Unit Test: 

This test states that the formula for constructing an index number should be independent of the units in which prices and quantities are expressed. All methods, except simple aggregative method, satisfy this test. Except for unweighted aggregative index number, all other indices satisfy this test.

Time Reversal Test:

This test guides whether the method works both ways in time forward and backward. According to Prof. Fisher, the formula for calculating an index number should be such that it gives the same ratio between one point of time and the other, no matter which of the two time is taken as the base. In other words, when the data for any two years are treated by the same method, but with the base reversed, the two index numbers should be reciprocals of each other.

Symbolically the test is represented as: P₀₁ X P₁₀ = 1

Where, P₀₁ is the index for time “1” on time “0” as base and P₁₀ is the index for time “0” on time “1” as the base. If the product is not unity, the method suffers from time bias. The multiplying factor 100 should not be considered during the test. Time reversal test is satisfied by

1. Simple aggregative method
2. Fisher’s method
3. Marshall Edgeworth’s method and
4. Kelly’s method.

Let us see how Fisher’s ideal method satisfies the test.

Thus Fisher’s method satisfies the time reversal test

Factor Reversal Test :

It says that the product of a price index and the quantity index should be equal to value index. In the words of Fisher, each formula permit interchanging the prices and quantities without giving inconsistent results which means two results multiplied together should give the true value ratio. The test says that the change in price multiplied by change in quantity should be equal to total change in value. If P01 is a price index for the current year with reference to base year and Q₀₁is the quantity index for the current year. This test is satisfied by Fisher’s method only.Then by factor reversal test

Symbolically the test is represented as P₀₁ x Q₀₁ =  V₀₁, Where V₀₁ is the value index.

Let us see how Fishers ideal method satisfies the test.

Thus Fisher’s method satisfies the factor reversal test

Note: Fisher’s method satisfies both the time reversal test and factor reversal test. Hence it is called the ideal index number.

Circular Test:

Another test of the adequacy of the index number formula is what is known as ‘circular test’. If in the use of index numbers interest attaches not merely to a comparison of two years, but to the measurement of price changes over a period of years. It is frequently desirable to shift the base. Clearly, the desirability of this property is that it enables us to adjust the index values from period to period without referring each time to the original base. A test of this shift ability of base. A test of this shift ability of base is called to the circular test. This test is just an extension of the time reversal test.

According to this, if indices are constructed for year one based on year zero, for year two based on year one and for year zero based on year two, the product of all the indices should be equal to 1.

Symbolically the test is represented as: P₀₁ X P₁₂ X P₂₀ = 1, This test is satisfied by

1. Simple aggregative method and
2. Kelly’s method.

Let us see how the simple aggregative method satisfies the test.

Thus the simple aggregative method satisfies the factor reversal test

Similarly, let us see how the fixed weight aggregative method satisfies the test.

Thus the fixed weight aggregative method satisfies the factor reversal test

An index which satisfies this test has the advantages of reducing the computations every time a change in the base year has to be made. Such index numbers can be adjusted from year to year without referring each time to the original bases. The circular test is not met by the ideal index or by any of weighted aggregative with changing weights. This test is met by the simple geometric mean of price relatives and the weighted aggregative fixed weights.

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Kelly’s Method:

Truman L. Kelly has suggested the following formula for constructing index number. Here weights are the quantities which may refer to some period, not necessarily the base year or current year. Thus the average quantity of two or more years may be used as weights. This method is known as a fixed-weighted aggregative index and is currently in great favour n the construction of index number series.

Steps Involved:

1. Denote prices of the commodity in the current year as P1 and its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P₀ and its quantity consumed in that year by Q₀.
3. Find q = (Q₀ + Q₁)/2 for each commodity
4. Find the quantities P₀q and P₁q for each commodity.
5. Find sum of each column of P₀q and P₁q and denote the sums by ∑ P₀q and ∑ P₁q respectively.
6. Use following formula to find the Price index number

Example – 01:

Compute Price index by Kelly’s Method from the following data.

Solution:

By Kelly’s method the price index number is 157.00

Walsh’s Method:

Here weights are the geometric mean of quantities of two or more years may be used as weights.

