Suppose the firm’s demand curve is given by the relation P = f(Q). Because the firm possesses monopoly power, it faces a downward-sloping demand curve: f ’(Q) < 0. Revenue is R(Q) = QP = Qf(Q) and marginal revenue is dR(Q)/dQ = R’(Q). Using the product rule, R’(Q) = f(Q) + Qf ’(Q). Noting that P = f(Q), R’(Q) = P + Qf ’(Q). The second term is clearly negative, indicating that MR = R’(Q) < P as was asserted.

What is the meaning of the term Qf ’(Q) in MR? Consider what happens when a monopolist wishes to sell an additional unit of output. The price falls by dP/dQ = f ’(Q), and this price decrease applies to the first Q units sold. Hence, the increase in revenue is equal to the price at which the last unit is sold, P, minus the loss in revenue from selling Q units at this lower price, Qf ’(Q).

Consider the special case of a linear, downward-sloping demand curve: f(Q) = a - bQ where a is the price at which zero units are sold and -b = f ’(Q) is the slope of the demand curve. Using our formula for marginal revenue, MR = f(Q) + Qf ’(Q) = (a - bQ) - bQ = a - 2bQ. We see that the marginal revenue relation is also linear, has the same price-axis intercept, and is twice as steep. Graphically, this implies that the marginal revenue curve is everywhere half the distance between the price axis and the demand curve. (Note that we could also form R(Q) directly as PQ = (a - bQ)Q = aQ - bQ² and R’(Q) = a - 2bQ as before.)