Given a linear demand curve $q(p) = a - bp$, how would one find the price elasticity of demand at $p = 0$? The quantity that would be demanded is given: $Q_0 = a$
The formula for the price elasticity of demand then dictates:
$\epsilon_{p=0} = \frac{dq}{dp} *\frac {p}{q} = -b * \frac{p}{a} = -b * \frac{0}{a}= 0$,
this would mean perfectly inelastic. Thus, no matter the change in price there should be no change in quantity. This does not seem to make sense as at this point demand should be the most elastic. Where did I go wrong?
As an example one could think of a free good (e.g. straws), the demand for which can be observed. Then a levy is imposed and the new demand is being observed, allowing one to derive a linear demand function. How would one find the price elasticity for straws when they are free ($p=0$)?