Marginal Analysis

Definition

The marginal value of an economic function — marginal revenue, marginal cost, marginal propensity to consume — is its derivative with respect to the relevant quantity: MR = d(TR)/dQ, MC = d(TC)/dQ. It measures the instantaneous rate at which the total changes, and closely approximates (but is not identical to) the older textbook definition of "the change caused by a one-unit increase." A firm maximises profit exactly at the output where marginal revenue equals marginal cost: "marginal revenue = marginal cost."

In the Book

Section 4.3 introduces marginal revenue via a demand-driven total revenue function TR = 100Q − 2Q², showing MR = 100 − 4Q, and then contrasts the calculus definition against the older discrete approximation: increasing Q from 15 to 16 raises TR by 38, close to but not exactly the derivative's value of 40 at Q = 15. The book makes the geometry explicit — MR is the slope of the tangent to the TR curve at a point, while the discrete approximation is the slope of the chord to a point one unit over, and the two converge as the step size shrinks (Figure 4.12). The same definition is then applied to marginal cost, marginal propensity to consume, and marginal product. Later in the section, a worked profit-maximisation example (TR = 30Q − Q², TC = ½Q² + 6Q + 7) finds the maximum at Q = 8 by setting dπ/dQ = 0, and separately computes MR = 14 and MC = 14 at that same output — demonstrating that profit-maximising output is exactly where MR = MC, a result the book states holds "for any profit function."

Why It Matters

Marginal analysis replaces "is this good on average?" with "is the next unit worth it?" — the question that actually governs optimal decisions, because average and marginal quantities can point in opposite directions (a firm can have positive average profit while its marginal unit is destroying value). The MR = MC rule is really an instance of a much more general principle — a quantity is optimised where its rate of change crosses zero, i.e., where the marginal benefit of one more unit of input exactly equals its marginal cost — which is why the same logic reappears anywhere a continuous resource (time, budget, effort) must be allocated to maximise or minimise something.