# Index Numbers.

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Index Numbers

Contents Definition Classification Methods of construction
Unweighted Indices Weighted Indices Tests of adequacy of indices formulae Value Index number Consumer Price Index number

Definition An index number measures the relative change in price, quantity, value, or some other item of interest from one time period to another. A simple index number measures the relative change in one or more than one variable. According to Blair “Index numbers are a specialized type of averages.”

“Index numbers are used to measure the changes in some quantity which we cannot observe directly”.
- Bowley

Classification of index numbers
Price index numbers: It measures the changes in the prices of the commodities produced, consumed or sold in a given period with reference to the base period. Quantity index numbers: These help to measure and compare the changes in the physical volume of goods produced, sold and purchased in a given period compared to some other given period. Value index numbers: These indexes show changes in the value of any commodity in a given period in reference to the base period.

Classification (contd.)
Consumer price index: These indexes measure the average over time in the prices paid by the consumers for a specific group of goods and services. Special purpose index numbers: These indexes are framed for a special study relating to a particular variable or aspect.

Methods of construction of index numbers
Simple or unweighted Simple aggregative method Simple Price relative method Weighted Weighted aggregative method Weighted price relative method

Unweighted Indices

Simple average - example

Simple Aggregate Index – Example

Weighted Indices When all commodities are not of equal importance. We assign weight to each commodity relative to its importance and index number computed from these weights is called weighted index numbers. Laspeyre’s Index Number: In this index number the base year quantities are used as weights, so it also called base year weighted index.

Weighted Indexes (contd.)
Paasche’s Index Numbers: In this index number, the current (given) year quantities are used as weights, so it is also called current year weighted index. Marshal-Edgeworth Index Number: In this index number, the average of the base year and current year quantities are used as weights. This index number is proposed by two English economists Marshal and Edgeworth.

Weighted Indexes (contd.)
Marshal-Edgeworth Index Number (contd.): Fisher’s Ideal Index Numbers: Geometric mean of Laspeyre’s and Paasche’s index numbers is known as Fisher’s ideal index number. It is called ideal because it satisfies the time reversal and factor reversal test.

Weighted Indexes (contd.)
Dorbish and Bowley’s index numbers: This method takes into account base as well as current period for the construction of index numbers. It is the average of Laspeyer’s and Paasche’s method. Kelley’s index numbers: Kelly believes that a ratio of aggregates with selected weights (not necessarily of base year or current year) gives the base index number.

Laspeyres vs. Paasche Index
When is Laspeyres most appropriate & when is Paasche the better choice? Laspeyre’s: Advantages: Requires quantity data from only the base period. This allows a more meaningful comparison over time. The changes in the index can be attributed to changes in the price. Disadvantages: Does not reflect changes in buying patterns over time. Also, it may overweight goods whose prices increase. Ernst Louis Étienne Laspeyres  (1834 – 1913)

Paasche’s: Advantages: Because it uses quantities from the current period, it reflects current buying habits. Disadvantages: It requires quantity data for the current year. Because different quantities are used each year, it is impossible to attribute changes in the index to changes in price alone. It tends to overweight the goods whose prices have declined. It requires the prices to be recomputed each year. Hermann Paasche (1851–1925)

Fisher’s Ideal Index Laspeyres’ index tends to
overweight goods whose prices have increased. Paasche’s index, on the other hand, tends to overweight goods whose prices have gone down. Fisher’s ideal index was developed in an attempt to offset these shortcomings. It is the geometric mean of the Laspeyres and Paasche indexes. Sir Ronald Aylmer Fisher ( )

Weighted Indexes - Example

Weighted Indexes – Example (contd.)

Weighted Indexes – Example (contd.)

Time reversal test It is a test to determine whether a given method will work both ways in time, forward and backward. In the words of Fisher, “The test is that the formula for calculating the index number should be such that it will give the same ratio between one point of comparison and the other, no matter which of the two is taken as base.” Symbolically, the following relation should be satisfied:    P01 X P10 = 1 Where P01 is the index for time “I” on time “0” as base and P10 is the index for time “0” on time “I” as base. If the product is not unity, there is said to be a time bias in the method.

Time reversal test (contd.)
The test is not satisfied by Laspeyres method and the Paasche method as can be seen below: When Laspeyres method is used- and the test is not satisfied. When Paasche method is used- and the test is not satisfied.

Time reversal test (contd.)
Let us now see how Fisher’s Ideal formula satisfies the test. Proof: Changing time, i.e., 0 to 1 and 1 to 0. Since P01 X P10 = 1, the Fisher’s ideal index satisfies the test.

Factor reversal test Another test suggested by Fisher is known as factor reversal test. It holds that the product of a price index and the quantity index should be equal to the corresponding value index. In the words of Fisher, “Just as each formula should permit the interchange the prices and quantities without giving inconsistent results, so it ought to permit the interchange of the two times without giving inconsistent results, i.e., the two results multiplied together should give the true value ratio.”

Factor reversal test (contd.)
The factor reversal test is satisfied only by the Fisher’s Ideal Index. Proof: Change p to q and q to p:

Factor reversal test (contd.)
Since   the factor reversal test is satisfied by the Fisher’s Ideal index. This means, of course, that the formula serves equally well for constructing indices of quantities as for constructing indices of prices, the quantity index being derived by interchanging p and q in the ideal formulae.

Circular test This test is just an extension of the time reversal test. The test requires that if an index is constructed for the year a on base year b, and for the year b on base year c, we ought to get the same result as if we calculated direct an index for a on base year c without going through b as an intermediary. The Laspeyres index does not satisfy the test as can be seen from the following: If the three years are 0, 1, 2, the index by Laspeyres method will be:

Circular test (contd.) The product of all these is not equal to 1. Hence the test is not satisfied. Similarly, it can be shown that the Paasche’s index and Fisher’s index do not satisfy the test. The circular test is not met by the ideal index or by any of weighted aggregative with changing weights. This test is met by simple geometric mean of price relatives and the weighted aggregative fixed weights.

Value Index A value index measures changes in both the price and quantities involved. A value index, such as the index of department store sales, needs the original base-year prices, the original base-year quantities, the present-year prices, and the present year quantities for its construction. Its formula is:

Value Index - Example The prices and quantities sold at the Waleska Clothing Emporium for various items of apparel for May 2000 and May 2005 are: What is the index of value for May 2005 using May 2000 as the base period?

Value Index – Example (contd.)

Consumer Price Index numbers
Also known as cost of living index numbers, these are generally intended to represent the average change over time in the prices paid by the ultimate consumer of a specified basket of goods and services. Uses : 1.Serves the basis for wage negotiations & contracts. 2.Helps in formulation of wage policy, price policy, etc. 3.Also measures varying purchasing power of currency.

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