Steps Involved:

1. Denote prices of commodity in the current year as P₁ and its quantity consumed in that year by Q₁.
2. Denote prices of commodity in the base year as P₀ and its quantity consumed in that year by Q₀.
3. Find q = (Q₀ x Q₁)^1/2 = Square root of (Q₀ x Q₁) for each commodity
4. Find the quantities P₀q and P₁q for each commodity.
5. Find sum of each column of P₀q and P₁q and denote the sums by ∑ P₀q and ∑ P₁q respectively.
6. Use following formula to find the Price index number

Example – 02:

Compute Price index by Walsh’s Method from the following data.

Solution:

Thus Walsh’s price index number is 155.84

An important advantage of these formulae is that like Laspeyres index it does not demand yearly changes in the weights. Moreover, the base period can be changed without necessitating a corresponding change in the weights. This is very important because the construction of appropriate quantity weights for a general propose index usually requires a considerable amount of work.

Previous Topic: Marshall Edgeworth Method

Next Topic: Test of Adequacy of Index Number

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The Marshall Edgeworth Method for the index number, credited to Marshall (1887) and Edgeworth (1925), is a weighted relative of the current period to base period sets of prices. This index uses the arithmetic average of the current and based period quantities for weighting.

Price Index by Marshall Edgeworth Method:

Steps involved:

1. Denote prices of the commodity in the current year as P1 its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P0 and its quantity consumed in that year by Q0.
3. Find the quantities P0Q0, P1Q0, P0Q1, and p1Q1 for each commodity.
4. Find the sum of each column of P0Q0, P1Q0 , P0Q1, and p1Q1 denote the sums by ∑ P0Q0, ∑ P1Q0, ∑ P0Q1, and ∑ P1Q1 respectively.
5. Use following formula to calculate the index number by Marshall Edgeworth Method.

Example – 01:

Compute Price index by Marshall Edgeworth Method from the following data.

Solution:

Marshall Edgeworth price index is 155.92

Quantity Index by Marshall Edgeworth Method:

Steps involved:

1. Denote prices of the commodity in the current year as P1 its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P0 and its quantity consumed in that year by Q0.
3. Find the quantities P0Q0, P1Q0, P0Q1, and p1Q1 for each commodity.
   Find the sum of each column of P0Q0, P1Q0 , P0Q1, and p1Q1 denote the sums by ∑ P0Q0, ∑ P1Q0, ∑ P0Q1, and ∑ P1Q1 respectively.
4. Use the following formula to find Marshall Edgeworth index number.

Example – 02:

Compute Price index by Marshall Edgeworth Method from the following data.

Marshall Edgworth quantity index is 104.73

Previous Topic: Fisher’s Ideal Index Number

Next Topic: Kelly’s Method

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Laspeyre’s method is based on fixed weights of the base year. For price index base year’s quantities are used as weights. Paasche’s method is based on fixed weights of the current year. For price index, current year’s quantities are used as weights. Fisher has suggested a geometric mean of the two indices (Laspeyres and Paasche) mentioned above so as to take into account the influence of both the periods, i.e., current as well as base periods.  Thus Fisher’s Method is a geometric mean of Laspeyre’s index and Passche’s index.

Price Index by Fisher’s Method:

Steps involved:

1. Denote prices of the commodity in the current year as P1 its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P0 and its quantity consumed in that year by Q0.
3. Find the quantities P0Q0, P1Q0, P0Q1, and p1Q1 for each commodity.
4. Find the sum of each column of P0Q0, P1Q0 , P0Q1, and p1Q1 denote the sums by ∑ P0Q0, ∑ P1Q0, ∑ P0Q1, and ∑ P1Q1 respectively.
5. Find the Laspeyre’s Price index number
6. Find the Paasche’s Price index number
7. To find the Fisher’s index number calculate the geometric mean of Laspeyre’s index and Passche’s index.

Laspeyre’s price index

LP₀₁ = (∑ P₁ x Q₀) / ( ∑ P₀ x Q₀) × 100

Paasche’s index

PP₀₁ = (∑ P₁ x Q₁) / ( ∑ P₀ x Q₁ ) × 100

Fisher’s index number

Example – 01:

Compute Price index by Dorbish and Browley’s Method from the following data.

