Marginal cost and marginal revenue

Cost, revenue and profit functions

Let x be the number of units produced and sold.

• Total cost C(x) = fixed cost + variable cost.
• Average cost AC = C(x)/x.
• Total revenue R(x) = p × x, where p is the price per unit. If price depends on demand, p = p(x) and R(x) = x·p(x).
• Profit P(x) = R(x) − C(x).

Marginal functions

Marginal cost (MC) is the rate of change of total cost with respect to output: MC = dC/dx. It is approximately the extra cost of producing one more unit. Marginal revenue (MR) is MR = dR/dx, approximately the extra revenue from selling one more unit. Marginal profit = dP/dx = MR − MC.

Examples

• C(x) = 0.005x³ − 0.02x² + 30x + 5000. MC = 0.015x² − 0.04x + 30. At x = 3, MC = 0.135 − 0.12 + 30 = ₹30.015.
• R(x) = 3x² + 36x + 5. MR = 6x + 36; at x = 5, MR = 66.
• Demand p = 50 − 2x gives R = 50x − 2x² and MR = 50 − 4x. Revenue is maximum where MR = 0, that is, at x = 12.5.

Marginal and average cost

Average cost is minimum where d(AC)/dx = 0, and at that output MC = AC. Example: C(x) = x² + 4x + 100, so AC = x + 4 + 100/x. Setting 1 − 100/x² = 0 gives x = 10, where AC = 24 and MC = 2x + 4 = 24.

Profit maximisation

Profit is maximum where MR = MC (first condition) and d²P/dx² < 0 (second condition).

Key points

• MC = C′(x); MR = R′(x).
• Fixed cost does not affect marginal cost (its derivative is 0).
• At minimum average cost, MC = AC.
• Maximum profit: MR = MC with P″(x) < 0.

Exam tips

• Read whether the question wants total, average or marginal values.
• When price is given as a function of demand, build R(x) = x·p(x) before differentiating.