Unit 2: Index Numbers
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Unit 2: Index Numbers Index Number Index numbers are statistical devices designed to measrure the relative changes in the level of phenomenon with respect to time, geographical location or other characteristics such as income, profession etc Price Relative Price relative is the price in current year (p 1) expressed in the percentage of the price in the base year (p0) πππππ π ππππ‘ππ£π (π) = π1 × 100 π0 Quantity Relative It is the ratio of quantity of a commodity in the current year (q1) expressed as percentage of quantity in the base year (q0) and is given as ππ’πππ‘ππ‘π¦ π ππππ‘ππ£π (π) = π1 × 100 π0 Value Relative It is the ration of value of commodity in the current year (v1) as expressed as percentage of the value in base year (v0) and is given as: ππππ’π π ππππ‘ππ£π (π) = π£1 π1 π1 × 100 = × 100 π£0 π0 π0 Characteristics of Index Numbers 1. 2. 3. 4. Index numbers are specialized type of averages. They are expressed in ratios but for comparison purposes are expressed as a percentage. Index numbers study the effect of factors that cannot be measured directly. Index numbers facilitate the comparison Uses of Index Numbers 1. 2. 3. 4. 5. Index numbers simplifies the data thus facilitates comparative study.] Index numbers are used in studying trends and tendencies. Index numbers help in measuring purchasing power of money and also used for deflation. It measures change in cost of living of different groups of people. Index numbers are used by the government to frame suitable policies regarding fixation of bank rates, allowances to employees, license to new firms, imports, exports, taxation. Limitations of Index Numbers 1. 2. 3. 4. 5. Index numbers are based on sample data, it is only an approximate indicator. There is a chance of error introduced at each stage of the construction of index numbers. While constructing index numbers, quality of product is not taken into consideration. Like other statistical techniques, index numbers can be misused to get desired conclusions. As customs, tradition and habits fo people vary from time to ti me, it is difficult to assign proper weights to various items. 6. Many formulae are used in construction of index numbers and different formulae give different answers. Steps in Construction of Index Numbers 1. Purpose of the Index Number: Before constructing an index number, it should be decided the purpose for which it is needed. An index number constructed for one category or purpose cannot be used for others. A cost of living index of working classes cannot be used for farmers because the items entering into their consumption will be different. 2. Selection of the Base Period: The selection of the base period is the most important step in the construction of an index number. It is a period against which comparisons are made. The base period should be economically stable. The base period should not be too distant from the current period. a) If the period of comparison is kept fixed of current years, it is called fixed-base period. b) If the relative changes in the level of phenomenon for any period are compared with that of the immediate preceding period, it is called chain-base period. 3. Selection of Commodities: Commodities to be selected depend upon the purpose or objective of the index number to be constructed. But the number of commodities should neither be too large nor too small. Moreover, commodities to be selected must be broadly representative of the group of commodities. They should also be comparable in the sense that standard or graded items should be taken. 4. Selection of Prices: The next step is to select the prices of these commodities. For this purpose, care should be taken to select prices from representative persons, places or journals or other sources. But they must be reliable. 5. Selection of an Average: Since index numbers are averages, the problem is how to select an appropriate average. The two important and common averages are the arithmetic mean and geometric mean. The arithmetic mean is the simpler of the two. But geometric mean is more accurate. However, the average prices should be reduced to price relatives (percentages) either on the basis of the fixed base method or the chain base method. 