Marginal cost is the cost of producing one more unit, the derivative of total cost; average cost is total cost per unit. Learn why the marginal-cost curve crosses the average-cost curve exactly at its minimum, and how to use calculus to find the output that minimizes average cost.
Marginal cost and average cost
Marginal cost is the cost of producing one more unit — the derivative of total cost, MC = dC/dq. Average cost is the total cost spread over every unit made, AC = C/q. Studying how they move together is a classic use of calculus in economics.
The key relationship
Average cost falls while marginal cost sits below it and rises once marginal cost climbs above it. So the two curves cross exactly at the minimum of average cost.
Marginal cost crosses average cost precisely at the lowest point of the average-cost curve.
This is the same logic as a test score: a new score below your average pulls it down, a score above pulls it up, and your average is flat exactly when the new score equals it.
Minimizing average cost
To find the output that minimizes average cost, set the derivative of AC to zero and solve — a standard optimization problem. The solution is a critical number of the average-cost function, and it always satisfies MC = AC.
Worked idea
If total cost is C(q) = q² + 100, then AC = q + 100/q and MC = 2q. Setting AC = MC gives q + 100/q = 2q, so q² = 100 and q = 10 units — the same result you would get by minimizing revenue-side quantities in marginal and average revenue.