Science:Math Exam Resources/Courses/MATH104/December 2014/Question 03 (a)/Solution 1

Recall that the price elasticity of a product is given by $\epsilon ={\frac {dq}{dp}}{\frac {p}{q}}$, where $p$ and $q$, are price and demand respectively. We implicitly differentiate the given relation with respect to $p$ giving:

${\begin{aligned}3p^{2}+{\frac {dq}{dp}}+3q^{2}{\frac {dq}{dp}}&=0\\{\frac {dq}{dp}}+3q^{2}{\frac {dq}{dp}}&=-3p^{2}\\{\frac {dq}{dp}}(1+3q^{2})&=-3p^{2}\\{\frac {dq}{dp}}&={\frac {-3p^{2}}{1+3q^{2}}}\end{aligned}}$

Substitute into the formula for elasticity and obtain:

$\epsilon ={\frac {-3p^{2}}{1+3q^{2}}}{\frac {p}{q}}={\frac {-3p^{3}}{q+3q^{3}}}$.