Price of maximum profit

Problem

For Gelmini & Cremonini’s drug, with demand $D(p)=10\,000\bigl(1-\tfrac{p}{28}\bigr)$ and unit cost $5$ €, compute the price that maximises the profit $(p-5)\cdot D(p)$.

Solution

The profit is $\pi(p) = (p-5)\cdot 10\,000\Bigl(1-\frac{p}{28}\Bigr) = \frac{10\,000}{28}\,(p-5)(28-p).$ It is a parabola in $p$ opening downwards, which vanishes at $p=5$ and $p=28$. The vertex — the maximum point — lies halfway between the two zeros: $p^* = \frac{5+28}{2} = \boxed{16{,}5\ \text{€}}.$ At this price $D(16{,}5)=10\,000\bigl(1-\tfrac{16{,}5}{28}\bigr)\approx 4\,107$ boxes, and the maximum profit is $\pi(16{,}5)=(16{,}5-5)\cdot 4\,107\approx 47\,200$ €/month. Note that $p^*=16{,}5$ is different from the equilibrium price ($\approx 14{,}58$ €): maximising profit and clearing the market are distinct objectives.

Links

Topics: Functions and properties
Concepts: Demand curve
Functions: Parabola
Skills: Modelling · Using formulae
Exercise types: Modelling problem