Marginal revenue for a monopolist

Marginal revenue and the demand function

Denote the inverse demand function by P(y). (That is, for any output y, P(y) is the price such that the aggregate demand at p is equal to y.)

The monopolist's total revenue is TR(y) = yP(y), so its marginal revenue function is given by

MR(y) = P(y) + yP'(y).

We conclude that if P'(y) < 0 (as we normally assume),

MR(y) < P(y) if y > 0:

when output is positive, marginal revenue is less than the price. (When a monopolist sells an extra unit, the price falls, not only for the extra unit, but for all the units it sells.) Thus the relation between MR and P is like that shown in the following figure. (Notice that average revenue is just P(y); we refer to P alternatively as AR.)

(AR and MR are related in the same way that any average and marginal curves are related.)

Marginal revenue and the elasticity of demand

The price elasticity of demand is defined by

(y) = P(y)/yP'(y).

Thus

MR(y) = P(y)[1 + 1/(y)].

Now,

(y) is negative (assuming that the demand function is downward-sloping), so we can write

MR(y) = P(y)[1 1/|(y)|].

Hence

MR(y) > 0 if |(y)| > 1 (demand is elastic)
MR(y) < 0 if |(y)| < 1 (demand is inelastic).

Examples and exercises on the marginal revenue function