Which of the following is true if a monopolist’s marginal revenue is negative at the current level of output?

Question

BlackTom AI · Accepted Answer

For a monopolist marginal revenue is given by MR = P(1 + 1/ε), where ε is the price elasticity of demand (ε = (dQ/dP)*(P/Q)). This relationship links the sign of MR directly to whether demand is elastic or inelastic. 1) Relevant concepts - Price elasticity of demand (ε) measures responsiveness of Q to P; for a downward‑sloping demand curve ε is negative. - Total revenue TR = P(Q)*Q, and MR = dTR/dQ = P + Q dP/dQ. - Using ε one can rewrite MR in compact form: MR = P(1 + 1/ε). 2) Step‑by‑step derivation Start from MR = P + Q dP/dQ. Define elasticity ε = (dQ/dP)*(P/Q). Invert: (Q/P) dP/dQ = 1/ε. Substitute into MR: MR = P + P*(Q/P) dP/dQ = P[1 + (Q/P) dP/dQ] = P(1 + 1/ε). 3) Interpretation and sign logic (intermediate steps shown) - If demand is elastic (|ε| > 1, e.g., ε = -2), then 1/ε is between -1 and 0 (1/(-2) = -0.5) so MR = P(1 - 0.5) = 0.5P > 0. - If demand is unit elastic (|ε| = 1, ε = -1), then 1/ε = -1 and MR = P(1 - 1) = 0. - If demand is inelastic (|ε| < 1, e.g., ε = -0.5), then 1/ε < -1 (1/(-0.5) = -2) so MR = P(1 - 2) = -P < 0. Thus MR is negative when demand is price inelastic. 4) Why the correct answer is correct Negative MR means reducing output (or lowering price to sell one more unit) reduces total revenue — a hallmark of inelastic demand. Algebraically MR < 0 ⇔ 1 + 1/ε < 0 ⇔ 1/ε < -1 ⇔ |ε| < 1 (inelastic, remembering ε < 0). 5) Why other choices are incorrect - Unit elastic: gives MR = 0, not negative. - Price elastic: yields MR > 0, not negative. - Marginal cost = price: not implied by MR negative; monopolists typically set MR = MC to maximize profit, but MR being negative does not equate MC to price. - MR = price: true under perfect competition only, not for a monopolist and not when MR is negative. Conclusion: MR negative at the current output implies demand is price inelastic. This matches the provided answer.

Which of the following is true if a monopolist’s marginal revenue is negative at the current level of output?单项选择题

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