A single-price monopolist’s marginal revenue is

Question

BlackTom AI · Accepted Answer

For a single-price monopolist marginal revenue is P(Q) + Q·dP/dQ. Starting from the inverse demand P(Q), total revenue is TR = P(Q)·Q. Differentiating:
TR = P(Q)Q
MR = d(TR)/dQ = P(Q) + Q·P′(Q)
Because demand is downward sloping for a monopolist, P′(Q) < 0, so Q·P′(Q) is negative and therefore MR < P(Q). This formal derivative shows why marginal revenue lies below price for any nonhorizontal (downward) demand curve.

An alternative expression uses elasticity. Define price elasticity of demand ε = (dQ/dP)·(P/Q). Rearranging gives (dP/dQ)·(Q/P) = 1/ε, so
MR = P + Q·P′(Q) = P[1 + (1/ε)].
Since ε < 0 for a downward-sloping demand, 1 + 1/ε < 1, making MR less than P. (MR can be positive, zero, or negative depending on elasticity magnitude, but it is always below the price.)

Why the correct answer is "less than its price":
A single-price monopolist must lower the market price on all units to sell an additional unit; the extra revenue gained on the marginal unit is partly offset by the price reduction on prior units, so MR is lower than the stated price.

Why the other options are incorrect:
- "Equal to its price": true for a price taker (perfect competition), not for a single-price monopolist with downward demand.
- "Greater than its price": impossible under downward-sloping demand because dP/dQ < 0 makes MR < P.
- "Negative when it maximizes revenues": revenue is maximized when MR = 0, not when MR is negative.
- "Zero when it maximizes profit": profit maximization requires MR = MC, which need not be zero.

Conclusion: Marginal revenue for a single-price monopolist is less than its price, matching the provided correct answer.

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