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.

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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

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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

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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

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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.

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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.

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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.

Previous Topic: Index Number by Simple Aggregative Method

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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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According to the Spiegel: “An index number is a statistical measure, designed to measure changes in a variable, or a group of related variables with respect to time, geographical location or other characteristics such as income, profession, etc.”

According to Patternson: “In its simplest form, an index number is the ratio of two index numbers expressed as a percent. An index is a statistical measure, a measure designed to show changes in one variable or a group of related variables over time, with respect to geographical location or other characteristics”.

According to Tuttle: “Index number is a single ratio (or a percentage) which measures the combined change of several variables between two different times, places or situations”.

Index numbers is a statistical tool for measuring relative change in a group of related variables over two or more different times. Index number expresses the relative change in price, quantity, or value compared to a base period. An index number is used to measure changes in prices paid for raw materials; numbers of employees and customers, annual income and profits, etc.

Thus index numbers are economic barometers to judge the inflation (increase in prices) or deflationary (decrease in prices) tendencies of the economy. They help the government in adjusting its policies in case of inflationary situations.

Types of Index Numbers:

Three are three types of principal indices. They are: 1. the price index, 2. the quantity index 3. the value index.

Price Index:

It is the most frequently used form of index numbers. A price index compares charges in the price of commodities over a period. If an attempt is being made to compare the prices of the particular commodity this year to the prices of edible oils last year, it involves, firstly, a comparison of two price situations over time and secondly, the heterogeneity of the commodity given the various varieties of the commodity. The Whole Price Index (WPI). Consumer Price Index (CPI) are some of the popularly used price indices.

Quantity Index:

A quantity index measures the changes in quantity from one period to another. If in the above example, instead of the price of the commodity, we are interested in the quantum of production or consumption of the commodity in those years, then we are comparing quantities in two different years or over a period of time. The index of industrial production (IIP) is a commonly used quantity index.

Value Index:

The value index is a combination index. It combines price and quantity changes to present a more spatial comparison. The value index as such measures changes in net monetary worth. Though the value index enables comparison of the value of a commodity in a year to the value of that commodity in a base year, it has limited use. Generally, the value index is used in sales, inventories, foreign trade, etc.

Characteristics of Index Number:

Expressed in Number:

Index numbers are expressed in terms of numbers to show the extent of relative change· in price, output, consumption, etc.

Expressed as Percentage:

Index numbers are expressed in terms of percentage to show the extent of relative change· in price, output, consumption, etc. the % sign is not used.

Measures Relative Changes:

Index numbers measure the relative change in the value of a variable or a group of related variables over a period of time or between places.

Measures Changes which are not Directly Measurable:

Index numbers measures changes which are not directly measurable. The cost of living, the price level or the business activity in a country are not directly measurable but it is possible to study relative changes in these activities by measuring the changes in the values of variables/factors which effect these activities.

Expressed on an Average:

Index number presents the changes on an average bases. These are generally used to express the changes in a group of variables and not of a particular variable or commodity, if the prices of some commodities are increasing there may be at constant level. But on an average what other is j the change in prices of a group of items will be represented by index numbers. Index numbers are of special type of average.

They are Specified Averages:

Index numbers represent special case of averages, as in case of weighted averages.

They Has Universal Utility:

Generally index numbers are used to measure the changes in prices in the group of items but the changes in the quantity of agricultural production, industrial production, imports, and exports can also be measured through index numbers.

Advantages of Index Number

Used for the measurement of change in the price level or the value of money: 

Index number helps to measure the quantitative changes in variables by comparing the data of current year with the data of base year. Hence the index number can be used to know the impact of the change in the value of money on different sections of the society due to different policies. Thus, we can prepare an index number of wages, imports, exports, industrial production, unemployment, profit, area under cultivation, enrolment in a college, etc.

Used for Comparing data:

We can compare, with the help of index numbers, economic conditions of a class of people at two different periods.

Gives the knowledge of the exact change in standard of living: 

It helps to measure standard of living by comparing cost of living and price level. So, the exact living standard of people can be known. Income may increase but if the index number shows a decrease in the value in money, then there is decrease in the living standard.

Helps in the measurement of the inflation rate:

Index number helps to know the inflation rate by measuring price level and value of money between two periods. It helps Central Bank in deciding credit policy.

Helps in adjustment in salaries and allowances: 

Index number helps to determine wages and allowance of employees. Adjustment in the wages and salaries are made according to the cost of living. Cost of living index number is a useful guide to the government and private enterprises to make necessary adjustment in salaries and allowances of the workers. The extent of dearness allowance is decided by the cost of living index.

It is useful to business community: 

Index number is very useful in planning and decision making process of the business firm. 

It provides information regarding foreign trade: 

It is used to compare imports and exports of two different periods. So, it provides key information about the volume of foreign trade. Index of exports and imports provides useful information regarding foreign trade. Using which Government can decide EXIM policy.

Limitation of Index Number:

Not completely true: 

Index number not fully true because they are simply rough indications (approximations) of the relative changes. They cannot be taken as infallible guides. Their data are open to question and they lead to different interpretations. The index number simply indicate arithmetical tendency of the temporal changes in the variable. The choice of representative commodities may lead to fallacious conclusions as they are based on samples. If there are errors in the choice of base periods or weights, etc. then result is not reliable.

International comparison is not possible: 

International comparisons are difficult, if not impossible, on account of the different bases, different sets of commodities or difference in their quality or quantity. Hence index numbers do not help international comparisons.

Difference of time: 

Comparison between different times are is not easy. Over long periods, some popular commodities are replaced by others or consumption habits of people got changed. Entirely new types of commodities come to figure in. With the passage of time, it is difficult to make comparison of index number. Thus comparisons of changes in variables over long periods are not reliable.

Limited use: 

Index numbers are prepared with certain specific objective. If they are used for another purpose they may lead to wrong conclusion. Index numbers measure only changes in the sectional price levels. For another section an entirely different set of commodities will have to be selected. Different people use different things and hold different assets. Therefore, different classes of people are affected differently by a given change in the price level. Hence, the same index number cannot throw light on the effects of price changes on all sections of society. Thus they may be useful for one purpose but not for another.

Lack of retail price index number: 

Most of the index numbers are prepared on the basis of wholesale prices. But in real life, retail prices are most relevant, but it is difficult to collect retail prices.

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