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.

Answer 1

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")`