Question

Make a use of calculus to prove that the price elasticity of demand is a constant ε everywhere along the demand curve whose function is Q = Apε , where A is a positive constant and p is the market price

Answer 1

Differentiating the demand function as Q = Ap^ε with respect to p, we find that $\frac{dQ}{dp} =ε Apε^{-1}$ . To get the elasticity, we multiply $\frac{dQ}{dp}$ by $\frac{p}{Q} = \frac{p}{Apε^{-1}} =\frac{1}{Apε^{-1}}$ That is, the elasticity is

$ε Apε^{-1} \times \frac{1}{Apε^{-1}} =ε .$ Because this result holds for any p, the elasticity is the same, ε, at every point along the demand curve.