Exercise 125Calculus › Single-variable calculus › Differentiation › Optimisation and curve sketchingL1Added 4 Nov 2026

Maximising profit

#optimisation#economics#marginal cost#modelling

Statement

A company sells $q$ units at price $p(q)=80-0.5q$ (€ per unit). Producing $q$ units costs $C(q)=200+20q+0.25q^2$ .

Write the profit $\Pi(q)$ and find the production level that maximises it.
Check that, at the optimum, the marginal revenue equals the marginal cost.

Hint

Revenue is $R(q)=q\,p(q)$ . Marginal revenue and marginal cost are $R'(q)$ and $C'(q)$ .

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Solution

1. The price is nonnegative for $0\le q\le160$ . $R(q)=80q-0.5q^2$ , so
$\Pi(q)=R(q)-C(q)=60q-0.75q^2-200.$
$\Pi'(q)=60-1.5q=0$ gives $q=40$ , which lies in $[0,160]$ ; the parabola is concave, so this is the global maximum. Then $p=80-20=60$ € and $\Pi(40)=2400-1200-200=1000$ €.

2. $R'(q)=80-q=40$ and $C'(q)=20+0.5q=40$ at $q=40$ . They are equal.

Remark. $\Pi'=0$ is exactly $R'=C'$ : it is worth producing one more unit as long as the extra revenue exceeds the extra cost. This "marginal" reasoning is the heart of microeconomics.

Maximising profit

Statement

Solution

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