Solution:

Lapeyre’s Price Index = LP₀₁ = (∑ P₁ x Q₀) / (∑ P₀ x Q₀) × 100

LP₀₁ = (395 / 260) × 100

LP₀₁ = 151.92

Paasche’s Price Index = PP₀₁ = (∑ P₁ x Q₁) / (∑ P₀ x Q₁) × 100

PP₀₁ = (422 / 264) × 100

PP₀₁ = 159.85

Direct Calculation:

Fisher’s price index is 155.89

Quantity Index by Fisher’s Method:

Steps involved:

1. Denote prices of the commodity in the current year as P1 its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P0 and its quantity consumed in that year by Q0.
3. Find the quantities P0Q0, P1Q0, P0Q1, and p1Q1 for each commodity.
   Find the sum of each column of P0Q0, P1Q0 , P0Q1, and p1Q1 denote the sums by ∑ P0Q0, ∑ P1Q0, ∑ P0Q1, and ∑ P1Q1 respectively.
4. Find the Laspeyre’s Quantity index number
5. Find the Paasche’s Quantity index number
6. To find the Fisher’s index number calculate the geometric mean of Laspeyre’s index and Passche’s index.

Laspeyre’s quantity index

LQ₀₁ = (∑ Q₁ x P₀) / ( ∑ Q₀ x P₀) × 100

Paasche’s quanity index

PQ₀₁ = (∑ Q₁ x P₁) / ( ∑ Q₁ x P₀ ) × 100

Fisher’s quantity index number

Example – 02:

Compute Price index by Dorbish and Browley’s Method from the following data.

Laspeyre’s Quantity Index = LQ₀₁ = (∑ Q₁ x P₀) / (∑ Q₀ x P₀) × 100

LQ₀₁ = (264 / 260) × 100

LQ₀₁ = 101.54

Pasche’s Quantity Index = PQ₀₁ = (∑ Q₁ x P₁) / ( ∑ Q₀ x P₁) × 100

PQ₀₁ = (422 / 395) × 100

PQ₀₁ = 106.84

Direct Calculation:

Fisher’s quantity index is 104.16

Advantages of Fisher’s Method:

• It is free from bias. It reduces the influence of high and low values of the data.
• This method considers values of both, the current year and the base year.
• Fisher’s index lies between the other two indexes.  It is referred to as an “ideal” index because it correctly predicts the expenditure index and it satisfies both the time reversal test as well as factor reversal test.

Disadvantages of Fisher’s Method:

• It is tedious and time-consuming.
• As the data of the current year and the base year is required, the data collection is costly and time-consuming.

Previous Topic: Dorbish and Browley’s Method

Next Topic: Marshall Edgeworth Method

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Laspeyre’s method is based on fixed weights of the base year. For price index base year’s quantities are used as weights. Paasche’s method is based on fixed weights of the current year. For price index, current year’s quantities are used as weights. Dorbish and Bowley have suggested simple arithmetic mean of the two indices (Laspeyres and Paasche) mentioned above so s to take into account the influence of both the periods, i.e., current as well as base periods.  Thus Dorbish and Browley’s Method is arithmetic mean of Laspeyre’s index and Passche’s index.

Price Index by Dorbish and Browley’s Method:

Steps involved:

1. Denote prices of the commodity in the current year as P1 its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P0 and its quantity consumed in that year by Q0.
3. Find the quantities P0Q0, P1Q0, P0Q1, and p1Q1 for each commodity.
4. Find the sum of each column of P0Q0, P1Q0 , P0Q1, and p1Q1 denote the sums by ∑ P0Q0, ∑ P1Q0, ∑ P0Q1, and ∑ P1Q1 respectively.
5. Find the Laspeyre’s Price index number
6. Find the Paasche’s Price index number
7. To find the Dorbish and Browley’s index number calculate the arithmetic mean of Laspeyre’s index and Passche’s index.

Laspeyre’s price index

LP₀₁ = (∑ P₁ x Q₀) / ( ∑ P₀ x Q₀) × 100

Paasche’s index

PP₀₁ = (∑ P₁ x Q₁) / ( ∑ P₀ x Q₁ ) × 100

Dorbish and Browley’s index number

P₀₁ = ( LP₀₁ + PP₀₁ ) /2

Example – 01:

Compute Price index by Dorbish and Browley’s Method from the following data.

