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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 |
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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`
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