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Minimum unit cost

The unit cost is the cost per unit, that is the cost function divided by the number. It is smallest where it equals the marginal cost. For a cost of the form a times x squared plus a fixed term, with no linear term, the minimum is where x is the square root of the fixed term divided by a. You insert that x-value into the unit cost to get the amount itself.

A(x) = ax + \frac{c}{x}

when C(x) = ax^2 + c

x = \sqrt{\frac{c}{a}}

minimum unit cost

Symbols

$a$	coefficient
$c$	fixed cost	NOK
$A$	unit cost	NOK

Example

$C(x) = x^2 + 100$ . $A(x) = x + \dfrac{100}{x}$ is smallest at $x = \sqrt{100} = 10$ .

Then $A(10) = 10 + \dfrac{100}{10} = 20$ NOK.

The minimum is where unit cost equals marginal cost.

Practise cost, revenue and profit for free →

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Part of Mathematics S1: Cost, revenue and profit.