Solution:

Lapeyre’s Price Index = LP₀₁ = (∑ P₁ x Q₀) / (∑ P₀ x Q₀) × 100

LP₀₁ = (395 / 260) × 100

LP₀₁ = 151.92

Paasche’s Price Index = PP₀₁ = (∑ P₁ x Q₁) / (∑ P₀ x Q₁) × 100

PP₀₁ = (422 / 264) × 100

PP₀₁ = 159.85

Now, P₀₁ = ( LP₀₁ + PP₀₁ ) /2 = (151.92 + 159.85)/2 = 311.77/2 = 155.89

Dorbish and Browley’s price index is 155.89

Quantity Index by Dorbish and Browley’s Method:

Steps involved:

1. Denote prices of the commodity in the current year as P1 its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P0 and its quantity consumed in that year by Q0.
3. Find the quantities P0Q0, P1Q0, P0Q1, and p1Q1 for each commodity.
   Find the sum of each column of P0Q0, P1Q0 , P0Q1, and p1Q1 denote the sums by ∑ P0Q0, ∑ P1Q0, ∑ P0Q1, and ∑ P1Q1 respectively.
4. Find the Laspeyre’s Quantity index number
5. Find the Paasche’s Quantity index number
6. To find the Dorbish and Browley’s index number calculate the arithmetic mean of Laspeyre’s index and Passche’s index.

Laspeyre’s quantity index

LQ₀₁ = (∑ Q₁ x P₀) / ( ∑ Q₀ x P₀) × 100

Paasche’s quanity index

PQ₀₁ = (∑ Q₁ x P₁) / ( ∑ Q₁ x P₀ ) × 100

Dorbish and Browley’s index number

Q₀₁ = ( LQ₀₁ + PQ₀₁ ) /2

Example – 02:

Compute Price index by Dorbish and Browley’s Method from the following data.

Laspeyre’s Quantity Index = LQ₀₁ = (∑ Q₁ x P₀) / (∑ Q₀ x P₀) × 100

LQ₀₁ = (264 / 260) × 100

LQ₀₁ = 101.54

Pasche’s Quantity Index = PQ₀₁ = (∑ Q₁ x P₁) / ( ∑ Q₀ x P₁) × 100

PQ₀₁ = (422 / 395) × 100

PQ₀₁ = 106.84

Now, Q₀₁ = ( LQ₀₁ + PQ₀₁ ) /2 = (101.54 + 106.84)/2 = 208.38/2 = 104.19

Dorbish and Browley’s quantity index is 104.19

Advantages of Dorbish and Browley’s Method:

• It is free from bias. It reduces the influence of high and low values of the data.
• This method considers values of both, the current year and the base year.

Disadvantages of Dorbish and Browley’s Method:

• It is tedious and time-consuming.
• As the data of the current year and the base year is required, the data collection is costly and time-consuming.

Previous Topic: Paasche’s Method

Next Topic: Fisher’s Method

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Price Index Number by Paasche’s Method:

Paasche’s method is based on fixed weights of the current year. For price index current year’s quantities are used as weights.

Steps involved:

1. Denote prices of the commodity in the current year as P1 and its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P₀.
3. Find the quantities P₀Q₁ and P₁Q₁ for each commodity.
4. Find the sum of each column of P₀Q₁ and P₁Q₁ and denote the sums by ∑ P₀Q₁ and ∑ P₁Q₁ respectively.
5. Use the following formula to find the Price index number

PP₀₁ = (∑ P₁ x Q₁) / ( ∑ P₀ x Q₁ ) × 100

Example – 01:

Compute Price index by Paasche’s Method from the following data .

Solution:

PP₀₁ = (∑ P₁ x Q₁) / (∑ P₀ x Q₁) × 100

PP₀₁ = (422 / 264) × 100

PP₀₁ = 159.85

Thus Paasche’s price index number is 159.85

Quantity Index by Paasche’s Method

Paasche’s method is based on fixed weights of the current year. For quantity index, current year’s prices are used as weights.

