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Question
Find the elasticity of demand in terms of x for the following demand laws and also find the value of x where elasticity is equal to unity.
p = a – bx2
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Solution
p = a – bx2
`= "dp"/"dx" = 0 - "b" "d"/"dx" (x^2)`
= - b(2x)
= - 2bx
Elasticity of demand: ηd = `- "p"/x * "dx"/"dp"`
`= (- p)/x xx 1/("dp"/"dx")`
`= (- ("a" - "b"x)^2)/x xx 1/(- 2"b"x)`
ηd = `("a" - "b"x^2)/(2"b"x^2)`
When elasticity is equals to unit,
`("a" - "b"x^2)/(2"b"x^2)` = 1
a – bx2 = 2bx2
2bx2 = a – bx2
2bx2 + bx2 = a
3bx2 = a
`x^2 = "a"/"3b"`
x = `sqrt("a"/"3b")`
∴ The value of x when elasticity is equal to unity is `sqrt("a"/"3b")`
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