6. Selection of Weights: While constructing an index number due weightage or importance should be given to the various commodities. Commodities which are more important in the consumption of consumers should be given higher weightage than other commodities. 7. Selection of Formula: A number of formulas have been devised to construct an index number. But the selection of an appropriate formula depends upon the availability of data and purpose of the index number. No single formula may be used for all types of index numbers. Price Index Numbers (Unweighted) Simple Aggregative Method π01 = ∑ π1 × 100 ∑ π0 Simple Unweighted Arithmetic Mean π01 = ∑π π Unweighted Simple Geometric Mean π01 = π΄ππ‘ππππ ( ∑ ππππ ) π Price Index Numbers (Weighted) Laspeyre’s Price Index In Laspeyre’s method base year quantities (q0) are used as weight. πΏ π01 = ∑ π1 π0 × 100 ∑ π0 π0 Paasche’s Price Index In Paasche’s method, current year quantities (q1) are used as weight. π π01 = ∑ π1 π1 × 100 ∑ π0 π1 Marshall-Edgeworth Price Index In this method, base year and current year quantities are used to find the index numbers. Weights assigned are the average of the base year and current year quantities. ππΈ π01 = ∑ π1 π0 + ∑ π1 π1 × 100 ∑ π0 π0 + ∑ π0 π1 Dorbish – Bowley’s Price Index It is the AM of Laspeyre’s and Paasche’s index numbers. It takes into account the base and current year quantities as weights. π·π΅ π01 = ∑ π1 π0 ∑ π1 π1 1 ) × 100 ×( + ∑ π0 π0 ∑ π0 π1 2 Fisher’s Price Index It is the GM of Laspeyre’s and Paasche’s index numbers. It takes into account the base and current year quantities as weights. πΉ π01 =√ ∑ π1 π0 ∑ π1 π1 × × 100 ∑ π0 π0 ∑ π0 π1 Why Fisher’s Index Number is Ideal 1. 2. 3. 4. It is based on geometric mean which considers appropriate average for averaging ratio. It takes into account both current year as well as base year quantities. It satisfies both Time Reversal Test and Factor Reversal Test, It is free of bias. Kelly’s Price Index / Fixed Weight Index πΎ π01 = ∑ π1 π × 100 ∑ π0 π Weighted Average of Price Relatives a) Weighted AM π01 = ∑ ππ π b) Weighted GM π01 = π΄ππ‘ππππ ( ∑ πππππ ) ∑π Quantity Index Numbers (Weighted) Laspeyre’s Quantity Index πΏ π01 = ∑ π1 π0 × 100 ∑ π0 π0 π π01 = ∑ π1 π1 × 100 ∑ π0 π1 Paasche’s Quantity Index Marshall-Edgeworth Quantity Index ππΈ π01 = ∑ π1 π0 + ∑ π1 π1 × 100 ∑ π0 π0 + ∑ π0 π1 Dorbish – Bowley’s Quantity Index ∑ π1 π0 ∑ π1 π1 1 π·π΅ ) × 100 π01 = ×( + ∑ π0 π0 ∑ π0 π1 2 Fisher’s Quantity Index It is the GM of Laspeyre’s and Paasche’s index numbers. It takes into account the base and current year quantities as weights. πΉ π01 =√ ∑ π1 π0 ∑ π1 π1 × × 100 ∑ π0 π0 ∑ π0 π1 Value Index Numbers π01 = ∑ π1 π1 × 100 ∑ π0 π0 Test for Adequacy of Index Number Unit Test This test requires index numbers must be independent of the units in which price or quantities of various commodities are quoted. It is satisfied by all index numbers discussed except Aggregative Price Index formula Time Reversal Test This was proposed by Irving Fisher. According to this, an index number P01 should be such that when base year and current year are interchanged (P10) the resulting inex number should be the reciprocal of the other. TRT requires that the index number computed backward should be reciprocal of the index number computed forward, except for constant proportionality. π10 = 1 = π01 × π10 = 1 π01 Here P01 and P10 are ratios and must not be expressed as percentages. 