Steps involved:

1. Denote prices of the commodity in the current year as P1 and its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P0 and its quantity consumed in that year by Q0.
3. Find the quantities Q₀P₁ and Q₁P₁ for each commodity.
4. Find the sum of each column of Q₀P₁ and Q₁P₁ and denote the sums by ∑ Q₀P₁ and ∑ Q₁P₁ respectively.
5. Use the following formula to find the Quantity index number

PQ₀₁ = (∑ Q₁ x P₁) / ( ∑ Q₀ x P₁) × 100

Example – 02:

Compute Quantity index by Laspeyre’s Method from the following data by Laspeyre’s method .

Solution:

PQ₀₁ = (∑ Q₁ x P₁) / ( ∑ Q₀ x P₁) × 100

PQ₀₁ = (422 / 395) × 100

PQ₀₁ = 106.84

Thus Paasche’s quantity index number is 106.84

Example – 03:

Compute Price index and Quantity index by Paasche’s Method from the following data.

Solution:

Price Index:

PP₀₁ = (∑ P₁ x Q₁) / (∑ P₀ x Q₁) × 100

PP₀₁ = (8745 / 6457) × 100

PP₀₁ = 135.43

Thus Paasche’s price index number is 135.43

Quantity Index:

PQ₀₁ = (∑ Q₁ x P₁) / ( ∑ Q₀ x P₁) × 100

PQ₀₁ = (8745 / 7810) × 100

PQ₀₁ = 111.97

Thus Paasche’s quantity index number is 111.97

Merits of Paasche’s Method :

• Uses current quantities (weights) and so takes changes in consumption patterns into account.  
• Does not overstate price increases.

Demerits of Paasche’s Method :

• Not a pure index as price and quantities change.
• Long and expensive to update weights.
• The indexes for each year cannot be compared directly since the quantities change
• Paasche index tends to underestimate the rise in prices or has a downward bias.
• Paasche index is not frequently used in practice when the number of commodities is large. This is because, for Paasche index, revised weights or quantities must be computed for each year examined. Such information is either unavailable or hard to gather. It makes the data gathering expensive.

Previous Topic: Laspeyre’s Method

Next Topic: Dorbish and Bowley’s method

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When all commodities are not of equal importance, we assign a weight to each commodity relative to its importance and the index number computed from these weights is called a weighted index number. For example, when calculating the price index number if the price of a unit of rice is twice the price of a unit sugar then the rice will be weighed in as ‘2’ whereas sugar will be weighed in as ‘1’. Hence it is a relatively average measure. It is more realistic in comparison to simple index number because it accurately reflects the change over time. Example of the weighted index number is that obtained by Laspeyre’s method, or by Paasche’s method, or by Fisher method.

If ‘w’ is the weight attached to a commodity, then price index is given by

P₀₁ = (∑ P₁ x w) / ( ∑ P₀ x w) × 100

Example – 01:

Compute the weighted price index from the following data .

Solutions:

P₀₁ = (∑ P₁ x w) / ( ∑ P₀ x w) × 100

P₀₁ = (41.65 / 34.4) × 100

P₀₁ = 121.07

Hence the weighted price index is 121.07

Different Weighted Index Methods:

• Laspeyre’s Method
• Paasche’s Method
• Dorbish and Browley’s Method
• Fisher’s Ideal Index Method

Price Index by Laspeyre’s Method:

Laspeyre’s method is based on fixed weights of the base year. For price index base year’s quantities are used as weights.

Steps involved:

1. Denote prices of the commodity in the current year as P1.
2. Denote prices of the commodity in the base year as P0 and its quantity consumed in that year by Q0.
3. Find the quantities P0Q0 and P1Q0 for each commodity.
4. Find the sum of each column of P0Q0 and P1Q0 and denote the sums by ∑ P0Q0 and ∑ P1Q0 respectively.
5. Use the following formula to find the Price index number

LP₀₁ = (∑ P₁ x Q₀) / ( ∑ P₀ x Q₀) × 100

Example – 02:

Compute Price index by Laspeyre’s Method from the following data.

Solution:

LP₀₁ = (∑ P₁ x Q₀) / (∑ P₀ x Q₀) × 100

LP₀₁ = (395 / 260) × 100

LP₀₁ = 151.92

Thus Laspeyre’s price index number is 151.92

Quantity Index by Laspeyre’s Method

Laspeyre’s method is based on fixed weights of the base year. For quantity index base year’s prices are used as weights.

