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Find x in the following table if Laspeyre’s and Paasche’s Price Index Numbers are equal.

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Question

Find x in the following table if Laspeyre’s and Paasche’s Price Index Numbers are equal.

Commodity Base Year Current year
Price Quantity Price Quantity
A 2 10 2 5
B 2 5 x 2
Sum
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Solution

Commodity Base Year Current year p0q0 p1q0 p0q1 p1q1
p0 q0 p1 q1
A 2 10 2 5 20 20 10 10
B 2 5 x 2 10 5x 4 2x
Total - - - - 30 20+5x 14 10+2x

From the table,

∑ p0q0 = 30, ∑ p1q0 = 20 + 5x

∑ p0q1 = 14, ∑ p1q1 = 10 + 2x

`"P"_01("L") = (sum "p"_1"q"_0)/(sum "p"_0"q"_0) xx 100`

∴ `"P"_01("L") = (20 + 5 x)/30 xx 100`   ...(i)

`"P"_01("P") = (sum "p"_1"q"_1)/(sum "p"_0"q"_1) xx 100`

∴ `"P"_01("P") = (10 + 2x)/14 xx 100`     ....(ii)

Since P01(L) = P01(P),

`(20 + 5x)/30 xx 100 = (10 + 2x)/14 xx 100`     ....[From (i) and (ii)]

∴ 14(20 + 5x) = 30(10 + 2x)

∴ 280 + 70x = 300 + 60x

∴ 70x - 60x = 300 - 280

∴ 10x = 20

∴ x = `20/10 = 2`

shaalaa.com
Construction of Index Numbers - Weighted Aggregate Method
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Chapter 5: Index Numbers - Exercise 5.2 [Page 82]

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