Question

Find x if Laspeyre’s Price Index Number is same as Paasche’s Price Index Number for the following data

Commodity	Base Year		Current Year
	Price p0	Quantity q0	Price p1	Quantity q1
A	3	x	2	5
B	4	6	3	5

Answer 1

Commodity	Base Year		Current Year		p0q0	p0q1	p1q0	p1q1
	p0	q0	p1	q1
A	3	x	2	5	3x	15	2x	10
B	4	6	3	5	24	20	18	15
Total	–	–	–	–	= 24 + 3x	= 35	= 18 + 2x	= 25

From the table,

`sump_0q_0` = 3x + 24,

`sump_0q_1` = 35

`sump_1q_0` = 2x + 18,

`sump_1q_1` = 25

Laspeyre’s Price Index Number:

P₀₁(L) = `(sump_1q_0)/(sump_0q_0) xx 100`

= `(2x + 18)/(3x + 24) xx 100`      ...(i)

Paasche’s Price Index Number:

P₀₁(P) = `(sump_1q_1)/(sump_0q_1) xx 100`

= `(25)/(35) xx 100`

= `(5)/(7) xx 100`    ...(ii)

Since P₀₁(L) = P₀₁(P),

`(2x + 18)/(3x + 24) xx 100 = (5)/(7) xx 100`    ...[From (i) and (ii)]

∴ `(2x + 18)/(3x + 24) = (5)/(7)`

∴ 14x + 126 = 15x + 120

∴ 126 – 120 = 15x – 14x

∴ x = 6.