Steps involved:

1. Denote prices of the commodity in the current year as P1 and its quantity consumed in that year by Q1.
2. Denote prices of the commodity in the base year as P₀.
3. Find the quantities Q₀P₀ and Q₁P₀ for each commodity.
4. Find the sum of each column of Q₀P₀ and Q₁P₀ and denote the sums by ∑ Q₀P₀ and ∑ Q₁P₀ respectively.
5. Use following formula to find the Price index number

LQ₀₁ = (∑ Q₁ x P₀) / ( ∑ Q₀ x P₀) × 100

Example – 03:

Compute Quantity index by Laspeyre’s Method from the following data.

Solution:

LQ₀₁ = (∑ Q₁ x P₀) / (∑ Q₀ x P₀) × 100

LQ₀₁ = (264 / 260) × 100

LQ₀₁ = 101.54

Thus Laspeyre’s quantity index number is 101.54

Example – 04:

Compute Price index and Quantity index by Laspeyre’s Method from the following data.

Solution:

Price Index:

LP₀₁ = (∑ P₁ x Q₀) / (∑ P₀ x Q₀) × 100

LP₀₁ = (7810 / 5830) × 100

LP₀₁ = 133.96

Thus Laspeyre’s price index number is 133.96

Quantity Index:

LQ₀₁ = (∑ Q₁ x P₀) / (∑ Q₀ x P₀) × 100

LQ₀₁ = (6457 / 5830) × 100

LQ₀₁ = 110.75

Thus Laspeyre’s quantity index number is 110.75

Merits of Laspeyre’s Method :

• Weights (the quantities) are only needed for one year, the base year.
• Due to above reason this it is cheaper to construct.
• It is easy to calculate and commonly used.
• Quantities for future years do not need to be calculated – only base year quantities (weightings) are used
• The indexes for each year can be compared directly. It provides A meaningful comparison as changes in the index are attributed to the changes.

Demerits of Laspeyre’s Method :

• More expensive new goods that cause an upward bias in prices.
• Does not take account of changes in demand.
• Substituting goods or services that have become relatively cheaper for those that have become relatively more expensive.
• Price increases solely due to quality improvements. It should not be considered inflation.

Previous Topic: Simple Average of Relative Method

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Simple Average of Relative Method Using Arithmetic Mean:

In this method, average of price relative of commodity is calculated.

Steps involved

1. Find price relative for each commodity for the current year using the formula R = (P1 / P0) × 100.
2. Add all price relatives of all the commodities.
3. Divide sum obtained in step 2 by the number of commodities (N).
4. Overall formula for the method is.

Example – 01:

Prices of commodities for the year 2000 and 2004 are as given in the table. Find the price index by a simple average of relative method and using the arithmetic mean from the data given in the table.

Solution:

The price index number by simple average of relative method  using arithmetic mean for 2004 taking 2000 as base year is given by

P₀₁ = (1/N)(∑ R)

P₀₁ = (1/5)(730)

P₀₁ = 146.0

Simple Average Relative Method Using Geometric Mean:

Steps involved

1. Find price relative for each commodity for the current year using the formula R = (P1 / P0) × 100.
2. Find log R in each case.
3. Add all logs of price relatives of all the commodities.
4. Divide sum obtained in step 3 by the number of commodities (N).
5. Find antilog of the number obtained in step 4
6. Overall formula for the method is.

P₀₁ =Antilog ((1/N) (∑ Log R)) = Antilog [(1/N)(∑ log ((P₁ / P₀) × 100)))]

Example – 02:

Prices of commodities for the year 2000 and 2004 are as given in table. Find the price index by simple average of relative method and using geometric mean from the data given in the table.

Solution:

The price index number by simple average of relative method 

using geometric mean for 2004 taking 2000 as base year is given by

P₀₁ = Antilog [(1/N)(∑ log R)]

P₀₁ = Antilog [(1/5)(10.6704)]

P₀₁ = Antilog 2.1341

P₀₁ = 136.2

Example – 03:

Prices of commodities for the year 2002 and 2003 are as given in table. Find the price index by a simple average of relative method and using a) arithmetic mean and b) geometric mean.

