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Sum

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 – bx^{2}

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#### Solution

p = a – bx^{2 }

`= "dp"/"dx" = 0 - "b" "d"/"dx" (x^2)`

= - b(2x)

= - 2bx

Elasticity of demand: η_{d} = `- "p"/x * "dx"/"dp"`

`= (- p)/x xx 1/("dp"/"dx")`

`= (- ("a" - "b"x)^2)/x xx 1/(- 2"b"x)`

η_{d} = `("a" - "b"x^2)/(2"b"x^2)`

When elasticity is equals to unit,

`("a" - "b"x^2)/(2"b"x^2)` = 1

a – bx^{2} = 2bx^{2}

2bx^{2} = a – bx^{2}

2bx^{2} + bx^{2} = a

3bx^{2} = a

`x^2 = "a"/"3b"`

x = `sqrt("a"/"3b")`

∴ The value of x when elasticity is equal to unity is `sqrt("a"/"3b")`

Concept: Applications of Differentiation in Business and Economics

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