Business Mathematics and Statistics · Ch 6 — Applications of Differentiation (incl. Business/Economics applications, Maxima/Minima, Partial Derivatives)

Applications: Maximising Profit and Minimising Average Cost

The maxima/minima tests of the previous section become genuinely useful business tools once applied to profit and cost.

Maximising profit. Profit is $P(x) = R(x) - C(x)$ . Differentiating, $P'(x) = R'(x) - C'(x) = MR - MC$ . Setting $P'(x)=0$ gives the well-known rule:

Note

Profit-Maximising Condition

Profit is maximised (not minimised) at the output where $MR = MC$ , provided $P''(x) = MR' - MC' < 0$ there — i.e. marginal revenue is falling faster (or rising slower) than marginal cost at that output. This second condition is exactly the second-derivative test of the previous section applied to $P(x)$ ; skipping it risks mistaking a minimum-profit output (a point where cost is briefly cheaper to produce more, e.g. a small output level) for the true optimum.

Minimising average cost. $AC(x) = C(x)/x$ . Differentiating using the quotient rule and setting $\dfrac{d(AC)}{dx}=0$ leads to a second well-known result:

Note

Average-Cost-Minimising Condition

Average cost is minimised at the output where $MC = AC$ — marginal cost crosses average cost exactly at AC's lowest point. This can be shown directly: $\dfrac{d}{dx}\!\left(\dfrac{C(x)}{x}\right) = \dfrac{xC'(x)-C(x)}{x^2} = \dfrac{MC - AC}{x}$ , which is zero exactly when $MC=AC$ (for $x>0$ ). …

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