Solution:

Solution:

Simple average of relative method and using arithmetic mean:

The price index number by simple average of relative method 

using arithmetic mean for 2003 taking 2002 as base year is given by

P₀₁ = (1/N)(∑ R)

P₀₁ = (1/3)(310.44)

P₀₁ = 103.48

Simple average of relative method and using geometric mean:

The price index number by simple average of relative method 

using geometric mean for 2003 taking 2002 as base year is given by

P₀₁ = Antilog [(1/N)(∑ log R)]

P₀₁ = Antilog [(1/3)( 6.0424)]

P₀₁ = Antilog 2.0141

P₀₁ = 103.30

Merits of Simple Average of Relative Method:

• It is not affected by units in which prices are quoted.
• As prices are converted into price relatives, it is not affected by absolute values of prices.
• It gives equal importance to all items and hence extreme values of certain items do not unduly affect the index number.
• The index number calculated by this method satisfies the unit test.

Demerits of Simple Average of Relative Method:

• As it is unweighted average, the importance of all the items is assumed to be the same.
• The index number constructed by this method does not satisfy the criteria laid down for index number.
• The index number is unduly influenced by high or low prices when the arithmetic mean is used.
• The index number constructed using geometric mean is tedious and time-consuming.

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In Simple Aggregative Method, the total price of commodities in a given (current) year is divided by the total price of commodities in a base year and expressed as a percentage.

Steps involved in Simple Aggregative Method:

1. Add the prices of all the commodities in the current year. Denote the sum as ∑ P₁
2. Add the prices of all the commodities in the base year. Denote the sum as ∑ P_o
3. Use the following formula to find simple price index number of current year based on the base year.

Problems Based on Fixed Base Year Method:

Example – 01:

Prices of commodities for the year 2000 and 2004 are as given in the table. Find the simple aggregative price index from the data displayed in the table.

Solution:

The price index number for 2004 taking 2000 as base year is given by

P₀₁ = (∑P₁ / ∑P₀) × 100

P₀₁ = (120 / 105) × 100

P₀₁ = 114.3

It indicates that the prices in the year 2004 had increased by 14.3 % as compared to the year 2000.

Example – 02:

Prices of commodities for the year 2000 and 2004 are as given in table. Find the simple aggregative price index from the data displayed in the table.

Solution:

The price index number for 2004 taking 2000 as base year is given by

P₀₁ = (∑P₁ / ∑P_o) × 100

P₀₁ = (800 / 500) × 100

P₀₁ = 160

It indicates that the prices in the year 2002 had increased by 60 % as compared to the year 2001.

Example – 03:

Prices of commodities for the year 2000, 2001, 2002, and 2003 are as given in table. Find the simple aggregative price index from the data displayed in the table taking 2000 as base year.

Solution:

The price index number for 2001 taking 2000 as base year is given by

P₀₁ = (∑P₁ / ∑P_o) × 100

P_{2000, 2001} = (175 / 153) × 100

P_{2000, 2001} = 114.4

The price index number for 2002 taking 2000 as base year is given by

P₀₁ = (∑P₁ / ∑P_o) × 100

P_{2000, 2002} = (198 / 153) × 100

P_{2000, 2002} = 129.4

The price index number for 2003 taking 2000 as base year is given by

P₀₁ = (∑P₁ / ∑P_o) × 100

P_{2000, 2003} = (226 / 153) × 100

P_{2000, 2003} = 147.7

Problems Based on Chain Base Year Method:

Example – 04:

Prices of commodities for the year 2000, 2001, 2002, and 2003 are as given in table. Find the simple aggregative price index from the data displayed in the table using chain base method.

Solution:

Solution:

The price index number for 2001 taking 2000 as base year is given by

P₀₁ = (∑P₁ / ∑P_o) × 100

P_{2000, 2001} = (175 / 153) × 100

P_{2000, 2001} = 114.4

The price index number for 2002 taking 2001 as base year is given by

P₀₁ = (∑P₁ / ∑P_o) × 100

P_{2001, 2002} = (198 / 175) × 100

P_{2001, 2002} = 113.1

The price index number for 2003 taking 2002 as base year is given by

P₀₁ = (∑P₁ / ∑P_o) × 100

P_{2002, 2003} = (226 / 198) × 100

P_{2002, 2003} = 114.1

Merits and Demerits of Simple Aggregative Method:

Merits of Simple Aggregative Method:

• This is the simplest method of constructing index number.
• It is very easy to understand.
• It is very simple calculate.

