Here, P01 and P10 are mere ratios – they should not be expressed as percentages. Time reversal test is not satisfied by Laspeyre’s and Paasche’s index numbers. But it is satisfied by Marshall – Edgeworth and Fischer’s ideal index numbers. Factor Reversal Test This test also proposed by Irving Fisher. Here, the argument is that the index number (formula) should be such that the price index and quantity index computed according to the formula should both be quality effective in indicating changes. Factor reversal test requires that the product of the index number of price (with quantities as weights) and the index number of quantity (with prices as weights) should indicate net change in value taking place in between the two periods.
8. 8. Thus if, P01 and Q01 are mere ratios – they should not be expressed as percentages. Fisher’s index number satisfies factor reversal test. But, Laspeyre’s, Paasche’s and Marshall – Edgeworth index numbers do not satisfy this test. BIAS IN AN INDEX NUMBER Generally, if price of a commodity shows significantly high increase, its use will decrease. The consumers lessen the use of such commodities. Thus, if base year quantities are used as weights, the greater variation of price will get greater weightage than needed. Therefore such an index number will be an overestimate of the actual situation. Thus, Laspeyre’s index number which uses the base year quantities as weights, is generally an over estimate. It shows upward bias. On the other hand, if current year quantities are used as weights, the greater variations will be paid lesser importance than needed. This leads to a downward bias. Thus, Paasche’s index number, which uses current year quantities as weights, is generally an under estimate. It shows downward bias. However, fisher’s and Marshall – Edgeworth index numbers make use of base as well as current year quantities and so, they are free of bias. FISHER’S INDEX NUMBER IS ‘IDEAL’. Fisher’s index number is called ‘Ideal Index Number’ because of the following reasons. o It is a geometric mean which is considered as the appropriate average for averaging ratios. o It takes into account the base year quantities as well as the current year quantities. o It is free of bias. o It satisfies both time reversal and factor reversal test. CONSUMER PRICE INDEX NUMBER Consumer Price Index Number is an index number of the cost met by a specified class of consumers in buying a ‘Basket of goods and services’. Here, Basket of goods and services’ means goods and services needed in day to day life of the specified class of consumers. The
9. 9. pattern of consumption of goods is different in different classes. And so, the general index numbers fail to indicate the changes in costs with regard to various classes of consumers. Here, ‘Class of consumers’ means group of consumers having almost identical pattern of consumption. Generally, the classes are those of workers of a factory, people belonging to a particular community, government employees, etc. USES OF CONSUMER PRICE INDEX NUMBERS 1. Consumer Price Index Numbers indicate the changes in the consumer prices. And so, they help governments in formulating policies regarding control of price, taxation, imports and exports of commodities, etc. 2. They are used in granting allowances and other facilities to employees. 3. They are used for the evaluation of purchasing power of money. They are used for deflating money. 4. They are used for comparing changes in the coat of living of different classes of people. STEPS IN THE CONSTRUCTION OF CONSUMER PRICE INDEX NUMBER The steps in the construction of a consumer price index number are – 1. Defining Scope and Coverage At the very outset, it is necessary to decide the class of consumers for which the index number is required. The class may be that of bank employees, government employees, merchants, farmers etc. In any case the geographical coverage should also be decided. That is, the locally, city or town where the class dwells should be mentioned. Anyhow the consumers in the class should have almost the same pattern of consumption. 2. Conducting family budget enquiry and selecting the weights. Having decided about the scope and coverage, the next step is to conduct a sample survey of consumer families regarding their budget on various items. The survey should cover a reasonably good number of representative families. It should be conducted during a period of economic
