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Question
The demand for a quantity A is q = `13 - 2"p"_1 - 3"p"_2^2`. Find the partial elasticities `"E"_"q"/("E"_("p"_1))` and `"E"_"q"/("E"_("p"_2))` when p1 = p2 = 2.
Solution
q = `13 - 2"p"_1 - 3"p"_2^2`
`(del"q")/(del"p"_1)` = 0 - 2(1) - 0 = - 2
`(del"q")/(del"p"_2)`= 0 - 0 - 3(2)p2 = - 6p2
`"E"_"q"/("E"_("p"_1)) = - "p"_1/"q" (del"q")/(del"p"_1)`
`= (- "p"_1)/(13 - 2"p"_1 - 3"p"_2^2)`(- 2)
When p1 = p2 = 2,
`"E"_"q"/("E"_("p"_1)) = ((-2)/(13 - 2 xx 2 - 3 xx 2^2))`(-2)
`= (- 2 xx -2)/(13 - 4 - 3 xx 4)`
`= 4/(13 - 4 - 12)`
`= 4/(13 - 16)`
`= 4/(- 3) = (-4)/3`
`"E"_"q"/("E"_("p"_2)) = - "p"_2/"q" (del"q")/(del"p"_2)`
`= (- "p"_2)/(13 - 2"p"_1 - 3"p"_2^2) (- 6"p"_2)`
When p1 = p2 = 2,
`"E"_"q"/("E"_("p"_2)) = ((-2)/(13 - 2 xx 2 - 3 xx 2^2)) (- 6 xx 2)`
`= (- 2 xx -12)/(13 - 4 - 12)`
`= 24/(-3)` = -8
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