Demerits of Simple Aggregative Method:

• It is affected by the magnitude of the prices of the different commodities.
• It is influenced by the units of the articles through which the prices are quoted.
• It is based on the assumption that the various items and their prices are expressed in the same unit.
• It ignores the relative importance of the different commodities included in the index number as no consideration is given to the relative importance of the commodities.
• It is unduly affected by high and low values of the commodities selected.
• It is not capable of being calculated through other averages viz. geometric mean, median, etc.

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Following are the steps involved in the construction of index number

Purpose of index number: 

The steps which are taken in the construction of index number generally depend on the purpose of the index number. Hence the purpose of index numbers must be defined clearly, perfectly and unambiguously. There are various types of an index number, constructed with different objectives. Thus before constructing an index number, one must define the objective. An index number, which is designed keeping, the specific objective in mind, is a very powerful tool. For example, an index whose purpose is to measure consumer price 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 base year: 

Selection of base year is another problem in the construction of the index number. The index number for a particular future year is compared against a year in the near past, which is called a base year. It may be kept in mind that the base year should be a normal year and economically stable year. The period of abnormalities should not be considered as the base year. The base year should not be too distant in the past. Base year should be that year for which reliable figures (data) are available.

There are three types f base periods.

Fixed Base (A Single-Period): In a fixed base, the base period must be a normal period. A normal period means that the period must be free from all sorts of abnormalities of random causes such as financial crisis, floods, famines, earthquakes, strikes of labourers, wars, etc.

Fixed Base (An Average of Selected Periods): When it is difficult to choose one single period as a normal, then a better choice is to take an average of several periods as a base.

Chain Base: If the comparison is a required year on year, a system of chain base is used.

Selection of goods and services (Commodities): 

After the objective of construction of index numbers is defined, only those items which are related to and are relevant with the purpose should be included. The consumption pattern of consumers might change and thereby make the index number useless. It is better to use a consumption pattern at the time of the setting of the index number. Guidelines for the selection of commodities are as follows:

• The number of commodities should neither be too large nor too small.
• The commodities to be selected must be broadly representative of the group of commodities.
• Selection should contain Standard or graded items and there must be no significant variation in the quality.
• Selection should represent the real tastes, habits, and customs of the people.
• The selection must be easily recognizable.
• The selection should be tangible.
• The economic and social importance of various items should be considered
• All those varieties of a commodity which are in common use and are stable in character should be included,

Selection of price: 

Whether wholesale or retail prices to be used is also a problem in the construction of index number. For a consumer price index, wholesale prices are required, while for a cost of living index, retail prices are needed. Care should be taken to avoid mixing of the two types of prices. Care should be taken to select prices from representative persons, places or journals or other sources. But they must be reliable. Prices may be quoted in money terms. Guidelines for the selection of price are as follows:

• prices are to be collected from those places where a particular commodity is traded in large quantities,
• Published information regarding the prices should also be utilized but information should be reliable,
• The institutions supplying price quotations, should not be biased.
• Take care to consider wholesale price or retail price as per the requirement.
• Prices collected from various places should be averaged.

Choice of Average:

As index numbers are themselves specialized averages, it has to be decided first as to which average should be used for their construction of the index number. The arithmetic mean, being easy to use and calculate, is preferred over other averages (median, mode or geometric mean). The selection of average depends on the relative merits and demerits of the various averages. The average may be weighted or unweighted.

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. It is universally agreed that wheat is the most important cereal as against other cereals, and hence should be given due importance. The weights are determined with reference to the relative amounts of income spent on commodities by consumers. Weights may be given in terms of value or quantity. There are two methods for assigning weights.

Implicit Method: In this method, several varieties of a certain type of commodity under study are used. Such weights are called implicit weight.

Explicit Weight: In this method, the weights are laid down on the basis of one outward evidence or importance of commodities.

These weights may be fixed or fluctuating. The importance of commodities also changes with the change in the tastes and incomes of consumers and also with the passage of time. Therefore, weights are to be revised from time to time and not fixed arbitrarily.

Selection of appropriate 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. After deciding all the factors appropriate formula (Fisher’s or Laspeyres) is selected to represent the data correctly.

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