Question

The supply function of certain goods is given by x = a${\displaystyle \sqrt{\text{p}-\text{b}}}$ where p is unit price, a and b are constants with p > b. Find elasticity of supply at p = 2b.

Answer 1

Given that x = a${\displaystyle \sqrt{\text{p}-\text{b}}}$

Elasticity of supply: η_s = ${\displaystyle \frac{\text{p}}{x}\cdot \frac{\text{dx}}{\text{dp}}}$

x = a${\displaystyle \sqrt{\text{p}-\text{b}}}$

${\displaystyle \frac{\text{dx}}{\text{dp}}=\text{a}\left(\frac{1}{2\sqrt{\text{p - b}}}\right)}$

η_s = ${\displaystyle \frac{\text{p}}{x}\cdot \frac{\text{dx}}{\text{dp}}}$

${\displaystyle =\frac{\text{p}}{\text{a}\sqrt{\text{p}-\text{b}}}\times \text{a}\times \frac{1}{2\sqrt{\text{p}-\text{b}}}}$

${\displaystyle =\frac{\text{p}}{2\left(\text{p - b}\right)}}$

Hint for differentiation

Use y = ${\displaystyle \sqrt{x}}$

${\displaystyle \frac{\text{dy}}{\text{dx}}=\frac{1}{2\sqrt{x}}}$

(or) x = ${\displaystyle \text{a}\sqrt{\text{p - b}}}$

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

${\displaystyle \frac{\text{dx}}{\text{dp}}=\text{a}\cdot \frac{1}{2}{\left(\text{p - b}\right)}^{\frac{1}{2}-1}}$

${\displaystyle =\frac{\text{a}}{2}{\left(\text{p - b}\right)}^{-\frac{1}{2}}}$

${\displaystyle =\frac{\text{a}}{2}\frac{1}{\sqrt{\text{p - b}}}}$

When p = 2b, Elasticity of supply: η_s = ${\displaystyle \frac{\text{2b}}{2(\text{2b}-\text{b})}=\frac{\text{2b}}{\text{2b}}=1}$