1. Laspeyre’s Method πΏ π01 = ∑ π1 π0 × 100 ∑ π0 π0 Accoridng to TRT πΏ π10 = πΏ πΏ π01 × π10 = Laspeyre’s number does not satisfy TRT. ∑ π0 π1 ∑ π1 π1 ∑ π1 π0 ∑ π0 π1 × ≠1 ∑ π0 π0 ∑ π1 π1 2. Paasche’s Method π π01 = ∑ π1 π1 ∑ π0 π1 π π10 = ∑ π0 π0 ∑ π1 π0 When reversed it is π π π01 × π10 = ∑ π1 π1 ∑ π0 π0 × ≠1 ∑ π0 π1 ∑ π1 π0 Paasche’s index number does not satisfy TRT. 3. Marshall-Edgeworth Method ππΈ π01 = (∑ π1 π0 + ∑ π1 π1 ) (∑ π0 π0 + ∑ π0 π1 ) When reversed it is: ππΈ π10 = ππΈ ππΈ π01 × π10 = (∑ π0 π1 + ∑ π0 π0 ) (∑ π1 π1 + ∑ π1 π0 ) (∑ π1 π0 + ∑ π1 π1 ) (∑ π0 π1 + ∑ π0 π0 ) × =1 (∑ π0 π0 + ∑ π0 π1 ) (∑ π1 π1 + ∑ π1 π0 ) Marshall-Edgeworth Index number satisfied TRT. 4. Dorbish Bowley’s Index Number ∑ π1 π0 ∑ π1 π1 1 π·π΅ ) π01 = ×( + ∑ π0 π0 ∑ π0 π1 2 When reversed ∑ π0 π1 ∑ π0 π0 1 π·π΅ ) π10 = ×( + ∑ π1 π1 ∑ π1 π0 2 ∑ π1 π0 ∑ π1 π1 ∑π π ∑ π0 π0 1 1 π·π΅ π·π΅ )× ×( 0 1+ )≠1 π01 × π10 = ×( + ∑ π0 π0 ∑ π0 π1 ∑ π1 π1 ∑ π1 π0 2 2 Dorbish Bowley index number does not satisfy TRT. 5. Fisher’s Method ∑ π1 π0 ∑ π1 π1 πΉ π01 =√ × ∑ π0 π0 ∑ π0 π1 When reversed ∑ π0 π1 ∑ π0 π0 πΉ π10 =√ × ∑ π1 π1 ∑ π1 π0 ∑ π1 π0 ∑ π1 π1 ∑ π0 π1 ∑ π0 π0 πΉ πΉ π01 × π10 =√ × ×√ × =1 ∑ π0 π0 ∑ π0 π1 ∑ π1 π1 ∑ π1 π0 Fisher index number satisfies TRT 6. Kelly’s Method πΎ π01 = ∑ π1 π ∑ π0 π πΎ π10 = ∑ π0 π ∑ π1 π When reversed πΎ πΎ π01 × π10 = ∑ π1 π ∑ π0 π × =1 ∑ π0 π ∑ π1 π Factor Reversal Test This was also made by Irving Fisher. It requires that the product of price index and quantity index number should be equal to corresponding value index number. According to this π01 × π01 = π01 Only Fisher index number satisfies this test. Fisher Index Number: ∑ π1 π0 ∑ π1 π1 πΉ π01 =√ × ∑ π0 π0 ∑ π0 π1 ∑ π1 π0 ∑ π1 π1 πΉ π01 =√ × ∑ π0 π0 ∑ π0 π1 According to FRT π01 × π01 = π01 = π1 π1 π0 π0 Consider ∑ π1 π0 ∑ π1 π1 ∑ π1 π0 ∑ π1 π1 π01 × π01 = √ × ×√ × ∑ π0 π0 ∑ π0 π1 ∑ π0 π0 ∑ π0 π1 π1 π1 2 ) π01 × π01 = √( π0 π0 π01 × π01 = π1 π1 = π01 π0 π0 Circular Test This was suggested by Westergaard, It is an extension of Time Reversal Test for more than two periods and is based on the ability to shift base period. This requires that the index number is to work in a circular manner and this property enables us to find the index numbers from period to period without referring back to original base each time. For 3 years; 0, 1, and 2 circular test is π01 × π12 × π20 = 1 Bias in an Index Number Laspeyre’s index which is based on base year quantities gives relatively more weightage for such commodities than needed and consequently the numerator is relatively larger. Hence Laspeyre’s index number has ‘Upward Bias’ Paasche’s index number, which uses the current year quantities as weights is generally an underestimate. Because greater variations of price will be paid lesser importance needed and it shows ‘Downward bias’ Cost of Living Index Number CPI is the index number of the cost met by a specified class of consumers in buying a basket of goods and services. Basket of goods and services means goods and services needed in day to day life of the specified class of consumers. Class of consumers means group of consumers who have almost identical pattern of consumption. Steps involved in Construction of CPI 1. The fixed basket of goods and services is defined. This requires figuring out where the typical consumer spends his or her money. The Bureau of Labor Statistics surveys consumers to gather this information. 2. The prices for every item in the fixed basket are found. Since the same basket of goods and services is used across a number of time periods to determine changes in the CPI, the price for every item in the fixed basket must be found for every point in time. 3. The cost of the fixed basket of goods and services must be calculated for each time period. Like computing GDP, the cost of the fixed basket of goods and services is found by multiplying the quantity of each item times its price. 4. A base year is chosen and the index is computed. The price of the fixed basket of goods and services for each comparison year is then divided by the price of the fixed basket of goods in the base year. The result is multiplied by 100 to give the relative level of the cost of living between the base year and the comparison years. Computing CPI Aggregative Expenditure Method The base year quantities (q0) for various commodities consumed by a particular class of people are taken as weights. πΆππΌ ππ πΆππ π‘ ππ πΏππ£πππ πΌππππ₯ ππ = ∑ π1 π0 ∑ π0 π0 Family Budget Method CPI is constructed by using weighted arithmetic mean of price relatives, the weight being values of the items consumed in base year. ππππβπ‘ (π€) = π0 π0 πΆππΌ = ∑ ππ ∑π
