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

Answer 1

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`