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Question
The demand for a quantity A is q = `80 - "p"_1^2 + 5"p"_2 - "p"_1"p"_2`. Find the partial elasticities `"E"_"q"/("E"_("p"_1))` and `"E"_"q"/("E"_("p"_2))` when p1 = 2, p2 = 1.
Solution
q = `80 - "p"_1^2 + 5"p"_2 - "p"_1"p"_2`
`(del"q")/(del"p"_1) = 0 - 2"p"_1 + 0 - (1)"p"_2`
`= - 2"p"_1 - "p"_2`
`(del"q")/(del"p"_2)`= 0 - 0 + 5(1) - p1 (1)
= 5 - p1
`therefore "E"_"q"/(del_("p"_1)) = - "p"_1/"q" (del"q")/(del"p"_1)`
`= (- "p"_1)/(80 - "p"_1^2 + 5"p"_2 - "p"_1"p"_2) xx (- 2"p"_1 - "p"_2)`
When p1 = 2, p2 = 1,
`"E"_"q"/(del_("p"_1)) = ((-2)/(80 - 2^2 + 5(1) - 2 xx 1)) (- 2 xx 2 - 1)`
`= (-2)/(80 - 4 + 5 - 2) xx (- 4 - 1) = 10/79`
`"E"_"q"/("E"_("p"_2)) = - "p"_2/"q" (del"q")/(del"p"_2)`
`= (- "p"_2)/(80 - "p"_1^2 + 5"p"_2 - "p"_1"p"_2) (- 5"p"_1)`
When p1 = 2, p2 = 1,
`"E"_"q"/("E"_("p"_2)) = (-1)/(80 - 2^2 + 5 xx 1 - 2xx1) (5 - 2)`
`= (-1)/79 xx (3) = (-3)/79`
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