All courses › Mathematics S1 › Production and profit

Production and profit

The profit is largest where marginal revenue equals marginal cost. With a fixed price the marginal revenue is the price itself. You differentiate the cost, set the price equal to the derivative and solve for the number of units. Then you calculate the profit by subtracting the cost from the revenue. That the derivative is zero is not enough on its own: for this cost, which curves upwards, it is a maximum of the profit.

P(x)=px−C(x)P(x) = px - C(x)profit
p=C′(x)p = C'(x)largest profit

Symbols

ppprice per unitNOK
CCcostNOK
xxnumber

Example

C(x)=0.02x2+20x+5000C(x) = 0.02x^2 + 20x + 5000 and the price is 100 NOK.

x=100−200.04=2000x = \dfrac{100 - 20}{0.04} = 2000. P(2000)=200 000−(80 000+40 000+5000)=75 000P(2000) = 200\,000 - (80\,000 + 40\,000 + 5000) = 75\,000 NOK.

Keep producing as long as one more unit brings in more than it costs.
Practise cost, revenue and profit for free →

← Finding the growth factor · Minimum unit cost →

Part of Mathematics S1: Cost, revenue and profit.