Lapeyre’s Index Number

Management > Managerial Statistics > Index Number By Larpeyre’s Method

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 .

Commodity	1995	2000	Weights
A	0.50	0.75	2
B	0.60	0.75	5
C	2.00	2.40	4
D	1.80	2.10	8
E	8.00	10.00	1

Solutions:

Commodity	1995	2000	Weights	P0w	P1w
A	0.50	0.75	2	1	1.5
B	0.60	0.75	5	3	3.75
C	2.00	2.40	4	8	9.6
D	1.80	2.10	8	14.4	16.8
E	8.00	10.00	1	8	10
Total				34.4	41.65

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.

Commodity	Base Year		Current Year
	Price	Quantity	Price	Quantity
A	3	25	5	28
B	1	50	3	60
C	2	30	1	30
D	5	15	6	12

Solution:

Commodity	Base Year		Current Year
	Price (P0)	Quantity (Q0)	Price (P1)	Quantity(Q1)	P0Q0	P1Q0
A	3	25	5	28	75	125
B	1	50	3	60	50	150
C	2	30	1	30	60	30
D	5	15	6	12	75	90
Total					∑P0Q0=260	∑P1Q0=395

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.

Commodity	Base Year		Current Year
	Price	Quantity	Price	Quantity
A	3	25	5	28
B	1	50	3	60
C	2	30	1	30
D	5	15	6	12

Solution:

Commodity	Base Year		Current Year
	Price (P0)	Quantity (Q0)	Price (P1)	Quantity(Q1)	Q0P0	Q1P0
A	3	25	5	28	75	84
B	1	50	3	60	50	60
C	2	30	1	30	60	60
D	5	15	6	12	75	60
Total					∑Q0P0=260	∑Q1P0=264

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.

Commodity	Base Year 1997		Current Year 2005
	Price	Quantity	Price	Quantity
A	16	110	25	132
B	5	220	5	264
C	10	132	15	165
D	25	66	30	55

Solution:

Commodity	Base Year		Current Year
	(P0)	(Q0)	(P1)	(Q1)	Q0P0	Q0P1	Q1P0
A	16	110	25	132	1760	2750	2112
B	5	220	5	264	1100	1100	1320
C	10	132	15	165	1320	1980	1650
D	25	66	30	55	1650	1980	1375
Total					5830	7810	6457

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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