10. 10. stability. In the survey, information regarding commodities consumed by the families, their quality, and the respective budget are collected. The items included in the index number are classified generally under the heads (1) Food, (2) Clothing, (3) Fuel and lighting, (4) Miscellaneous. Sufficiently large number of representative items is included under each head. 3. Obtaining price quotations The quotations of retail prices of different commodities are collected from local market. The quotation is collected from different agencies and from different places. Then, they are averaged and the averages are made use of. The price quotations of the current period and that of the base periods should be collected. 4. Computing the index number. There are two methods of computation of consumer price index number. They are – a. Aggregative expenditure method. b. Family budget method. Aggregative expenditure method Here the quantities used in the base year are taken as weights. Thus, the consumer price index number by this method is: P01 = (Total expenditure in the current year / Total expenditure in the base year) x 100 Family budget method: Consumer price index number by this method is the weighted arithmetic mean of the price relatives. The weights assigned are the expenditure in a normal period. Thus, the consumer price index number is: P01 = (∑WI / ∑W) where W = P0Q0 and I = (P1/P0) METHODS (along with formulas) o Simple (Unweighted) Aggregate Method: ∑ p1 ∑ q1 *100 *100 ∑ p0 ∑ q0
11. 11. P01 = Q01 = o Weighted Aggregate Method: ∑ wp1 P01 = ∑ wp0 *100 o Lapeyre’s Price Index or Base Year Method: ∑p1q0 La P01 = *100 ∑p0q0 o Paasche’s Price Index: P01Pa = ∑p1q1 *100 ∑p0q1 o Fisher’s Price Index P01F= [P01Pa *P01La]1/2 o Marshall Edgeworth Price Index Number: ∑p1q1 + ∑p1q0 P01Ma = *100 ∑p0q1 + ∑p0q0 Problem 1: From the following compute Price Index Numbers using all four methods. 1970 1980 Commodities Price Quantity Price Quantity A 20 8 40 6 B 50 10 60 5 C 40 15 50 15 D 20 20 20 25
12. 12. Answer: 1970 1980 Commodities p0q0 p0q1 p1q0 p1q1 p0 q0 p1 q1 A 20 8 40 6 160 120 320 240 B 50 10 60 5 500 250 600 300 C 40 15 50 15 600 600 750 750 D 20 20 20 25 400 500 400 500 1660 1470 2070 1790 Answer: Laspeyre’s Index Number: ∑p1q0 2070 *100 *100 124.699 ∑p0q0 1660 Paasche’s Index number: ∑p1q1 1790 121.77 *100 *100 ∑p0q1 1470
13. 13. Fisher’s Ideal Index Number: [P01F= P01Pa *P01La]1/2 [124.699*121.77 ]1/2 123.32 Marshall Edgeworth Index Number: ∑p1q1 + ∑p1q0 1790 + 2070 *100 *100 123.23 ∑p0q1 + ∑p0q0 1470 + 1660 Problem 2: From the following construct index number of the group of four commodities by using Fishers Ideal method Base Year Current Year Commodities Price Expenditure Price Expenditure A 2 40 5 75 B 4 16 8 40 C 1 10 2 24 D 5 25 10 60 Answer: Base Year Current Year Commodities q0 q1 p1q0 p0q1 p0 p0q0 p1 p1q1 A 2 40 5 75 20 15 100 30 B 4 16 4 8 4 5 32 20 C 1 10 2 24 10 12 20 12 D 5 25 10 60 5 6 50 30 91 199 202 92
14. 14. Fisher’s Ideal Price Index √ 202 * 199 219.12 [P01F= P01Pa *P01La]1/2 *100 91 * 92 TEST OF CONSISTENCY o Unit Test: This test requires that the Index Number formula should be independent of the units in which the prices or the quantities of various commodities are quoted. All those formulas which were discussed earlier other than Simple Aggregate of Prices (Quantities) satisfy this test. o Time Reversal Test : P01 * P10 = 1 Other than Laspeyre’s & Paasche’s Index Numbers all others satisfy this test. o Factor Reversal Test: P01 * Q01 = [∑p1q1/ ∑p0q0] Problem 3: From the following check whether (i) Laspeyre’s (ii) Paasche’s (iii) Fishers Index Numbers satisfy the Time & factor Reversal Tests commodities Base Year Current Year Price Quantity Price Quantity A 6.5 500 10.8 560 B 2.8 124 2.9 148 C 4.7 69 8.2 78 D 10.9 38 13.4 24 E 8.6 49 10.8 27 Answer:
15. 15. Commodities p0q0 p1q0 p0q1 p1q1 A 3250 5400 3640 6048 B 347.2 359.6 414.4 429.2 Laspeyre’s Price Index Number: 154.80 Laspeyre’s C 324.3 565.8 366.6 639.6 Quantity Index Number: 101.21 D 414.2 509.2 261.6 321.6 E 421.4 529.2 232.2 291.6 Paasche’s Price Index Number: 4757.1 7363.8 4914.8 7730 157.28 Paasche’s Quantity Index Number: 104.97 Fisher’s Ideal Price Index Number: 156.03 Fisher’s Ideal Quantity Index Number: 103.01 By trail we can find that Fisher’s Index Number satisfies both the tests. Problem 4: From the following calculate Cost Of Living Index Number Commodities Base Year Price Current Year Price Weights A 30 47 4 B 8 12 1 C 14 18 3 D 22 15 2 E 25 30 1 Answer: Commodities P WP A 156.67 626.67 B 150 150 C 128.57 385.71 D 68.18 136.36 E 120 120 1418.74
16. 16. 1418.74/11 = 128.98 1418.74/11 = 128.98