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Question
If Laspeyre's Price Index Number is four times Paasche's Price Index Number, then find the relation between Dorbish-Bowley's and Fisher's Price Index Numbers.
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Solution
Laspeyre’s Price Index Number:
`"P"_01("L") = (sum "p"_1"q"_0)/(sum "p"_0"q"_0) xx 100`
Paasche’s Price Index Number:
`"P"_01("P") = (sum "p"_1"q"_1)/(sum "p"_0"q"_1) xx 100`
It is given that
P01(L) = 4 × P01(P)
∴ `(sum "p"_1"q"_0)/(sum "p"_0"q"_0) xx 100 = 4 xx (sum "p"_1"q"_1)/(sum "p"_0"q"_1) xx 100`
∴ `(sum "p"_1"q"_0)/(sum "p"_0"q"_0) = 4 xx (sum "p"_1"q"_1)/(sum "p"_0"q"_1)`
If we denote `(sum "p"_1"q"_0)/(sum "p"_0"q"_0) = "A", (sum "p"_1"q"_1)/(sum "p"_0"q"_1) = "B"`,
then A = 4B
Dorbish-Bowley’s Price Index Number:
`"P"_01("D - B") = ("P"_01("L") + "P"_01("P"))/2`
`"P"_01("D - B") = ((sum "p"_1"q"_0)/(sum "p"_0"q"_0) + (sum "p"_1"q"_1)/(sum "p"_0"q"_1))/2 xx 100`
`= ("A + B")/2 xx 100`
`= (4"B" + "B")/2 xx 100` ....[∵ A = 4B]
`= "5B"/2 xx 100`
= 250 B
∴ P01(D-B) = 250 B ....(i)
Fisher’s Price Index Number:
`"P"_01 ("F") = sqrt((sum "p"_1"q"_0)/(sum "p"_0"q"_0) xx (sum "p"_1"q"_1)/(sum "p"_0"q"_1)) xx 100`
`= sqrt("A" xx "B") xx 100`
`= sqrt("4B" xx "B") xx 100`
`= sqrt("4B"^2) xx 100`
= 2B × 100
∴ P01 (F) = 200 B ...(ii)
Dividing (i) by (ii), we get
`("P"_01 ("D - B"))/("P"_01 ("F")) = (250"B")/(200 "B")`
∴ `("P"_01 ("D - B"))/("P"_01 ("F")) = 5/4`
∴ `"P"_01 ("D - B") = 5/4 xx "P"_01 ("F")`
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