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

Answer 1

p = a – bx² 

${\displaystyle =\frac{\text{dp}}{\text{dx}}=0-\text{b}\frac{\text{d}}{\text{dx}}\left(x^2\right)}$

= - b(2x)

= - 2bx

Elasticity of demand: η_d = ${\displaystyle -\frac{\text{p}}{x}\cdot \frac{\text{dx}}{\text{dp}}}$

${\displaystyle =\frac{-p}{x}\times \frac{1}{\frac{\text{dp}}{\text{dx}}}}$

${\displaystyle =\frac{-{(\text{a}-\text{b}x)}^2}{x}\times \frac{1}{-2\text{b}x}}$

η_d = ${\displaystyle \frac{\text{a}-\text{b}{x}^2}{2\text{b}{x}^2}}$

When elasticity is equals to unit,

${\displaystyle \frac{\text{a}-\text{b}{x}^2}{2\text{b}{x}^2}}$ = 1

a – bx² = 2bx²

2bx² = a – bx²

2bx² + bx² = a

3bx² = a

${\displaystyle x^2=\frac{\text{a}}{\text{3b}}}$

x = ${\displaystyle \sqrt{\frac{\text{a}}{\text{3b}}}}$

∴ The value of x when elasticity is equal to unity is ${\displaystyle \sqrt{\frac{\text{a}}{\text{3b}}}}$