Price of maximum profit for hang gliders

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Your company sells hang gliders at a production cost of $100$ € each. In 2025 the demand law makes $20$ gliders/month sell at a price of $300$ €, and $2$ gliders/month at $500$ €.

• (a) Find the selling price that maximises profit, and the maximum monthly profit.
• (b) In 2026 a $25\%$ tax is introduced: for each glider sold you keep only $\tfrac34$ of the price. How does the optimal price change?

Solution

Linear demand. Through $(300;20)$ and $(500;2)$ the slope is $\dfrac{2-20}{500-300}=-\dfrac{9}{100}$, so $q(p)=20-\tfrac{9}{100}(p-300)=47-\tfrac{9}{100}p.$ (a) Profit is $\pi(p)=(p-100)\,q(p)=(p-100)\left(47-\tfrac{9}{100}p\right)$: a downward parabola vanishing at $p=100$ and $p=\tfrac{4700}{9}\approx 522{,}2$. The maximum is at the vertex, halfway between the zeros: $p^**=\frac{100+\tfrac{4700}{9}}{2}=\frac{2800}{9}\approx 311{,}11\ \text{€}.$ There $q=19$ and $\pi=\left(\tfrac{2800}{9}-100\right)\cdot 19=\tfrac{36100}{9}\approx 4011{,}11$ €/month. (b) With the tax the unit revenue is $\tfrac34 p$, so $\pi(p)=\left(\tfrac34 p-100\right)\left(47-\tfrac{9}{100}p\right)$. Zeros at $p=\tfrac{400}{3}\approx 133{,}3$ and $p=\tfrac{4700}{9}$; the vertex is at $p^**=\frac{\tfrac{400}{3}+\tfrac{4700}{9}}{2}=\frac{2950}{9}\approx 327{,}78\ \text{€}.$

$\boxed{p_{2025}\approx 311{,}11\ \text{€},\quad \pi_{max}\approx 4011{,}11\ \text{€},\quad p_{2026}\approx 327{,}78\ \text{€}}$