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Break-even point

Profit is income minus cost. The break-even point is where the profit is zero, that is where the income just covers the cost. With a fixed price and a linear cost you divide the fixed cost by the contribution margin, which is price minus cost per unit. If you sell fewer, you make a loss. If you sell more, the profit becomes positive.

P(x) = px - (ax + b)

profit

x = \frac{b}{p - a}

break-even when p > a

Symbols

$p$	price per unit	NOK
$a$	cost per unit	NOK
$b$	fixed cost	NOK

Example

Fixed cost 5000 NOK, 40 NOK per unit, price 90 NOK.

$x = \dfrac{5000}{90 - 40} = 100$ . Then the profit is $90 \cdot 100 - (40 \cdot 100 + 5000) = 0$ NOK.

The contribution margin must be positive, otherwise sales never cover the fixed costs.

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Part of Mathematics S1: Functions and models.